Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Benjamin Gess

doi:10.1016/j.crma.2012.02.004

Probability Theory/Partial Differential Equations

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Presented by: Alain Bensoussan

Benjamin Gess ¹

¹ Faculty of Mathematics, University of Bielefeld, 33615 Bielefeld, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 299-302.

Abstract
Résumé

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in $L^{1} (O)$ . The generation of a continuous, order-preserving random dynamical system (RDS) on $L^{1} (O)$ and the existence of a “small” random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. Uniform $L^{\infty}$ bounds and uniform space–time continuity of solutions is shown. General noise including fractional Brownian Motion for all Hurst parameters is contained. A pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained.

Lʼexistence et lʼunicité des solutions des équations aux milieux poreux pilotés par des « rough paths », continus, linéaire multiplicatifs et distribués dans lʼespace et le temps sont démontrées pour des conditions initiales dans $L^{1} (O)$ . On obtient la génération dʼun système dynamique aléatoire continu et monotone dans $L^{1} (O)$ ainsi que lʼexistence dʼun « petit » attracteur aléatoire pour des équations aux milieux poreux stochastiques perturbés par un bruit linéaire multiplicatif, distribué dans lʼespace et le temps. Des bornes uniformes dans $L^{\infty} (O)$ et la continuité uniforme des solutions dans lʼespace et le temps sont démontrées. Le cas dʼun bruit généralisé, y compris le mouvement Brownien fractionnaire pour tous les paramétres de Hurst est contenu. Un résultat trajectoire du type Wong–Zakai pour un bruit mené par une semimartingale continue est obtenu.

Received: 2011-08-13
Accepted: 2012-02-07
Published online: 2012-02-23

DOI: 10.1016/j.crma.2012.02.004

Author's affiliations:

Benjamin Gess ¹

¹ Faculty of Mathematics, University of Bielefeld, 33615 Bielefeld, Germany

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Benjamin Gess. Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 299-302. doi : 10.1016/j.crma.2012.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.02.004/

References
Cited by

[1] V. Barbu; M. Röckner On a random scaled porous media equation, J. Differential Equations, Volume 251 (2011) no. 9, pp. 2494-2514

[2] I. Chueshov Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002

[3] E. DiBenedetto Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., Volume 32 (1983) no. 1, pp. 83-118

[4] B. Gess Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise, 2011 | arXiv

[5] S.-E.A. Mohammed; T. Zhang; H. Zhao The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., Volume 196 (2008) no. 917 (pp. vi+105)

[6] E. Wong; M. Zakai On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., Volume 36 (1965), pp. 1560-1564

Cited by Sources:

^☆ A more detailed account of the results presented here can be found in (Gess, 2011 [4]).

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