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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

Davit Martirosyan 1,

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

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 and 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, Amsterdam, 1992.

[2]

Y. Bakhtin, E. Cator and K. Khanin, Space-time stationary solutions for the Burgers equation, J. Amer. Math. Soc., 27 (2014), 193-238. doi: 10.1090/S0894-0347-2013-00773-0.

[3]

V. Barbu and G. Da Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim., 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4.

[4]

J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing of the 2D stochastic Navier-Stokes dynamics, Comm. Math. Phys., 230 (2002), 87-132. doi: 10.1007/s00220-002-0708-1.

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of AMS Coll. Publ. AMS, Providence, 2002.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[8]

A. Debussche, Ergodicity results for the stochastic Navier-Stokes equations: An introduction, In Topics in Mathematical Fluid Mechanics, 2073 (2013), 23-108. doi: 10.1007/978-3-642-36297-2_2.

[9]

A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation, J. Evol. Equ., 5 (2005), 317-356. doi: 10.1007/s00028-005-0195-x.

[10]

A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing,, , (). 

[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), 777-796 (electronic). doi: 10.1137/040611896.

[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), 83-106. doi: 10.1007/s002201224083.

[13]

W. E, K. Khanin, A. Mazel and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2), 151 (2000), 877-960. doi: 10.2307/121126.

[14]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Comm. Math. Phys., 172 (1995), 119-141. doi: 10.1007/BF02104513.

[15]

T. Girya and I. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb., 186 (1995), 29-46. doi: 10.1070/SM1995v186n01ABEH000002.

[16]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091. doi: 10.1214/08-AOP392.

[17]

A. Haraux, Two remarks on hyperbolic dissipative problems, In Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983-1984), Res. Notes in Math., Pitman, Boston, MA, 122 (1985), 161-179.

[18]

R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys., 232 (2003), 377-428.

[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), 389-423. doi: 10.1007/s40072-013-0010-6.

[20]

S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Comm. Math. Phys., 213 (2000), 291-330. doi: 10.1007/s002200000237.

[21]

S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139137119.

[22]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[23]

C. Mueller, Coupling and invariant measures for the heat equation with noise, Ann. Probab., 21 (1993), 2189-2199. doi: 10.1214/aop/1176989016.

[24]

C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case, Probab. Theory Related Fields, 140 (2008), 41-82. doi: 10.1007/s00440-007-0057-2.

[25]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247. doi: 10.1007/s00440-005-0427-6.

[26]

A. Shirikyan, Exponential mixing for randomly forced partial differential equations: Method of coupling, In Instability in models connected with fluid flows. II, Int. Math. Ser. (N. Y.), pages. Springer, New York, 7 (2008), 155-188. doi: 10.1007/978-0-387-75219-8_4.

[27]

M. I. Vishik, A. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210.

[28]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal, 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing, Amsterdam, 1992.

[2]

Y. Bakhtin, E. Cator and K. Khanin, Space-time stationary solutions for the Burgers equation, J. Amer. Math. Soc., 27 (2014), 193-238. doi: 10.1090/S0894-0347-2013-00773-0.

[3]

V. Barbu and G. Da Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim., 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4.

[4]

J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing of the 2D stochastic Navier-Stokes dynamics, Comm. Math. Phys., 230 (2002), 87-132. doi: 10.1007/s00220-002-0708-1.

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of AMS Coll. Publ. AMS, Providence, 2002.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[7]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.

[8]

A. Debussche, Ergodicity results for the stochastic Navier-Stokes equations: An introduction, In Topics in Mathematical Fluid Mechanics, 2073 (2013), 23-108. doi: 10.1007/978-3-642-36297-2_2.

[9]

A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation, J. Evol. Equ., 5 (2005), 317-356. doi: 10.1007/s00028-005-0195-x.

[10]

A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing,, , (). 

[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), 777-796 (electronic). doi: 10.1137/040611896.

[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), 83-106. doi: 10.1007/s002201224083.

[13]

W. E, K. Khanin, A. Mazel and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2), 151 (2000), 877-960. doi: 10.2307/121126.

[14]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Comm. Math. Phys., 172 (1995), 119-141. doi: 10.1007/BF02104513.

[15]

T. Girya and I. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb., 186 (1995), 29-46. doi: 10.1070/SM1995v186n01ABEH000002.

[16]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091. doi: 10.1214/08-AOP392.

[17]

A. Haraux, Two remarks on hyperbolic dissipative problems, In Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983-1984), Res. Notes in Math., Pitman, Boston, MA, 122 (1985), 161-179.

[18]

R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys., 232 (2003), 377-428.

[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), 389-423. doi: 10.1007/s40072-013-0010-6.

[20]

S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Comm. Math. Phys., 213 (2000), 291-330. doi: 10.1007/s002200000237.

[21]

S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139137119.

[22]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[23]

C. Mueller, Coupling and invariant measures for the heat equation with noise, Ann. Probab., 21 (1993), 2189-2199. doi: 10.1214/aop/1176989016.

[24]

C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case, Probab. Theory Related Fields, 140 (2008), 41-82. doi: 10.1007/s00440-007-0057-2.

[25]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247. doi: 10.1007/s00440-005-0427-6.

[26]

A. Shirikyan, Exponential mixing for randomly forced partial differential equations: Method of coupling, In Instability in models connected with fluid flows. II, Int. Math. Ser. (N. Y.), pages. Springer, New York, 7 (2008), 155-188. doi: 10.1007/978-0-387-75219-8_4.

[27]

M. I. Vishik, A. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210.

[28]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal, 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921.

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