Partial Differential Equations
A partial differential equation (PDE) relates a function of several variables to its partial derivatives. The fundamental equations of physics are of this kind: the wave equation, the heat equation, the Laplace and Poisson equations, and the Schrödinger equation are all linear second-order PDEs, and their boundary- and initial-value problems occupy the rest of this treatise.
Three facts organise the whole subject and are established below. The first is that a second-order equation carries an invariant — the inertia of its principal coefficient matrix — which sorts the equations of physics into three families with entirely different behaviour, and which no change of coordinates can alter. The second is that each family admits its own kind of side condition and no other: asking an elliptic equation for initial data, or a parabolic one for a solution running backwards in time, is not merely difficult but ill posed in Hadamard's precise sense. The third is that a linear problem is solved once the response to a point source is known, which is what a Green's function is.
The general theory followed here — classification, characteristics, well-posedness and the canonical equations — is that of Courant–Hilbert [Courant:1962]; the operator-by-operator catalogue of separable coordinates and Green's functions is Morse–Feshbach [Morse:1953].
Throughout, \(\Omega\) denotes a nonempty open subset of \(\R^{N}\), repeated indices are summed from \(1\) to \(N\), and \(\pp_{i} = \pp/\pp x^{i}\). The Laplacian is written \(\nabla^{2} = \pp_{i}\pp_{i}\) in Cartesian coordinates, and the wave operator of speed \(c\) is
Definitions
Let \(u : \Omega \longrightarrow \R\) be the unknown. For a multi-index \(\alpha = (\alpha_{1},\dots,\alpha_{N}) \in \N^{N}\) write \(\abs{\alpha} = \alpha_{1}+\cdots+\alpha_{N}\) and
A partial differential equation of order \(m\) on \(\Omega\) is a relation
in which \(F\) depends nontrivially on at least one derivative of order exactly \(m\). A classical solution is a function \(u \in C^{m}(\Omega)\) satisfying Equation (10.3) at every point of \(\Omega\). Rests on Definitions 6.2 and 7.66.
The equation Equation (10.3) is
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linear if it reads \(\sum_{\abs{\alpha}\le m} a_{\alpha}(x)\,\pp^{\alpha}u = f(x)\), and homogeneous if in addition \(f \equiv 0\);
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semilinear if the terms of order \(m\) are linear with coefficients depending on \(x\) alone, the remaining dependence being arbitrary;
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quasilinear if it is linear in the derivatives of order \(m\) with coefficients that may depend on \(x\), on \(u\) and on derivatives of order below \(m\);
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fully nonlinear otherwise.
Rests on Definition 10.1.
Linearity is what makes superposition available, and superposition is what every method of this chapter rests on: if \(u_{1}\) and \(u_{2}\) solve a homogeneous linear equation, so does \(\lambda_{1}u_{1} + \lambda_{2}u_{2}\), because a linear differential operator is a linear map on the vector space \(C^{m}(\Omega)\) (Linear Algebra and Representation Theory). The last section of the chapter shows what breaks when it is unavailable.
For a linear operator \(L = \sum_{\abs{\alpha}\le m} a_{\alpha}(x)\,\pp^{\alpha}\) the principal part is \(\sum_{\abs{\alpha}=m}a_{\alpha}\pp^{\alpha}\) and the principal symbol is the homogeneous polynomial
Rests on Definition 10.2 and Equation (10.2).
The symbol is obtained by the substitution \(\pp^{\alpha} \longmapsto \xi^{\alpha}\), which is exactly what the Fourier transform performs up to a factor \(\ii\) per derivative (Equation (17.38)): for constant coefficients, \(\widehat{Lu}(\xi) = \sigma_{L}(\ii\xi)\,\hat u(\xi)\) modulo the lower-order terms. Everything in this section is therefore a statement about a polynomial.
The general linear second-order operator on \(\Omega\) is
with real coefficients and \(a^{ij} = a^{ji}\). Its principal symbol is the quadratic form \(\sigma_{L}(x,\xi) = a^{ij}(x)\,\xi_{i}\xi_{j}\), and \(A(x)\) denotes the symmetric matrix \(\bigl(a^{ij}(x)\bigr)\). Rests on Definition 10.3 and Proposition 7.73.
The symmetry \(a^{ij}=a^{ji}\) costs nothing. For \(u \in C^{2}\) one has \(\pp_{i}\pp_{j}u = \pp_{j}\pp_{i}u\) (Proposition 7.73), so only the symmetric part \(\tfrac{1}{2}(a^{ij}+a^{ji})\) of the coefficient array acts on the solution; the antisymmetric part is invisible and is discarded. This is why the classification below is a statement about a symmetric matrix, and hence about its inertia.
Classification and characteristics
Let \(\bigl(n_{+},n_{-},n_{0}\bigr)\) be the inertia of \(A(x)\) — the numbers of positive, negative and zero eigenvalues, so that \(n_{+}+n_{-}+n_{0} = N\). At the point \(x\) the operator \(L\) of Equation (10.5) is
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elliptic if \(n_{0}=0\) and \(n_{+}n_{-}=0\), that is if \(A(x)\) is definite;
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hyperbolic if \(n_{0}=0\) and \(\min(n_{+},n_{-})=1\);
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ultrahyperbolic if \(n_{0}=0\) and \(\min(n_{+},n_{-})\ge2\);
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parabolic if \(n_{0}\ge1\), that is if \(A(x)\) is singular.
\(L\) is elliptic (hyperbolic, …) on \(\Omega\) if it is so at every point of \(\Omega\). Rests on Definition 10.4 and Theorem 5.112.
Case (iv) as stated is only degeneracy of the principal part; the heat operator \(\pp_{t}-D\nabla^{2}\) is the case that matters, in which \(n_{0}=1\) exactly and the first-order part supplies a derivative in the one direction the principal part cannot see. An equation with \(n_{0}\ge1\) and no such first-order term — \(\pp_{x}^{2}u=0\) in two variables — is degenerate rather than parabolic. Ultrahyperbolic operators do not occur in this treatise: they need at least two time-like directions, and the observed world has one.
The classification would be worthless if it depended on the coordinates used to write the equation down. It does not, and the reason is Sylvester's law of inertia.
Let \(y = y(x)\) be a \(C^{2}\) change of coordinates on \(\Omega\) with invertible Jacobian \(J^{a}{}_{i} = \pp y^{a}/\pp x^{i}\), and let \(\tilde A\) be the principal coefficient matrix of the transformed operator. Then
and the inertia \((n_{+},n_{-},n_{0})\) of \(A\) is unchanged. The type of a second-order operator at a point is therefore an attribute of the equation, not of the chart in which it is written. Rests on Definition 10.6, Theorem 5.112 and Proposition 7.31.
Derives Theorem 10.8. Write \(u(x) = \tilde u(y(x))\) and let \(\tilde\pp_{a} = \pp/\pp y^{a}\). The chain rule (Proposition 7.31) gives \(\pp_{i}u = J^{a}{}_{i}\,\tilde\pp_{a}\tilde u\) and, differentiating once more,
The second term carries only one derivative of \(\tilde u\), so it joins the first-order part and cannot affect the principal coefficients. Contracting the first term with \(a^{ij}\) gives \(\tilde a^{ab} = J^{a}{}_{i}\,a^{ij}\,J^{b}{}_{j}\), which is Equation (10.6). Since \(J\) is invertible, this is a congruence of symmetric matrices, and by Sylvester's law of inertia (Theorem 5.112) congruent real symmetric matrices have the same triple \((n_{+},n_{-},n_{0})\); Equation (5.142) is the normal form both reach.
∎In \(\R^{4}\) with coordinates \((ct,x^{1},x^{2},x^{3})\) the wave operator Equation (10.1) has \(A = \diag(1,-1,-1,-1)\), inertia \((1,3,0)\): hyperbolic. The Laplacian \(\nabla^{2}\) on \(\R^{3}\) has \(A = \diag(1,1,1)\), inertia \((3,0,0)\): elliptic. The heat operator \(\pp_{t}-D\nabla^{2}\) on \(\R\times\R^{3}\) has \(A = \diag(0,-D,-D,-D)\), inertia \((0,3,1)\): parabolic, with the single null direction the time direction, exactly as in Remark 10.7. Rests on Definition 10.6 and Equation (10.1).
Let \(\phi \in C^{1}(\Omega)\) with \(\nabla\phi \neq 0\), and let \(S = \set{x \in \Omega \mid \phi(x) = 0}\). The hypersurface \(S\) is characteristic for \(L\) at \(x \in S\) if and only if
Let \(S\) be a \(C^{2}\) hypersurface and let Cauchy data — the values of \(u\) and of its normal derivative on \(S\) — be prescribed. Every second-order derivative of \(u\) on \(S\) is then determined by the data together with the equation \(Lu = f\) if and only if \(S\) is non-characteristic at the point in question. Rests on Definition 10.10 and Theorem 10.8.
Derives Proposition 10.11. Choose coordinates \(y = (y^{1},\dots,y^{N})\) near the point with \(y^{N} = \phi\), so that \(S\) is the slice \(\set{y^{N}=0}\); this is admissible because \(\nabla\phi \neq 0\). The Cauchy data are \(\tilde u\) and \(\tilde\pp_{N}\tilde u\) on the slice. Differentiating those two functions along the slice determines every derivative \(\tilde\pp_{a}\tilde\pp_{b}\tilde u\) with \(a,b < N\) and every \(\tilde\pp_{a}\tilde\pp_{N}\tilde u\) with \(a < N\). The only second derivative not so obtained is \(\tilde\pp_{N}^{2}\tilde u\), and the equation reads
because every other principal term carries at least one index below \(N\). By Equation (10.6), \(\tilde a^{NN} = a^{ij}\pp_{i}\phi\,\pp_{j}\phi = \sigma_{L}(x,\nabla\phi)\). Hence Equation (10.9) determines \(\tilde\pp_{N}^{2}\tilde u\) if and only if that symbol is nonzero, which by Equation (10.8) is exactly non-characteristic.
∎An elliptic operator has no real characteristic surface. The characteristics of \(\Box\) are the surfaces with \(\abs{\nabla\phi}^{2} = c^{-2}\bigl(\pp_{t}\phi\bigr)^{2}\), that is the light cones \(\abs{\vect{x}-\vect{x}_{0}} = c\,\abs{t-t_{0}}\). The characteristics of \(\pp_{t}-D\nabla^{2}\) are the slices \(t = \) constant. Rests on Proposition 10.11 and Equation (10.1).
Derives Corollary 10.12. Ellipticity means \(A\) definite, so \(a^{ij}\xi_{i}\xi_{j} = 0\) forces \(\xi = 0\), which \(\nabla\phi \neq 0\) excludes. For \(\Box\) the symbol is \(c^{-2}(\pp_{t}\phi)^{2} - \abs{\nabla\phi}^{2}\), and the stated cones are the level sets of \(\phi = c(t-t_{0}) - \abs{\vect{x}-\vect{x}_{0}}\), which annihilate it. For the heat operator the principal symbol is \(-D\abs{\vect{\xi}}^{2}\), with no dependence on the time component of the conormal; it vanishes precisely when \(\nabla_{\vect{x}}\phi = 0\), i.e. when \(\phi\) is a function of \(t\) alone.
∎That the slices of constant time are characteristic for the heat operator is not a technicality: it is the reason a parabolic problem takes one initial condition and not two, and it will reappear as the one-sided character of diffusion in Section 10.5.2.
Let \(L u = a\,u_{xx} + 2b\,u_{xy} + c\,u_{yy} + \cdots\) have \(C^{1}\) coefficients near a point where \((a,b,c) \neq (0,0,0)\), and set \(\Delta = b^{2}-ac\). There are new coordinates \((\xi,\eta)\) in which the principal part becomes
up to a nonvanishing factor. The three cases are respectively hyperbolic, parabolic and elliptic in the sense of Definition 10.6. Rests on Theorem 10.8 and Definition 10.10.
Derives Proposition 10.13. Here the principal coefficient matrix is
so the sign of \(\Delta\) is the sign pattern of the eigenvalues of \(A\), which is Definition 10.6 for \(N = 2\): two eigenvalues of opposite sign when \(\Delta>0\), one zero eigenvalue when \(\Delta=0\), and two of the same sign when \(\Delta<0\). Under \(\xi = \xi(x,y)\), \(\eta = \eta(x,y)\) with Jacobian \(J_{0} = \xi_{x}\eta_{y}-\xi_{y}\eta_{x} \neq 0\), Equation (10.6) gives
with \(\tilde b\) the associated polarisation, and taking determinants of Equation (10.6) yields \(\tilde b^{2}-\tilde a\tilde c = J_{0}^{2}\,\Delta\).
Now \(\tilde a\) vanishes exactly when \(\xi = \) constant is a characteristic, by Equation (10.8). Suppose first that \(a \neq 0\). On such a curve \(\xi_{y} \neq 0\), since \(\xi_{y}=0\) would reduce Equation (10.12) to \(a\xi_{x}^{2}=0\) and force \(\nabla\xi = 0\), which Definition 10.10 excludes; dividing Equation (10.12) by \(\xi_{y}^{2}\) and writing \(m = \dd y/\dd x = -\xi_{x}/\xi_{y}\) for the slope of such a curve, the condition is the quadratic
If \(a = 0\) the closed form is empty and the families must be read off Equation (10.12) directly, which then factorises as \(\tilde a = \xi_{y}\bigl(2b\,\xi_{x}+c\,\xi_{y}\bigr)\): the vertical family \(x = \) constant, of infinite slope and so invisible to \(m\), is always characteristic, and if \(b \neq 0\) a second family of slope \(m = c/(2b)\) accompanies it — two families, as \(\Delta = b^{2} > 0\) demands. If \(a = b = 0\) then \(\Delta = 0\), only the vertical family survives, and the principal part is already \(c\,u_{yy}\) with \(c \neq 0\), which is the parabolic normal form. The archetype of the first of these degenerate cases is \(L u = 2u_{xy}\), whose two characteristic families are the coordinate lines themselves and which is already in hyperbolic canonical form. With the vertical family admitted, the count of real families in each of the three cases is the one the sign of \(\Delta\) dictates, and the constructions below apply verbatim.
If \(\Delta > 0\) there are two real families; take \(\xi\) and \(\eta\) to be first integrals of the two, so \(\tilde a = \tilde c = 0\) and \(\tilde b^{2} = J_{0}^{2}\Delta > 0\), leaving \(2\tilde b\,u_{\xi\eta}\) as the principal part. If \(\Delta = 0\) the two families coincide; take \(\xi\) to be a first integral of it and \(\eta\) any independent function, so \(\tilde a = 0\) and then \(\tilde b^{2} = \tilde a\tilde c = 0\), leaving \(\tilde c\,u_{\eta\eta}\) with \(\tilde c \neq 0\): otherwise the whole transformed principal matrix would vanish, and by Equation (10.6) with \(J\) invertible so would \(A\), contradicting \((a,b,c) \neq (0,0,0)\). If \(\Delta < 0\) then \(ac > b^{2} \ge 0\), so \(a \neq 0\) and the roots of Equation (10.13) are a complex-conjugate pair. Assume for this branch that the coefficients are real analytic. Complexifying \(y\) and integrating \(\dd y/\dd x = \bigl(b+\ii\sqrt{-\Delta}\bigr)/a\) then supplies a complex first integral \(\zeta\) with \(a\zeta_{x}^{2}+2b\zeta_{x}\zeta_{y}+c\zeta_{y}^{2}=0\) and with \(\Re\zeta\), \(\Im\zeta\) functionally independent; setting \(\xi = \Re\zeta\), \(\eta = \Im\zeta\) turns that one complex condition into the real pair \(\tilde a = \tilde c\), \(\tilde b = 0\), which is \(\tilde a\,(u_{\xi\xi}+u_{\eta\eta})\). Remark 10.14 records how the same branch is settled at \(C^{1}\).
∎The hyperbolic and parabolic branches of the proof are complete at \(C^{1}\): their characteristic equations are real, and Theorem 9.8 integrates them under exactly that hypothesis. The elliptic branch is not. A complex first integral is not delivered by any real ordinary differential equation theorem, because the object sought is a function \(\zeta\) of two real variables solving a Beltrami equation with non-vanishing Jacobian; complexifying the independent variable and integrating the characteristic equation constructs it when the coefficients are real analytic, and that is the hypothesis under which the classical argument of Courant and Hilbert is stated [Courant:1962]. For merely \(C^{1}\) — hence locally Hölder — coefficients the existence of \(\zeta\) is a separate and much harder statement, the Korn–Lichtenstein theorem on the local solvability of the Beltrami equation. The conclusion of Proposition 10.13 does hold at \(C^{1}\), and the appendix supplies it: the reduction of the canonical form to a Beltrami equation, and the construction of its solution, are carried out there from a single quoted singular-integral estimate. Two things should be carried away from that section. The map \((\xi,\eta)\) it produces is \(C^{1,\alpha}\) and not \(C^{2}\), so at merely Hölder coefficients the canonical form is a statement about the principal part alone — which is all Equation (10.6) ever needed, since that congruence involves the Jacobian and no second derivative; and one more derivative on the coefficients buys a \(C^{2,\alpha}\) map and the classical reading of Equation (10.10), first-order remainder included.
Full derivation in Appendix A.
Derives Proposition 10.13.
Let \(u\) be a \(C^{1}\) solution of the quasilinear equation
Then along every solution of the characteristic system \(\dot x = a\), \(\dot y = b\), \(\dot u = c\) the graph of \(u\) is swept out; the equation is thus reduced to a system of ordinary differential equations. Rests on Definition 10.2 and Proposition 7.31.
Derives Proposition 10.15. Let \(s \longmapsto (x(s),y(s))\) solve \(\dot x = a(x,y,u(x,y))\), \(\dot y = b(x,y,u(x,y))\) and put \(U(s) = u(x(s),y(s))\). The chain rule (Proposition 7.31) gives \(\dot U = \pp_{x}u\,\dot x + \pp_{y}u\,\dot y = a\,\pp_{x}u + b\,\pp_{y}u = c\), which is the third equation. The converse is the statement that the surface swept by the characteristic curves through a non-characteristic initial curve is a solution surface, since its tangent plane contains the vector \((a,b,c)\) at every point.
∎The equation of a mechanical principal function is not of the quasilinear form Equation (10.14): the Hamiltonian is quadratic in the derivatives. What covers that case is Cauchy's method, in which the derivatives are carried along the curve as unknowns of their own, so that a curve in \(\R^{n}\) is replaced by a strip — a point, the value there, and the tangent plane there.
Let \(F(\vect{x},u,\vect{p})\) be of class \(C^{2}\) on an open subset of \(\R^{n}\times\R\times\R^{n}\) and let \(u\) be a \(C^{2}\) solution of the fully nonlinear first-order equation
Fix \(\vect{x}_{0}\) and let \(s\longmapsto(\vect{x}(s),U(s),\vect{p}(s))\) solve the characteristic strip system
with data \(\vect{x}(0)=\vect{x}_{0}\), \(U(0)=u(\vect{x}_{0})\) and \(\vect{p}(0)=\nabla u(\vect{x}_{0})\). Then \(U(s)=u(\vect{x}(s))\) and \(\vect{p}(s)=\nabla u(\vect{x}(s))\) for as long as the solution exists, and \(F\) keeps the value zero along the strip. The system Equation (10.16) is closed: its right-hand sides involve only \((\vect{x},U,\vect{p})\), so it can be integrated without knowing \(u\) beforehand, and that is what makes it a method of solution rather than an identity satisfied by a solution already found. Rests on Proposition 10.15, Theorem 9.8 and Proposition 7.73.
Derives Proposition 10.16. Let \(s\longmapsto\vect{x}(s)\) solve \(\dot{x}^{i}=\pp_{p_{i}}F(\vect{x},u(\vect{x}),\nabla u(\vect{x}))\) with \(\vect{x}(0)=\vect{x}_{0}\); the right-hand side is \(C^{1}\) in \(\vect{x}\), so Theorem 9.8 supplies a unique solution on some interval about \(s=0\). Put \(U(s)=u(\vect{x}(s))\) and \(p_{i}(s)=\pp_{i}u(\vect{x}(s))\). The chain rule (Proposition 7.72) gives \(\dot{U}=\pp_{j}u\,\dot{x}^{j}=p_{j}\,\pp_{p_{j}}F\), which is the second equation, and \(\dot{p}_{i}=\pp_{j}\pp_{i}u\,\dot{x}^{j} =\pp_{i}\pp_{j}u\,\pp_{p_{j}}F\), the order of the two derivatives being immaterial by Proposition 7.73. Differentiating Equation (10.15) with respect to \(x^{i}\),
so that \(\pp_{i}\pp_{j}u\,\pp_{p_{j}}F =-\pp_{x^{i}}F-p_{i}\,\pp_{u}F\), which is the third equation. The triple \((\vect{x},U,\vect{p})\) therefore solves the closed system Equation (10.16) with the stated data; that system has a \(C^{1}\) right-hand side because \(F\) is \(C^{2}\), so Theorem 9.8 applied to it says the triple is the only solution with those data, which is the claim. Finally, along any solution of Equation (10.16),
so \(F\) retains its initial value, which is zero.
∎For \(F=a^{i}(\vect{x},u)\,p_{i}-c(\vect{x},u)\) the first two equations of Equation (10.16) read \(\dot{x}^{i}=a^{i}\) and \(\dot{U}=c\), and the third is not needed: that is Proposition 10.15 again, in \(n\) variables. For \(F=p_{0}+H(\vect{x},\vect{p})\) — the shape the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy takes when the time is written as one further coordinate \(x^{0}\), so that \(p_{0}=\pp S/\pp t\) — one has \(\pp_{u}F=0\) and \(\pp_{p_{0}}F=1\), so \(s\) may be identified with \(t\) and the remaining equations of Equation (10.16) are \(\dot{x}^{i}=\pp H/\pp p_{i}\) and \(\dot{p}_{i}=-\pp H/\pp x^{i}\): Hamilton's equations themselves. The characteristics of the Hamilton–Jacobi equation are the trajectories of the mechanical system, and this is the calculation that establishes it.
A complete integral is a family of solutions large enough to generate the rest by a purely algebraic operation. The operation is the formation of an envelope, and the one thing that has to be proved about it is that it does not leave the equation.
Let \(u=\varphi(\vect{x};\vect{a})\) be a family of \(C^{2}\) solutions of Equation (10.15), one for each \(\vect{a}\) in an open set \(A\subseteq\R^{n}\), with \(\varphi\) of class \(C^{2}\) in both arguments jointly. The family is a complete integral when the \(n\times(n+1)\) matrix whose rows are \(\bigl(\pp\varphi/\pp a_{k},\; \pp^{2}\varphi/\pp a_{k}\pp x^{1},\dots, \pp^{2}\varphi/\pp a_{k}\pp x^{n}\bigr)\) has rank \(n\) everywhere, so that distinct parameters give genuinely distinct solutions. Given a \(C^{1}\) map \(\vect{x}\longmapsto\vect{a}(\vect{x})\) into \(A\) with
the function \(v(\vect{x})=\varphi(\vect{x};\vect{a}(\vect{x}))\) is an envelope of the family: at each point it touches one member of the family and has the same tangent plane there. Rests on Equation (10.15) and Proposition 7.72.
Every envelope in the sense of Definition 10.18 solves Equation (10.15). Rests on Definition 10.18 and Proposition 7.72.
Derives Proposition 10.19. By the chain rule (Proposition 7.72),
the second group of terms vanishing by Equation (10.19). Hence the triple \((\vect{x},v(\vect{x}),\nabla v(\vect{x}))\) is exactly the triple \((\vect{x},\varphi,\nabla_{\vect{x}}\varphi)\) formed from the single member \(\varphi(\cdot\,;\vect{a}(\vect{x}))\) of the family, and \(F\) vanishes on it because that member is a solution.
∎Proposition 10.19 is what makes a complete integral worth having. Restricting the parameters by \(n-1\) arbitrary relations and enveloping over what is left produces, from the \(n\)-parameter family, a solution containing an arbitrary function — the general integral of the equation; enveloping over all \(n\) parameters at once produces the singular solution, which belongs to no member of the family and is where the family's members touch one another. In Hamilton–Jacobi Theory and the Optical–Mechanical Analogy this is the sense in which a caustic is the envelope of a family of solutions, and Equation (10.19) is the condition that chapter writes as the vanishing of a Jacobian. The construction and its use for first-order equations are Cauchy's; the classical account, with the strip system Equation (10.16) and the envelope theorem side by side, is [Courant:1962], chapter II.
Proposition 10.16 is the mechanism behind the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy: there the first-order PDE for the principal function is integrated along the characteristic strips, and those strips are, by Remark 10.17, the trajectories of the mechanical system carried together with their momenta. Its quasilinear predecessor Proposition 10.15 does not cover that equation, and the distinction matters: a Hamiltonian is quadratic in \(\pp S/\pp q^{a}\), so the derivatives do not enter linearly. The quasilinear construction reappears below as the mechanism of shock formation (Section 10.6.3), where the characteristics cross.
If the coefficients and the data of a normal Cauchy problem are real analytic near a point, and the initial surface is non-characteristic there, then the problem has a solution analytic in a neighbourhood of that point, and it is unique among analytic solutions. Rests on Proposition 10.11 and Definition 10.2.
Full derivation in Appendix A.
Derives Theorem 10.22.
The theorem is due to Kovalevskaya [Kowalevsky:1875], and its hypotheses cannot be relaxed casually. Analyticity is essential: Lewy exhibited a linear equation of first order with polynomial coefficients and a smooth right-hand side possessing no solution whatever, in any neighbourhood of any point [Lewy:1957]. Nor does the theorem say anything useful about physics even when it applies, because analytic data cannot be prescribed independently in disjoint regions: the Cauchy problem it solves for the Laplace equation is the ill-posed one of Section 10.1.2. Existence and well-posedness are different questions, and only the second is physical.
Well-posedness in the sense of Hadamard
A problem consisting of a differential equation together with side conditions, with data \(d\) in a normed space \(\mathcal{D}\) and solutions \(u\) in a normed space \(\mathcal{U}\), is well posed in the sense of Hadamard if and only if
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a solution exists for every admissible \(d\);
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it is unique;
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it depends continuously on the data: \(d_{n} \to d\) in \(\mathcal{D}\) implies \(u_{n} \to u\) in \(\mathcal{U}\).
A problem failing any one of the three is ill posed. Rests on Definitions 6.24 and 10.1.
Hadamard introduced the three conditions together, and insisted on the third [Hadamard:1902]; the systematic development is in his Yale lectures [Hadamard:1923]. The third condition is the one that carries the physics. Data are measured, hence known only to within an experimental uncertainty, and a problem that fails (iii) converts an arbitrarily small uncertainty into an arbitrarily large one: its solution is not a prediction. The following is the standard example, and it is the heat equation run backwards.
Consider \(\pp_{t}u = D\,\pp_{x}^{2}u\) on \((0,\pi)\times(0,T)\) with \(u(0,t)=u(\pi,t)=0\), and let the datum be the final profile \(u(\cdot,T) = g\), the unknown being the initial profile \(u(\cdot,0)\). With the supremum norm on both, this problem violates condition (iii) of Definition 10.24. Rests on Definitions 10.6 and 10.24.
Derives Proposition 10.25. For \(n \in \N\) put
Then \(\pp_{t}u_{n} = -Dn^{2}u_{n}\) and \(D\pp_{x}^{2}u_{n} = -Dn^{2}u_{n}\), so each \(u_{n}\) solves the equation and vanishes at \(x = 0,\pi\). Its datum is \(g_{n} = u_{n}(\cdot,T) = n^{-1}\sin(nx)\), with \(\norm{g_{n}}_{\infty} = 1/n \longrightarrow 0\), so \(g_{n}\) tends to the zero datum, whose solution is \(u \equiv 0\). But \(\norm{u_{n}(\cdot,0)}_{\infty} = n^{-1}\ee^{Dn^{2}T} \longrightarrow \infty\). Data differing by \(\varepsilon\) therefore admit initial profiles differing by any prescribed amount, and no continuity estimate can hold.
∎The same phenomenon forbids Cauchy data for an elliptic equation. Take \(\pp_{x}^{2}u+\pp_{y}^{2}u = 0\) on the half-plane \(y>0\) with \(u(x,0) = 0\) and \(\pp_{y}u(x,0) = n^{-1}\sin(nx)\). The solution is \(u = n^{-2}\sin(nx)\sinh(ny)\), and while the Cauchy data tend uniformly to zero, \(\sup_{x}\abs{u(x,y)} = n^{-2}\sinh(ny)\) diverges for every fixed \(y>0\). An elliptic equation accepts data on a closed boundary — Dirichlet or Neumann — and nothing else; a hyperbolic equation accepts Cauchy data on a non-characteristic surface; a parabolic equation accepts one initial condition and boundary conditions, forward in time only. The match between type and side condition is not a convention but a theorem about which problems are well posed.
Ill-posed problems cannot simply be avoided, because some of them are the questions one wants to ask: reconstructing a source from its field, deconvolving an instrumental response from a measured spectrum, or inferring a past state from a present one. The Fourier analysis of the same difficulty is Remark 17.55: the forward map annihilates high frequencies, so its inverse amplifies them without bound, and the practical answer is regularisation — solving a nearby well-posed problem and quantifying the difference.
The equations of physics
The equations this chapter serves are five, and between them they cover almost every field equation of the treatise.
The wave equation
with \(c\) a speed in \(\mathrm{m}/\mathrm{s}\), is hyperbolic (Example 10.9). It was written down by d'Alembert for the vibrating string [dAlembert:1747] and returns for sound (Oscillations and Mechanical Waves), for light (Electromagnetic Waves and Optics) and for gravitational radiation (Gravitational-Wave Theory).
The heat or diffusion equation
with diffusivity \(D\) in \(\mathrm{m}^{2}/\mathrm{s}\), is parabolic. It is Fourier's [Fourier:1822], and it governs conduction, molecular diffusion (Experiment: Brownian Motion and Avogadro's Number) and the transport coefficients of Nonequilibrium Thermodynamics and Transport.
The Laplace and Poisson equations
are elliptic. The homogeneous form is the potential equation of Laplace's celestial mechanics [Laplace:1799]; Poisson added the source term valid inside the distribution [Poisson:1813]. They govern electrostatics (Electrostatics) and Newtonian gravity.
The Helmholtz equation
with \(k\) a wavenumber in \(/\mathrm{m}\), is elliptic and is the common time-reduced form of the first two: separating the time dependence out of Equation (10.22) or Equation (10.23) leaves Equation (10.25) for the spatial factor, as Section 10.2 shows. Its separable coordinate systems are catalogued by Morse and Feshbach [Morse:1953].
The Schrödinger equation
with \(\hbar\) in \(\mathrm{J}\,\mathrm{s}\) and \(m\) in \(\mathrm{kg}\), is the equation of The Postulates of Quantum Mechanics [Schroedinger:1926a]. Its principal coefficients are imaginary, so the real trichotomy of Definition 10.6 does not apply to it directly: it has the shape of a diffusion equation with imaginary diffusivity \(\ii\hbar/2m\), and behaves like neither, propagating wave packets that spread without damping. Its relativistic counterparts in The Klein–Gordon Equation are genuinely hyperbolic.
From partial to ordinary differential equations
The link between this chapter and the preceding one is the method of separation of variables: writing the unknown as a product of single-variable factors decomposes a linear PDE, in coordinates adapted to the boundary, into ordinary differential equations of Sturm–Liouville type. It is by this route that the classical families of Ordinary Differential Equations and Sturm–Liouville Theory enter physics: the one-dimensional Helmholtz equation and its Fourier system arise on intervals (Section 9.7), the Bessel functions in cylindrical problems (Section 9.8), and the Legendre functions and spherical harmonics in spherical ones (Section 9.9) [Jackson:1999].
The whole method rests on one elementary observation.
Let \(I_{1},I_{2}\) be intervals and let \(F : I_{1}\to\R\), \(G : I_{2}\to\R\) satisfy \(F(s) = G(t)\) for every \((s,t) \in I_{1}\times I_{2}\). Then \(F\) and \(G\) are one and the same constant. Rests on Definition 10.2.
Derives Lemma 10.28. Fix \(t_{0} \in I_{2}\). Then \(F(s) = G(t_{0})\) for every \(s \in I_{1}\), so \(F\) is the constant \(G(t_{0})\); symmetrically \(G\) is the constant \(F(s_{0})\), and the two constants agree.
∎Let \(u(\vect{x},t) = T(t)\,\psi(\vect{x})\) with \(T,\psi\) not identically zero. Then \(u\) solves Equation (10.23) if and only if there is a constant \(\lambda\) with
and it solves Equation (10.22) if and only if
In both cases the spatial factor obeys the Helmholtz equation Equation (10.25) with \(k^{2} = \lambda\). Rests on Lemma 10.28 and Equation (10.25).
Derives Proposition 10.29. Substituting \(u = T\psi\) into Equation (10.23) and dividing by \(DT\psi\) where it is nonzero gives
The left side depends on \(t\) alone and the right on \(\vect{x}\) alone, so by Lemma 10.28 both equal a constant, written \(-\lambda\); that is Equation (10.27). The wave case is identical with \(T'/D\) replaced by \(T''/c^{2}\). Conversely a pair satisfying Equation (10.27) reassembles into a solution by direct substitution, and likewise for Equation (10.28).
∎The product form is not an assumption about the solution sought. It is a device for manufacturing a supply of exact solutions, each of which is then a legitimate solution of the linear homogeneous equation; the solution of the actual problem is assembled from them by superposition, and the assembly is what has to be justified. Two things can go wrong. The separated solutions may fail to span the data — the question of completeness, settled for a regular Sturm–Liouville problem in Ordinary Differential Equations and Sturm–Liouville Theory and in the abstract in Section 12.2; and the boundary may fail to be a coordinate surface of any system in which the operator separates, in which case the method does not start at all. Superposition also requires linearity, so nothing in this section survives into Section 10.6.3.
Separable coordinate systems
The three systems this treatise uses are worked out here; the general classification follows.
Put \(\psi = X(x)Y(y)Z(z)\) in Equation (10.25) and divide by \(\psi\):
Each quotient depends on one variable alone, so each is a constant by Lemma 10.28: \(X'' = -\alpha^{2}X\), \(Y'' = -\beta^{2}Y\), \(Z'' = -\gamma^{2}Z\) with \(\alpha^{2}+\beta^{2}+\gamma^{2} = k^{2}\). Each factor obeys the one-dimensional Helmholtz equation Equation (9.106), whose eigenfunctions on an interval are the Fourier system (Section 9.7.4). Rests on Lemma 10.28 and Equation (9.106).
In circular cylindrical coordinates \((\rho,\varphi,z)\),
With \(\psi = R(\rho)\Phi(\varphi)Z(z)\), single-valuedness forces \(\Phi'' = -m^{2}\Phi\) with \(m \in \Z\), and \(Z'' = -\gamma^{2}Z\); the radial factor then obeys
which is Bessel's equation Equation (9.125) of order \(m\) in the variable \(\sqrt{k^{2}-\gamma^{2}}\,\rho\). Rests on Lemma 10.28 and Equation (9.125).
In spherical coordinates \((r,\theta,\varphi)\),
With \(\psi = R(r)\,Y(\theta,\varphi)\) the bracket acting on \(Y\) must be a constant multiple of \(Y\); regularity on the whole sphere fixes that constant to be \(-l(l+1)\) with \(l \in \N\), and \(Y\) is a spherical harmonic Equation (9.163), its polar factor obeying the associated Legendre equation Equation (9.157) in \(\cos\theta\). The radial factor obeys
the spherical Bessel equation, which the substitution \(R = r^{-1/2}\,W\) turns into Equation (9.125) of order \(l+\tfrac{1}{2}\). For \(k = 0\) this is the Laplace equation and Equation (10.34) is solved by \(r^{l}\) and \(r^{-l-1}\): the multipole expansion of Electrostatics, and, with the Coulomb potential restored, the radial problem of The Hydrogen Atom. Rests on Lemma 10.28, Equation (9.157) and Equation (9.163).
In three dimensions the Helmholtz equation Equation (10.25) separates in exactly eleven orthogonal coordinate systems: Cartesian, circular cylindrical, elliptic cylindrical, parabolic cylindrical, spherical, prolate spheroidal, oblate spheroidal, parabolic, conical, ellipsoidal and paraboloidal. Separation holds if and only if the metric coefficients of the system satisfy the Stäckel conditions, which state that the reciprocals of the scale factors are expressible through a matrix whose rows depend each on one coordinate alone. Rests on Equation (10.25) and Lemma 10.28.
Full derivation in Appendix A.
Derives Theorem 10.34.
The Stäckel conditions are derived there — both their necessity and their sufficiency — by substituting the product ansatz into the Laplacian of an orthogonal system and demanding that the three separation constants enter the separated equations linearly; the ellipsoidal system, of which the three examples above are degenerate limits, is verified against them outright, and the eleven systems are obtained as the degenerations of one confocal quadric family. Only the completeness of that list — that no orthogonal system outside the confocal scheme separates — is quoted, from the catalogue of Morse and Feshbach [Morse:1953], whose tables give the separated equations system by system.
The Laplace equation, being the case \(k = 0\), separates in strictly more systems than the Helmholtz equation — bispherical and toroidal coordinates among them — because it also admits R-separation, in which the product ansatz is preceded by division by a fixed function of position. A Stäckel matrix of the same shape governs the separability of the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, which is not a coincidence: both ask when a single operator falls apart along the coordinate directions. The two statements are nevertheless different, the Hamilton–Jacobi equation being of first order and nonlinear, and it is Theorem 23.17 that carries that case; Remark 23.19 sets the two side by side and explains why the eleven systems above coincide with the separable systems of a free particle. In practice the coordinate system is chosen by the boundary, not by the operator: one separates in the system whose surfaces the boundary is made of, and the worked boundary-value problems appear in Electrostatics and The Hydrogen Atom.
Eigenfunction expansions
Separation supplies a family of solutions indexed by the separation constants; the data are matched by superposing them, which is possible exactly when the separated eigenfunctions are complete.
Let \(f\) be continuous on \([0,L]\) with \(f(0)=f(L)=0\) and let
Then
converges for \(t>0\) together with all its derivatives, solves Equation (10.23) on \((0,L)\times(0,\infty)\) with \(u(0,t)=u(L,t)=0\), and satisfies \(u(\cdot,t) \to f\) in the mean square as \(t \to 0^{+}\). Rests on Proposition 10.29 and Remark 9.53.
Derives Theorem 10.36. Each summand is a separated solution: by Proposition 10.29 with \(\lambda = n^{2}\pi^{2}/L^{2}\) the spatial factor \(\sin(n\pi x/L)\) solves the one-dimensional Helmholtz equation with the stated boundary values and the temporal factor solves \(T' = -D\lambda T\). Bessel's inequality Equation (9.53) bounds the coefficients, \(\abs{c_{n}} \le C\) uniformly in \(n\); for \(t \ge t_{0}>0\) the factor \(\ee^{-Dn^{2}\pi^{2}t/L^{2}}\) then dominates any fixed power of \(n\), so the series and every series obtained from it by differentiating term by term in \(x\) or \(t\) converge absolutely and uniformly on \([0,L]\times[t_{0},\infty)\). Term-by-term differentiation is therefore legitimate there and each term satisfies the equation, so \(u\) does. Since \(t_{0}>0\) was arbitrary, this holds for every \(t>0\). Finally the partial sums of Equation (10.36) at \(t = 0\) are the partial sums of the sine series of \(f\), which converge to \(f\) in the mean square by completeness of the Sturm–Liouville eigenfunctions (Remark 9.53); the exponential factors are bounded by \(1\) and tend to \(1\) for each fixed \(n\), whence the stated limit.
∎The proof shows more than it was asked for. The initial datum was merely continuous, yet for every \(t>0\) the solution is infinitely differentiable, because the factor \(\ee^{-Dn^{2}\pi^{2}t/L^{2}}\) kills the high harmonics faster than differentiation can excite them. Diffusion destroys the information carried by the fine structure of the datum, and that destruction is exactly the obstruction met in Proposition 10.25.
The same separation applied to Equation (10.22) on \([0,L]\) with fixed ends gives, by Equation (10.28),
with \(a_{n}\) and \(n\pi c\,b_{n}/L\) the sine coefficients of the initial displacement and velocity. Here the exponentials of Theorem 10.36 are replaced by oscillations of the same magnitude for all \(n\), so nothing is damped and no smoothing occurs: the wave equation remembers. That the general motion is a superposition of such modes is Daniel Bernoulli's claim [Bernoulli:1753], contested at the time and vindicated only by Fourier's theory of the coefficients [Fourier:1822]; the physical reading, in which the \(n\)-th term is a normal mode, is developed in Oscillations and Mechanical Waves. Rests on Proposition 10.29 and Equation (10.28).
Integral-transform methods
On an unbounded domain the eigenfunctions are not discrete and the sum of Section 10.2.2 becomes an integral. The machinery is that of Fourier Analysis and Integral Transforms, and the principle is one line: a transform that converts differentiation into multiplication converts a PDE in \(N+1\) variables into a problem in one.
Let \(P(\pp_{x})\) be a constant-coefficient operator in \(x\) and consider \(\pp_{t}u = P(\pp_{x})u\) for \(x \in \R\), \(t>0\), with \(u(\cdot,0)=f\) absolutely integrable. Then the transform \(\hat u(\xi,t)\) of \(u\) in \(x\) satisfies the ordinary differential equation
one equation for each \(\xi\), whose solution is \(\hat u = \hat f\,\ee^{P(\ii\xi)t}\). Rests on Equation (17.38) and Definition 17.29.
Derives Proposition 10.39. Each \(x\)-derivative contributes a factor \(\ii\xi\) under the transform (Equation (17.38)), so \(\widehat{P(\pp_{x})u} = P(\ii\xi)\hat u\); the \(t\)-derivative passes through the \(x\)-integral. What remains is a linear first-order ordinary differential equation in \(t\) with \(\xi\) as a parameter, and Equation (10.38) is its statement. The inversion Equation (17.34) returns \(u\).
∎When the variable runs over a half-line — as time does, a system being switched on at \(t=0\) — the Laplace transform of Definition 17.65 is the right tool, because its derivative rule Equation (17.87) does not merely multiply by \(s\) but absorbs the initial values into the transformed equation as polynomial terms. The initial conditions are thus imposed automatically rather than fitted afterwards, and the solution is recovered from the Bromwich integral Theorem 17.69 by residues. The choice of transform is therefore dictated by the domain and the side conditions, not by taste: Fourier for a whole line with decay at both ends, Laplace for a half-line with data at its origin, a series for a bounded interval.
Substituting the plane wave \(u = \ee^{\ii(\vect{k}\cdot\vect{x}-\omega t)}\) into a constant-coefficient linear equation reduces it to an algebraic relation between \(\omega\) and \(\vect{k}\) — the dispersion relation, which is the symbol Equation (10.4) evaluated on the wave. For Equation (10.22) it is \(\omega = \pm c\abs{\vect{k}}\): all wavelengths travel at the same speed and a signal keeps its shape. For Equation (10.23) it is \(\omega = -\ii D\abs{\vect{k}}^{2}\), purely imaginary, so nothing propagates and everything decays, the finer structure faster. For Equation (10.26) with \(V=0\) it is \(\hbar\omega = \hbar^{2}\abs{\vect{k}}^{2}/2m\), real but nonlinear in \(\abs{\vect{k}}\): propagation with spreading. The three types of Definition 10.6 are visible in these three answers, and the physical consequences are drawn in Oscillations and Mechanical Waves and Matter Waves.
Green's functions
For an inhomogeneous linear problem, the Green's function — the response of the differential operator to a point source — converts the equation into an integral representation of the solution. Green introduced both the function and the reciprocity it obeys, for the potential equation, in a privately printed essay [Green:1828].
Let \(L\) be a linear differential operator on \(\Omega\) together with homogeneous boundary conditions. A Green's function for the pair is a function \(G(\vect{x},\vect{x}')\) satisfying, in the sense of distributions in \(\vect{x}\) for each fixed \(\vect{x}' \in \Omega\),
and the boundary conditions in the variable \(\vect{x}\). Rests on Definitions 10.4 and 17.40.
If \(G\) is a Green's function for \(L\) with the given homogeneous boundary conditions and \(f\) is such that the integral converges, then
solves \(Lu = f\) with those boundary conditions. Rests on Definitions 10.2 and 10.42.
Derives Proposition 10.43. \(L\) acts on \(\vect{x}\) and the integration is over \(\vect{x}'\), so \(L\) may be taken under the integral sign: \(Lu(\vect{x}) = \int_{\Omega}\bigl[L G(\vect{x},\vect{x}')\bigr]f(\vect{x}')\dd^{3}x' = \int_{\Omega}\delta(\vect{x}-\vect{x}')f(\vect{x}')\dd^{3}x' = f(\vect{x})\) by Equation (10.39) and the defining property of the Dirac distribution (Definition 17.38). Each \(G(\cdot, \vect{x}')\) satisfies the homogeneous boundary conditions, and so therefore does the superposition.
∎The content of the method is thus entirely in the construction of \(G\), to which the next two subsections are devoted. Note what has been gained: one function of two points, computed once, solves the problem for every source.
The Green's function of a boundary-value problem
Let \(V \subset \R^{3}\) be a bounded region with piecewise-smooth boundary and let \(u,v \in C^{2}(\overline{V})\), \(p \in C^{1}\). Then
where \(\pp_{\nu}\) is the outward normal derivative. Rests on Theorem 7.101 and Definition 7.66.
Derives Lemma 10.44. Apply the divergence theorem Equation (7.123) to the vector field \(\vect{A} = p\left(v\nabla u - u\nabla v\right)\). Its divergence is \(v\,\nabla\cdot(p\nabla u) - u\,\nabla\cdot(p\nabla v)\), the two terms \(p\nabla v\cdot\nabla u\) produced by the product rule cancelling; and \(\vect{A}\cdot\dd\vect{S} = p(v\pp_{\nu}u-u\pp_{\nu}v)\dd S\).
∎Let \(L u = \nabla\cdot(p\nabla u) + q\,u\) with \(p,q\) real, and let the boundary conditions be homogeneous of the form \(\alpha u + \beta\,\pp_{\nu}u = 0\) on \(\pp V\) with real \(\alpha,\beta\) not both zero. Then the Green's function is symmetric,
Rests on Lemma 10.44 and Definition 10.42.
Derives Theorem 10.45. Put \(u(\vect{x}) = G(\vect{x},\vect{x}_{1})\) and \(v(\vect{x}) = G(\vect{x},\vect{x}_{2})\) in Equation (10.41). The terms in \(q\) are symmetric in \(u\) and \(v\) and cancel, so the left-hand side is
On the boundary both \(u\) and \(v\) satisfy \(\alpha w + \beta\pp_{\nu}w = 0\). If \(\beta \neq 0\) then \(\pp_{\nu}w = -(\alpha/\beta)w\) for both, and \(v\pp_{\nu}u - u\pp_{\nu}v = -(\alpha/\beta)(vu-uv) = 0\); if \(\beta = 0\) then \(u = v = 0\) there and the same combination vanishes. So the right-hand side of Equation (10.41) is zero, and Equation (10.43) gives Equation (10.42).
∎The hypothesis is exactly that \(L\) is formally self-adjoint and that the boundary conditions make it symmetric on its domain — the Sturm–Liouville form Equation (9.54) of Ordinary Differential Equations and Sturm–Liouville Theory, in one dimension. The symmetry of \(G\) is therefore not an accident of the potential problem but the kernel-level shadow of self-adjointness, and it is the statement Green called reciprocity: the field at \(\vect{x}_{1}\) due to a unit source at \(\vect{x}_{2}\) equals the field at \(\vect{x}_{2}\) due to a unit source at \(\vect{x}_{1}\). The operator-theoretic setting, in which a self-adjoint operator is diagonalised by the spectral theorem, is Hilbert Spaces.
Let \(\set{\psi_{n}}\) be a complete orthonormal system of eigenfunctions of \(L\) on \(V\) satisfying the boundary conditions, with \(L\psi_{n} = \lambda_{n}\psi_{n}\) and \(\lambda_{n} \neq \mu\) for all \(n\). Then the Green's function of \(L-\mu\) is
Rests on Definition 10.42 and Remark 9.53.
Derives Proposition 10.47. Apply \(L-\mu\) term by term in \(\vect{x}\): the \(n\)-th term becomes \(\psi_{n}(\vect{x})\overline{\psi_{n}(\vect{x}')}\), and completeness of the system is the closure relation \(\sum_{n}\psi_{n}(\vect{x})\overline{\psi_{n}(\vect{x}')} = \delta(\vect{x}-\vect{x}')\) (Remark 9.53), which is Equation (10.39) for \(L-\mu\). Each term satisfies the boundary conditions, hence so does the sum.
∎Equation (10.44) displays the Green's function as a meromorphic function of the parameter \(\mu\) whose poles are exactly the eigenvalues and whose residues are the eigenprojections. This is the resolvent operator of Hilbert Spaces written as a kernel, and it is why the analytic structure of a response function encodes a spectrum — the observation that drives the contour techniques of Complex Analysis and, in Radiation and Scattering of Electromagnetic Waves, the identification of resonances with poles.
For \(L = \nabla^{2}\) on the half-space \(z>0\) with \(G = 0\) on the plane \(z=0\), write \(\vect{x}'^{\,*}\) for the mirror image of \(\vect{x}'\) in that plane. Then
The second term is harmonic throughout \(z>0\), since its singularity lies outside; so Equation (10.39) holds by Proposition 10.67, proved below, which identifies \(-1/4\pi r\) as the fundamental solution of the Laplacian. On \(z=0\) the two distances are equal and \(G\) vanishes. The construction succeeds whenever the boundary is a plane or a sphere, and fails otherwise; the electrostatic reading is in Electrostatics [Jackson:1999], and the catalogue of Green's functions for the standard operators and domains is Morse–Feshbach [Morse:1953]. Rests on Proposition 10.67 and Definition 10.42.
Causal Green's functions and Duhamel's principle
The retarded and advanced Green's functions of the wave operator Equation (10.1) in three space dimensions are
each satisfying \(\Box G_{\pm} = \delta(\vect{x})\,\delta(t)\), with \(G_{+}\) supported on the future light cone of the source and \(G_{-}\) on its past cone. Rests on Definition 10.42 and Equation (10.1).
Both solve the same equation, so the equation does not choose between them; the choice is a physical postulate, that an effect follows its cause. Selecting \(G_{+}\) turns Equation (10.40) into the retarded potentials of The Maxwell Equations and the radiation fields of Radiation and Scattering of Electromagnetic Waves: the field at \((\vect{x},t)\) is the source evaluated at the earlier time \(t - \abs{\vect{x}-\vect{x}'}/c\), integrated over space.
Transforming Equation (10.39) for \(\Box\) in space and time gives \(\bigl(\abs{\vect{k}}^{2}-\omega^{2}/c^{2}\bigr)\hat G = 1\), whose right-hand side is finite while the left vanishes on the light cone \(\omega = \pm c\abs{\vect{k}}\): the inverse transform is an integral through two poles on the real \(\omega\) axis and is undefined until they are avoided. Which way to close the contour is fixed by the synthesis kernel of Equation (17.34), which in the convention of Definition 17.29 is \(\ee^{+\ii\omega t}\): on a large semicircle at \(\Im\omega = \tau\) its modulus is \(\ee^{-\tau t}\), so the contour must be closed in the upper half-plane for \(t>0\) and in the lower for \(t<0\). Displacing the poles by \(\omega \to \omega - \ii\varepsilon\) in the denominator moves them to \(\omega = \pm c\abs{\vect{k}} + \ii\varepsilon\), both in the upper half-plane, so closing downward encloses nothing and gives zero for \(t<0\) (Corollary 8.15) while closing upward gives the retarded function for \(t>0\); the opposite displacement gives the advanced one. Causality is thus one of the two possible pole prescriptions, and the Sokhotski–Plemelj formula Equation (17.54) is the statement that the difference between them is a delta function on the light cone. The residue calculus that evaluates either is Theorem 8.24.
The sign of \(\varepsilon\) is convention-dependent and is quoted the other way round in much of the physics literature, which synthesises with \(\ee^{-\ii\omega t}\) — the kernel of the plane-wave ansatz \(\ee^{\ii(\vect{k}\cdot\vect{x}-\omega t)}\) of Remark 10.41 — rather than with the \(\ee^{+\ii\omega t}\) fixed for this treatise in Definition 17.29. Under that reading the half-planes are exchanged and the retarded prescription is \(\omega \to \omega + \ii\varepsilon\). What is convention-independent, and is the content of the remark, is that the retarded function is the one whose poles lie on the side of the real axis that the closure for \(t>0\) encircles.
Let \(\mathcal{A}\) be a linear operator acting on the space variables only. Suppose that for each \(s \ge 0\) the function \(v(\cdot,t;s)\) solves the homogeneous problem \(\pp_{t}v = \mathcal{A}v\) for \(t>s\) with \(v(\cdot,s;s) = f(\cdot,s)\). Then
solves the driven problem \(\pp_{t}u = \mathcal{A}u + f\) with \(u(\cdot,0)=0\). Rests on Definition 10.2 and Theorem 7.42.
Derives Theorem 10.52. Differentiating Equation (10.47) in \(t\), the upper limit contributes the integrand evaluated at \(s=t\) and the integrand contributes its own \(t\)-derivative (Theorem 7.42):
using the initial condition of \(v\) in the first term and its equation in the second. Since \(\mathcal{A}\) does not act on \(s\), the remaining integral is \(\mathcal{A}u\). That \(u(\cdot,0)=0\) is immediate from Equation (10.47).
∎If \(w(\cdot,t;s)\) solves \(\pp_{t}^{2}w = c^{2}\nabla^{2}w\) for \(t>s\) with \(w(\cdot,s;s)=0\) and \(\pp_{t}w(\cdot,s;s)=f(\cdot,s)\), then \(u = \int_{0}^{t}w(\vect{x},t;s)\dd s\) solves \(\pp_{t}^{2}u - c^{2}\nabla^{2}u = f\) with vanishing initial data. Rests on Theorem 10.52 and Equation (10.22).
Derives Corollary 10.53. The first derivative is \(\pp_{t}u = w(\vect{x},t;t) + \int_{0}^{t}\pp_{t}w\,\dd s = \int_{0}^{t}\pp_{t}w\,\dd s\), since \(w\) vanishes on the diagonal. Differentiating once more, \(\pp_{t}^{2}u = \pp_{t}w(\vect{x},t;t) + \int_{0}^{t}\pp_{t}^{2}w\,\dd s = f + c^{2}\nabla^{2}u\), by the second initial condition and the equation for \(w\).
∎Duhamel's principle is superposition in the time variable: a distributed source is a continuum of impulses, and the response is the integral of the impulse responses. It is the same statement as Proposition 10.43 with the causal Green's function inserted, and in the language of Fourier Analysis and Integral Transforms it is the convolution Equation (17.66) of the source with the impulse response of a time-invariant linear system.
The canonical equations and their fundamental solutions
The wave equation
Let \(f \in C^{2}(\R)\) and \(g \in C^{1}(\R)\). The unique \(C^{2}\) solution of \(\pp_{t}^{2}u = c^{2}\pp_{x}^{2}u\) on \(\R\times\R\) with \(u(x,0)=f(x)\) and \(\pp_{t}u(x,0)=g(x)\) is
Rests on Proposition 10.13 and Theorem 7.43.
Derives Theorem 10.55. Introduce the characteristic coordinates \(\xi = x-ct\), \(\eta = x+ct\) of Proposition 10.13. Then \(\pp_{x} = \pp_{\xi}+\pp_{\eta}\) and \(\pp_{t} = c\left(\pp_{\eta}-\pp_{\xi}\right)\), so
The equation is therefore \(\pp_{\xi}\pp_{\eta}u = 0\). Integrating in \(\eta\) shows \(\pp_{\xi}u\) is a function of \(\xi\) alone, and integrating in \(\xi\) (Theorem 7.43) gives \(u = F(\xi)+G(\eta)\) with \(F,G \in C^{2}\); conversely every such \(u\) solves the equation, so this is the general solution.
Imposing the data at \(t=0\), where \(\xi=\eta=x\), gives \(F(x)+G(x) = f(x)\) and \(c\left(G'(x)-F'(x)\right) = g(x)\). Integrating the second from a fixed \(x_{0}\), \(G(x)-F(x) = c^{-1}\int_{x_{0}}^{x}g + A\) for a constant \(A\). Solving the pair,
and \(u = F(x-ct)+G(x+ct)\) is Equation (10.49), the constant \(A\) cancelling and the two partial integrals combining into one over \([x-ct,x+ct]\). Uniqueness follows because every solution has the form \(F(\xi)+G(\eta)\) and the data determine \(F\) and \(G\) up to the constant that cancels.
∎The value \(u(x_{0},t_{0})\) with \(t_{0}>0\) depends on the data only through their restriction to the interval \([x_{0}-ct_{0},\,x_{0}+ct_{0}]\). Conversely the data near a point \(x_{1}\) influence the solution only inside the forward cone \(\abs{x-x_{1}} \le c\,t\). In particular no signal travels faster than \(c\). Rests on Theorem 10.55.
Derives Corollary 10.56. Read Equation (10.49): \(f\) enters only at the two endpoints of the interval and \(g\) only through its integral over the interval. The converse statement is the contrapositive, since \(x_{1}\) lies in the interval of dependence of \((x,t)\) if and only if \(\abs{x-x_{1}} \le ct\).
∎The same conclusion holds in three space dimensions, and is proved there without any formula, by an energy estimate that will also give uniqueness.
Let \(u \in C^{2}\) solve \(\pp_{t}^{2}u = c^{2}\nabla^{2}u\) on \(\R^{3}\times(0,\infty)\), fix \((\vect{x}_{0},t_{0})\) and set
Then \(e\) is non-increasing. Consequently, if \(u\) and \(\pp_{t}u\) vanish at \(t=0\) on the ball \(B(\vect{x}_{0},ct_{0})\), then \(u\) vanishes throughout the backward cone, and solutions of the Cauchy problem are unique. Rests on Theorem 7.101 and Equation (10.22).
Derives Proposition 10.57. Write \(R(t) = c(t_{0}-t)\) and differentiate Equation (10.52). The integrand's own \(t\)-derivative gives, on using the equation and the divergence theorem Equation (7.123) on \(\nabla\cdot\left(\pp_{t}u\,\nabla u\right)\),
because the remaining volume term is \(\int\pp_{t}u\left(\pp_{t}^{2}u-c^{2}\nabla^{2}u\right) = 0\). The shrinking of the ball, \(\dot R = -c\), contributes \(-\frac{c}{2}\oint_{\pp B}\left[(\pp_{t}u)^{2} +c^{2}\abs{\nabla u}^{2}\right]\dd S\). Now for real numbers \(a = \pp_{t}u\) and \(b = \abs{\nabla u}\) one has \(c^{2}ab \le \frac{c}{2}\left(a^{2}+c^{2}b^{2}\right)\), since the difference is \(\frac{c}{2}(a-cb)^{2} \ge 0\); and \(\abs{\pp_{\nu}u} \le \abs{\nabla u}\). The surface term of Equation (10.53) is therefore dominated by the negative one, so \(\dot e(t) \le 0\).
If the data vanish on \(B(\vect{x}_{0},ct_{0})\) then \(e(0)=0\), and since \(e \ge 0\) and \(e\) is non-increasing, \(e \equiv 0\): both \(\pp_{t}u\) and \(\nabla u\) vanish in the cone, so \(u\) is constant there, and the constant is zero. Uniqueness follows by applying this to the difference of two solutions, whose data vanish everywhere.
∎For \(h \in C^{2}(\R^{3})\) define the spherical mean
the integral running over the unit sphere with total solid angle \(4\pi\). Then \(M_{h}(\vect{x},r) \to h(\vect{x})\) as \(r \to 0\) and
Rests on Theorem 7.101 and Definition 7.66.
Derives Lemma 10.58. Differentiating Equation (10.54) under the integral sign, \(\pp_{r}M_{h} = \frac{1}{4\pi}\int_{S^{2}}\vect{n}\cdot\nabla h(\vect{x}+r\vect{n}) \dd\Omega\), which on rewriting the solid angle as a surface element, \(\dd S = r^{2}\dd\Omega\), is \(\frac{1}{4\pi r^{2}}\oint_{\pp B_{r}}\pp_{\nu}h\,\dd S\); the divergence theorem Equation (7.123) turns this into the first half of Equation (10.55). Multiplying by \(r^{2}\) and differentiating again,
which is the second half after dividing by \(r^{2}\). The last equality in Equation (10.55) holds because \(\nabla_{\vect{x}}^{2}\) commutes with the average over \(\vect{n}\). The limit \(r\to0\) is the continuity of \(h\).
∎Let \(f \in C^{3}(\R^{3})\) and \(g \in C^{2}(\R^{3})\). The solution of \(\pp_{t}^{2}u = c^{2}\nabla^{2}u\) with \(u(\cdot,0)=f\) and \(\pp_{t}u(\cdot,0)=g\) is
with \(M\) the spherical mean Equation (10.54). Rests on Lemma 10.58 and Theorem 10.55.
Derives Theorem 10.59. Fix \(\vect{x}\) and write \(M(r,t) = M_{u(\cdot,t)}(\vect{x},r)\). Averaging the equation over the sphere of radius \(r\) and using Equation (10.55),
Set \(W(r,t) = r\,M(r,t)\). Then \(\pp_{r}^{2}W = r\left(\pp_{r}^{2}M + \frac{2}{r}\pp_{r}M\right)\), so Equation (10.58) becomes the one-dimensional wave equation \(\pp_{t}^{2}W = c^{2}\pp_{r}^{2}W\) on \(r>0\), with \(W(0,t)=0\) and data \(W(r,0) = r\,M_{f}(\vect{x},r)\), \(\pp_{t}W(r,0) = r\,M_{g}(\vect{x},r)\). Extending both data as odd functions of \(r\) — which is consistent with the boundary condition at the origin — d'Alembert's formula Equation (10.49) applies. Writing \(\tilde F(s) = s\,M_{f}(\vect{x},s)\) and \(\tilde G(s) = s\,M_{g}(\vect{x},s)\) for the odd extensions,
for \(0<r<ct\), the oddness having been used to replace \(\tilde F(r-ct)\) by \(-\tilde F(ct-r)\). Now \(u(\vect{x},t) = \lim_{r\to0^{+}}M(r,t) = \lim_{r\to0^{+}}W(r,t)/r\). The first term of Equation (10.59) divided by \(r\) tends to \(\tilde F'(ct)\) by the definition of the derivative, and the second to \(\tilde G(ct)/c\) by continuity of \(\tilde G\). Hence \(u = \tilde F'(ct) + t\,M_{g}(\vect{x},ct)\), and since \(\pp_{t}\left[t\,M_{f}(\vect{x},ct)\right] = M_{f}(ct) + ct\,\pp_{s}M_{f}(s)\bigl|_{s=ct} = \tilde F'(ct)\), this is Equation (10.57).
∎The value of \(u\) at \((\vect{x},t)\) depends on the initial data only through their values on the sphere \(\abs{\vect{y}-\vect{x}} = ct\), and not at all on their values inside it. A signal emitted at one instant from one point is therefore received at one instant, with no trailing wake. Rests on Theorem 10.59 and Corollary 10.12.
Derives Corollary 10.60. Every term of Equation (10.57) is an average over the sphere of radius \(ct\), or a \(t\)-derivative of such an average, and no integration over the interior occurs.
∎Huygens stated the sharp propagation of light as a construction of wave fronts [Huygens:1690]; Kirchhoff turned it into the formula Equation (10.57) [Kirchhoff:1883], and Hadamard identified the property as special — it holds for the wave operator in three space dimensions and fails in general [Hadamard:1923]. It is what makes hearing and vision possible: sound arrives when it arrives, and the acoustic and optical consequences are drawn in Oscillations and Mechanical Waves and Electromagnetic Waves and Optics. The support of the solution on the characteristic cone of Corollary 10.12 is the same statement in the language of the classification.
The heat equation
Let \(f\) be bounded and continuous on \(\R\). Then
satisfies \(\int_{\R}K(x,t)\dd x = 1\) for every \(t>0\), and \(u = K(\cdot,t) * f\) solves Equation (10.23) on \(\R\times(0,\infty)\) with \(u(\cdot,t) \to f\) pointwise as \(t \to 0^{+}\). In three dimensions the kernel is the product of three such factors, \(K_{3}(\vect{x},t) = (4\pi Dt)^{-3/2}\exp\left(-\abs{\vect{x}}^{2}/4Dt\right)\). Rests on Proposition 10.39, Equation (17.42) and Theorem 17.53.
Derives Theorem 10.62. By Proposition 10.39 with \(P(\pp_{x}) = D\pp_{x}^{2}\), the transform of the solution is
The convolution theorem Equation (17.66) says that a product of transforms is \((2\pi)^{-1/2}\) times the transform of the convolution, so \(u = K*f\) with \(\hat K(\xi) = (2\pi)^{-1/2}\ee^{-D t\xi^{2}}\). The Gaussian pair Equation (17.42) with \(a = \sqrt{2Dt}\) reads \(\widehat{\ee^{-x^{2}/(4Dt)}}(\xi) = \sqrt{2Dt}\,\ee^{-Dt\xi^{2}}\), whence \(K = (2\pi)^{-1/2}(2Dt)^{-1/2}\ee^{-x^{2}/(4Dt)}\), which is Equation (10.60). Its integral is \(\hat K(0)\sqrt{2\pi} = 1\). That \(u\) solves the equation may now be checked directly: \(K\) is smooth for \(t>0\) and differentiation under the integral sign is justified by the Gaussian decay, while \(\pp_{t}K = D\pp_{x}^{2}K\) is a computation from Equation (10.60). For the initial values, \(K(\cdot,t)\) is a nonnegative kernel of unit mass concentrating on the origin, so \((K*f)(x) \to f(x)\) wherever \(f\) is continuous. In three dimensions the transformed solution is \(\hat f\,\ee^{-D\abs{\vect{\xi}}^{2}t}\), which factorises over the three coordinates, and so does the kernel.
∎Let \(f \ge 0\) be continuous, not identically zero, and supported in a bounded set. Then the solution of Equation (10.23) given by Theorem 10.62 satisfies \(u(x,t)>0\) for every \(x \in \R\) and every \(t>0\). Rests on Theorem 10.62.
Derives Proposition 10.63. \(u(x,t) = \int_{\R}K(x-y,t)f(y)\dd y\) with \(K>0\) everywhere for \(t>0\) by Equation (10.60), and \(f \ge 0\) is strictly positive on some interval. The integrand is therefore nonnegative and strictly positive on a set of positive measure.
∎The disturbance is felt everywhere immediately, in flat contradiction with Corollary 10.56 for the wave equation. The contradiction is with relativity, not with observation: at distance \(x\) and time \(t\) the response is suppressed by \(\exp\left(-x^{2}/4Dt\right)\), which for any laboratory value of \(D\) is unmeasurably small outside the diffusion length. The heat equation is a macroscopic limit of a kinetic description with a finite mean free path (Nonequilibrium Thermodynamics and Transport), and the limit is where the infinite speed comes from.
The kernel Equation (10.60) is a Gaussian of variance \(2Dt\), so a point release spreads as
and the diffusion length grows as \(\sqrt{t}\), not as \(t\): a random walk, not a flight. With \(D\) in \(\mathrm{m}^{2}/\mathrm{s}\) the dimensions are those of Equation (10.62) as written. This is Einstein's relation for Brownian motion, whose molecular origin and measurement are Experiment: Brownian Motion and Avogadro's Number. Time enters Equation (10.60) only through \(Dt\) and only for \(t>0\): the equation is not invariant under \(t \to -t\), and running it backwards is the ill-posed problem of Proposition 10.25.
The Laplace and Poisson equations
A function \(u \in C^{2}(\Omega)\) is harmonic on \(\Omega\) if and only if \(\nabla^{2}u = 0\) there; it is subharmonic if \(\nabla^{2}u \ge 0\) and superharmonic if \(\nabla^{2}u \le 0\). Rests on Equation (10.24) and Definition 7.66.
In \(\R^{3}\) the function
is harmonic away from the origin and satisfies \(\nabla^{2}\Phi = \delta(\vect{x})\) in the sense of distributions. Rests on Lemma 10.44 and Definition 17.38.
Derives Proposition 10.67. For \(r = \abs{\vect{x}} > 0\) the radial part of Equation (10.33) gives \(\nabla^{2}r^{-1} = r^{-2}\pp_{r}\left(r^{2}\pp_{r}r^{-1}\right) = r^{-2}\pp_{r}(-1) = 0\), so \(\Phi\) is harmonic there. Let \(\varphi\) be a test function (Definition 17.38). Then
the integral converging absolutely because \(\Phi\) is integrable near the origin. On the region \(r>\varepsilon\) apply Equation (10.41) with \(p=1\), \(u=\varphi\), \(v=\Phi\); the term \(\varphi\nabla^{2}\Phi\) vanishes there, and only the inner boundary contributes, with outward normal \(-\vect{n}\) pointing towards the origin. The two surface terms are
the first because the sphere has area \(4\pi\varepsilon^{2}\) and \(\nabla\varphi\) is bounded, the second because the factor is exactly the reciprocal of that area and \(\varphi\) is continuous. Hence Equation (10.64) equals \(\varphi(\vect{0})\), which by Equation (17.50) says \(\nabla^{2}\Phi = \delta\).
∎Combining Proposition 10.67 with Proposition 10.43, the solution of \(\nabla^{2}u = -\varrho\) on all of \(\R^{3}\) decaying at infinity is \(u(\vect{x}) = \frac{1}{4\pi}\int\varrho(\vect{x}') \abs{\vect{x}-\vect{x}'}^{-1}\dd^{3}x'\): the superposition of \(1/r\) potentials that Laplace and Poisson wrote down [Laplace:1799], [Poisson:1813], and the electrostatic potential of a charge density once \(\varrho\) is \(\rho/\varepsilon_{0}\) (Electrostatics).
Let \(u\) be harmonic on \(\Omega\) and let \(\overline{B(\vect{x},r)} \subset \Omega\). Then
Conversely a function of class \(C^{2}\) with the first of these properties for every admissible ball is harmonic. Rests on Lemma 10.58 and Definition 10.66.
Derives Theorem 10.69. By Equation (10.55), \(\pp_{r}M_{u}(\vect{x},r)\) is \((4\pi r^{2})^{-1}\) times the integral of \(\nabla^{2}u\) over the ball, which vanishes. So \(M_{u}(\vect{x},\cdot)\) is constant on \((0,r]\), and its limit as the radius tends to zero is \(u(\vect{x})\): that is the first equality. Multiplying it by \(4\pi s^{2}\) and integrating over \(s \in [0,r]\) converts the spherical average into the solid one, since \(\int_{0}^{r}4\pi s^{2}\dd s = \tfrac{4}{3}\pi r^{3}\): that is the second. For the converse, if \(\nabla^{2}u(\vect{x}_{0}) > 0\) then by continuity \(\nabla^{2}u>0\) on a small ball, and Equation (10.55) makes \(\pp_{r}M_{u}>0\) there, so \(M_{u}\) is not constant — contradicting the mean-value property; the case \(\nabla^{2}u<0\) is symmetric. Both \(\nabla^{2}u(\vect{x}_{0})\) and the appeal to Equation (10.55) use the hypothesis \(u \in C^{2}\), which is why the converse is stated for that class.
∎The converse in Theorem 10.69 is in fact true for a function only assumed continuous, but not by the argument above: that argument decides the sign of a Laplacian whose existence is what is to be proved. The route taken in the appendix first establishes smoothness — averaging a continuous function with the mean-value property against a radial bump reproduces the function itself, exactly and on every compact subset, so it is automatically infinitely differentiable — and only then runs the sign argument. That smoothing step needs a mollifier, which is built there from scratch, together with the one calculus lemma it consumes; nothing is quoted, and the whole construction stays inside the Riemann integral of Real Analysis, every integrand being continuous on a compact region. Everything the chapter subsequently uses, including Theorem 10.77, needs only the \(C^{2}\) form; what the continuous form adds is that the mean-value property may be taken as the definition of harmonicity, and with it the stability of the class under locally uniform limits.
Full derivation in Appendix A.
Derives Theorem 10.69.
Let \(V\) be a bounded region with piecewise-smooth boundary and let \(u \in C^{2}(\overline{V})\). Then for \(\vect{x}\) in the interior,
with \(\Phi\) the Newtonian potential Equation (10.63) and the normal derivatives taken in \(\vect{y}\). Rests on Proposition 10.67 and Lemma 10.44.
Derives Proposition 10.71. Apply Equation (10.41) with \(p = 1\) on the region \(V\) punctured by a ball of radius \(\varepsilon\) about \(\vect{x}\), where \(\Phi(\cdot-\vect{x})\) is harmonic. The inner boundary contributes, in the limit \(\varepsilon\to0\), exactly the two terms evaluated in Equation (10.65), the first vanishing and the second producing \(-u(\vect{x})\) with the orientation of the punctured region; rearranging gives Equation (10.67).
∎Equation (10.67) is not yet a solution of the Dirichlet problem, because it uses both \(u\) and \(\pp_{\nu}u\) on the boundary while the problem prescribes only the first. Adding to \(\Phi\) a harmonic correction that cancels it on \(\pp V\) produces the Green's function of Definition 10.42 and removes the unwanted term — Equation (10.45) is the case where the correction can be written down. That such a correction always exists is the solvability of the Dirichlet problem, argued by Riemann from the existence of a minimiser of the energy \(\int\abs{\nabla v}^{2}\), criticised by Weierstrass for assuming the minimum attained, and rehabilitated by Hilbert [Hilbert:1904]. The variational side of the story is Calculus of Variations, and the modern form of the argument is Section 10.6.2.
The Helmholtz decomposition
The Newtonian potential solves Poisson's equation one scalar at a time. Applied to each Cartesian component of a vector field it yields a structural statement about vector fields themselves, and it is the statement most often wanted in physics: that a field splits into a part carrying all of its divergence and a part carrying all of its curl. That statement is Theorem 7.105, and this chapter is where its engine lives: each Cartesian component of the vector potential Equation (7.130) is a Newtonian potential, so \(\nabla^{2}\vect{w}=\vect{u}\) by Proposition 10.67, and the identity \(\nabla\times(\nabla\times\vect{w}) =\nabla(\nabla\cdot\vect{w})-\nabla^{2}\vect{w}\) read backwards turns that Laplacian into a gradient plus a curl. The analysis the step needs — convergence of the singular integral, differentiation under it, and the decay that forces uniqueness — is The Helmholtz Decomposition.
Theorem 7.105 is stated for a field that is smooth and of compact support, and both words are load-bearing: smoothness is inherited by the potentials, which the theorem asserts to be smooth as well, and compact support is what makes the singular integral converge and lets it be differentiated under the sign. Compact support is nonetheless a convenience rather than a necessity, and Remark A.527 says exactly how much of it is needed: the same argument runs for a smooth field decaying faster than \(\abs{\vect{x}}^{-2}\) whose first derivatives decay faster than \(\abs{\vect{x}}^{-3}\), that is, for some \(\varepsilon>0\),
What that relaxation does not buy is a drop in differentiability: the proof obtains the potentials from Proposition A.521, which returns \(C^{k}\) potentials for a \(C^{k}\) field, so a merely \(C^{2}\) field would give a \(C^{1}\) vector potential and not the smooth one the theorem states. Some condition at infinity is indispensable, and the failure is immediate rather than technical — the constant field \(\hat{\vect{z}}\) is both a gradient and a curl, so the split is not unique for it. That case is worked in the same appendix remark. Rests on Theorem 7.105 and Proposition 10.67.
Theorem 7.105 is the reason a wave in an elastic solid splits into one longitudinal and two transverse polarizations (Continuum Mechanics and Elasticity): the irrotational part and the solenoidal part obey wave equations with different speeds, so a general disturbance separates into a pressure wave and a shear wave. It is also what lies behind the Coulomb gauge of The Maxwell Equations, where \(\vect{A}\) is chosen divergence-free, and behind the split of a fluid velocity field into a potential flow and a vortical part (Fluid Dynamics). Some hypothesis at infinity is not decoration: without one the uniqueness fails outright, since a constant field is both a gradient and a curl, and on a bounded region the statement acquires boundary terms and a third, harmonic, summand. Both the compact support of Theorem 7.105 and the rate Equation (10.68) are convenient sufficient conditions, and both are comfortably met by the fields the physical chapters decompose, which are compactly supported or exponentially localised. Sharper hypotheses exist and are not needed here. Rests on Theorem 7.105.
Maximum principles
Elliptic maximum principles
Let \(\Omega \subset \R^{3}\) be bounded and open and let \(u \in C^{2}(\Omega)\) be continuous on \(\overline{\Omega}\) with \(\nabla^{2}u \ge 0\). Then
The same holds for the minimum of a superharmonic function, and a harmonic function attains both its extremes on the boundary. Rests on Definition 10.66 and Theorem 6.11.
Derives Theorem 10.75. Suppose first that \(\nabla^{2}u > 0\) strictly. If \(u\) attained an interior maximum at \(\vect{x}_{0}\), then each second pure derivative \(\pp_{i}^{2}u(\vect{x}_{0})\) would be non-positive, since \(x^{i} \longmapsto u\) has an interior maximum along each coordinate line; summing, \(\nabla^{2}u(\vect{x}_{0}) \le 0\), a contradiction. So the maximum is attained on the boundary, which is where it must be, both sets being compact and \(u\) continuous.
In general set \(u_{\varepsilon} = u + \varepsilon\abs{\vect{x}}^{2}\) for \(\varepsilon>0\). Then \(\nabla^{2}u_{\varepsilon} = \nabla^{2}u + 6\varepsilon > 0\), so the strict case applies: \(\max_{\overline{\Omega}}u_{\varepsilon} = \max_{\pp\Omega}u_{\varepsilon}\). Since \(\Omega\) is bounded there is \(R\) with \(\abs{\vect{x}} \le R\) on \(\overline{\Omega}\), and therefore
Letting \(\varepsilon \to 0\) gives \(\le\) in Equation (10.69), and the reverse inequality is trivial. The minimum statement follows on replacing \(u\) by \(-u\).
∎Let \(\Omega\) be bounded and let \(u_{1},u_{2}\) solve \(\nabla^{2}u_{i} = f\) in \(\Omega\) with boundary values \(g_{i}\). Then
In particular the solution is unique, and the Dirichlet problem is well posed in the sense of Definition 10.24 with respect to the supremum norm. Rests on Theorem 10.75 and Definition 10.24.
Derives Corollary 10.76. \(w = u_{1}-u_{2}\) is harmonic, so by Equation (10.69) applied to \(w\) and to \(-w\) its maximum and its minimum are attained on \(\pp\Omega\); hence \(\max\abs{w}\) is attained there and equals \(\max_{\pp\Omega}\abs{g_{1}-g_{2}}\). With \(g_{1}=g_{2}\) this gives \(w \equiv 0\).
∎Let \(\Omega\) be open and connected and let \(u\) be harmonic on \(\Omega\). If \(u\) attains its supremum at some point of \(\Omega\), then \(u\) is constant. Rests on Theorem 10.69 and Definition 6.13.
Derives Theorem 10.77. Let \(M = \sup_{\Omega}u\) and \(E = \set{\vect{x}\in\Omega \mid u(\vect{x}) = M}\), nonempty by hypothesis and closed in \(\Omega\) by continuity. Let \(\vect{x}_{0} \in E\) and let \(\overline{B(\vect{x}_{0},r)} \subset \Omega\). By the solid mean-value property Equation (10.66), \(M = u(\vect{x}_{0})\) is the average of \(u\) over the ball, and \(u \le M\) there; a continuous function no greater than its own average must equal it everywhere on the ball, since otherwise \(u<M\) on a nonempty open subset and the average would be strictly less than \(M\). So \(B(\vect{x}_{0},r) \subseteq E\) and \(E\) is open. Being nonempty, open and closed in the connected set \(\Omega\) (Definition 6.13), \(E = \Omega\).
∎Hopf proved the strong principle for general second-order elliptic operators, without the mean-value property, by a barrier construction [Hopf:1927]. The boundary-point lemma that usually travels with it — at a boundary point where the maximum is attained, the outward normal derivative is strictly positive unless the solution is constant — is not in that paper, whose content is the strong principle alone; Hopf published the boundary lemma a quarter of a century later [Hopf:1952], and Oleinik obtained it independently the same year, in a paper this bibliography does not carry (Matematicheskii Sbornik 30(72) (1952) 695–702). The physical reading of both results is that a static temperature or potential field has no interior hot spot: an extreme in the interior requires a source there, and where there is no source the field is controlled entirely by its boundary values. It is also why Equation (10.66) is sometimes taken as the definition of harmonicity — legitimately, by the converse of Theorem 10.69 for twice-differentiable functions and, with the smoothing step quoted in Remark 10.70, for merely continuous ones.
Parabolic maximum principles
For \(\Omega \subset \R^{3}\) bounded and open and \(T>0\) put \(\Omega_{T} = \Omega\times(0,T]\). The parabolic boundary is
the initial slice together with the lateral wall. The final slice \(\Omega\times\set{T}\) belongs to \(\Omega_{T}\) and not to \(\Gamma_{T}\). Rests on Definitions 6.3 and 10.6.
The asymmetry is the content: a parabolic problem is posed on the parabolic boundary and predicts the final slice, never the reverse.
Let \(u\) be continuous on \(\overline{\Omega_{T}}\), with \(\pp_{t}u\) and the second space derivatives continuous in \(\Omega_{T}\), and suppose \(\pp_{t}u - D\nabla^{2}u \le 0\) there. Then
Rests on Definition 10.79 and Theorem 10.75.
Derives Theorem 10.80. Assume first the strict inequality \(\pp_{t}u - D\nabla^{2}u < 0\) in \(\Omega_{T}\), and suppose the maximum over the compact set \(\overline{\Omega_{T}}\) were attained at a point \((\vect{x}_{0},t_{0}) \in \Omega_{T}\), so \(\vect{x}_{0}\) interior and \(0 < t_{0} \le T\). Spatially the point is an interior maximum, so \(\nabla^{2}u(\vect{x}_{0},t_{0}) \le 0\) by the argument of Theorem 10.75. In time, if \(t_{0}<T\) then \(\pp_{t}u = 0\) there, while if \(t_{0}=T\) then \(\pp_{t}u \ge 0\), because \(u\) cannot decrease on approach to a maximum at the end of the interval. Either way \(\pp_{t}u - D\nabla^{2}u \ge 0\), contradicting the strict inequality.
For the general case put \(u_{\varepsilon} = u - \varepsilon t\) with \(\varepsilon>0\). Then \(\pp_{t}u_{\varepsilon} - D\nabla^{2}u_{\varepsilon} = \left(\pp_{t}u - D\nabla^{2}u\right) - \varepsilon < 0\), so \(\max_{\overline{\Omega_{T}}}u_{\varepsilon} = \max_{\Gamma_{T}}u_{\varepsilon}\), whence \(\max_{\overline{\Omega_{T}}}u \le \max_{\Gamma_{T}}u + \varepsilon T\). Let \(\varepsilon \to 0\).
∎The initial-boundary-value problem \(\pp_{t}u = D\nabla^{2}u + f\) in \(\Omega_{T}\), with \(u\) prescribed on \(\Gamma_{T}\), has at most one solution, and two solutions whose data differ by at most \(\varepsilon\) on \(\Gamma_{T}\) differ by at most \(\varepsilon\) everywhere. Rests on Theorem 10.80 and Definition 10.24.
Derives Corollary 10.81. The difference \(w\) of two solutions satisfies \(\pp_{t}w - D\nabla^{2}w = 0\), so Equation (10.73) applies to \(w\) and to \(-w\), giving \(\max_{\overline{\Omega_{T}}}\abs{w} = \max_{\Gamma_{T}}\abs{w}\).
∎Theorem 10.80 says that the extremes of a temperature field can only shrink: no interior point ever becomes hotter than the hottest point of the initial datum or of the wall. Together with Remark 10.37 this is the analytic face of the irreversibility of diffusion — the equation has a preferred direction of time built into its type, and the statistical account of why the world's diffusion equations run that way rather than the other is Statistical Mechanics. The analogy should not be pressed into an identification: the maximum principle is a theorem about a parabolic PDE, while entropy growth is a statement about ensembles.
Weak solutions
Distributional solutions and Sobolev spaces
Every construction above has assumed a classical solution, differentiable as many times as the equation demands. Three things force that requirement to be relaxed: the fundamental solutions themselves are not twice differentiable at the source; the data of a physical problem may be discontinuous; and, as Section 10.6.3 shows, a perfectly smooth initial condition may produce a solution that ceases to be differentiable in finite time. The remedy is to move the derivatives off the solution and onto a test function.
Let \(L = \sum_{\abs{\alpha}\le m}a_{\alpha}\pp^{\alpha}\) have smooth coefficients and let its formal adjoint be
A locally integrable \(u\) is a weak solution of \(Lu = f\) on \(\Omega\) if and only if
for every smooth \(\varphi\) with compact support in \(\Omega\). Rests on Definitions 10.3 and 17.38.
Every classical solution is a weak solution. Conversely a weak solution of class \(C^{m}\) is a classical solution. Rests on Definition 10.83 and Equation (7.28).
Derives Proposition 10.84. For \(u \in C^{m}\), integrating by parts \(\abs{\alpha}\) times (Equation (7.28), applied in each variable) moves each \(\pp^{\alpha}\) from \(u\) onto \(a_{\alpha}\varphi\) with a sign \((-1)^{\abs{\alpha}}\); the boundary terms vanish because \(\varphi\) has compact support. Hence \(\int u L^{*}\varphi = \int (Lu)\varphi\), and Equation (10.75) says \(\int (Lu-f)\varphi = 0\) for all test \(\varphi\). It remains to show that a continuous \(h\) with \(\int h\varphi = 0\) for all such \(\varphi\) vanishes. If \(h(\vect{x}_{0}) > 0\), continuity gives a ball on which \(h > h(\vect{x}_{0})/2\); choosing \(\varphi \ge 0\) smooth, supported in that ball and not identically zero makes \(\int h\varphi > 0\), a contradiction, and \(h(\vect{x}_{0})<0\) is symmetric. This proves both directions at once.
∎For \(\Omega \subseteq \R^{N}\) open, \(H^{1}(\Omega)\) is the space of square-integrable \(u\) whose first-order distributional derivatives \(\pp_{i}u\) are again square-integrable, with the norm
and \(H^{1}_{0}(\Omega)\) is the closure in this norm of the smooth functions of compact support — the functions “vanishing on \(\pp\Omega\)” in the only sense available. Rests on Definitions 10.83 and 12.2.
\(H^{1}\) is complete, hence a Hilbert space (Hilbert Spaces); the generalised derivatives are Sobolev's, introduced precisely to handle the Cauchy problem for hyperbolic equations [Sobolev:1936], and the systematic theory in which they and the Dirac distribution are the same kind of object is Schwartz's [Schwartz:1950], developed in this treatise at Section 17.2.3. In that language every Green's function of Section 10.3 is a genuine solution of its equation: \(\Phi\) of Equation (10.63) has no second derivative at the origin as a function, yet Proposition 10.67 exhibits \(\nabla^{2}\Phi\) as the Dirac distribution, and \(G_{\pm}\) of Equation (10.46) is not a function at all. A fundamental solution is a distribution or it is nothing.
Variational formulation
For \(p,q\) bounded and measurable on a bounded \(\Omega\), define on \(H^{1}_{0}(\Omega)\) the bilinear form and the linear functional
A weak solution of \(-\nabla\cdot(p\nabla u) + qu = f\) with vanishing boundary values is a \(u \in H^{1}_{0}(\Omega)\) with \(a(u,v) = \ell(v)\) for every \(v \in H^{1}_{0}(\Omega)\). Rests on Definitions 10.83 and 10.85.
The form is obtained from the equation by multiplying by \(v\) and integrating by parts once: the divergence theorem Equation (7.123) moves one derivative from \(u\) to \(v\) and produces a boundary term \(\oint p\,v\,\pp_{\nu}u\), which vanishes because \(v\) vanishes on the boundary. Only one derivative of \(u\) now appears, which is why \(H^{1}\) is the natural space.
Let \(H\) be a real Hilbert space and \(a : H\times H \to \R\) bilinear with
for constants \(C\) and \(\alpha>0\), and let \(\ell\) be a bounded linear functional on \(H\). Then there is exactly one \(u \in H\) with \(a(u,v) = \ell(v)\) for every \(v \in H\), and \(\norm{u} \le \norm{\ell}/\alpha\). Rests on Definitions 10.87 and 12.2.
Derives Theorem 10.88. For fixed \(u\) the map \(v \longmapsto a(u,v)\) is a bounded linear functional, so by the Riesz representation theorem (Hilbert Spaces) there is a unique \(Au \in H\) with \(a(u,v) = \gen{Au,v}\) for all \(v\); uniqueness makes \(A\) linear, and the first hypothesis of Equation (10.78) gives \(\norm{Au} \le C\norm{u}\).
Coercivity gives \(\alpha\norm{u}^{2} \le a(u,u) = \gen{Au,u} \le \norm{Au}\,\norm{u}\), hence
So \(A\) is injective. Its range is closed: if \(Au_{n} \to w\) then Equation (10.79) applied to \(u_{n}-u_{m}\) shows \((u_{n})\) is Cauchy, so \(u_{n} \to u\) by completeness, and \(Au = w\) by continuity. The range is also dense: if \(w \perp A(H)\) then in particular \(\gen{Aw,w} = 0\), so \(\alpha\norm{w}^{2} \le a(w,w) = 0\) and \(w = 0\). A closed dense subspace is the whole space, so \(A\) is a bijection.
Riesz again gives \(F\) with \(\ell(v) = \gen{F,v}\), and \(u = A^{-1}F\) is the unique solution. Finally \(\alpha\norm{u}^{2} \le a(u,u) = \ell(u) \le \norm{\ell}\,\norm{u}\).
∎If \(p \ge p_{0} > 0\) and \(q \ge q_{0} > 0\) almost everywhere and \(f\) is square-integrable, the problem of Definition 10.87 has exactly one weak solution \(u \in H^{1}_{0}(\Omega)\), and \(\norm{u}_{H^{1}} \le \norm{f}/\min(p_{0},q_{0})\). Rests on Theorem 10.88 and Definition 10.87.
Derives Corollary 10.89. Boundedness of \(a\) follows from the boundedness of \(p\) and \(q\) and the Cauchy–Schwarz inequality; coercivity is immediate, \(a(u,u) \ge \min(p_{0},q_{0})\norm{u}_{H^{1}}^{2}\) by Equation (10.76); and \(\ell\) is bounded because \(\abs{\ell(v)} \le \norm{f}\,\norm{v} \le \norm{f}\,\norm{v}_{H^{1}}\). Theorem 10.88 applies with \(\alpha = \min(p_{0},q_{0})\).
∎The hypothesis \(q \ge q_{0}>0\) was used only to control the term \(\int u^{2}\) in Equation (10.76). For the Dirichlet problem proper, \(q = 0\), coercivity instead follows from the Poincaré inequality — on a bounded \(\Omega\) the \(H^{1}_{0}\) norm is equivalent to \(\norm{\nabla u}\) alone — and the conclusion is unchanged.
Let \(a\) be symmetric in addition to the hypotheses of Theorem 10.88, and set \(J(v) = \tfrac{1}{2}a(v,v) - \ell(v)\). Then \(u\) is the weak solution if and only if \(u\) minimises \(J\) over \(H\). Rests on Theorem 10.88 and Definition 10.87.
Derives Proposition 10.91. For \(v \in H\) and \(t \in \R\), bilinearity and symmetry give
If \(a(u,v) = \ell(v)\) for all \(v\), the linear term vanishes and \(a(v,v) \ge \alpha\norm{v}^{2} \ge 0\), so \(J(u+tv) \ge J(u)\) with equality only for \(v=0\): \(u\) is the strict minimiser. Conversely if \(u\) minimises then the derivative of Equation (10.80) at \(t=0\) vanishes for every \(v\), which is \(a(u,v) = \ell(v)\).
∎The three conclusions of Theorem 10.88 are exactly Hadamard's three conditions (Definition 10.24): existence, uniqueness, and the energy estimate \(\norm{u} \le \norm{\ell}/\alpha\), which is continuous dependence because the map \(\ell \longmapsto u\) is linear. Coercivity is thus the hypothesis that carries well-posedness, and the Euler–Lagrange correspondence of Proposition 10.91 is the one developed in general in Calculus of Variations. The functional-analytic ingredients — completeness, the Riesz theorem, orthogonality — are Hilbert Spaces.
Shocks and conservation laws
Weak solutions were introduced above to accommodate irregular data. For nonlinear equations they are unavoidable for a stronger reason: the classical solution can destroy itself.
Given a flux \(F \in C^{1}(\R)\), the associated conservation law is
equivalently \(\pp_{t}u + F'(u)\,\pp_{x}u = 0\) for smooth \(u\). The case \(F(u) = \tfrac{1}{2}u^{2}\), that is \(\pp_{t}u + u\,\pp_{x}u = 0\), is Burgers' equation. Rests on Definition 10.2 and Proposition 10.15.
The name is the content: integrating Equation (10.81) over an interval shows that \(\int u\,\dd x\) changes only through the flux at the two ends, so \(u\) is the density of a conserved quantity.
Let \(u_{0} \in C^{1}(\R)\) and let \(u\) solve Burgers' equation with \(u(\cdot,0)=u_{0}\). Then \(u\) is constant along each straight characteristic \(x = x_{0} + u_{0}(x_{0})\,t\), and
If \(u_{0}' < 0\) somewhere, no \(C^{1}\) solution exists beyond
Rests on Proposition 10.15 and Definition 10.93.
Derives Proposition 10.94. By Proposition 10.15 with \(a=1\) in the time variable, \(b = u\) and \(c = 0\), the solution is constant along the curves \(\dd x/\dd t = u\); being constant there, those curves have constant slope and are the stated straight lines. Hence \(u\left(x_{0}+u_{0}(x_{0})t,\,t\right) = u_{0}(x_{0})\). Differentiating this identity with respect to \(x_{0}\),
which is Equation (10.82). Where \(u_{0}'(x_{0})<0\) the denominator vanishes at \(t = -1/u_{0}'(x_{0})\), and the earliest such time is Equation (10.83); as \(t \to t_{*}^{-}\) the slope diverges. Equivalently, two characteristics with \(u_{0}\) decreasing between them meet at that time, and the classical solution would have to take two values at once.
∎A bounded measurable \(u\) on \(\R\times[0,\infty)\) is a weak solution of Equation (10.81) with datum \(u_{0}\) if and only if
for every smooth \(\varphi\) of compact support in \(\R\times[0,\infty)\). Rests on Definitions 10.83 and 10.93.
Let \(u\) be a weak solution which is \(C^{1}\) on either side of a \(C^{1}\) curve \(x = s(t)\) and has one-sided limits \(u_{\pm}\) across it. Then at every point of the curve
Rests on Definition 10.95 and Theorem 7.99.
Derives Theorem 10.96. Let \(\varphi\) be supported in a small disc \(B\) meeting the curve and no other singularity, and away from \(t=0\). Split \(B\) into \(B_{-} = B\cap\set{x<s(t)}\) and \(B_{+} = B\cap\set{x>s(t)}\). On \(B_{-}\),
and the last integral vanishes because \(u\) is a classical solution there. The first is a divergence in the plane \((t,x)\), so by the divergence form of Green's theorem (Theorem 7.99) it equals the flux of \(\left(u\varphi,\ F(u)\varphi\right)\) through \(\pp B_{-}\); only the part of the boundary lying on the curve contributes, since \(\varphi\) vanishes on \(\pp B\). There the outward normal of \(B_{-}\) is proportional to \(\left(-\dot s,\,1\right)\) and the arclength element is \(\sqrt{1+\dot s^{2}}\,\dd t\), so the contribution is \(\int\left[F(u_{-}) - \dot s\,u_{-}\right]\varphi\,\dd t\). The region \(B_{+}\) gives the same expression with \(u_{+}\) and the opposite sign. Adding, and using Equation (10.85),
for every such \(\varphi\), and the bracket, being continuous, must vanish identically; that is Equation (10.86).
∎For \(F(u)=u^{2}/2\), Equation (10.86) gives
the shock travels at the arithmetic mean of the states it separates. Riemann obtained the same relation for finite-amplitude sound waves and identified the steepening that produces it [Riemann:1860]; the jump conditions across a shock in a compressible fluid are Rankine's [Rankine:1870] and Hugoniot's [Hugoniot:1889], and the gas-dynamical form is developed in Fluid Dynamics. Rests on Theorem 10.96 and Definition 10.93.
Take Burgers' equation with \(u_{0}=0\) for \(x<0\) and \(u_{0}=1\) for \(x>0\). Both of
are weak solutions in the sense of Definition 10.95: the first is continuous and solves the equation classically wherever it is smooth, and the second satisfies Equation (10.89) with \(u_{-}=0\), \(u_{+}=1\). The weak formulation alone therefore does not determine the evolution. Rests on Theorem 10.96 and Definition 10.95.
A discontinuity of a weak solution of Equation (10.81) is admissible if and only if
that is, if the characteristics on both sides run into the discontinuity as time increases. For Burgers' equation this reads \(u_{-} > u_{+}\). Rests on Theorem 10.96 and Definition 10.95.
In Example 10.98 the jump of \(u_{\mathrm{s}}\) has \(u_{-}=0 < 1 = u_{+}\) and is inadmissible, so the entropy condition rejects it and keeps the rarefaction \(u_{\mathrm{r}}\). The criterion is not an extra postulate but a limit: adding a viscous term \(\varepsilon\,\pp_{x}^{2}u\) to Equation (10.81) makes the problem parabolic and its solution unique, and as \(\varepsilon \to 0^{+}\) the solutions converge to the weak solution satisfying Equation (10.91) and to no other [Courant:1962]. The name records that for the equations of gas dynamics the admissible jumps are exactly those across which the entropy of a fluid element does not decrease, which is why a compression shock exists in nature and a rarefaction shock does not (Fluid Dynamics).
An entropy pair for Equation (10.81) is a pair \((\eta,q)\) with \(\eta\) of class \(C^{2}\) and convex and \(q\) of class \(C^{1}\) with \(q' = \eta'F'\). A weak solution \(u\) in the sense of Definition 10.95 is an entropy solution if and only if
for every entropy pair and every non-negative smooth \(\varphi\) of compact support in \(\R\times[0,\infty)\); briefly, \(\pp_{t}\eta(u)+\pp_{x}q(u) \le 0\) in the sense of distributions. Rests on Definitions 10.95 and 10.99.
Equation (10.91) is a statement about the one-sided limits across a curve of discontinuity, so it presupposes that those limits exist. A bounded measurable weak solution need not be piecewise \(C^{1}\) and its set of discontinuities need not be a union of curves at all, so for such a solution the jump condition selects nothing; Equation (10.92) is meaningful for every weak solution and says the same thing wherever both do.
Let \(F\) be convex and let \(u\) be a weak solution which is \(C^{1}\) on either side of a \(C^{1}\) curve \(x = s(t)\), with one-sided limits \(u_{\pm}\). Then Equation (10.92) holds for every test function supported near the curve and away from \(t=0\) if and only if
for every entropy pair; and for a strictly convex flux that is in turn equivalent to \(u_{-} \ge u_{+}\), hence to Equation (10.91). Rests on Definition 10.101 and Theorem 10.96.
Derives Proposition 10.102. Off the curve \(u\) is a classical solution, so there \(\pp_{t}\eta(u)+\pp_{x}q(u) = \eta'(u)\left[\pp_{t}u+F'(u)\pp_{x}u\right] = 0\). Repeating the computation of Theorem 10.96 with \(\bigl(\eta(u),q(u)\bigr)\) in place of \(\bigl(u,F(u)\bigr)\) therefore reduces the left-hand side of Equation (10.92) to \(\int\left\{q(u_{-})-q(u_{+}) - \dot s\left[\eta(u_{-})-\eta(u_{+})\right]\right\}\varphi\,\dd t\), and requiring that to be non-negative for every non-negative \(\varphi\) is Equation (10.93).
For the second equivalence write \(\sigma = \dd s/\dd t\) and
the difference between the flux and the chord joining the two states. It is convex and vanishes at \(v = u_{-}\) and, by Equation (10.86), at \(v = u_{+}\), so \(G \le 0\) between them. Since \(q' = \eta'F'\) one has \(\dd\bigl[q(v)-\sigma\eta(v)\bigr]/\dd v = \eta'(v)\,G'(v)\), and integrating that by parts between the two states, the boundary term vanishing with \(G\),
With \(\eta''\ge0\) and \(G\le0\) the integrand is non-positive on the interval between the two states, so the integral in Equation (10.95), whose limits run backwards precisely when \(u_{-} > u_{+}\), is non-negative exactly in that case; and for strictly convex \(F\) one has \(G<0\) strictly between the states, so a strictly convex \(\eta\) makes the inequality fail whenever \(u_{-}<u_{+}\). Finally \(F'\) is non-decreasing and, by Equation (10.86), \(\sigma\) is a mean value of \(F'\) between the two states, so Equation (10.91) is likewise equivalent to \(u_{-} \ge u_{+}\).
∎For a convex flux \(F\) and bounded measurable datum \(u_{0}\), the conservation law Equation (10.81) has exactly one entropy solution in the sense of Definition 10.101, and it depends continuously on \(u_{0}\) in the mean. Where that solution is \(C^{1}\) on either side of a \(C^{1}\) curve of discontinuity, its jumps satisfy Equation (10.91) by Proposition 10.102. Rests on Definitions 10.95 and 10.101.
Full derivation in Appendix A.
Derives Theorem 10.103.
The uniqueness half is Kruzhkov's doubling of variables, carried out there in full: the entropy inequality Equation (10.92) is first extended, by approximation from the \(C^{2}\) family admitted in Definition 10.101, to the entropies \(\eta(u)=\abs{u-k}\); each of two solutions is then tested against the other, in two sets of variables at once, and the \(L^{1}\) contraction is read off. Convexity of \(F\) turns out not to be needed for that half. The existence half is the vanishing-viscosity limit described above, whose entropy estimate is derived there and whose two compactness inputs are named. What the appendix does not build is the Lebesgue theory the subject is written in — a bounded measurable weak solution is not an object of Real Analysis — and its section on quoted inputs says so item by item.
With Theorem 10.103 the pattern of the chapter closes on itself. A nonlinear equation loses its classical solution in finite time (Proposition 10.94); weak solutions restore existence but cost uniqueness (Example 10.98); and an admissibility criterion drawn from the physics restores uniqueness and with it Hadamard's three conditions (Definition 10.24). Which class of solutions to admit is thus not a technical preference but part of the statement of the problem, and it is settled by asking what the equation is a model of.