The Annals of Probability

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

Benjamin Gess

Full-text: Open access

Abstract

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in $L^{1}(\mathcal{O})$ on bounded domains $\mathcal{O} $. The generation of a continuous, order-preserving random dynamical system on $L^{1}(\mathcal{O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal{O})$ norm. Uniform $L^{\infty}$ bounds and uniform space–time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space–time rough signals, existence of solutions is proven for initial data in $L^{m+1}(\mathcal{O})$.

Article information

Source
Ann. Probab., Volume 42, Number 2 (2014), 818-864.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251304

Digital Object Identifier
doi:10.1214/13-AOP869

Mathematical Reviews number (MathSciNet)
MR3178475

Zentralblatt MATH identifier
06288295

Subjects
Primary: 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 76S05: Flows in porous media; filtration; seepage
Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 37L30: Attractors and their dimensions, Lyapunov exponents

Keywords
Stochastic partial differential equations stochastic porous medium equation stochastic fast diffusion equation random dynamical system random attractor Wong–Zakai approximation

Citation

Gess, Benjamin. Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise. Ann. Probab. 42 (2014), no. 2, 818--864. doi:10.1214/13-AOP869. https://projecteuclid.org/euclid.aop/1393251304


Export citation

References

  • [1] Anderson, J. R. (1991). Local existence and uniqueness of solutions of degenerate parabolic equations. Comm. Partial Differential Equations 16 105–143.
  • [2] Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
  • [3] Arnold, L. and Scheutzow, M. (1995). Perfect cocycles through stochastic differential equations. Probab. Theory Related Fields 101 65–88.
  • [4] Barbu, V., Da Prato, G. and Röckner, M. (2008). Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57 187–211.
  • [5] Barbu, V., Da Prato, G. and Röckner, M. (2008). Some results on stochastic porous media equations. Boll. Unione Mat. Ital. (9) 1 1–15.
  • [6] Barbu, V., Da Prato, G. and Röckner, M. (2009). Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37 428–452.
  • [7] Barbu, V., Da Prato, G. and Röckner, M. (2009). Finite time extinction for solutions to fast diffusion stochastic porous media equations. C. R. Math. Acad. Sci. Paris 347 81–84.
  • [8] Barbu, V., Da Prato, G. and Röckner, M. (2009). Stochastic porous media equations and self-organized criticality. Comm. Math. Phys. 285 901–923.
  • [9] Barbu, V. and Röckner, M. (2011). On a random scaled porous media equation. J. Differential Equations 251 2494–2514.
  • [10] Bénilan, P., Crandall, M. G. and Pierre, M. (1984). Solutions of the porous medium equation in $\mathbf{R}^{N}$ under optimal conditions on initial values. Indiana Univ. Math. J. 33 51–87.
  • [11] Beyn, W.-J., Gess, B., Lescot, P. and Röckner, M. (2011). The global random attractor for a class of stochastic porous media equations. Comm. Partial Differential Equations 36 446–469.
  • [12] Brzeźniak, Z. and Li, Y. (2006). Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains. Trans. Amer. Math. Soc. 358 5587–5629 (electronic).
  • [13] Buckdahn, R. and Ma, J. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl. 93 181–204.
  • [14] Buckdahn, R. and Ma, J. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stochastic Process. Appl. 93 205–228.
  • [15] Caruana, M., Friz, P. K. and Oberhauser, H. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 27–46.
  • [16] Chueshov, I. (2002). Monotone Random Systems Theory and Applications. Lecture Notes in Math. 1779. Springer, Berlin.
  • [17] Crauel, H., Debussche, A. and Flandoli, F. (1997). Random attractors. J. Dynam. Differential Equations 9 307–341.
  • [18] Crauel, H. and Flandoli, F. (1994). Attractors for random dynamical systems. Probab. Theory Related Fields 100 365–393.
  • [19] Da Prato, G. and Röckner, M. (2004). Weak solutions to stochastic porous media equations. J. Evol. Equ. 4 249–271.
  • [20] Da Prato, G., Röckner, M., Rozovskii, B. L. and Wang, F.-Y. (2006). Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity. Comm. Partial Differential Equations 31 277–291.
  • [21] Diaz, J. I. and Kersner, R. (1987). On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. J. Differential Equations 69 368–403.
  • [22] DiBenedetto, E. (1983). Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J. 32 83–118.
  • [23] Flandoli, F. and Lisei, H. (2004). Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications. Stoch. Anal. Appl. 22 1385–1420.
  • [24] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [25] Gess, B. (2012). Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise. C. R. Math. Acad. Sci. Paris 350 299–302.
  • [26] Gess, B. (2012). Strong solutions for stochastic partial differential equations of gradient type. J. Funct. Anal. 263 2355–2383.
  • [27] Gess, B. (2013). Random attractors for degenerate stochastic partial differential equations. J. Dynam. Differential Equations 25 121–157.
  • [28] Gess, B., Liu, W. and Röckner, M. (2011). Random attractors for a class of stochastic partial differential equations driven by general additive noise. J. Differential Equations 251 1225–1253.
  • [29] Gilding, B. H. (1989). Improved theory for a nonlinear degenerate parabolic equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 165–224.
  • [30] Gubinelli, M. and Tindel, S. (2010). Rough evolution equations. Ann. Probab. 38 1–75.
  • [31] Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math. 64 1547–1585.
  • [32] Kim, J. U. (2006). On the stochastic porous medium equation. J. Differential Equations 220 163–194.
  • [33] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Amer. Math. Soc., Providence, RI.
  • [34] Li, Y. and Guo, B. (2008). Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. J. Differential Equations 245 1775–1800.
  • [35] Lieberman, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ.
  • [36] Lions, P.-L. and Souganidis, P. E. (2000). Fully nonlinear stochastic pde with semilinear stochastic dependence. C. R. Acad. Sci. Paris Sér. I Math. 331 617–624.
  • [37] Lions, P.-L. and Souganidis, P. E. (2002). Viscosity solutions of fully nonlinear stochastic partial differential equations. Sūrikaisekikenkyūsho Kōkyūroku 1287 58–65.
  • [38] Mohammed, S.-E. A., Zhang, T. and Zhao, H. (2008). The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Mem. Amer. Math. Soc. 196 vi+105.
  • [39] Protter, P. E. (2004). Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [40] Ren, J., Röckner, M. and Wang, F.-Y. (2007). Stochastic generalized porous media and fast diffusion equations. J. Differential Equations 238 118–152.
  • [41] Röckner, M. and Wang, F.-Y. (2008). Non-monotone stochastic generalized porous media equations. J. Differential Equations 245 3898–3935.
  • [42] Sacks, P. E. (1983). The initial and boundary value problem for a class of degenerate parabolic equations. Comm. Partial Differential Equations 8 693–733.
  • [43] Schmalfuss, B. (1992). Backward cocycle and attractors of stochastic differential equations. International Seminar on Applied Mathematics—Nonlinear Dynamics: Attractor Approximation and Global Behavior (V. Reitmann, T. Riedrich and N. Koksch, eds.) 185–192. Technische Universität Dresden.
  • [44] van Neerven, J. and Veraar, M. C. (2006). On the stochastic Fubini theorem in infinite dimensions. In Stochastic Partial Differential Equations and Applications—VII. Lect. Notes Pure Appl. Math. 245 323–336. Chapman & Hall/CRC, Boca Raton, FL.
  • [45] Vázquez, J. L. (2007). The Porous Medium Equation: Mathematical Theory. Clarendon, Oxford.