Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations

We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an LP-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns Out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view its the analog of Doeblin's condition. We then proceed to Study Situations where the behavior is not SO Uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokes equations. even in situations where the forcing is extremely de generate. Using the convergence result, we show that the stochastic Navier-Stokes equations' invariant measures depend continuously on the viscosity and the structure of the forcing.

[1]  C. Villani Optimal Transport: Old and New , 2008 .

[2]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[3]  Jonathan C. Mattingly,et al.  Malliavin calculus for infinite-dimensional systems with additive noise , 2006, math/0610754.

[4]  A. Majda,et al.  The emergence of large‐scale coherent structure under small‐scale random bombardments , 2006 .

[5]  Bohdan Maslowski,et al.  Exponential ergodicity for stochastic Burgers and 2D Navier–Stokes equations , 2005 .

[6]  Andrey Sarychev,et al.  Navier–Stokes Equations: Controllability by Means of Low Modes Forcing , 2005 .

[7]  Jonathan C. Mattingly On recent progress for the stochastic Navier Stokes equations , 2004, math/0409194.

[8]  Jonathan C. Mattingly,et al.  Malliavin calculus for the stochastic 2D Navier—Stokes equation , 2004, math/0407215.

[9]  M. Röckner,et al.  A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations , 2004 .

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

[11]  C. Odasso Ergodicity for the stochastic Complex Ginzburg-Landau equations , 2004, math/0405519.

[12]  G. Burton TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[13]  C. Villani Topics in Optimal Transportation , 2003 .

[14]  James C. Robinson Stability of random attractors under perturbation and approximation , 2002 .

[15]  Jonathan C. Mattingly Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics , 2002 .

[16]  M. Romito Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise , 2002, math/0210082.

[17]  Sean P. Meyn,et al.  The ODE Method and Spectral Theory of Markov Operators , 2002, math/0209277.

[18]  L. Young,et al.  Ergodic Theory of Infinite Dimensional Systems¶with Applications to Dissipative Parabolic PDEs , 2002 .

[19]  Jonathan C. Mattingly,et al.  Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation , 2001 .

[20]  Weinan E,et al.  Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation , 2001 .

[21]  J. Bricmont,et al.  Ergodicity of the 2D Navier--Stokes Equations¶with Random Forcing , 2001 .

[22]  Martin Hairer,et al.  Exponential mixing properties of stochastic PDEs through asymptotic coupling , 2001, math/0109115.

[23]  A. Shirikyan,et al.  Ergodicity for the Randomly Forced 2D Navier–Stokes Equations , 2001 .

[24]  Armen Shirikyan,et al.  A Coupling Approach¶to Randomly Forced Nonlinear PDE's. I , 2001 .

[25]  J. Bricmont,et al.  Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics , 2000 .

[26]  J.Bricmont,et al.  Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics , 2000, math-ph/0010028.

[27]  J. Eckmann,et al.  Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise , 2000, nlin/0009028.

[28]  Jonathan C. Mattingly Ergodicity of 2D Navier–Stokes Equations with¶Random Forcing and Large Viscosity , 1999 .

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

[30]  G. Fayolle,et al.  Topics in the Constructive Theory of Countable Markov Chains , 1995 .

[31]  Hubert Hennion,et al.  Sur un théorème spectral et son application aux noyaux lipchitziens , 1993 .

[32]  J. Freeman Probability Metrics and the Stability of Stochastic Models , 1991 .

[33]  L. E. Fraenkel,et al.  NAVIER-STOKES EQUATIONS (Chicago Lectures in Mathematics) , 1990 .

[34]  D. W. Robinson ONE‐PARAMETER SEMIGROUPS (London Mathematical Society Monographs, 15) , 1982 .

[35]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[36]  D. Williams,et al.  PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON STOCHASTIC DIFFERENTIAL EQUATIONS , 1980 .

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

[38]  J. Yorke,et al.  On the existence of invariant measures for piecewise monotonic transformations , 1973 .

[39]  R. Nussbaum The radius of the essential spectrum , 1970 .

[40]  L. Hörmander Hypoelliptic second order differential equations , 1967 .

[41]  C. Ionescu,et al.  THEORIE ERGODIQUE POUR DES CLASSES D'OPERATIONS NON COMPLETEMENT CONTINUES , 1950 .

[42]  J. Doob Asymptotic properties of Markoff transition prababilities , 1948 .

[43]  Ádám Gyenge Malliavin calculus and its applications , 2010 .

[44]  Carlangelo Liverani,et al.  Invariant measures and their properties. A functional analytic point of view , 2004 .

[45]  A. Shirikyan,et al.  Coupling approach to white-forced nonlinear PDEs , 2002 .

[46]  J.,et al.  Ergodicity of the 2 D Navier-Stokes Equations with Random Forcing , 2000 .

[47]  G. Fayolle,et al.  Topics in the Constructive Theory of Countable Markov Chains: Ideology of induced chains , 1995 .

[48]  G. Constantain,et al.  Probability Metrics and the Stability of Stochastic Models , 1995 .

[49]  N. Nadirashvili,et al.  JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES , 1994 .

[50]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[51]  Hantaek Bae Navier-Stokes equations , 1992 .

[52]  J. Norris Simplified Malliavin calculus , 1986 .

[53]  E. Davies,et al.  One-parameter semigroups , 1980 .

[54]  R. Melrose,et al.  JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES , 1979 .

[55]  P. Malliavin Stochastic calculus of variation and hypoelliptic operators , 1978 .

[56]  O. H. Lowry Academic press. , 1972, Analytical chemistry.

[57]  C. Foiaș,et al.  Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$ , 1967 .

[58]  T. E. Harris The Existence of Stationary Measures for Certain Markov Processes , 1956 .

[59]  W. Doeblin Sur les propriétés asymptotiques de mouvements régis par certains types de chaînes simples , 1938 .