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Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise. (English) Zbl 1329.60220
Summary: In this paper, we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications, our main results are applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier-Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDEs.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 35R60 PDEs with randomness, stochastic partial differential equations
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##### References:
 [1] Aldous, D., Stopping times and tightness, Ann. Probab., 6, 335-340, (1978) · Zbl 0391.60007 [2] Blömker, D.; Flandoli, F.; Romito, M., Markovianity and ergodicity for a surface growth PDE, Ann. Probab., 37, 275-313, (2009) · Zbl 1184.60024 [3] Brzézniak, Z.; Debbi, L.; Goldys, B., Ergodic properties of fractional stochastic Burgers equations, Glob. Stoch. Anal., 1, 2, 149-174, (2011) · Zbl 1296.58025 [4] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171, 3, 1903-1930, (2010) · Zbl 1204.35063 [5] Chae, D.; Constantin, P.; Cordoba, D.; Gancedo, F.; Wu, J., Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math., 65, 8, 1037-1066, (2012) · Zbl 1244.35108 [6] Da Prato, G.; Debussche, A.; Temam, R., Stochastic burgers’ equation, Nonlinear Differential Equations Appl., 1, 4, 389-402, (1994) · Zbl 0824.35112 [7] Da Prato, G.; Flandoli, F.; Priola, E.; Röckner, M., Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., 41, 5, 3306-3344, (2013) · Zbl 1291.35455 [8] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052 [9] Da Prato, G.; Zabczyk, J., (Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, vol. 229, (1996), Cambridge University Press) · Zbl 0849.60052 [10] Flandoli, F., Random pertubation of PDE and fluid dynamic models, (2010), Springer [11] Flandoli, F.; Gatarek, D., Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102, 367-391, (1995) · Zbl 0831.60072 [12] Flandoli, F.; Gubinelli, M.; Priola, E., Well posedness of the transport equation by stochastic perturbation, Invent. Math., 180, 1-53, (2010) · Zbl 1200.35226 [13] Glatt-Holtz, N.; Vicol, V., Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann. Probab., 42, 1, 1-430, (2014) [14] Hairer, M., Solving the KPZ equation, Ann. of Math., 178, 559-664, (2013) · Zbl 1281.60060 [15] Ju, N., Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys., 251, 365-376, (2004) · Zbl 1106.35061 [16] Kardar, M.; Parisi, G.; Zhang, Y.-C., Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56, 9, 889-892, (1986) · Zbl 1101.82329 [17] Kiselev, A.; Nazarov, F., Variation on a theme of caffarelli and vasseur, J. Math. Sci., 166, 1, 31-39, (2010) · Zbl 1288.35393 [18] Kiselev, A.; Nazarov, F.; Shterenberg, R., Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5, 3, 211-240, (2008) · Zbl 1186.35020 [19] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 445-453, (2007) · Zbl 1121.35115 [20] Kurtz, T. G., The Yamada-Watanabe-engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab., 12, 951-965, (2007) · Zbl 1157.60068 [21] Liu, W.; Röckner, M., Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254, 2, 725-755, (2013) · Zbl 1264.60046 [22] Liu, W.; Röckner, M.; Zhu, X., Large deviation principles for the stochastic quasi-geostrophic equation, Stoch. Proc. Appl., 123, 3299-3327, (2013) · Zbl 1291.60133 [23] Ondreját, M., Brownian representations of cylindrical local martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces, Czechoslovak Math. J., 55, 130, 1003-1039, (2005) · Zbl 1081.60049 [24] Prevot, C.; Röckner, M., (A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1905, (2007), Springer) [25] S. Resnick, Danamical Problems in Non-linear Advective Partial Differential Equations, Ph.D. Thesis, University of Chicago, Chicago, 1995. [26] M. Röckner, R.-C. Zhu, X.-C. Zhu, Sub- and supercritical stochastic quasi-geostrophic equation, Ann. Probab. (in press). arXiv:1110.1984v3. [27] M. Röckner, R.-C. Zhu, X.-C. Zhu, A note on stochastic semilinear equations and their associated Fokker-Planck equations, J. Math. Anal. Appl. (in press). http://dx.doi.org/10.1016/j.jmaa.2014.01.058. [28] Stein, E., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501 [29] Temam, R., Navier-Stokes equations, (1984), North-Holland Amsterdam · Zbl 0572.35083 [30] Wu, J., Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263, 803-831, (2005) · Zbl 1104.35037 [31] Zhang, X., Stochastic Lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311, 1, 133-155, (2012) · Zbl 1251.35203 [32] R.-C. Zhu, X.-C. Zhu, Random attractor associated with the quasi-geostrophic equation, arXiv:1303.5970. · Zbl 1368.60069
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