Finite Speed of Propagation for Stochastic Porous Media Equations

We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding random attractor has infinite fractal dimension.

[1]  Benjamin Gess,et al.  Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise , 2011, 1108.2413.

[2]  Michael Röckner,et al.  Some Results on Stochastic Porous Media Equations , 2008 .

[3]  Sergey Zelik,et al.  Finite- and infinite-dimensional attractors for porous media equations , 2008 .

[4]  Viorel Barbu,et al.  Localization of solutions to stochastic porous media equations: finite speed of propagation , 2012 .

[5]  Tomás Caraballo,et al.  Stabilisation of linear PDEs by Stratonovich noise , 2004, Syst. Control. Lett..

[6]  M. Marcus,et al.  Markov Processes, Gaussian Processes, and Local Times: Contents , 2006 .

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

[8]  Hans Crauel,et al.  The effect of noise on the chafee-infante equation : A nonlinear case study , 2006 .

[9]  Feng-Yu Wang,et al.  Stochastic generalized porous media and fast diffusion equations , 2006, math/0602369.

[10]  Juan Luis Vázquez.  The interfaces of one-dimensional flows in porous media , 1983 .

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

[12]  J. U. Kim,et al.  On the stochastic porous medium equation , 2006 .

[13]  Harald Oberhauser,et al.  Rough path stability of (semi-)linear SPDEs , 2010, 1005.1781.

[14]  Massimiliano Gubinelli,et al.  Rough evolution equations , 2008, 0803.0552.

[15]  Zdzislaw Brzezniak,et al.  Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains , 2006 .

[16]  H. Alt.  Lineare Funktionalanalysis : eine anwendungsorientierte Einführung , 2002 .

[17]  S. V. Lototsky,et al.  A RANDOM CHANGE OF VARIABLES AND APPLICATIONS TO THE STOCHASTIC POROUS MEDIUM EQUATION WITH MULTIPLICATIVE TIME NOISE , 2007, 0707.3155.

[18]  Michael Röckner,et al.  On a random scaled porous media equation , 2011 .

[19]  Michael Röckner,et al.  General extinction results for stochastic partial differential equations and applications , 2011, J. Lond. Math. Soc..

[20]  Kening Lu,et al.  PATHWISE SOLUTIONS TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[21]  Emmanuele DiBenedetto,et al.  Continuity of Weak Solutions to a General Porous Media Equation. , 1981 .

[22]  Benjamin Gess,et al.  Strong Solutions for Stochastic Partial Differential Equations of Gradient Type , 2011, 1104.4243.

[23]  Michael Röckner,et al.  The Global Random Attractor for a Class of Stochastic Porous Media Equations , 2010, 1010.0551.

[24]  Michael Röckner,et al.  Stochastic Porous Media Equations and Self-Organized Criticality , 2008, 0801.2478.

[25]  Michael Röckner,et al.  Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions , 2011, 1102.3593.

[26]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[27]  Michael Röckner,et al.  Non-monotone stochastic generalized porous media equations☆ , 2008 .

[28]  Harald Oberhauser,et al.  A (rough) pathwise approach to a class of non-linear stochastic partial differential equations , 2009, 0902.3352.

[29]  É. Pardoux,et al.  Équations aux dérivées partielles stochastiques non linéaires monotones : étude de solutions fortes de type Ito , 1975 .

[30]  Giuseppe Da Prato,et al.  Existence and uniqueness of nonnegative solutions to the stochastic porous media equation , 2007 .

[31]  Michael Röckner,et al.  Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity , 2005, math/0512259.

[32]  Giuseppe Da Prato,et al.  Existence of strong solutions for stochastic porous media equation under general monotonicity conditions , 2007, math/0703421.

[33]  Michael Röckner,et al.  Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise , 2012 .

[34]  Benjamin Gess,et al.  Random Attractors for Degenerate Stochastic Partial Differential Equations , 2012, 1206.2329.

[35]  K. Elworthy.  RANDOM DYNAMICAL SYSTEMS (Springer Monographs in Mathematics) , 2000 .

[36]  Michael Röckner,et al.  Weak solutions to stochastic porous media equations , 2004 .

[37]  Wei Liu,et al.  Random attractors for a class of stochastic partial differential equations driven by general additive noise , 2010, 1010.4641.

[38]  Panagiotis E. Souganidis,et al.  Fully nonlinear stochastic PDE with semilinear stochastic dependence , 2000 .

[39]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[40]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

[41]  K. Kadlec,et al.  Stochastic Evolution Equations , 2013 .