Stability of solutions for wave equations Mikko Salo Dept. of - - PowerPoint PPT Presentation

Stability of solutions for wave equations

Mikko Salo

• Dept. of Mathematics and Statistics / RNI

University of Helsinki

Wave equation

Cauchy problem

     (∂2

t − c(x)2∆)u(t, x) = 0

in {t > 0} × Rn,

u(0, x) = f(x)

in Rn,

∂tu(0, x) = g(x)

in Rn where ∂t = ∂

∂t, ∆ = n j=1 ∂2 xj, and

c(x) > 0: sound speed f(x): initial position g(x): initial velocity.

Stability of solutions for wave equations – p.1

Motivation

A model for

n = 1: vibrating string n = 2: vibrating membrane (drum) n = 3: acoustic waves in the earth

For n = 3, we consider an inverse problem in seismic

• imaging. This is used in finding oil within the earth.

Stability of solutions for wave equations – p.2

Inverse problem

Determine the structure of the earth by sending acoustic waves and measuring reflected waves on the surface. Let c(x) be the sound speed within the earth. An explosion at point xs at time t = 0 causes an acoustic wave u(t, x) = u(t, x ; xs):

     (∂2

t − c(x)2∆)u(t, x) = 0

in {t > 0} × Rn,

u(0, x) = 0

in Rn,

∂tu(0, x) = δ(x − xs)

in Rn

Stability of solutions for wave equations – p.3

Inverse problem

Source points xs ∈ S and receiver points xr ∈ R, where S and R are subsets of the surface. Inverse problem: from the measurements

{u(t, xr ; xs) ; t ∈ [0, T], xs ∈ S, xr ∈ R},

determine the sound speed c(x) in the earth. Very little is known of the full problem. Partial results for the linearized problem if c ∈ C∞ (Rakesh 1988).

Stability of solutions for wave equations – p.4

Inverse problem

What if c is not smooth? For instance, c = c0 + p where c0 ∈ C∞ and p is a small nonsmooth perturbation. Useful for explaining "scale separation" in geophysics. Sharp estimates for linearized problem might lead to progress in the full problem. First step: show that u depends continuously on c.

Stability of solutions for wave equations – p.5

Stability

Theorem 1. Let ajk(x) ∈ C1,1(Rn) be positive definite, and let

     (∂2

t − n j,k=1 ajk(x)∂xj∂xk)u(t, x) = 0

in {t > 0} × Rn,

u(0, x) = f(x)

in Rn,

∂tu(0, x) = 0

in Rn where f ∈ L2(Rn) is fixed. Then the map

(ajk) → u(t, · )

is uniformly continuous C1,1(Rn) → L2(Rn).

Stability of solutions for wave equations – p.6

Stability

Here a ∈ C1,1(Rn) means that a and ∇a are Lipschitz

• continuous. The norm is

aC1,1 =

• |α|≤2

∂αaL∞.

The proof is based on constructing a solution operator using wave packet methods (Smith 1998).

Stability of solutions for wave equations – p.7

Wave packets

If x ∈ Rn and ξ ∈ Rn, a wave packet centered at (x, ξ) is given by

g(y; x, ξ) = eiξ·(y−x)ψ(y − x)

where ψ ∈ C∞ is supported in the unit ball. The shape of g is preserved under wave evolution. Applications: Image processing (curvelets) Numerical solution of hyperbolic equations

Stability of solutions for wave equations – p.8

Wave packets

A function f ∈ L2 may be written as a superposition of wave packets:

f(y) =

• F(x, ξ)g(y; x, ξ) dx dξ.

An approximate solution with initial data f is obtained by translating the wave packets along Hamilton flow:

u(t, y) ≈

• F(x, ξ)g(y; x(t), ξ(t))) dx dξ.

Stability of solutions for wave equations – p.9

Hamilton flow

The Hamilton flow is (x(t), ξ(t)) where

x′(t) = ∇ξp(x(t), ξ(t)), ξ′(t) = −∇xp(x(t), ξ(t))

with initial value (x(0), ξ(0)) = (x, ξ), and

p(x, ξ) =

• n
• j,k=1

ajk(x)ξjξk.

Stability of solutions for wave equations – p.10

Hamilton flow

Hörmander: singularities propagate along Hamilton flow. When the wave equation is factored into half-wave equations

∂2

t − n

• j,k=1

ajk(x)∂xj∂xk = (∂t − iP)(∂t + iP)

where P = p(x, −i∇), the Hamilton flow arises as the linearization of P around a wave packet.

Stability of solutions for wave equations – p.11