# American Institute of Mathematical Sciences

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

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.
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, arXiv:1310.3779. [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, arXiv:1310.3779. [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.
 [1] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [2] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [3] Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012 [4] Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61 [5] Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011 [6] Xuhui Peng, Jianhua Huang, Yan Zheng. Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4479-4506. doi: 10.3934/cpaa.2020204 [7] Sanchit Chaturvedi, Jonathan Luk. Phase mixing for solutions to 1D transport equation in a confining potential. Kinetic and Related Models, 2022, 15 (3) : 403-416. doi: 10.3934/krm.2022002 [8] Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713 [9] James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of ${\rm{PSL}}(2, \mathbb{R})$. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042 [10] Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315 [11] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [12] Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032 [13] David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298 [14] Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic and Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041 [15] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [16] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [17] Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 [18] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [19] Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 [20] Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277

2021 Impact Factor: 1.169