Local energy decay for the wave equation in a dissipative wave guide Julien Royer Institut de Math´ ematiques de Toulouse - France Differential Operators on Graphs and Waveguides TU Graz February 25, 2018
Local Energy for the Free Wave Equation Let u be the solution in R d of the free wave equation # B 2 t u ´ ∆ u “ 0 p u , B t u q| t “ 0 “ p u 0 , u 1 q . We have conservation of the energy : } ∇ u p t q} 2 L 2 ` }B t u p t q} 2 L 2 “ } ∇ u 0 } 2 L 2 ` } u 1 } 2 L 2 . If u 0 and u 1 are compactly supported and χ P C 8 0 p R d q : d ě 3 odd: propagation at speed 1 (Huygens’ principle) } χ p x q ∇ u p t , x q} 2 ` } χ p x qB t u p t , x q} 2 “ 0 for t large enough. d even: propagation at speed ď 1 } χ p x q ∇ u p t , x q} 2 ` } χ p x qB t u p t , x q} 2 À t ´ 2 d ` } ∇ u 0 } 2 L 2 ` } u 1 } 2 ˘ . L 2 Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 2
Generalizations for perturbations of the model case Wave equation in an exterior domain or for a Laplace-Beltrami operator (given by the refraction index) Uniform decay for the local energy under the non-trapping condition on the classical flow (assumption for high frequencies). " e ´ γ t * ` } χ p x q ∇ u p t , x q} 2 ` } χ p x qB t u p t , x q} 2 À } ∇ u 0 } 2 L 2 ` } u 1 } 2 ˘ . t ´ 2 d L 2 (Lax-Morawetz-Philipps ’63, Ralston ’69, Morawetz-Ralston-Strauss ’77, Bony-H¨ afner ’10, Bouclet ’11, Bouclet-Burq ’19) Without non-trapping: Logarithmic decay with loss of regularity (Burq ’98): } χ p x q ∇ u p t , x q} 2 ` } χ p x qB t u p t , x q} 2 À ln p 2 ` t q ´ 2 k ` } u 0 } 2 H k ` 1 ` } u 1 } 2 ˘ . H k Various intermediate settings . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 3
Stabilization in a bounded domain On Ω Ă R d bounded, we consider the damped wave equation $ B 2 t u ´ ∆ u ` a B t u “ 0 on Ω , ’ & + Dirichlet or Neumann at the boundary , ’ + initial conditions . % a p x q ě 0 is the absorption index The global energy decays: for t 1 ď t 2 ż t 2 ż a p x q |B t u p s , x q| 2 dx ds ď 0 . E p t 2 q ´ E p t 1 q “ ´ t 1 Ω Uniform decay under the Geometric Control Condition. We can also consider damping at the boundary: $ B 2 t u ´ ∆ u “ 0 on Ω , ’ & B ν u ` a B t u “ 0 on B Ω , ’ + initial conditions . % See Rauch-Taylor ’74, Bardos-Lebeau-Rauch ’92, Lebeau ’96,... Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 4
Damped wave equation on unbounded domains For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Aloui-Khenissi ’02, Khenissi ’03: damped wave equation in an exterior domain (compactly supported absorption index). Bouclet-R. ’14, R. ’16: asymptotically free damped wave equation. ù We recover the same rates of decay as for the undamped case under the non-trapping condition. Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 5
Damped wave equation with strong damping For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Absorption index effective at infinity (for instance a p x q Ñ 1 as | x | Ñ 8 ) Model case: B 2 t u ´ ∆ u ` B t u “ 0 ù The local energy decays like t ´ d ´ 2 . This can be slower than for the undamped wave. For low frequencies, the solution of the damped wave equation behaves like a solution of the heat equation ´ ∆ v ` B t v “ 0 . Matsumura ’76 (decay estimates) Orive-Pozato-Zuazua ’01 (periodic medium, constant damping) Marcati-Nishihara ’03, Nishihara ’03, Hosono-Ogawa ’04, Narazaki ’04 (diffusive phenomenon) Ikehata ’02, Aloui-Ibrahim-Khenissi ’15 (exterior domains) + time dependant damping + semi-linear equation + abstract results + . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 6
Damped wave equation with strong damping For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Absorption index effective at infinity (for instance a p x q Ñ 1 as | x | Ñ 8 ) Model case: B 2 t u ´ ∆ u ` B t u “ 0 ù The local energy decays like t ´ d ´ 2 . This can be slower than for the undamped wave. For low frequencies, the solution of the damped wave equation behaves like a solution of the heat equation ´ ∆ v ` B t v “ 0 . Matsumura ’76 (decay estimates) Orive-Pozato-Zuazua ’01 (periodic medium, constant damping) Marcati-Nishihara ’03, Nishihara ’03, Hosono-Ogawa ’04, Narazaki ’04 (diffusive phenomenon) Ikehata ’02, Aloui-Ibrahim-Khenissi ’15 (exterior domains) + time dependant damping + semi-linear equation + abstract results + . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 6
Damped wave equation with strong damping For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Absorption index effective at infinity (for instance a p x q Ñ 1 as | x | Ñ 8 ) Model case: B 2 t u ´ ∆ u ` B t u “ 0 ù The local energy decays like t ´ d ´ 2 . This can be slower than for the undamped wave. For low frequencies, the solution of the damped wave equation behaves like a solution of the heat equation ´ ∆ v ` B t v “ 0 . Matsumura ’76 (decay estimates) Orive-Pozato-Zuazua ’01 (periodic medium, constant damping) Marcati-Nishihara ’03, Nishihara ’03, Hosono-Ogawa ’04, Narazaki ’04 (diffusive phenomenon) Ikehata ’02, Aloui-Ibrahim-Khenissi ’15 (exterior domains) + time dependant damping + semi-linear equation + abstract results + . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 6
Damped wave equation with strong damping For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Absorption index effective at infinity (for instance a p x q Ñ 1 as | x | Ñ 8 ) Model case: B 2 t u ´ ∆ u ` B t u “ 0 ù The local energy decays like t ´ d ´ 2 . This can be slower than for the undamped wave. For low frequencies, the solution of the damped wave equation behaves like a solution of the heat equation ´ ∆ v ` B t v “ 0 . Matsumura ’76 (decay estimates) Orive-Pozato-Zuazua ’01 (periodic medium, constant damping) Marcati-Nishihara ’03, Nishihara ’03, Hosono-Ogawa ’04, Narazaki ’04 (diffusive phenomenon) Ikehata ’02, Aloui-Ibrahim-Khenissi ’15 (exterior domains) + time dependant damping + semi-linear equation + abstract results + . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 6
Damped wave equation with strong damping For the damped wave equation in an unbounded domain, the expected necessary and sufficient condition for uniform decay of the local energy is: Each bounded classical trajectory goes through the damping region. Absorption index effective at infinity (for instance a p x q Ñ 1 as | x | Ñ 8 ) Model case: B 2 t u ´ ∆ u ` B t u “ 0 ù The local energy decays like t ´ d ´ 2 . This can be slower than for the undamped wave. For low frequencies, the solution of the damped wave equation behaves like a solution of the heat equation ´ ∆ v ` B t v “ 0 . Matsumura ’76 (decay estimates) Orive-Pozato-Zuazua ’01 (periodic medium, constant damping) Marcati-Nishihara ’03, Nishihara ’03, Hosono-Ogawa ’04, Narazaki ’04 (diffusive phenomenon) Ikehata ’02, Aloui-Ibrahim-Khenissi ’15 (exterior domains) + time dependant damping + semi-linear equation + abstract results + . . . Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 6
Wave equation with strong damping in various settings On a wave guide Ω “ R d ˆ ω with dissipation at the boundary: 1 # B 2 t u ´ ∆ u “ 0 on R ` ˆ Ω , B ν u ` a B t u “ 0 on R ` ˆ B Ω . On a wave guide with dissipation at infinity (with M. Malloug, Sousse): 2 # B 2 t u ´ ∆ u ` a B t u “ 0 on R ` ˆ Ω , B ν u “ 0 on R ` ˆ B Ω , where a p x , y q Ñ 1 as | x | Ñ 8 . On an asymptotically periodic medium (with R. Joly, Grenoble) 3 B 2 t u ` Pu ` a p x qB t u “ 0 on R ` ˆ R d , where P “ ´ div G p x q ∇ , G p x q “ G per . p x q` G Ñ 0 p x q , a p x q “ a per . p x q` a Ñ 0 p x q Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 7
Wave equation with strong damping in various settings On a wave guide Ω “ R d ˆ ω with dissipation at the boundary: 1 # B 2 t u ´ ∆ u “ 0 on R ` ˆ Ω , B ν u ` a B t u “ 0 on R ` ˆ B Ω . On a wave guide with dissipation at infinity (with M. Malloug, Sousse): 2 # B 2 t u ´ ∆ u ` a B t u “ 0 on R ` ˆ Ω , B ν u “ 0 on R ` ˆ B Ω , where a p x , y q Ñ 1 as | x | Ñ 8 . On an asymptotically periodic medium (with R. Joly, Grenoble) 3 B 2 t u ` Pu ` a p x qB t u “ 0 on R ` ˆ R d , where P “ ´ div G p x q ∇ , G p x q “ G per . p x q` G Ñ 0 p x q , a p x q “ a per . p x q` a Ñ 0 p x q Julien Royer (Toulouse) Local energy decay in a dissipative wave guide 7
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