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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 |
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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