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Exponential mixing for the white-forced damped nonlinear wave equation

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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.
    Mathematics Subject Classification: 35L70, 35R60, 37A25, 60H15.


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  • [1]

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


    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.


    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.


    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.


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


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


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


    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.


    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.


    A. Debussche and J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing, arXiv:1310.3779.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    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.


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


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


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


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


    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.


    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.


    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.


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


    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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