×

zbMATH — the first resource for mathematics

Exponential mixing for the white-forced damped nonlinear wave equation. (English) Zbl 1304.35430
Summary: 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.

MSC:
35L71 Second-order semilinear hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
37A25 Ergodicity, mixing, rates of mixing
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L20 Initial-boundary value problems for second-order hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A. V. Babin, <em>Attractors of Evolution Equations</em>,, North-Holland Publishing, (1992) · Zbl 0778.58002
[2] Y. Bakhtin, Space-time stationary solutions for the Burgers equation,, J. Amer. Math. Soc., 27, 193, (2014) · Zbl 1296.37051
[3] V. Barbu, The stochastic nonlinear damped wave equation,, Appl. Math. Optim., 46, 125, (2002) · Zbl 1024.47025
[4] J. Bricmont, Exponential mixing of the 2D stochastic Navier-Stokes dynamics,, Comm. Math. Phys., 230, 87, (2002) · Zbl 1033.76011
[5] V. V. Chepyzhov, <em>Attractors for Equations of Mathematical Physics</em>, volume 49 of <em>AMS Coll. Publ.</em>, AMS, (2002) · Zbl 0986.35001
[6] G. Da Prato, <em>Stochastic Equations in Infinite Dimensions</em>,, Cambridge University Press, (1992) · Zbl 1140.60034
[7] G. Da Prato, <em>Ergodicity for Infinite Dimensional Systems</em>,, Cambridge University Press, (1996) · Zbl 0849.60052
[8] A. Debussche, Ergodicity results for the stochastic Navier-Stokes equations: An introduction,, In Topics in Mathematical Fluid Mechanics, 2073, 23, (2013) · Zbl 1301.35086
[9] A. Debussche, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation,, J. Evol. Equ., 5, 317, (2005) · Zbl 1091.60010
[10] A. Debussche, Invariant measure of scalar first-order conservation laws with stochastic forcing,, <a href= · Zbl 1331.60117
[11] N. Dirr, Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise,, SIAM J. Math. Anal., 37, 777, (2005) · Zbl 1099.35185
[12] W. E., Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation,, Comm. Math. Phys., 224, 83, (2001) · Zbl 0994.60065
[13] W. E., Invariant measures for Burgers equation with stochastic forcing,, Ann. of Math. (2), 151, 877, (2000) · Zbl 0972.35196
[14] F. Flandoli, Ergodicity of the 2D Navier-Stokes equation under random perturbations,, Comm. Math. Phys., 172, 119, (1995) · Zbl 0845.35080
[15] T. Girya, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems,, Mat. Sb., 186, 29, (1995) · Zbl 0851.60036
[16] M. Hairer, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations,, Ann. Probab., 36, 2050, (2008) · Zbl 1173.37005
[17] A. Haraux, Two remarks on hyperbolic dissipative problems,, In Nonlinear partial differential equations and their applications. Collège de France seminar, 122, 1983, (1985) · Zbl 0579.35057
[18] R. Iturriaga, Burgers turbulence and random Lagrangian systems,, Comm. Math. Phys., 232, 377, (2003) · Zbl 1029.76030
[19] S. Kuksin, Stochastic CGL equations without linear dispersion in any space dimension,, Stochastic Partial Differential Equations: Analysis and Computations, 1, 389, (2013) · Zbl 1287.60093
[20] S. Kuksin, Stochastic dissipative PDEs and Gibbs measures,, Comm. Math. Phys., 213, 291, (2000) · Zbl 0974.60046
[21] S. Kuksin, <em>Mathematics of Two-Dimensional Turbulence</em>,, Cambridge University Press, (2012) · Zbl 1333.76003
[22] J.-L. Lions, <em>Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires</em>,, Dunod; Gauthier-Villars, (1969) · Zbl 0189.40603
[23] C. Mueller, Coupling and invariant measures for the heat equation with noise,, Ann. Probab., 21, 2189, (1993) · Zbl 0795.60056
[24] C. Odasso, Exponential mixing for stochastic PDEs: The non-additive case,, Probab. Theory Related Fields, 140, 41, (2008) · Zbl 1137.60030
[25] A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE’s,, Probab. Theory Related Fields, 134, 215, (2006) · Zbl 1099.35188
[26] A. Shirikyan, Exponential mixing for randomly forced partial differential equations: Method of coupling,, In Instability in models connected with fluid flows. II, 7, 155, (2008) · Zbl 1296.37044
[27] M. I. Vishik, Some mathematical problems of statistical hydromechanics,, Uspekhi Mat. Nauk, 34, 135, (1979) · Zbl 0503.76045
[28] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal, 3, 921, (2004) · Zbl 1197.35168
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.