Exponential mixing for the white-forced damped nonlinear wave equation

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December 2014, 3(4): 645-670. doi: 10.3934/eect.2014.3.645

Exponential mixing for the white-forced damped nonlinear wave equation

1.	Department of Mathematics, CNRS UMR 8088, University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95300 Cergy-Pontoise, France

Received April 2014 Revised September 2014 Published October 2014

Abstract
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The paper is devoted to studying the stochastic nonlinear wave (NLW) equation $$ \partial_t^2 u + \gamma \partial_t u - \triangle u + f(u)=h(x)+\eta(t,x) $$ in a bounded domain $D\subset\mathbb{R}^3$. The equation is supplemented with the Dirichlet boundary condition. Here $f$ is a nonlinear term, $h(x)$ is a function in $H^1_0(D)$ and $\eta(t,x)$ is a non-degenerate white noise. We show that the Markov process associated with the flow $\xi_u(t)=[u(t),\dot u (t)]$ has a unique stationary measure $\mu$, and the law of any solution converges to $\mu$ with exponential rate in the dual-Lipschitz norm.

Keywords: stationary measure, NLW equation, exponential mixing..

Mathematics Subject Classification: 35L70, 35R60, 37A25, 60H1.

Citation: Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645

References:

[1]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing, (1992). Google Scholar
[2]	Y. Bakhtin, E. Cator and K. Khanin, Space-time stationary solutions for the Burgers equation,, J. Amer. Math. Soc., 27 (2014), 193. doi: 10.1090/S0894-0347-2013-00773-0. Google Scholar
[3]	V. Barbu and G. Da Prato, The stochastic nonlinear damped wave equation,, Appl. Math. Optim., 46 (2002), 125. doi: 10.1007/s00245-002-0744-4. Google Scholar
[4]	J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing of the 2D stochastic Navier-Stokes dynamics,, Comm. Math. Phys., 230 (2002), 87. doi: 10.1007/s00220-002-0708-1. Google Scholar
[5]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of AMS Coll. Publ., AMS, (2002). Google Scholar
[6]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar
[7]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511662829. Google Scholar
[8]	A. Debussche, Ergodicity results for the stochastic Navier-Stokes equations: An introduction,, In Topics in Mathematical Fluid Mechanics, 2073 (2013), 23. doi: 10.1007/978-3-642-36297-2_2. Google Scholar
[9]	A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation,, J. Evol. Equ., 5 (2005), 317. doi: 10.1007/s00028-005-0195-x. Google Scholar
[10]	A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing,, , (). Google Scholar
[11]	N. Dirr and P. Souganidis, Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise,, SIAM J. Math. Anal., 37 (2005), 777. doi: 10.1137/040611896. Google Scholar
[12]	W. E, J. C. Mattingly and Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation,, Comm. Math. Phys., 224 (2001), 83. doi: 10.1007/s002201224083. Google Scholar
[13]	W. E, K. Khanin, A. Mazel and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing,, Ann. of Math. (2), 151 (2000), 877. doi: 10.2307/121126. Google Scholar
[14]	F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations,, Comm. Math. Phys., 172 (1995), 119. doi: 10.1007/BF02104513. Google Scholar
[15]	T. Girya and I. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems,, Mat. Sb., 186 (1995), 29. doi: 10.1070/SM1995v186n01ABEH000002. Google Scholar
[16]	M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations,, Ann. Probab., 36 (2008), 2050. doi: 10.1214/08-AOP392. Google Scholar
[17]	A. Haraux, Two remarks on hyperbolic dissipative problems,, In Nonlinear partial differential equations and their applications. Collège de France seminar, 122* (1985), 1983. Google Scholar*
[18]	R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems,, Comm. Math. Phys., 232 (2003), 377. Google Scholar
[19]	S. Kuksin and V. Nersesyan, Stochastic CGL equations without linear dispersion in any space dimension,, Stochastic Partial Differential Equations: Analysis and Computations, 1 (2013), 389. doi: 10.1007/s40072-013-0010-6. Google Scholar
[20]	S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures,, Comm. Math. Phys., 213 (2000), 291. doi: 10.1007/s002200000237. Google Scholar
[21]	S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence,, Cambridge University Press, (2012). doi: 10.1017/CBO9781139137119. Google Scholar
[22]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires,, Dunod; Gauthier-Villars, (1969). Google Scholar
[23]	C. Mueller, Coupling and invariant measures for the heat equation with noise,, Ann. Probab., 21 (1993), 2189. doi: 10.1214/aop/1176989016. Google Scholar
[24]	C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case,, Probab. Theory Related Fields, 140 (2008), 41. doi: 10.1007/s00440-007-0057-2. Google Scholar
[25]	A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's,, Probab. Theory Related Fields, 134 (2006), 215. doi: 10.1007/s00440-005-0427-6. Google Scholar
[26]	A. Shirikyan, Exponential mixing for randomly forced partial differential equations: Method of coupling,, In Instability in models connected with fluid flows. II, 7 (2008), 155. doi: 10.1007/978-0-387-75219-8_4. Google Scholar
[27]	M. I. Vishik, A. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics,, Uspekhi Mat. Nauk, 34 (1979), 135. Google Scholar
[28]	S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal, 3 (2004), 921. doi: 10.3934/cpaa.2004.3.921. Google Scholar

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References:

[1]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing, (1992). Google Scholar
[2]	Y. Bakhtin, E. Cator and K. Khanin, Space-time stationary solutions for the Burgers equation,, J. Amer. Math. Soc., 27 (2014), 193. doi: 10.1090/S0894-0347-2013-00773-0. Google Scholar
[3]	V. Barbu and G. Da Prato, The stochastic nonlinear damped wave equation,, Appl. Math. Optim., 46 (2002), 125. doi: 10.1007/s00245-002-0744-4. Google Scholar
[4]	J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing of the 2D stochastic Navier-Stokes dynamics,, Comm. Math. Phys., 230 (2002), 87. doi: 10.1007/s00220-002-0708-1. Google Scholar
[5]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of AMS Coll. Publ., AMS, (2002). Google Scholar
[6]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar
[7]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511662829. Google Scholar
[8]	A. Debussche, Ergodicity results for the stochastic Navier-Stokes equations: An introduction,, In Topics in Mathematical Fluid Mechanics, 2073 (2013), 23. doi: 10.1007/978-3-642-36297-2_2. Google Scholar
[9]	A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation,, J. Evol. Equ., 5 (2005), 317. doi: 10.1007/s00028-005-0195-x. Google Scholar
[10]	A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing,, , (). Google Scholar
[11]	N. Dirr and P. Souganidis, Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise,, SIAM J. Math. Anal., 37 (2005), 777. doi: 10.1137/040611896. Google Scholar
[12]	W. E, J. C. Mattingly and Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation,, Comm. Math. Phys., 224 (2001), 83. doi: 10.1007/s002201224083. Google Scholar
[13]	W. E, K. Khanin, A. Mazel and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing,, Ann. of Math. (2), 151 (2000), 877. doi: 10.2307/121126. Google Scholar
[14]	F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations,, Comm. Math. Phys., 172 (1995), 119. doi: 10.1007/BF02104513. Google Scholar
[15]	T. Girya and I. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems,, Mat. Sb., 186 (1995), 29. doi: 10.1070/SM1995v186n01ABEH000002. Google Scholar
[16]	M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations,, Ann. Probab., 36 (2008), 2050. doi: 10.1214/08-AOP392. Google Scholar
[17]	A. Haraux, Two remarks on hyperbolic dissipative problems,, In Nonlinear partial differential equations and their applications. Collège de France seminar, 122* (1985), 1983. Google Scholar*
[18]	R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems,, Comm. Math. Phys., 232 (2003), 377. Google Scholar
[19]	S. Kuksin and V. Nersesyan, Stochastic CGL equations without linear dispersion in any space dimension,, Stochastic Partial Differential Equations: Analysis and Computations, 1 (2013), 389. doi: 10.1007/s40072-013-0010-6. Google Scholar
[20]	S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures,, Comm. Math. Phys., 213 (2000), 291. doi: 10.1007/s002200000237. Google Scholar
[21]	S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence,, Cambridge University Press, (2012). doi: 10.1017/CBO9781139137119. Google Scholar
[22]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires,, Dunod; Gauthier-Villars, (1969). Google Scholar
[23]	C. Mueller, Coupling and invariant measures for the heat equation with noise,, Ann. Probab., 21 (1993), 2189. doi: 10.1214/aop/1176989016. Google Scholar
[24]	C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case,, Probab. Theory Related Fields, 140 (2008), 41. doi: 10.1007/s00440-007-0057-2. Google Scholar
[25]	A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's,, Probab. Theory Related Fields, 134 (2006), 215. doi: 10.1007/s00440-005-0427-6. Google Scholar
[26]	A. Shirikyan, Exponential mixing for randomly forced partial differential equations: Method of coupling,, In Instability in models connected with fluid flows. II, 7 (2008), 155. doi: 10.1007/978-0-387-75219-8_4. Google Scholar
[27]	M. I. Vishik, A. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics,, Uspekhi Mat. Nauk, 34 (1979), 135. Google Scholar
[28]	S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal, 3 (2004), 921. doi: 10.3934/cpaa.2004.3.921. Google Scholar

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