Linear Evolution Equations

Abstract

Here we study linear PDE for which one poses an initial-value problem, also called a “Cauchy problem,” say at time t = t₀. The emphasis is on the wave and heat equations:

$$\frac{{\partial }^{2}u} {\partial {t}^{2}} -\Delta u = 0,\quad\frac{\partial u} {\partial t} -\Delta u = 0,$$

(0.1)

though some other sorts of PDE, such as symmetric hyperbolic systems, are also discussed.

Appendices

A Some Banach spaces of harmonic functions

If B is the unit ball in ${\mathbb{R}}^{k}$, consider the space ${\mathfrak{X}}_{j}$ of harmonic functions f on B such that

$${N}_{j}(f) {=\sup \limits_{x\in B}}\delta {(x)}^{j}\vert f(x)\vert$$

is finite, where δ(x) = 1 − |x| is the distance of x from ∂B. In case k = 2n and we identify ${\mathbb{R}}^{2n}$ with ${\mathbb{C}}^{n}$, via z_ℓ = x_ℓ + ix_{n + ℓ}, the space ${\mathfrak{H}}_{j}$ of holomorphic functions on B such that (A.1) is finite is a closed, linear subspace of ${\mathfrak{X}}_{j}$. For results in §3, it is useful both to know that

$$\frac{\partial } {\partial {z}_{{\ell}}} : {\mathfrak{H}}_{j}\rightarrow {\mathfrak{H}}_{j+1}$$

and to estimate its norm. It is just as convenient to estimate the norm of

$${\partial }_{{\ell}} : {\mathfrak{X}}_{j}\rightarrow {\mathfrak{X}}_{j+1},$$

where ∂_ℓ = ∂∕∂x_ℓ; then the desired estimate on (A.2) will follow from that on (A.3).

Given x ∈ B, let B_ρ(x) be the ball of radius ρ centered at x; take ρ ∈ (0, δ(x)). Then, as a consequence of the Poisson integral formula for functions harmonic on a ball (see (C.34) of Chap. 5), we have

$${\partial }_{{\ell}}u(x) =\frac{k - 1} {{\rho }^{2}}{\text{ Avg}}_{\partial {B}_{\rho }(x)}\ {\bigl\{({y}_{{\ell}} - {x}_{{\ell}})u(y)\bigr\}} $$

(A.4)

if u is harmonic on B. Now, for y ∈ ∂B_ρ(x),|y_ℓ − x_ℓ| ≤ ρ; furthermore, δ(y) ≥ δ(x) − ρ. If we take ρ = βδ(x), β ∈ (0,1), we obtain

$$\begin{array}{rcl}\vert {\partial }_{{\ell}}u(x)\vert &\leq &\frac{k - 1} {{\rho }^{2}}\cdot \rho \cdot {\bigl [ (1 -\beta )\delta {(x)}\bigr ]}^{-j}{N}_{ j}(u)\\ & =&\frac{k - 1} {\beta {(1 -\beta )}^{j}}\delta {(x)}^{-(j+1)}{N}_{ j}(u),\end{array}$$

(A.5)

and hence

$${N}_{j+1}({\partial }_{{\ell}}u)\leq \frac{k - 1} {\beta {(1 -\beta )}^{j}}\ {N}_{j}(u),$$

(A.6)

for $u\in{\mathfrak{X}}_{j}$. The factor on the right is minimized at β = 1∕(j + 1). Using the power series expansion of log(1 − ε), one readily verifies that

$${\Bigl (1 -{\frac{1} {j + 1}}\Bigr )}^{-j}\leq e,$$

so, for all j ≥ 0, $u\in{\mathfrak{X}}_{j}$,

$${N}_{j+1}({\partial }_{{\ell}}u)\leq{\gamma }_{k}\ (j + 1){N}_{j}(u),\quad {\gamma }_{k} = (k - 1)e.$$

Since ∂∕∂z_ℓ = (1∕2)(∂_ℓ − i∂_{n + ℓ}), we also have, for all j ≥ 0, $u\in{\mathfrak{H}}_{j}$,

$${N}_{j+1}{\Bigl (\frac{\partial u} {\partial {z}_{{\ell}}}\Bigr )}\leq{\gamma }_{2n}\ (j + 1){N}_{j}(u).$$

(A.8)

Note that repeated application of (A.7) yields

$${N}_{m}({D}^{\alpha }u)\leq{\gamma }_{ k}^{m}\ (m!)\ {N}_{ 0}(u),\quad\vert\alpha\vert = m,$$

for $u\in{\mathfrak{X}}_{0}$. This estimate of course implies the well-known real analyticity of harmonic functions. In order for such analyticity to follow from (A.7), it is crucial to have linear dependence in j of the factor on the right side of(A.7). The fact that we can establish (A.7) and (A.8) in this form also makes it an effective tool in the proof of the Cauchy–Kowalewsky theorem, in §4.

B The stationary phase method

The one-dimensional stationary phase method was derived in §7. Here we discuss the multidimensional case. If M is a Riemannian manifold, $F\in{C}_{0}^{\infty }(M)$, and ψ ∈ C^∞(M) is real-valued, with only nondegenerate critical points, there is a formula for the asymptotic behavior of

$$I(\tau ) =\int \nolimits\nolimits F(x)\ {e}^{i\tau\psi (x)}\ dV (x)$$

(B.1)

as τ → ∞, given by the stationary phase method, which we now derive. First, using a partition of unity supported on coordinate neighborhoods, we can write (B.1) as a finite sum of integrals of the form

$$J(\tau ) =\int \nolimits\nolimits f(x)\ {e}^{i\tau\varphi (x)}\ dx,$$

(B.2)

where $f\in{C}_{0}^{\infty }({\mathbb{R}}^{n})$ and φ has either no critical points on supp f or only one critical point, located at x = 0.

Lemma B.1.

If φ has no critical point on supp f, then J(τ) is rapidly decreasing as τ →∞.

Proof.

Cover supp f with open sets on which, by a change of variable, φ(x) becomes linear, that is, φ(x) = ξ ⋅x + c, ξ ≠ 0. Then J(τ) is converted to a sum of integrals of the form

$$\int \nolimits\nolimits {f}_{j}(x){e}^{i\tau x\cdot\xi +\mathit{ic}\tau }\ dx = {e}^{\mathit{ic}\tau }\tilde{{f}}_{ j}(\tau\xi ),$$

with $\tilde{{f}}_{j}\in\mathcal{S}({\mathbb{R}}^{n})$. If ξ ≠ 0, the rapid decrease as τ → ∞ is clear.

It remains to consider the case of (B.2) when φ(x) has a single critical point, at x = 0, which is nondegenerate. In such a case, there exists a coordinate chart near 0 such that

$$\varphi (x) = Ax\cdot x + c,$$

where A is the nonsingular, real, symmetric matrix (A_jk) = (1∕2)(∂_j ∂_kφ(0)), and $c\in \mathbb{R}$. We can assume this holds on supp f. That this can be done is known as the Morse lemma; a proof is given in §8 of Appendix C. Thus it remains to consider

$${e}^{\mathit{ic}\tau }\ K(\tau ) = {e}^{\mathit{ic}\tau }\int \nolimits\nolimits g(x)\ {e}^{i\tau Ax\cdot x}\ dx,$$

(B.3)

as τ → + ∞, where $g\in{C}_{0}^{\infty }({\mathbb{R}}^{n})$. Using a rotation, we could assume Ax ⋅x = ∑a_j x_j², where the factors a_j are the eigenvalues of A.

Note that if P(ξ) = Bξ ⋅ξ, where B is an invertible, symmetric, real matrix, then

$${e}^{\mathit{itP(D)}}\delta (x) = {(2\pi )}^{-n/2}\mathcal{F}{\bigl ({e}^{itP}\bigr )}(x).$$

(B.4)

By diagonalizing B and looking at the one-dimensional cases, ${e}^{itb{\xi }^{2} }$, via techniques used in (6.42) of Chap. 3, we obtain

$${e}^{-itP(D)}\delta (x) =\text{ det }{\bigl (4\pi i{B}\bigr )}^{-1/2}\ {t}^{-n/2}\ {e}^{iAx\cdot x/t},\quad A = {(4B)}^{-1},$$

(B.5)

for t > 0, where the determinant is calculated as

$${}\lim\limits_{\epsilon\searrow 0}\text{ det }{\bigl (4\pi i{(B - i\epsilon )}\bigr )}^{-1/2},$$

(B.6)

using analytic continuation, and the convention that det(+4πεI)^−1∕2 > 0, for real ε > 0.

Thus, for K(τ) in (B.3), we have

$$K({t}^{-1}) = C(A)\ {t}^{n/2}\ u(t,0),$$

(B.7)

where C(A) = det(4πiB)^1∕2, 4B = A⁻¹, and u(t, x) solves a generalized Schrödinger equation:

$$u(t,x) = {e}^{-\mathit{itP(D)}}g(x). $$

(B.8)

Given $g\in{C}_{0}^{\infty }({\mathbb{R}}^{n})$, we know from material of Chap. 3, §5, that

$$u\in{C}^{\infty }{\bigl ([0,\infty ),\mathcal{S}({\mathbb{R}}^{n})\bigr )}\subset{C}^{\infty }{\bigl ([0,\infty )\times{\mathbb{R}}^{n}\bigr )}.$$

Thus we have, for t ↘ 0, an expansion

$$u(t,0)\sim \sum\limits_{j\geq 0}{a}_{j}{t}^{j},$$

(B.9)

with

$${a}_{j} =\frac{1} {j!}{\Bigl ({\frac{\partial } {\partial t}}\Bigr )}^{j}u(0,0) =\frac{{(-i)}^{j}} {j!} P{(D)}^{j}g(0).$$

(B.10)

Consequently, for (B.3) we have

$${e}^{\mathit{ic}\tau }\ K(\tau )\sim C{\tau }^{-n/2}{\Bigl ({a}_{ 0} + {a}_{1}{\tau }^{-1} + {a}_{ 2}{\tau }^{-2} +\cdots\,\Bigr )}{e}^{\mathit{ic}\tau },\quad\tau \rightarrow +\infty,$$

(B.11)

where C = C(A) is as in (B.7) and the factors a_j are given by (B.10). We can conclude that I(τ) in (B.1) is asymptotic to a finite sum of such expansions, under the hypotheses made on F(x) and ψ (x). Let us summarize what has been established.

Proposition B.2.

If F ∈ C₀^∞ (M) and ψ ∈ C^∞ (M) is real-valued, with only non-degenerate critical points, at ${x}_{1},\ldots,{x}_{k}$, then, as τ → + ∞, the integral (B.1) has the asymptotic behavior

$$\begin{array}{rcl} I(\tau )&\sim & \sum\limits_{j=1}^{k}{A}_{ j}(\tau ){\tau }^{-n/2}{e}^{i\tau\psi ({x}_{j})},\\ {A}_{j}(\tau )&\sim & {a}_{j0} + {a}_{j1}{\tau }^{-1} + {a}_{ j2}{\tau }^{-2} +\cdots\,.\end{array}$$

(B.12)