Large deviation principle for occupation measures of two dimensional stochastic convective Brinkman-Forchheimer equations

The present work is concerned about the two-dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations perturbed by a white noise (non degenerate) in smooth bounded domains in $\R^{2}$. We establish two important properties of the Markov semigroup associated with the solutions of the 2D SCBF equations (for the absorption exponent $r=1,2,3$), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss about the ergodic behavior of the 2D SCBF equations by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence.

[1]  Hongjun Gao,et al.  Ergodicity and dynamics for the stochastic 3D Navier–Stokes equations with damping , 2018 .

[2]  K. Elworthy,et al.  Formulae for the Derivatives of Heat Semigroups , 1994, 1911.10971.

[3]  G. Prato.  Kolmogorov Equations For Stochastic Pdes , 2004 .

[4]  G. Kallianpur,et al.  Stochastic Differential Equations in Infinite Dimensional Spaces , 1995 .

[5]  A. Shirikyan,et al.  Large Deviations from a Stationary Measure for a Class of Dissipative PDEs with Random Kicks , 2012, 1212.0527.

[6]  Mathieu Gourcy.  Large deviation principle of occupation measure for stochastic Burgers equation , 2007 .

[7]  Tusheng Zhang,et al.  Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles , 2012 .

[8]  Benedetta Ferrario,et al.  Ergodic results for stochastic navier-stokes equation , 1997 .

[9]  D. Stroock.  An Introduction to the Theory of Large Deviations , 1984 .

[10]  B. Maslowski.  On probability distributions of solutions of semilinear stochastic evolution equations , 1993 .

[11]  Lihu Xu,et al.  Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion , 2017, 1701.01204.

[12]  Sri Sritharan,et al.  Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion , 1998 .

[13]  Yoshikazu Giga,et al.  Solutions in Lr of the Navier-Stokes initial value problem , 1985 .

[14]  M. T. Mohan,et al.  Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise , 2018, Mathematische Nachrichten.

[15]  P. Dupuis,et al.  A VARIATIONAL REPRESENTATION FOR POSITIVE FUNCTIONALS OF INFINITE DIMENSIONAL BROWNIAN MOTION , 2000 .

[16]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[17]  Richard B. Sowers,et al.  Large Deviations for a Reaction-Diffusion Equation with Non-Gaussian Perturbations , 1992 .

[18]  M. T. Mohan.  Asymptotic log-Harnack inequality for the stochastic convective Brinkman-Forchheimer equations with degenerate noise , 2020, 2008.00955.

[19]  Mathieu Gourcy.  A large deviation principle for 2D stochastic Navier–Stokes equation , 2007 .

[20]  M. T. Mohan,et al.  Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman–Forchheimer equations in bounded domains , 2020, Stochastics.

[21]  S. Varadhan.  Large Deviations and Applications , 1984 .

[22]  A. Shirikyan,et al.  Large Deviations and Gallavotti–Cohen Principle for Dissipative PDEs with Rough Noise , 2015 .

[23]  Roger Temam,et al.  Navier–Stokes Equations and Nonlinear Functional Analysis: Second Edition , 1995 .

[24]  On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities , 2016, 1611.00734.

[25]  Franco Flandoli,et al.  Dissipativity and invariant measures for stochastic Navier-Stokes equations , 1994 .

[26]  J. Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Invariant measures for stochastic evolution equations , 1996 .

[27]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 2014 .

[28]  Franco Flandoli,et al.  Ergodicity of the 2-D Navier-Stokes equation under random perturbations , 1995 .

[29]  Liming Wu.  Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems , 2001 .

[30]  P. Chow,et al.  Large deviation problem for some parabolic itǒ equations , 1992 .

[31]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[32]  M. T. Mohan.  Wentzell–Freidlin Large Deviation Principle for Stochastic Convective Brinkman–Forchheimer Equations , 2020, Journal of Mathematical Fluid Mechanics.

[33]  Hongjun Gao,et al.  Stochastic 3D Navier–Stokes Equations with Nonlinear Damping: Martingale Solution, Strong Solution and Small Time LDP , 2019, Stochastic PDEs and Modelling of Multiscale Complex System.

[34]  Jonathan C. Mattingly,et al.  Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing , 2004, math/0406087.

[35]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[36]  Hongjun Gao,et al.  Well-posedness and invariant measures for a class of stochastic 3D Navier-Stokes equations with damping driven by jump noise , 2019, Journal of Differential Equations.

[37]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[38]  J. Xiong,et al.  Large deviation principle of occupation measures for non-linear monotone SPDEs , 2016, Science China Mathematics.

[39]  M. T. Mohan.  Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations , 2020, Stochastics.

[40]  Giuseppe Da Prato,et al.  Ergodicity for the 3D stochastic Navier–Stokes equations , 2003 .

[41]  R. Temam.  Navier-Stokes Equations , 1977 .

[42]  S. Varadhan,et al.  Large deviations , 2019, Graduate Studies in Mathematics.

[43]  Daisuke Fujiwara,et al.  An L_r-theorem of the Helmholtz decomposition of vector fields , 1977 .

[44]  M. T. Mohan.  Stochastic convective Brinkman-Forchheimer equations , 2020, 2007.09376.

[45]  C. Kenig,et al.  Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle , 1993 .

[46]  Arnaud Debussche,et al.  Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction , 2013 .

[47]  Sri Sritharan,et al.  Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise , 2006 .

[48]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[49]  P. G. Ciarlet,et al.  Linear and Nonlinear Functional Analysis with Applications , 2013 .

[50]  Tusheng Zhang,et al.  Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations , 2010 .