Ergodicity and local limits for stochastic local and nonlocal \(p\)-Laplace equations. (English) Zbl 1364.60082
Summary: Ergodicity for local and nonlocal stochastic singular \(p\)-Laplace equations is proven, without restriction on the spatial dimension and for all \(p\in[1,2)\). This generalizes previous results from [the authors, J. Math. Pures Appl. (9) 101, No. 6, 789–827 (2014; Zbl 1295.47042); W. Liu and the second author, Electron. Commun. Probab. 16, 447–457 (2011; Zbl 1244.60062); W. Liu, J. Evol. Equ. 9, No. 4, 747–770 (2009; Zbl 1239.60058)]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow, which solves an open problem raised in [V. Barbu et al., SIAM J. Math. Anal. 41, No. 3, 1106–1120 (2009; Zbl 1203.60079)]. Moreover, under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic \(p\)-Laplace equation to the unique invariant measure of the local stochastic \(p\)-Laplace equation is proven.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35K55 | Nonlinear parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 49J40 | Variational inequalities |
| 49J55 | Existence of optimal solutions to problems involving randomness |
| 37L15 | Stability problems for infinite-dimensional dissipative dynamical systems |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
stochastic partial differential equations; \(p\)-Laplacian; ergodicity; local limits; stochastic variational inequality; invariant measureReferences:
| [1] | S. Albeverio, A. Debussche, and L. Xu, {\it Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises}, Appl. Math. Optim., 66 (2012), pp. 273-308. · Zbl 1303.60051 |
| [2] | F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, {\it A nonlocal \(p\)-Laplacian evolution equation with Neumann boundary conditions}, J. Math. Pures Appl. (9), 90 (2008), pp. 201-227. · Zbl 1165.35322 |
| [3] | F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, {\it A nonlocal \(p\)-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions}, SIAM J. Math. Anal., 40 (2009), pp. 1815-1851, . · Zbl 1183.35034 |
| [4] | F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, {\it Local and nonlocal weighted \(p\)-Laplacian evolution equations with Neumann boundary conditions}, Publ. Mat., 55 (2011), pp. 27-66. · Zbl 1213.35282 |
| [5] | F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, {\it Nonlocal Diffusion Problems}, Math. Surveys Monogr. 165, AMS, Providence, RI, 2010. · Zbl 1214.45002 |
| [6] | V. Barbu, {\it Analysis and Control of Nonlinear Infinite-Dimensional Systems}, Math. Sci. Engrg. 190, Academic Press, Boston, MA, 1993. · Zbl 0776.49005 |
| [7] | V. Barbu, V. I. Bogachev, G. Da Prato, and M. Röckner, {\it Weak solutions to the stochastic porous media equation via Kolmogorov equations: The degenerate case}, J. Funct. Anal., 237 (2006), pp. 54-75. · Zbl 1104.60034 |
| [8] | V. Barbu and G. Da Prato, {\it Ergodicity for nonlinear stochastic equations in variational formulation}, Appl. Math. Optim., 53 (2006), pp. 121-139. · Zbl 1109.35123 |
| [9] | V. Barbu and G. Da Prato, {\it Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation}, Stochastic Process. Appl., 120 (2010), pp. 1247-1266. · Zbl 1201.60062 |
| [10] | V. Barbu, G. Da Prato, and M. Röckner, {\it Stochastic nonlinear diffusion equations with singular diffusivity}, SIAM J. Math. Anal., 41 (2009), pp. 1106-1120, . · Zbl 1203.60079 |
| [11] | V. Barbu and M. Röckner, {\it Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise}, Arch. Ration. Mech. Anal., 209 (2013), pp. 797-834. · Zbl 1286.35202 |
| [12] | Y. Belaud and J. I. Díaz, {\it Abstract results on the finite extinction time property: Application to a singular parabolic equation}, J. Convex Anal., 17 (2010), pp. 827-860. · Zbl 1213.35273 |
| [13] | V. Bogachev, {\it Measure Theory}, Vol. 1, Springer, Berlin, 2007. · Zbl 1120.28001 |
| [14] | A. Boritchev, {\it Multidimensional potential Burgers turbulence}, Comm. Math. Phys., 342 (2016), pp. 441-489. · Zbl 1338.60153 |
| [15] | H. Brézis, {\it How to recognize constant functions. Connections with Sobolev spaces}, Uspekhi Mat. Nauk, 57 (2002), pp. 59-74 (in Russian); translated in Russian Math. Surveys, 57 (2002), pp. 693-708. · Zbl 1072.46020 |
| [16] | I. Ciotir and J. M. Tölle, {\it Convergence of invariant measures for singular stochastic diffusion equations}, Stochastic Process. Appl., 122 (2012), pp. 1998-2017. · Zbl 1252.60059 |
| [17] | I. Ciotir and J. M. Tölle, {\it Corrigendum to “Convergence of invariant measures for singular stochastic diffusion equations”} [Stochastic Process. Appl., 122 (2012), pp. 1998-2017], Stochastic Process. Appl., 123 (2013), pp. 1178-1181. · Zbl 1252.60059 |
| [18] | G. Da Prato and M. Röckner, {\it Invariant measures for a stochastic porous medium equation}, in Stochastic Analysis and Related Topics in Kyoto, Adv. Stud. Pure Math. 41, Math. Soc. Japan, Tokyo, 2004, pp. 13-29. · Zbl 1102.76074 |
| [19] | G. Da Prato, M. Röckner, B. L. Rozovskii, and F.-Y. Wang, {\it Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity}, Comm. Partial Differential Equations, 31 (2006), pp. 277-291. · Zbl 1158.60356 |
| [20] | G. Da Prato and J. Zabczyk, {\it Ergodicity for Infinite-Dimensional Systems}, London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, UK, 1996. · Zbl 0849.60052 |
| [21] | A. Debussche and J. Vovelle, {\it Scalar conservation laws with stochastic forcing}, J. Funct. Anal., 259 (2010), pp. 1014-1042. · Zbl 1200.60050 |
| [22] | J. L. Doob, {\it Asymptotic properties of Markoff transition probabilities}, Trans. Amer. Math. Soc., 63 (1948), pp. 393-421. · Zbl 0041.45406 |
| [23] | A. Es-Sarhir and M.-K. von Renesse, {\it Ergodicity of stochastic curve shortening flow in the plane}, SIAM J. Math. Anal., 44 (2012), pp. 224-244, . · Zbl 1251.47041 |
| [24] | R. Ferreira and J. D. Rossi, {\it Decay estimates for a nonlocal \(p\)-Laplacian evolution problem with mixed boundary conditions}, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), pp. 1469-1478. · Zbl 1307.35046 |
| [25] | J. Földes, N. Glatt-Holtz, G. Richards, and E. Thomann, {\it Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing}, J. Funct. Anal., 269 (2015), pp. 2427-2504. · Zbl 1354.37056 |
| [26] | B. Gess, {\it Random attractors for degenerate stochastic partial differential equations}, J. Dynam. Differential Equations, 25 (2013), pp. 121-157. · Zbl 1264.37047 |
| [27] | B. Gess, {\it Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise}, Ann. Probab., 42 (2014), pp. 818-864. · Zbl 1385.37082 |
| [28] | B. Gess, W. Liu, and M. Röckner, {\it Random attractors for a class of stochastic partial differential equations driven by general additive noise}, J. Differential Equations, 251 (2011), pp. 1225-1253. · Zbl 1228.35062 |
| [29] | B. Gess and M. Röckner, {\it Singular-degenerate multivalued stochastic fast diffusion equations}, SIAM J. Math. Anal., 47 (2015), pp. 4058-4090, . · Zbl 1330.60080 |
| [30] | B. Gess and M. Röckner, {\it Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations}, Trans. Amer. Math. Soc., 2016, pp. 1-29, . · Zbl 1360.60123 |
| [31] | B. Gess and J. M. Tölle, {\it Multi-valued, singular stochastic evolution inclusions}, J. Math. Pures Appl., 101 (2014), pp. 789-827. · Zbl 1295.47042 |
| [32] | B. Gess and J. M. Tölle, {\it Stability of solutions to stochastic partial differential equations}, J. Differential Equations, 260 (2016), pp. 4973-5025. · Zbl 1338.60157 |
| [33] | B.-l. Guo and G.-l. Zhou, {\it Exponential stability of stochastic generalized porous media equations with jump}, Appl. Math. Mech. (Engl. Ed.), 35 (2014), pp. 1067-1078. · Zbl 1298.60085 |
| [34] | M. Hairer and J. C. Mattingly, {\it Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing}, Ann. of Math. (2), 164 (2006), pp. 993-1032. · Zbl 1130.37038 |
| [35] | M. Hairer and J. C. Mattingly, {\it Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations}, Ann. Probab., 36 (2008), pp. 2050-2091. · Zbl 1173.37005 |
| [36] | R. Z. Has\textquoterightminskiĭ, {\it Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations}, Teor. Verojatnost. i Primenen., 5 (1960), pp. 196-214. · Zbl 0093.14902 |
| [37] | M. A. Herrero and J. L. Vázquez, {\it Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem}, Ann. Fac. Sci. Toulouse Math. (5), 3 (1981), pp. 113-127. · Zbl 0498.35013 |
| [38] | L. I. Ignat, D. Pinasco, J. D. Rossi, and A. San Antolin, {\it Decay estimates for nonlinear nonlocal diffusion problems in the whole space}, J. Anal. Math., 122 (2014), pp. 375-401. · Zbl 1297.35038 |
| [39] | L. I. Ignat and J. D. Rossi, {\it Decay estimates for nonlocal problems via energy methods}, J. Math. Pures Appl. (9), 92 (2009), pp. 163-187. · Zbl 1173.35363 |
| [40] | J. Jaroszewska, {\it The Asymptotic Strong Feller Property Does Not Imply the e-Property for Markov-Feller Semigroups}, preprint, , 2013. |
| [41] | R. Kapica, T. Szarek, and M. Ślȩczka, {\it On a unique ergodicity of some Markov processes}, Potential Anal., 36 (2011), pp. 589-606. · Zbl 1244.60074 |
| [42] | J. U. Kim, {\it On the stochastic porous medium equation}, J. Differential Equations, 220 (2006), pp. 163-194. · Zbl 1099.35187 |
| [43] | T. Komorowski, S. Peszat, and T. Szarek, {\it On ergodicity of some Markov processes}, Ann. Probab., 38 (2010), pp. 1401-1443. · Zbl 1214.60035 |
| [44] | S. B. Kuksin, {\it The Eulerian limit for 2D statistical hydrodynamics}, J. Statist. Phys., 115 (2004), pp. 469-492. · Zbl 1157.76319 |
| [45] | S. B. Kuksin, {\it On distribution of energy and vorticity for solutions of 2D Navier-Stokes equation with small viscosity}, Commun. Math. Phys., 284 (2008), pp. 407-424. · Zbl 1168.35034 |
| [46] | W. Liu, {\it Harnack inequality and applications for stochastic evolution equations with monotone drifts}, J. Evol. Equations, 9 (2009), pp. 747-770. · Zbl 1239.60058 |
| [47] | W. Liu, {\it On the stochastic \(p\)-Laplace equation}, J. Math. Anal. Appl., 360 (2009), pp. 737-751. · Zbl 1180.60055 |
| [48] | W. Liu, {\it Ergodicity of transition semigroups for stochastic fast diffusion equations}, Front. Math. China, 6 (2011), pp. 449-472. · Zbl 1286.60067 |
| [49] | W. Liu and J. M. Tölle, {\it Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts}, Electron. Commun. Probab., 16 (2011), pp. 447-457. · Zbl 1244.60062 |
| [50] | W. Liu and F.-Y. Wang, {\it Harnack inequality and strong Feller property for stochastic fast-diffusion equations}, J. Math. Anal. Appl., 342 (2008), pp. 651-662. · Zbl 1151.60032 |
| [51] | Z.-M. Ma and M. Röckner, {\it Introduction to the theory of (non-symmetric) Dirichlet forms}, Universitext, Springer-Verlag, Berlin, Heidelberg, New York, 1992. · Zbl 0826.31001 |
| [52] | C. Marinelli and G. Ziglio, {\it Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise}, Dyn. Partial Differ. Equ., 7 (2010), pp. 1-23. · Zbl 1203.60088 |
| [53] | S.-X. Ouyang, {\it Harnack inequalities and applications for multivalued stochastic evolution equations}, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14 (2011), pp. 261-278. · Zbl 1225.60110 |
| [54] | M. M. Porzio, {\it On decay estimates}, J. Evol. Equ., 9 (2009), pp. 561-591. · Zbl 1239.35023 |
| [55] | M. M. Porzio, {\it Existence, uniqueness and behavior of solutions for a class of nonlinear parabolic problems}, Nonlinear Anal., 74 (2011), pp. 5359-5382. · Zbl 1222.35110 |
| [56] | C. Prévôt and M. Röckner, {\it A Concise Course on Stochastic Partial Differential Equations}, Lecture Notes in Math. 1905, Springer, Berlin, 2007. · Zbl 1123.60001 |
| [57] | M. Romito and L. Xu, {\it Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise}, Stochastic Process. Appl., 121 (2011), pp. 673-700. · Zbl 1369.76048 |
| [58] | A. Shirikyan, {\it Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations}, J. Funct. Anal., 249 (2007), pp. 284-306. · Zbl 1221.35287 |
| [59] | A. Shirikyan, {\it Mixing for the Burgers Equation Driven by a Localised Two-Dimensional Stochastic Forcing}, preprint, arXiv:1606.07763v1 [math.AP], 2016. · Zbl 1351.35082 |
| [60] | R. E. Showalter, {\it Monotone Operators in Banach Space and Nonlinear Partial Differential Equations}, Math. Surveys Monogr. 49, AMS, Providence, RI, 1997. · Zbl 0870.35004 |
| [61] | T. Szarek and D. T. H. Worm, {\it Ergodic measures of Markov semigroups with the e-property}, Ergodic Theory Dynam. Systems, 32 (2012), pp. 1117-1135. · Zbl 1261.37007 |
| [62] | A. van der Vaart and J. Wellner, {\it Weak Convergence and Empirical Processes}, Springer Ser. Statist., Springer, New York, 1996. · Zbl 0862.60002 |
| [63] | F.-Y. Wang, {\it Harnack inequalities and applications for multivalued stochastic evolution equations}, Ann. Probab., 35 (2007), pp. 1333-1350. · Zbl 1129.60060 |
| [64] | F.-Y. Wang, {\it Harnack Inequalities for Stochastic Partial Differential Equations}, Springer Briefs Math., Springer, New York, 2013. · Zbl 1332.60006 |
| [65] | F.-Y. Wang, {\it Asymptotic couplings by reflection and applications for nonlinear monotone SPDEs}, Nonlinear Anal., 117 (2015), pp. 169-188. · Zbl 1319.60144 |
| [66] | F.-Y. Wang, {\it Exponential convergence of non-linear monotone SPDEs}, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), pp. 5239-5253. · Zbl 1335.60115 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
