Applications of dispersive estimates to the acoustic pressure waves - PowerPoint PPT Presentation

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Initial conditions: u ε 0 = u ε ( · , 0) − → u 0 = u ( · , 0) strongly in L 2 (Ω) √ εp ε 0 = √ εp ε ( · , 0) − → 0 strongly in L 2 (Ω) . “initial layer”phenomenon for the pressure initial datum Dispersive estimates for acoustic waves – p.8/46

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Boundary condition: Dispersive estimates for acoustic waves – p.8/46

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Boundary condition : u ε | ∂ Ω = 0 Dispersive estimates for acoustic waves – p.8/46

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Boundary condition : u ε | ∂ Ω = 0 � t � � t � p ε ( x, s ) φ ( x, s ) dxds + u ε ( x, s ) ∇ φ ( x, s ) dxds ε 0 Ω 0 Ω � t � � ( u ε · n )( x, s ) φ ( x, s ) dσdt + ε p ε − 0 ( x ) φ ( x, 0) dx = 0 . 0 ∂ Ω Ω Dispersive estimates for acoustic waves – p.8/46

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Boundary condition : u ε | ∂ Ω = 0 � t � � t � p ε ( x, s ) φ ( x, s ) dxds + u ε ( x, s ) ∇ φ ( x, s ) dxds ε 0 Ω 0 Ω � t � � ( u ε · n )( x, s ) φ ( x, s ) dσdt + ε p ε − 0 ( x ) φ ( x, 0) dx = 0 0 ∂ Ω Ω p ε ( x, t ) = p ε a.e. in ∂ Ω 0 ( x ) Dispersive estimates for acoustic waves – p.8/46

Artificial compressibility approximation  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 Initial conditions u ε 0 = u ε ( · , 0) − → u 0 = u ( · , 0) strongly in L 2 (Ω) √ εp ε 0 = √ εp ε ( · , 0) − → 0 strongly in L 2 (Ω) . “initial layer”phenomenon for the pressure initial datum Boundary conditions u ε ( x, t ) = 0 x ∈ ∂ Ω , t ≥ 0 p ε ( x, t ) = p ε 0 ( x ) x ∈ ∂ Ω , t ≥ 0 Dispersive estimates for acoustic waves – p.8/46

Notations Nonhomogenous Sobolev Spaces: W k,p (Ω) = ( I − ∆) − k 2 L p (Ω) H k (Ω) = W k, 2 (Ω) Dispersive estimates for acoustic waves – p.9/46

Notations Nonhomogenous Sobolev Spaces: W k,p (Ω) = ( I − ∆) − k 2 L p (Ω) H k (Ω) = W k, 2 (Ω) L p L p t L q x = L p ([0 , T ]; L q (Ω)) t W k,q = L p ([0 , T ]; W k,q (Ω)) x Dispersive estimates for acoustic waves – p.9/46

Notations Nonhomogenous Sobolev Spaces: W k,p (Ω) = ( I − ∆) − k 2 L p (Ω) H k (Ω) = W k, 2 (Ω) L p L p t L q x = L p ([0 , T ]; L q (Ω)) t W k,q = L p ([0 , T ]; W k,q (Ω)) x Leray Projectors Q = ∇ ∆ − 1 projection on gradient vector fields N div projection on divergence - free vector fields P = I − Q Dispersive estimates for acoustic waves – p.9/46

Main Theorem Let ( u ε , p ε ) be a sequence of weak solution in Ω of the previous system, then (i) u ε ⇀ u t ˙ weakly in L 2 H 1 x Dispersive estimates for acoustic waves – p.10/46

Main Theorem Let ( u ε , p ε ) be a sequence of weak solution in Ω of the previous system, then (i) u ε ⇀ u t ˙ weakly in L 2 H 1 x (ii) Qu ε − t L p strongly in L 2 → 0 x , for any p ∈ [4 , 6) Dispersive estimates for acoustic waves – p.10/46

Main Theorem Let ( u ε , p ε ) be a sequence of weak solution in Ω of the previous system, then (i) u ε ⇀ u t ˙ weakly in L 2 H 1 x (ii) Qu ε − t L p strongly in L 2 → 0 x , for any p ∈ [4 , 6) (iii) Pu ε − strongly in L 2 t L 2 → Pu = u loc x Dispersive estimates for acoustic waves – p.10/46

Main Theorem Let ( u ε , p ε ) be a sequence of weak solution in Ω of the previous system, then (i) u ε ⇀ u t ˙ weakly in L 2 H 1 x (ii) Qu ε − t L p strongly in L 2 → 0 x , for any p ∈ [4 , 6) (iii) Pu ε − strongly in L 2 t L 2 → Pu = u loc x (iv) The sequence { p ε } will converge in the sense of distribution to p = ∆ − 1 div (( u · ∇ ) u ) = ∆ − 1 tr (( Du ) 2 ) . Dispersive estimates for acoustic waves – p.10/46

(v) u = Pu is a Leray weak solution to the incompressible Navier Stokes equation in D ′ ([0 , T ] × Ω) , P ( ∂ t u − ∆ u + ( u · ∇ ) u ) = 0 u ( x, 0) = u 0 ( x ) u | ∂ Ω = 0 Dispersive estimates for acoustic waves – p.11/46

(v) u = Pu is a Leray weak solution to the incompressible Navier Stokes equation in D ′ ([0 , T ] × Ω) , P ( ∂ t u − ∆ u + ( u · ∇ ) u ) = 0 u ( x, 0) = u 0 ( x ) u | ∂ Ω = 0 (vi) The following energy inequality holds � � T � � 1 |∇ u ( x, t ) | 2 dxdt ≤ 1 | u ( x, t ) | 2 dx + | u ( x, 0) | 2 dx. 2 2 Ω 0 Ω Ω Dispersive estimates for acoustic waves – p.11/46

(v) u = Pu is a Leray weak solution to the incompressible Navier Stokes equation in D ′ ([0 , T ] × Ω) , P ( ∂ t u − ∆ u + ( u · ∇ ) u ) = 0 u ( x, 0) = u 0 ( x ) u | ∂ Ω = 0 (vi) The following energy inequality holds � � T � � 1 |∇ u ( x, t ) | 2 dxdt ≤ 1 | u ( x, t ) | 2 dx + | u ( x, 0) | 2 dx. 2 2 Ω 0 Ω Ω (!!) For u ε the trace operator commutes with the limit, this is not true for p ε . Dispersive estimates for acoustic waves – p.11/46

References Artificial Compressibility in exterior domain Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 D. Donatelli P.Marcati, A dispersive approach to the artificial compressibility approximations of the Navier Stokes equation, Journal of Hyperbolic Differential Equations , 3, no. 3, (2006), 575-588. Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 D. Donatelli P.Marcati, A dispersive approach to the artificial compressibility approximations of the Navier Stokes equation, Journal of Hyperbolic Differential Equations , 3, no. 3, (2006), 575-588. D. Donatelli, On the artificial compressibility method for the Navier Stokes Fourier system. Preprint 2008 (submitted) . Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 D. Donatelli P.Marcati, A dispersive approach to the artificial compressibility approximations of the Navier Stokes equation, Journal of Hyperbolic Differential Equations , 3, no. 3, (2006), 575-588. D. Donatelli, On the artificial compressibility method for the Navier Stokes Fourier system. Preprint 2008 (submitted) . Artificial Compressibility in bounded domain Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 D. Donatelli P.Marcati, A dispersive approach to the artificial compressibility approximations of the Navier Stokes equation, Journal of Hyperbolic Differential Equations , 3, no. 3, (2006), 575-588. D. Donatelli, On the artificial compressibility method for the Navier Stokes Fourier system. Preprint 2008 (submitted) . Artificial Compressibility in bounded domain Numerical methods: Vladimirova, Kuznecov, Yanenko (’66), Chorin (’68, ’69), Oskolkov (’71), Kuznecov and Smagulov (’75), Smagulov (’79), Yanenko, Kuznecov and Smagulov (’84) Dispersive estimates for acoustic waves – p.12/46

References Artificial Compressibility in exterior domain D. Donatelli and P.Marcati, preprint Artificial Compressibility in R 3 D. Donatelli P.Marcati, A dispersive approach to the artificial compressibility approximations of the Navier Stokes equation, Journal of Hyperbolic Differential Equations , 3, no. 3, (2006), 575-588. D. Donatelli, On the artificial compressibility method for the Navier Stokes Fourier system. Preprint 2008 (submitted) . Artificial Compressibility in bounded domain Numerical methods: Vladimirova, Kuznecov, Yanenko (’66), Chorin (’68, ’69), Oskolkov (’71), Kuznecov and Smagulov (’75), Smagulov (’79), Yanenko, Kuznecov and Smagulov (’84) Rigorous convergence results: Ghidaglia and Temam (’88), Temam (’69, ’01) Dispersive estimates for acoustic waves – p.12/46

Energy estimates  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 � 1 � � 2 | u ε ( x, t ) | 2 + ε 2 | p ε ( x, t ) | 2 E ( t ) = dx Ω Dispersive estimates for acoustic waves – p.13/46

Energy estimates  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 � 1 � � 2 | u ε ( x, t ) | 2 + ε 2 | p ε ( x, t ) | 2 E ( t ) = dx Ω � t � |∇ u ε ( x, s ) | 2 dxds = E (0) E ( t ) + 0 Ω Dispersive estimates for acoustic waves – p.13/46

Energy estimates  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 � 1 � � 2 | u ε ( x, t ) | 2 + ε 2 | p ε ( x, t ) | 2 E ( t ) = dx Ω � t � |∇ u ε ( x, s ) | 2 dxds = E (0) E ( t ) + 0 Ω ⇓ Dispersive estimates for acoustic waves – p.13/46

Energy estimates  ∂ t u ε + ∇ p ε = µ ∆ u ε − ( u ε · ∇ ) u ε − 1  2(div u ε ) u ε  ε∂ t p ε + div u ε = 0 � 1 � � 2 | u ε ( x, t ) | 2 + ε 2 | p ε ( x, t ) | 2 E ( t ) = dx Ω � t � |∇ u ε ( x, s ) | 2 dxds = E (0) E ( t ) + 0 Ω ⇓ √ εp ε bd. in L ∞ t L 2 εp ε relatively compact in H − 1 x , t t,x ∇ u ε bd. in L 2 u ε bd. in L ∞ t L 2 x ∩ L 2 t L 6 t,x , x , t L 3 / 2 t L 3 / 2 ( u ε ·∇ ) u ε bd. in L 2 t L 1 x ∩ L 1 (div u ε ) u ε bd. in L 2 t L 1 x ∩ L 1 x , x Dispersive estimates for acoustic waves – p.13/46

Estimates on Qu ε - Part 1 Qu ε = ∇ ∆ − 1 N div u ε Dispersive estimates for acoustic waves – p.14/46

Estimates on Qu ε - Part 1 Qu ε = ∇ ∆ − 1 ε∂ t p ε = − div u ε N div u ε Dispersive estimates for acoustic waves – p.14/46

Estimates on Qu ε - Part 1 Qu ε = ∇ ∆ − 1 ε∂ t p ε = − div u ε N div u ε Qu ε = ε ∇ ∆ − 1 N ∂ t p ε Dispersive estimates for acoustic waves – p.14/46

Pressure wave equation Differentiate in t the“pressure equation” ε∂ tt p ε + div ∂ t u ε = 0 Dispersive estimates for acoustic waves – p.15/46

Pressure wave equation Differentiate in t the“pressure equation” ε∂ tt p ε + div ∂ t u ε = 0 taking the divergence of the first equation � � ( u ε · ∇ ) u ε + 1 ε∂ tt p ε − ∆ p ε = − ∆ div u ε + div 2(div u ε ) u ε Dispersive estimates for acoustic waves – p.15/46

Pressure wave equation Differentiate in t the“pressure equation” ε∂ tt p ε + div ∂ t u ε = 0 taking the divergence of the first equation � � ( u ε · ∇ ) u ε + 1 ε∂ tt p ε − ∆ p ε = − ∆ div u ε + div 2(div u ε ) u ε changing the time scale: t = √ ετ u ( x, τ ) = u ε ( x, √ ετ ) , p ( x, τ ) = p ε ( x, √ ετ ) ˜ ¯ � � u + 1 p = − ∆ div ˜ ∂ ττ ¯ p − ∆¯ u + div (˜ u · ∇ ) ˜ 2(div ˜ u )˜ u Dispersive estimates for acoustic waves – p.15/46

Pressure wave equation Differentiate in t the“pressure equation” ε∂ tt p ε + div ∂ t u ε = 0 taking the divergence of the first equation � � ( u ε · ∇ ) u ε + 1 ε∂ tt p ε − ∆ p ε = − ∆ div u ε + div 2(div u ε ) u ε changing the time scale: t = √ ετ u ( x, τ ) = u ε ( x, √ ετ ) , p ( x, τ ) = p ε ( x, √ ετ ) ˜ ¯ � � u + 1 ∂ ττ ¯ p − ∆¯ p = − ∆div ˜ u − div (˜ u · ∇ ) ˜ 2(div ˜ u )˜ u ���� � �� � L 2 t,x t L 3 / 2 L 1 x Dispersive estimates for acoustic waves – p.15/46

Strichartz Estimates   w tt − ∆ w = F  ( x, t ) ∈ R d × [0 , T ] w (0 , · ) = f   ∂ t w (0 , · ) = g Dispersive estimates for acoustic waves – p.16/46

Strichartz Estimates   w tt − ∆ w = F  ( x, t ) ∈ R d × [0 , T ] w (0 , · ) = f   ∂ t w (0 , · ) = g � T sup ( � w � ˙ x + � w t � L 2 x ) ≤ � f � ˙ x + � g � L 2 x + � F � L 2 x ds. H 1 H 1 0 t ∈ [0 ,T ] Dispersive estimates for acoustic waves – p.16/46

Strichartz Estimates   w tt − ∆ w = F  ( x, t ) ∈ R d × [0 , T ] w (0 , · ) = f   ∂ t w (0 , · ) = g � T sup ( � w � ˙ x + � w t � L 2 x ) ≤ � f � ˙ x + � g � L 2 x + � F � L 2 x ds. H 1 H 1 0 t ∈ [0 ,T ] we would like to have γ small � w � L q x ≤ � f � ˙ x + � g � ˙ + � F � L ˜ t L r H γ H γ − 1 q ′ r ′ t L ˜ x x Dispersive estimates for acoustic waves – p.16/46

Strichartz Estimates   w tt − ∆ w = F  ( x, t ) ∈ R d × [0 , T ] w (0 , · ) = f   ∂ t w (0 , · ) = g � T sup ( � w � ˙ x + � w t � L 2 x ) ≤ � f � ˙ x + � g � L 2 x + � F � L 2 x ds. H 1 H 1 0 t ∈ [0 ,T ] we would like to have γ small � w � L q x ≤ � f � ˙ x + � g � ˙ + � F � L ˜ t L r H γ H γ − 1 q ′ r ′ t L ˜ x x Strichartz (1977) proved that � w � L 4 t,x ≤ � f � ˙ + � g � ˙ + � F � L 4 / 3 H 1 / 2 H − 1 / 2 x x t,x Dispersive estimates for acoustic waves – p.16/46

Idea of the proof Strichartz realized that these estimates are related to the restriction theorem for the Fourier transform. Dispersive estimates for acoustic waves – p.17/46

Idea of the proof Strichartz realized that these estimates are related to the restriction theorem for the Fourier transform. � ˆ R d e − ixξ dx Fourier transform: f ( ξ ) = S ⊂ R d hypersurface Rf = ˆ Restriction mapping: f | S Dispersive estimates for acoustic waves – p.17/46

Idea of the proof Strichartz realized that these estimates are related to the restriction theorem for the Fourier transform. � ˆ R d e − ixξ dx Fourier transform: f ( ξ ) = S ⊂ R d hypersurface Rf = ˆ Restriction mapping: f | S S has the ( p, 2) restriction property if � Rf � L 2 ( S ) ≤ � f � L p ( R d ) . Dispersive estimates for acoustic waves – p.17/46

Idea of the proof Strichartz realized that these estimates are related to the restriction theorem for the Fourier transform. � ˆ R d e − ixξ dx Fourier transform: f ( ξ ) = S ⊂ R d hypersurface Rf = ˆ Restriction mapping: f | S S has the ( p, 2) restriction property if � Rf � L 2 ( S ) ≤ � f � L p ( R d ) . If S = sphere ⊂ R 3 , then 1 ≤ p ≤ 4 3 Dispersive estimates for acoustic waves – p.17/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) Dispersive estimates for acoustic waves – p.18/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) � R d e ixξ e it | ξ | β ( ξ ) ˆ w ( t, x ) = f ( ξ ) dξ := Tf ( t, x ) Dispersive estimates for acoustic waves – p.18/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) � R d e ixξ e it | ξ | β ( ξ ) ˆ w ( t, x ) = f ( ξ ) dξ := Tf ( t, x ) � w � L q x ≤ � f � L 2 ⇐ ⇒ � Tf � L q x ≤ � f � L 2 ( ∗ ) t L r t L r ⇒ T ∗ : L q ′ ⇒ TT ∗ : L q ′ ξ → L q x → L q t L r ′ t L r ′ T : L 2 t L r x → L 2 t L r x ⇐ ξ ⇐ x T ∗ is the adjoint operator of T , ( q ′ , r ′ ) are the dual exponent of ( q, r ) Dispersive estimates for acoustic waves – p.18/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) � R d e ixξ e it | ξ | β ( ξ ) ˆ w ( t, x ) = f ( ξ ) dξ := Tf ( t, x ) � w � L q x ≤ � f � L 2 ⇐ ⇒ � Tf � L q x ≤ � f � L 2 ( ∗ ) t L r t L r ⇒ T ∗ : L q ′ ⇒ TT ∗ : L q ′ ξ → L q x → L q t L r ′ t L r ′ T : L 2 t L r x → L 2 t L r x ⇐ ξ ⇐ x T ∗ is the adjoint operator of T , ( q ′ , r ′ ) are the dual exponent of ( q, r ) � R d e ixξ β ( ξ ) ˜ T ∗ f ( ξ ) ≃ β ( ξ ) ˜ � T ∗ f ( x ) = f ( | ξ | , ξ ) dξ f ( | ξ | , ξ ) = Rf ( ξ ) Dispersive estimates for acoustic waves – p.18/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) � R d e ixξ e it | ξ | β ( ξ ) ˆ w ( t, x ) = f ( ξ ) dξ := Tf ( t, x ) � w � L q x ≤ � f � L 2 ⇐ ⇒ � Tf � L q x ≤ � f � L 2 ( ∗ ) t L r t L r ⇒ T ∗ : L q ′ ⇒ TT ∗ : L q ′ ξ → L q x → L q t L r ′ t L r ′ T : L 2 t L r x → L 2 t L r x ⇐ ξ ⇐ x T ∗ is the adjoint operator of T , ( q ′ , r ′ ) are the dual exponent of ( q, r ) � R d e ixξ β ( ξ ) ˜ T ∗ f ( ξ ) ≃ β ( ξ ) ˜ � T ∗ f ( x ) = f ( | ξ | , ξ ) dξ f ( | ξ | , ξ ) = Rf ( ξ ) � T ∗ f � L 2 = � Rf � L 2 (Λ) Λ = { ( τ, ξ ) | τ = | ξ | > 0 } =light cone Dispersive estimates for acoustic waves – p.18/46

� w � L q x ≤ � f � ˙ t L r H γ x then if β is a cutoff function (localizing in frequencies) � R d e ixξ e it | ξ | β ( ξ ) ˆ w ( t, x ) = f ( ξ ) dξ := Tf ( t, x ) � w � L q x ≤ � f � L 2 ⇐ ⇒ � Tf � L q x ≤ � f � L 2 ( ∗ ) t L r t L r ⇒ T ∗ : L q ′ ⇒ TT ∗ : L q ′ ξ → L q x → L q t L r ′ t L r ′ T : L 2 t L r x → L 2 t L r x ⇐ ξ ⇐ x T ∗ is the adjoint operator of T , ( q ′ , r ′ ) are the dual exponent of ( q, r ) � R d e ixξ β ( ξ ) ˜ T ∗ f ( ξ ) ≃ β ( ξ ) ˜ � T ∗ f ( x ) = f ( | ξ | , ξ ) dξ f ( | ξ | , ξ ) = Rf ( ξ ) � T ∗ f � L 2 = � Rf � L 2 (Λ) Λ = { ( τ, ξ ) | τ = | ξ | > 0 } =light cone ( ∗ ) is equivalent to R : L q ′ t L r ′ x → L 2 (Λ) is bounded for suitable (q,r) Dispersive estimates for acoustic waves – p.18/46

Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98)) ( x, t ) ∈ R d × [0 , T ] w tt − ∆ w = F w (0 , · ) = f, ∂ t w (0 , · ) = g, Dispersive estimates for acoustic waves – p.19/46

Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98)) ( x, t ) ∈ R d × [0 , T ] w tt − ∆ w = F w (0 , · ) = f, ∂ t w (0 , · ) = g, � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x x x Dispersive estimates for acoustic waves – p.19/46

Strichartz Estimates (Ginibre-Velo (’95), Keel-Tao(’98)) ( x, t ) ∈ R d × [0 , T ] w tt − ∆ w = F w (0 , · ) = f, ∂ t w (0 , · ) = g, � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x x x ( q, r ) , are wave admissible pairs if (˜ q, ˜ r ) 1/q ((d−3)/2(d−2),1/2) 1/2 q, r, ˜ q, ˜ r ≥ 2 � 1 � 2 2 − 1 q ≤ ( d − 1) r � 1 � 2 2 − 1 q ≤ ( d − 1) ˜ r ˜ 1 q + d r = d 2 − γ = 1 q ′ + d r ′ − 2 ˜ ˜ 1/2 1/r Dispersive estimates for acoustic waves – p.19/46

1/q Strichartz d=3 1/2 q, r, ˜ q, ˜ r ≥ 2 1 q ≤ 1 2 − 1 r 1 q ≤ 1 2 − 1 ˜ r ˜ q + 3 1 r = 3 2 − γ = 1 q ′ + 3 r ′ − 2 ˜ ˜ 1/2 1/r � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x x x Dispersive estimates for acoustic waves – p.20/46

1/q Strichartz d=3 1/2 q, r, ˜ q, ˜ r ≥ 2 1 q ≤ 1 2 − 1 r S q ≤ 1 1 2 − 1 ˜ r ˜ 1 q + 3 r = 3 2 − γ = 1 q ′ + 3 r ′ − 2 ˜ ˜ 1/r 1/2 � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ t W − 1 ,r t L r H γ H γ − 1 q ′ r ′ t L ˜ x x x � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 4 / 3 t W − 1 , 4 H 1 / 2 H − 1 / 2 x x x t,x Dispersive estimates for acoustic waves – p.20/46

1/q Strichartz d=3 1/2 q, r, ˜ q, ˜ r ≥ 2 1 q ≤ 1 2 − 1 r S q ≤ 1 1 2 − 1 ˜ r ˜ 1 q + 3 r = 3 2 − γ = 1 q ′ + 3 r ′ − 2 ˜ ˜ 1/r 1/2 � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ x , t W − 1 ,r t L r H γ H γ − 1 q ′ r ′ t L ˜ x x � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 4 / 3 t W − 1 , 4 H 1 / 2 H − 1 / 2 x x x t,x q ′ , ˜ r ′ ) = (1 , 3 / 2) , then γ = 1 / 2 and ( q, r ) = (4 , 4) if d = 3 , (˜ Dispersive estimates for acoustic waves – p.20/46

1/q Strichartz d=3 1/2 q, r, ˜ q, ˜ r ≥ 2 q ≤ 1 1 2 − 1 r S 1 q ≤ 1 2 − 1 ˜ r ˜ 1 q + 3 r = 3 2 − γ = 1 q ′ + 3 r ′ − 2 ˜ ˜ 1/r 1/2 � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ x , t W − 1 ,r t L r H γ H γ − 1 q ′ r ′ t L ˜ x x � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 4 / 3 t W − 1 , 4 H 1 / 2 H − 1 / 2 x x x t,x q ′ , ˜ r ′ ) = (1 , 3 / 2) , then γ = 1 / 2 and ( q, r ) = (4 , 4) if d = 3 , (˜ � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 1 t W − 1 , 4 H 1 / 2 H − 1 / 2 t L 3 / 2 x x x x Dispersive estimates for acoustic waves – p.20/46

1/q Strichartz d=3 1/2 q, r, ˜ q, ˜ r ≥ 2 q ≤ 1 1 2 − 1 r S 1 q ≤ 1 2 − 1 ˜ r ˜ q + 3 1 r = 3 2 − γ = 1 q ′ + 3 r ′ − 2 ˜ ˜ 1/r 1/2 � w � L q x + � ∂ t w � L q � � f � ˙ x + � g � ˙ + � F � L ˜ x , t W − 1 ,r t L r H γ H γ − 1 q ′ r ′ t L ˜ x x � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 4 / 3 t W − 1 , 4 H 1 / 2 H − 1 / 2 x x x t,x q ′ , ˜ r ′ ) = (1 , 3 / 2) , then γ = 1 / 2 and ( q, r ) = (4 , 4) if d = 3 , (˜ � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 1 t W − 1 , 4 H 1 / 2 H − 1 / 2 t L 3 / 2 x x x x � w � L 4 t,x + � ∂ t w � L 4 � � f � ˙ + � g � ˙ + � F � L 1 x . t W − 1 , 4 H 1 / 2 H 1 / 2 t L 2 x x x Dispersive estimates for acoustic waves – p.20/46

Strichartz estimates on exterior domain Ω (Smith, Sogge, Metcalf, Burq)  � � ∂ 2  t − ∆ w ( t, x ) = F ( t, x ) , ( t, x ) ∈ R + × Ω    H γ w (0 , · ) = f ( x ) ∈ ˙ D H γ − 1 ∂ t w (0 , x ) = g ( x ) ∈ ˙    D  w ( t, x ) = 0 , x ∈ ∂ Ω , � w � L q x + � ∂ t w � L q � � f � ˙ D + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x D x Dispersive estimates for acoustic waves – p.21/46

Strichartz estimates on exterior domain Ω (Smith, Sogge, Metcalf, Burq)  � � ∂ 2  t − ∆ w ( t, x ) = F ( t, x ) , ( t, x ) ∈ R + × Ω    H γ w (0 , · ) = f ( x ) ∈ ˙ D H γ − 1 ∂ t w (0 , x ) = g ( x ) ∈ ˙    D  w ( t, x ) = 0 , x ∈ ∂ Ω , � w � L q x + � ∂ t w � L q � � f � ˙ D + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x D x β ∈ C ∞ 0 ( R d ) , β ( x ) = 1 on {| x | ≤ R } ∆ j f | ∂ Ω = 0 , 2 j < γ � f � ˙ D = � βf � H γ (Ω) + � (1 − β ) f � ˙ H γ H γ ( R d ) Dispersive estimates for acoustic waves – p.21/46

Strichartz estimates on exterior domain Ω (Smith, Sogge, Metcalf, Burq)  � � ∂ 2  t − ∆ w ( t, x ) = F ( t, x ) , ( t, x ) ∈ R + × Ω    H γ w (0 , · ) = f ( x ) ∈ ˙ D H γ − 1 ∂ t w (0 , x ) = g ( x ) ∈ ˙    D  w ( t, x ) = 0 , x ∈ ∂ Ω , � w � L q x + � ∂ t w � L q � � f � ˙ D + � g � ˙ + � F � L ˜ t W − 1 ,r H γ H γ − 1 t L r q ′ r ′ t L ˜ x D x β ∈ C ∞ 0 ( R d ) , β ( x ) = 1 on {| x | ≤ R } ∆ j f | ∂ Ω = 0 , 2 j < γ � f � ˙ D = � βf � H γ (Ω) + � (1 − β ) f � ˙ H γ H γ ( R d ) Ω is nontrapping, there is L R , such that non geodesic of lenght L R is completely contained in {| x | ≤ R } ∩ Ω Dispersive estimates for acoustic waves – p.21/46

Sketch of the proof Smith and Sogge (1995) proved local Strichartz estimates Dispersive estimates for acoustic waves – p.22/46

Sketch of the proof Smith and Sogge (1995) proved local Strichartz estimates Smith and Sogge (2000) proved global Strichartz estimates in odd space dimension Dispersive estimates for acoustic waves – p.22/46

Sketch of the proof Smith and Sogge (1995) proved local Strichartz estimates Smith and Sogge (2000) proved global Strichartz estimates in odd space dimension exponential decay of the local energy of solutions of the wave equation with compactly supported initial data (Ω) ≤ Ce − α | t | ( � f � H γ � βu � H γ D (Ω) + � β∂ t u � H γ − 1 D (Ω) + � g � H γ D (Ω) ) D Dispersive estimates for acoustic waves – p.22/46

Sketch of the proof Smith and Sogge (1995) proved local Strichartz estimates Smith and Sogge (2000) proved global Strichartz estimates in odd space dimension exponential decay of the local energy of solutions of the wave equation with compactly supported initial data (Ω) ≤ Ce − α | t | ( � f � H γ � βu � H γ D (Ω) + � β∂ t u � H γ − 1 D (Ω) + � g � H γ D (Ω) ) D Burq (2004), Metcalfe (2003) proved global Strichartz estimates in even space dimension Dispersive estimates for acoustic waves – p.22/46

Sketch of the proof Smith and Sogge (1995) proved local Strichartz estimates Smith and Sogge (2000) proved global Strichartz estimates in odd space dimension exponential decay of the local energy of solutions of the wave equation with compactly supported initial data (Ω) ≤ Ce − α | t | ( � f � H γ � βu � H γ D (Ω) + � β∂ t u � H γ − 1 D (Ω) + � g � H γ D (Ω) ) D Burq (2004), Metcalfe (2003) proved global Strichartz estimates in even space dimension Local energy decay (Ω) ≤ C | t | − d/ 2 ( � f � H γ � βu � H γ D (Ω) + � β∂ t u � H γ − 1 D (Ω) + � g � H γ D (Ω) ) D Dispersive estimates for acoustic waves – p.22/46

Pressure wave equation  � � u + 1  ∂ ττ ¯ p − ∆¯ p = − ∆ div ˜ u + div (˜ u · ∇ ) ˜ 2 (div ˜ u )˜ u ,    p ( x, 0) = p ε ¯ 0 ( x ) , p ( x, 0) = ε − 1 / 2 div u ε  ∂ τ ¯ 0 ( x ) ,    p ( x, t ) | ∂ Ω = p ε ¯ 0 ( x ) | ∂ Ω for fixed ε smoothing of initial data Dispersive estimates for acoustic waves – p.23/46