Uniform Large Deviations for 2D Incompressible Magneto-hydrodynamics Equations Driven by Multiplicative Noises

Jin MA

Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (1) : 47-66.

PDF(612 KB)
PDF(612 KB)
Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (1) : 47-66. DOI: 10.12386/A20210065

Uniform Large Deviations for 2D Incompressible Magneto-hydrodynamics Equations Driven by Multiplicative Noises

  • Jin MA
Author information +
History +

Abstract

We establish a Freidlin- Wentzell type large deviation principle for twodimensional incompressible Magneto-hydrodynamics equations driven by multiplicative noises when the noises converge to zero that are uniform with respect to initial conditions in bounded subsets of the infinite dimensional Banach space. The proof is based on the weak convergence approach.

Key words

large deviation principle / uniform large deviation principle / 2D incompressible stochastic MHD equations / weak convergence approach

Cite this article

Download Citations
Jin MA. Uniform Large Deviations for 2D Incompressible Magneto-hydrodynamics Equations Driven by Multiplicative Noises. Acta Mathematica Sinica, Chinese Series, 2023, 66(1): 47-66 https://doi.org/10.12386/A20210065

References

[1] Adams R., Fournier J., Sobolev Spaces, Academic Press, Second Edition, Amsterdan, 2003.
[2] Barbu V., Da Prato G., Existence and ergodicity for the two-dimensional stochastic magneto-hydrodynamics equations, Appl. Math. Optim., 2007, 56(2):145-168.
[3] Boué M., Dupuis P., A variational representation for certain functionals of Brownian motion, Ann. Probab., 1998, 26(4):1641-1659.
[4] Brzézniak Z., Capiński M., Flandoli F., Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 1993, 95:87-102.
[5] Brzézniak Z., Cerrai S., Freidlin M., Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise, Probab. Theory Relat. Fields, 2015, 162(3-4):739-793.
[6] Brzeźniak Z., Li Y. H., Asymptotic behaviour of solutions to the 2D stochastic Navier-Stokes equations in unbounded domains:New developments, Recent Developments in Stochastic Analysis and Related Topics, 2004, 358(12):78-111.
[7] Brzeźniak Z., Li Y. H., Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 2006, 358(12):5587-5629.
[8] Budhiraja A., Dupuis P., A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 2000, 20(1):39-61.
[9] Budhiraja A., Dupuis P., Maroulas V., Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 2008, 36:1390-1420.
[10] Budhiraja A., Dupuis P., Maroulas V., Large deviations for stochastic flows of diffeomorphisms, Bernoulli, 2010, 16(1):234-257.
[11] Budhiraja A., Dupuis P., Maroulas V., Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat., 2011, 47(3):725-747.
[12] Budhiraja A., Dupuis P., Salins M., Uniform large deviation principles for Banach space valued stochastic differential equations, Trans. Amer. Math. Soc., 2019, 372:8363-8421.
[13] Cerrai S., Röckner M., Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipshitz reaction term, Ann. Probab., 2004, 32(1B):1100-1139.
[14] Chenal F., Millet A., Uniform large deviations for parabolic SPDEs and applications, Stoch. Process. Appl., 1997, 72:161-187.
[15] Chueshov I., Millet A., Stochastic 2D hydrodynamical type systems:Well-posedness and large deviations, Appl. Math. Optim., 2010, 61:379-420.
[16] Dembo A., Zeitouni O., Large Deviations Techniques and Applications, Academic Press, San Diego, Calif., 1989.
[17] Desjardins B., Le Bris C., Remarks on a nonhomogeneous model of magnetohydrodynamics, Differ. Integral Equ., 1998, 11(3):377-394.
[18] Freidlin M., Wentzell A., Random Perturbations of Dynamical Systems, Springer, New York, Berlin, 1984.
[19] Gautier E., Uniform large deviations for the nonlinear Schrödinger equation with multiplicative noise, Stoch. Process. Appl., 2005, 115(12):1904-1927.
[20] Gerbeau J., Le Bris C., Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differ. Equ., 1997, 2(3):427-452.
[21] Gong H. J., Li J. K., Global existence of strong solutions to incompressible MHD, Commun. Pure Appl. Anal., 2014, 13(4):1553-1561.
[22] Huang J. H., Shen T. L., Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations, Nonlinear Anal., 2016, 133:102-133.
[23] Lin F. H., Xu L., Zhang P., Global small solutions of 2-D incompressible MHD system, J. Differ. Equ., 2015, 259(10):5440-5485.
[24] Manna U., Mohan M., 2-D magneto-hydrodynamic system with jump processes:well-posedness and invariant measures. Commun. Stoch. Anal., 2013, 7:153-178.
[25] Sermange M., Temam R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 1983, 36(5):635-664.
[26] Shen T. L., Huang J. H., Ergodicity of stochastic magneto-hydrodynamic equations driven by α-stable noise, J. Math. Anal. Appl., 2017, 446:746-769.
[27] Sowers R., Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Ann. Probab., 1992, 20(1):504-537.
[28] Sritharan S., Sundar P., The stochastic magneto-hydrodynamic system, Infn. Dimens. Anal. Quantum Probab. Relat. Top., 1999, 2(2):241-265.
[29] Stupyalis L., An initial-boundary value problem for a system of equations of magnetohydrodynamics, Lithuanian Math. J., 2000, 40(2):176-196.
[30] Sundar P., Stochastic Magnetohydrodynamic system perturbed by general noise, Commun. Stoch. Anal., 2010, 4(2):253-269.
[31] Temam R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edn., Springer-Verlag, New York, 1997.
[32] Temam R., Navier-Stokes Equations:Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI:2001.
[33] Vrabie I., C0-Semigroups and Applications, North-Holland, Amsterdam, 2003.
[34] Yamazaki K., Stochastic hall-magneto-hydrodynamics system in three and two and a half dimensions, J. Stat. Phys., 2017, 166(2):368-397.
[35] Yamazaki K., Three-dimensional magnetohydrodynamics system forced by space-time white noise, arXiv:1910.04820, 2019.
[36] Zeng Z. R., Mild solutions of the stochastic MHD equations driven by fractional Brownian motions, J. Math. Anal. Appl., 2020, 491(1):124296, 18 pp.
[37] Zhao W. Q., Li Y. R., Asymptotic behavior of two-dimensional stochastic magneto-hydrodynamics equations with additive noises, J. Math. Phys., 2011, 52(7):072701, 18 pp.
PDF(612 KB)

360

Accesses

0

Citation

Detail

Sections
Recommended

/