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 } × R n , in R n , u (0 , x ) = f ( x ) in R n ∂ t u (0 , x ) = g ( x ) ∂t , ∆ = � n where ∂ t = ∂ j =1 ∂ 2 x j , 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 x s at time t = 0 causes an acoustic wave u ( t, x ) = u ( t, x ; x s ) : ( ∂ 2 t − c ( x ) 2 ∆) u ( t, x ) = 0 in { t > 0 } × R n , in R n , u (0 , x ) = 0 in R n ∂ t u (0 , x ) = δ ( x − x s ) Stability of solutions for wave equations – p.3
Inverse problem Source points x s ∈ S and receiver points x r ∈ R , where S and R are subsets of the surface. Inverse problem: from the measurements { u ( t, x r ; x s ) ; t ∈ [0 , T ] , x s ∈ S, x r ∈ 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 = c 0 + p where c 0 ∈ 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 a jk ( x ) ∈ C 1 , 1 ( R n ) be positive definite, and let t − � n ( ∂ 2 in { t > 0 } × R n , j,k =1 a jk ( x ) ∂ x j ∂ x k ) u ( t, x ) = 0 in R n , u (0 , x ) = f ( x ) in R n ∂ t u (0 , x ) = 0 where f ∈ L 2 ( R n ) is fixed. Then the map ( a jk ) �→ u ( t, · ) is uniformly continuous C 1 , 1 ( R n ) → L 2 ( R n ) . Stability of solutions for wave equations – p.6
Stability Here a ∈ C 1 , 1 ( R n ) means that a and ∇ a are Lipschitz continuous. The norm is � � ∂ α a � L ∞ . � a � C 1 , 1 = | α |≤ 2 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 ∈ R n and ξ ∈ R n , a wave packet centered at ( x, ξ ) is given by g ( y ; x, ξ ) = e iξ · ( 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 ∈ L 2 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 ) = ∇ ξ p ( x ( t ) , ξ ( t )) , ξ ′ ( t ) = −∇ x p ( x ( t ) , ξ ( t )) with initial value ( x (0) , ξ (0)) = ( x, ξ ) , and � n � � � p ( x, ξ ) = a jk ( x ) ξ j ξ k . � j,k =1 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 n � ∂ 2 t − a jk ( x ) ∂ x j ∂ x k = ( ∂ t − iP )( ∂ t + iP ) j,k =1 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
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