Multi-d shock waves and surface waves S. Benzoni-Gavage University - PowerPoint PPT Presentation

Multi-d shock waves stability Neutral stability and well-posedness Weakly nonlinear surface waves Multi-d shock waves and surface waves S. Benzoni-Gavage University of Lyon (Universit´ e Claude Bernard Lyon 1 / Institut Camille Jordan) HYP2008 conference, June 11, 2008. 1 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Outline Multi-d shock waves stability 1 Theory Examples Neutral stability and well-posedness 2 Weakly nonlinear surface waves 3 Derivation of amplitude equation Well-posedness for amplitude equation 2 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 Basic assumption: hyperbolicity in t -direction, i.e. for all u ∈ U ⊂ R n , the matrix A 0 ( u ) := d f 0 ( u ) is nonsingular, and for all ν ∈ R d , the matrix A 0 ( u ) − 1 A j ( u ) ν j only has real semisimple eigenvalues. 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 A 0 ( u , ν ) := A 0 ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 A 0 ( u , ν ) := A 0 ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Shock is noncharacteristic iff both matrices A 0 ( u ± , ∇ Φ) are nonsingular along Φ = 0. 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 A 0 ( u , ν ) := A 0 ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Shock is classical (or Laxian) iff dim E u ( A 0 ( u − , ∇ Φ)) + dim E s ( A 0 ( u + , ∇ Φ)) = n + 1, dim E u ( A 0 ( u + , ∇ Φ) + dim E s ( A 0 ( u − , ∇ Φ)) = n − 1. 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d Φ = 0 x 1 A 0 ( u , ν ) := A 0 ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Shock is undercompressive iff dim E u ( A 0 ( u − , ∇ Φ)) + dim E s ( A 0 ( u + , ∇ x Φ)) = n + 1 − p , dim E u ( A 0 ( u + , ∇ Φ) + dim E s ( A 0 ( u − , ∇ Φ)) = n − 1 + p . 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves General equations for a ‘shock wave’ t ∂ t f 0 ( u ) + ∂ j f j ( u ) = 0 n , Φ( t , x ) � = 0 , [ f 0 ( u )] ∂ t Φ + [ f j ( u )] ∂ j Φ = 0 n , Φ( t , x ) = 0 . x d [ g 0 ( u )] ∂ t Φ + [ g j ( u )] ∂ j Φ = 0 p , Φ( t , x ) = 0 . Φ = 0 x 1 A 0 ( u , ν ) := A 0 ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Shock is undercompressive iff dim E u ( A 0 ( u − , ∇ Φ)) + dim E s ( A 0 ( u + , ∇ x Φ)) = n + 1 − p , dim E u ( A 0 ( u + , ∇ Φ) + dim E s ( A 0 ( u − , ∇ Φ)) = n − 1 + p . 3 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Linear analysis Ref. planar shock t = σ t x d x d x 1 ı’70] , [Kreiss’70] , [Blokhin’82] , [Majda’83] , [Freist¨ uhler’98] . [Lopatinski˘ 4 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Linear analysis Ref. planar shock t Change frame = ⇒ σ = 0 . x d x 1 ı’70] , [Kreiss’70] , [Blokhin’82] , [Majda’83] , [Freist¨ uhler’98] . [Lopatinski˘ 4 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Linear analysis Ref. planar shock t Change frame = ⇒ σ = 0 . Change coordinates ( t , x ) �→ ( t , y ) := ( t , x 1 , . . . , x d − 1 , Φ( t , x )), Φ( t , x ) = x d − χ ( t , x 1 , . . . , x d − 1 ). x d x 1 ı’70] , [Kreiss’70] , [Blokhin’82] , [Majda’83] , [Freist¨ uhler’98] . [Lopatinski˘ 4 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Linear analysis Ref. planar shock t Change frame = ⇒ σ = 0 . Change coordinates ( t , x ) �→ ( t , y ) := ( t , x 1 , . . . , x d − 1 , Φ( t , x )), Φ( t , x ) = x d − χ ( t , x 1 , . . . , x d − 1 ). Linearize eqns about x d ( u , χ ) = ( u , 0) , u := u ± , y d ≷ 0 . x 1 ı’70] , [Kreiss’70] , [Blokhin’82] , [Majda’83] , [Freist¨ uhler’98] . [Lopatinski˘ 4 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Normal modes analysis Transmission problem: � A 0 ( u ) ∂ t u + A j ( u ) ∂ j u = 0 n , y d ≷ 0 , [ F 0 ( u )] ∂ t χ + [ F j ( u )] ∂ j χ = [ d F d ( u ) · u ] , y d = 0 . 5 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Normal modes analysis Transmission problem: � A 0 ( u ) ∂ t u + A j ( u ) ∂ j u = 0 n , y d ≷ 0 , [ F 0 ( u )] ∂ t χ + [ F j ( u )] ∂ j χ = [ d F d ( u ) · u ] , y d = 0 . Fourier-Laplace transform ( t , ˇ y ) ❀ ( τ, ˇ η ) ⇒ shooting ODE problem. 5 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Normal modes analysis Transmission problem: � A 0 ( u ) ∂ t u + A j ( u ) ∂ j u = 0 n , y d ≷ 0 , [ F 0 ( u )] ∂ t χ + [ F j ( u )] ∂ j χ = [ d F d ( u ) · u ] , y d = 0 . Fourier-Laplace transform ( t , ˇ y ) ❀ ( τ, ˇ η ) ⇒ shooting ODE problem. A d ( u , ν ) := A d ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) 5 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Normal modes analysis Transmission problem: � A 0 ( u ) ∂ t u + A j ( u ) ∂ j u = 0 n , y d ≷ 0 , [ F 0 ( u )] ∂ t χ + [ F j ( u )] ∂ j χ = [ d F d ( u ) · u ] , y d = 0 . Fourier-Laplace transform ( t , ˇ y ) ❀ ( τ, ˇ η ) ⇒ shooting ODE problem. A d ( u , ν ) := A d ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Normal modes: χ = X e τ t + i η j y j , u = U ( y d ) e τ t + i η j y j with U ∈ L 2 ( R ), R e ( τ ) > 0, U (0+) ∈ E u ( A d ( u , τ, i ˇ η )) and U (0 − ) ∈ E s ( A d ( u , τ, i ˇ η )). 5 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Normal modes analysis Transmission problem: � A 0 ( u ) ∂ t u + A j ( u ) ∂ j u = 0 n , y d ≷ 0 , [ F 0 ( u )] ∂ t χ + [ F j ( u )] ∂ j χ = [ d F d ( u ) · u ] , y d = 0 . Fourier-Laplace transform ( t , ˇ y ) ❀ ( τ, ˇ η ) ⇒ shooting ODE problem. A d ( u , ν ) := A d ( u ) − 1 ( A 0 ( u ) ν 0 + A j ( u ) ν j ) Neutral modes of finite energy, or surface waves: χ = X e i η 0 t + i η j y j , u = U ( y d ) e i η 0 t + i η j y j with still U ∈ L 2 ( R ), U (0+) ∈ E u ( A d ( u , i η 0 , i ˇ η )) and U (0 − ) ∈ E s ( A d ( u , i η 0 , i ˇ η )). 5 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Surface waves t x d x 1 u x d x 1 6 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Surface waves Isotropic elasticity t ∂ tt u = λ ∆ u + ( λ + µ ) ∇ div u , x 2 > 0 , ∂ 2 u 1 + ∂ 1 u 2 = 0 , x 2 = 0 , µ∂ 1 u 1 + (2 λ + µ ) ∂ 2 u 2 = 0 , x 2 = 0 . x d x 1 u x d [Rayleigh1885] (see also [Serre’06] ) x 1 6 / 26

Multi-d shock waves stability Theory Neutral stability and well-posedness Examples Weakly nonlinear surface waves Surface waves Isotropic elasticity t ∂ tt u = λ ∆ u + ( λ + µ ) ∇ div u , x 2 > 0 , ∂ 2 u 1 + ∂ 1 u 2 = 0 , x 2 = 0 , µ∂ 1 u 1 + (2 λ + µ ) ∂ 2 u 2 = 0 , x 2 = 0 . x d For λ > 0, λ + µ > 0, ∃ Rayleigh x 1 u waves, or ‘Surface Acoustic Waves’, √ of speed less than λ . x d [Rayleigh1885] (see also [Serre’06] ) x 1 6 / 26