Dans l’article [KNS20], nous avons étudié le problème de mélange pour une classe d’EDP avec un bruit borné très dégénéré et établi l’unicité de la mesure stationnaire et sa stabilité exponentielle pour la métrique dual-Lipschitz. L’une des hypothèses imposées au problème en question exigeait que l’équation non perturbée ait exactement un point d’équilibre globalement stable. Dans cet article, on assouplit cette hypothèse, en ne supposant que la contrôlabilité globale à un point donné. On prouve que l’unicité d’une mesure stationnaire et la convergence restent vraies, alors que le taux de convergence n’est pas nécessairement exponentiel. Le résultat est applicable aux EDP de type parabolique avec une perturbation aléatoire, à condition que la partie déterministe de la force extérieure soit en position générale, ce qui garantit une structure régulière pour l’attracteur du problème non perturbé.
In the paper [KNS20], we studied the problem of mixing for a class of PDEs with a very degenerate bounded noise and established the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric. One of the hypotheses imposed on the problem in question required that the unperturbed equation should have exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem.
CC-BY 4.0
@article{JEP_2020__7__871_0,
author = {Sergei Kuksin and Vahagn Nersesyan and Armen Shirikyan},
title = {Mixing via controllability for randomly~forced nonlinear dissipative {PDEs}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {871--896},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
doi = {10.5802/jep.130},
zbl = {07184227},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.130/}
}
TY - JOUR AU - Sergei Kuksin AU - Vahagn Nersesyan AU - Armen Shirikyan TI - Mixing via controllability for randomly forced nonlinear dissipative PDEs JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 871 EP - 896 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.130/ DO - 10.5802/jep.130 LA - en ID - JEP_2020__7__871_0 ER -
%0 Journal Article %A Sergei Kuksin %A Vahagn Nersesyan %A Armen Shirikyan %T Mixing via controllability for randomly forced nonlinear dissipative PDEs %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 871-896 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.130/ %R 10.5802/jep.130 %G en %F JEP_2020__7__871_0
Sergei Kuksin; Vahagn Nersesyan; Armen Shirikyan. Mixing via controllability for randomly forced nonlinear dissipative PDEs. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 871-896. doi : 10.5802/jep.130. https://jep.centre-mersenne.org/articles/10.5802/jep.130/
[AS05] - “Navier–Stokes equations: controllability by means of low modes forcing”, J. Math. Fluid Mech. 7 (2005) no. 1, p. 108-152 | DOI | MR | Zbl
[AS06] - “Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing”, Comm. Math. Phys. 265 (2006) no. 3, p. 673-697 | DOI | MR | Zbl
[BKL02] - “Exponential mixing of the 2D stochastic Navier–Stokes dynamics”, Comm. Math. Phys. 230 (2002) no. 1, p. 87-132 | DOI | MR | Zbl
[Bri02] - “Ergodicity and mixing for stochastic partial differential equations”, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, p. 567-585 | MR | Zbl
[BV92] - Attractors of evolution equations, North-Holland Publishing, Amsterdam, 1992 | Zbl
[CI74] - “A bifurcation problem for a nonlinear partial differential equation of parabolic type”, Applicable Anal. 4 (1974), p. 17-37 | DOI | MR | Zbl
[Deb13] - “Ergodicity results for the stochastic Navier–Stokes equations: an introduction”, in Topics in mathematical fluid mechanics, Lect. Notes in Math., vol. 2073, Springer, Heidelberg, 2013, p. 23-108 | DOI | MR | Zbl
[EMS01] - “Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation”, Comm. Math. Phys. 224 (2001) no. 1, p. 83-106 | DOI | MR | Zbl
[ES00] - “New results in mathematical and statistical hydrodynamics”, Russian Math. Surveys 55 (2000) no. 4, p. 635-666 | DOI | MR
[Fel68] - An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, New York, 1968 | Zbl
[FGRT15] - “Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing”, J. Funct. Anal. 269 (2015) no. 8, p. 2427-2504 | DOI | MR | Zbl
[FM95] - “Ergodicity of the 2D Navier–Stokes equation under random perturbations”, Comm. Math. Phys. 172 (1995) no. 1, p. 119-141 | DOI | Zbl
[FW84] - Random perturbations of dynamical systems, Springer, New York–Berlin, 1984 | DOI | Zbl
[HM06] - “Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing”, Ann. of Math. (2) 164 (2006) no. 3, p. 993-1032 | DOI | MR | Zbl
[HM11] - “A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs”, Electron. J. Probab. 16 (2011) no. 23, p. 658-738 | DOI | MR | Zbl
[JNPS15] - “Large deviations and Gallavotti–Cohen principle for dissipative PDE’s with rough noise”, Comm. Math. Phys. 336 (2015) no. 1, p. 131-170 | DOI | MR | Zbl
[KNS20] - “Exponential mixing for a class of dissipative PDEs with bounded degenerate noise”, Geom. Funct. Anal. 30 (2020) no. 1, p. 126-187 | DOI | MR | Zbl
[KS00] - “Stochastic dissipative PDEs and Gibbs measures”, Comm. Math. Phys. 213 (2000) no. 2, p. 291-330 | DOI | MR | Zbl
[KS12] - Mathematics of two-dimensional turbulence, Cambridge Tracts in Math., vol. 194, Cambridge University Press, Cambridge, 2012 | MR | Zbl
[Kuk82] - “Diffeomorphisms of function spaces that correspond to quasilinear parabolic equations”, Mat. Sb. (N.S.) 117(159) (1982) no. 3, p. 359-378, 431 | MR | Zbl
[Lam96] - Probability, John Wiley & Sons, New York, 1996 | DOI | MR
[Mar17] - “Large deviations for stationary measures of stochastic nonlinear wave equations with smooth white noise”, Comm. Pure Appl. Math. 70 (2017) no. 9, p. 1754-1797 | DOI | MR | Zbl
[Ner10] - “Controllability of 3D incompressible Euler equations by a finite-dimensional external force”, ESAIM Control Optim. Calc. Var. 16 (2010) no. 3, p. 677-694 | DOI | Numdam | MR | Zbl
[Ner20] - “Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension”, Math. Control Relat. Fields (2020), under revision
[Shi06] - “Approximate controllability of three-dimensional Navier–Stokes equations”, Comm. Math. Phys. 266 (2006) no. 1, p. 123-151 | DOI | MR | Zbl
[Shi15] - “Control and mixing for 2D Navier–Stokes equations with space-time localised noise”, Ann. Sci. École Norm. Sup. (4) 48 (2015) no. 2, p. 253-280 | DOI | MR
[Shi20] - “Controllability implies mixing II. Convergence in the dual-Lipschitz metric”, J. Eur. Math. Soc. (JEMS) (2020), accepted for publication
[Sma65] - “An infinite dimensional version of Sard’s theorem”, Amer. J. Math. 87 (1965), p. 861-866 | DOI | MR | Zbl
[Zab08] - Mathematical control theory, Modern Birkhäuser Classics, Birkhäuser, Boston, MA, 2008 | DOI | Zbl
Cité par Sources :