Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.

[1]  Geometry's Fundamental Role in the Stability of Stochastic Differential Equations , 2011 .

[2]  D. Nualart.  Analysis on Wiener space and anticipating stochastic calculus , 1998 .

[3]  K. Elworthy.  ERGODICITY FOR INFINITE DIMENSIONAL SYSTEMS (London Mathematical Society Lecture Note Series 229) By G. Da Prato and J. Zabczyk: 339 pp., £29.95, LMS Members' price £22.47, ISBN 0 521 57900 7 (Cambridge University Press, 1996). , 1997 .

[4]  Luc Rey-Bellet,et al.  Ergodic properties of Markov processes , 2006 .

[5]  Armen Shirikyan.  Exact controllability in projections for three-dimensional Navier-Stokes equations , 2007 .

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

[7]  S. Varadhan,et al.  Diffusion processes , 2018, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance.

[8]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

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

[10]  Donald L. Turcotte,et al.  Mantle Convection in the Earth and Planets , 2001 .

[11]  Hans Crauel,et al.  Random attractors , 1997 .

[12]  G. Richards,et al.  Hydrodynamic stability in the presence of a stochastic forcing:a case study in convection , 2017, 1704.03840.

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

[14]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

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

[16]  Jonathan C. Mattingly,et al.  A practical criterion for positivity of transition densities , 2014, 1407.3858.

[17]  Jerzy Zabczyk,et al.  Stochastic Equations in Infinite Dimensions: Foundations , 1992 .

[18]  A. Shirikyan.  Control theory for the Burgers equation: Agrachev-Sarychev approach , 2017, 1712.09807.

[19]  Jerzy Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Appendices , 1996 .

[20]  Luc Rey Bellet.  Ergodic Properties of Markov Processes , 2006 .

[21]  H. Nersisyan.  Controllability of 3D incompressible Euler equations by a finite-dimensional external force , 2008, 0810.3297.

[22]  V. Vicol,et al.  Inviscid Limits for a Stochastically Forced Shell Model of Turbulent Flow , 2014, 1404.1098.

[23]  Physical Review , 1965, Nature.

[24]  David P. Herzog,et al.  The transition from ergodic to explosive behavior in a family of stochastic differential equations , 2011, 1105.2378.

[25]  S. Varadhan,et al.  On the Support of Diffusion Processes with Applications to the Strong Maximum Principle , 1972 .

[26]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

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

[28]  Velimir Jurdjevic,et al.  Control systems on semi-simple Lie groups and their homogeneous spaces , 1981 .

[29]  V. Nersesyan.  Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force , 2014, 1403.5369.

[30]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[31]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[32]  N. U. Prabhu,et al.  Stochastic Processes and Their Applications , 1999 .

[33]  A. Agrachev,et al.  On finite-dimensional projections of distributions for solutions of randomly forced PDE's , 2006 .

[34]  G. Richards,et al.  Ergodicity in randomly forced Rayleigh–Bénard convection , 2015, 1511.01247.

[35]  Martin Hairer,et al.  A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs , 2008, 0808.1361.

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

[37]  Donald L. Turcotte,et al.  Mantle Convection in the Earth and Planets: Contents , 2001 .

[38]  H. Crauel,et al.  Attractors for random dynamical systems , 1994 .

[39]  Velimir Jurdjevic,et al.  Polynomial control systems , 1985 .

[40]  P. Constantin,et al.  Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations , 2013, 1304.2022.

[41]  Mario Pulvirenti,et al.  Mathematical Theory of Incompressible Nonviscous Fluids , 1993 .

[42]  Nathan Glatt-Holtz,et al.  Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing , 2013, 1311.3620.

[43]  L. Arnold.  Random Dynamical Systems , 2003 .

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

[45]  R. Léandre,et al.  Decroissance exponentielle du noyau de la chaleur sur la diagonale (I) , 1991 .

[46]  A. Shirikyan.  Euler equations are not exactly controllable by a finite-dimensional external force , 2007, 0708.0520.

[47]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[48]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[49]  H. Fédérer.  Geometric Measure Theory , 1969 .

[50]  Nathan Glatt-Holtz,et al.  Asymptotic Analysis for Randomly Forced MHD , 2016, SIAM J. Math. Anal..

[51]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[52]  Shige Peng,et al.  Stochastics An International Journal of Probability and Stochastic Processes : formerly Stochastics and Stochastics Reports , 2014 .

[53]  Jonathan C. Mattingly,et al.  Malliavin calculus for the stochastic 2D Navier Stokes equation , 2004 .

[54]  Andrey A. Agrachev,et al.  Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing , 2005 .

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

[56]  David L. Elliott,et al.  Geometric control theory , 2000, IEEE Trans. Autom. Control..

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

[58]  Peter Constantin,et al.  On the Euler equations of incompressible fluids , 2007 .

[59]  Jonathan C. Mattingly,et al.  Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations , 2006, math/0602479.

[60]  Wolfgang Kliemann,et al.  On unique ergodicity for degenerate diffusions , 1987 .