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On stochastic evolution equations for nonlinear bipolar fluids: well-posedness and some properties of the solution. (English) Zbl 1383.35161
Summary: We investigate the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise of Lévy type. We show that the system we consider has a unique global strong solution. We also give some results concerning the properties of the solution. We mainly prove that the unique solution satisfies the Markov-Feller property. This enables us to prove by means of some results from ergodic theory that the semigroup associated to the unique solution admits at least an invariant measure which is ergodic and tight on a subspace of the Lebesgue space \(L^2\).

MSC:
35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
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[1] Albeverio, S.; Brzeźniak, Z.; Wu, J. L., Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl., 371, 1, 309-322, (2010) · Zbl 1197.60050
[2] Albeverio, S.; Mandrekar, V.; Rüdiger, B., Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise, Stochastic Process. Appl., 119, 3, 835-863, (2009) · Zbl 1168.60014
[3] Babin, A. V.; Vishik, M. I., Attractors of evolution equations, Stud. Math. Appl., vol. 25, (1992), North-Holland Publishing Co. Amsterdam · Zbl 0778.58002
[4] Bellout, H.; Bloom, F.; Necas, J., Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math., 50, 559-583, (1992) · Zbl 0759.76004
[5] Bellout, H.; Bloom, F.; Necas, J., Solutions for incompressible non-Newtonian fluids, C. R. Math. Acad. Sci. Paris Sér I. Math., 317, 795-800, (1993) · Zbl 0784.76005
[6] Bellout, H.; Bloom, F.; Necas, J., Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations, 19, 11&12, 1763-1803, (1994) · Zbl 0840.35079
[7] Bellout, H.; Bloom, F.; Necas, J., Bounds for the dimensions of the attractors of nonlinear bipolar viscous fluids, Asymptot. Anal., 11, 2, 131-167, (1995) · Zbl 0861.35072
[8] Bellout, H.; Bloom, F.; Necas, J., Existence, uniqueness and stability of solutions to initial boundary value problems for bipolar fluids, Differential Integral Equations, 8, 453-464, (1995) · Zbl 0831.35135
[9] Bensoussan, A., Stochastic Navier-Stokes equations, Acta Appl. Math., 38, 267-304, (1995) · Zbl 0836.35115
[10] Bensoussan, A.; Temam, R., Equations stochastiques du type Navier-Stokes, J. Funct. Anal., 13, 195-222, (1973) · Zbl 0265.60094
[11] Breckner, H., Approximation and optimal control of the stochastic Navier-Stokes equation, (1999), Martin-Luther University Halle-Wittenberg, Dissertation
[12] Breckner, H., Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stoch. Anal., 13, 3, 239-259, (2000) · Zbl 0974.60045
[13] Brzezniak, Z.; Debbi, L., On stochastic Burgers equation driven by a fractional Laplacian and space-time white noise, (Stochastic Differential Equations: Theory and Applications, Interdiscip. Math. Sci., vol. 2, (2007), World Sci. Publ.), 135-167 · Zbl 1137.35085
[14] Brzeźniak, Z.; Hausenblas, E., Maximal regularity for stochastic convolutions driven by Lévy processes, Probab. Theory Related Fields, 145, 3-4, 615-637, (2009) · Zbl 1178.60046
[15] Brzeźniak, Z.; Hausenblas, E.; Zhu, J., 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal. Theory Methods Appl., 79, 122-139, (2013) · Zbl 1261.60061
[16] Brzezniak, Z.; Maslowski, B.; Seidler, J., Stochastic nonlinear beam equation, Probab. Theory Related Fields, 132, 2, 119-144, (2005)
[17] Caraballo, T.; Langa, J. A.; Taniguchi, T., The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J. Differential Equations, 179, 2, 714-737, (2002) · Zbl 0990.35138
[18] Caraballo, T.; Márquez-Durán, A. M.; Real, J., The asymptotic behaviour of a stochastic 3D LANS-α model, Appl. Math. Optim., 53, 2, 141-161, (2006) · Zbl 1097.60053
[19] Caraballo, T.; Real, J.; Taniguchi, T., On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462, 2066, 459-479, (2006) · Zbl 1149.35397
[20] Chow, P.-L.; Khasminskii, R. Z., Stationary solutions of nonlinear stochastic evolution equations, Stoch. Anal. Appl., 15, 5, 671-699, (1997) · Zbl 0899.60056
[21] Chueshov, I.; Millet, A., Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim., 61, 3, 379-420, (2010) · Zbl 1196.49019
[22] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. Amer. Math. Soc., 53, 314, (1985), vii+67 pp. · Zbl 0567.35070
[23] Constantin, P.; Foias, C.; Temam, R., Phys. D Nonlinear Phenom., 30, 3, 284-296, (1988)
[24] Da Prato, G.; Debussche, A., 2D stochastic Navier-Stokes equations with a time-periodic forcing term, J. Dynam. Differential Equations, 20, 2, 301-335, (2008) · Zbl 1154.35099
[25] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052
[26] Deugoue, G.; Sango, M., On the strong solution for the 3D stochastic Leray-alpha model, Bound. Value Probl., 2010, (2010), 31 pages · Zbl 1187.35011
[27] Dong, Z.; Jianliang, Z., Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump, J. Differential Equations, 250, 2737-2778, (2011) · Zbl 1218.60056
[28] Fabian, M.; Habala, P.; Hajek, P.; Montesinos, V.; Zizler, V., Banach space theory. the basis for linear and nonlinear analysis, CMS Books Math./Ouvrages Math. SMC, (2011), Springer New York · Zbl 1229.46001
[29] Flandoli, F.; Gubinelli, M.; Hairer, M.; Romito, M., Rigourous remarks about scaling laws in turbulent fluid, Comm. Math. Phys., 278, 1-29, (2008) · Zbl 1140.76011
[30] Freshe, J.; Ruzicka, M., Non-homogeneous generalized Newtonian fluids, Math. Z., 260, 2, 353-375, (2008)
[31] Guo, B.; Guo, C.; Zhang, J., Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23, 3&4, 303-326, (2010) · Zbl 1240.60184
[32] Gyöngy, I.; Krylov, N. V., On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces, Stochastics, 6, 3-4, 153-173, (1981/1982) · Zbl 0481.60060
[33] Hasselblatt, B.; Katok, A., A first course in dynamics. with a panorama of recent developments, (2003), Cambridge University Press New York · Zbl 1027.37001
[34] Hausenblas, E.; Razafimandimby, P. A.; Sango, M., Martingale solution to differential type fluids of grade two driven by random force of Lévy type, Potential Anal., 38, 4, 1291-1331, (2013) · Zbl 1317.60078
[35] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, N.-Holl. Math. Libr., vol. 24, (1989), North-Holland Publishing Co. Amsterdam · Zbl 0684.60040
[36] Kupiainen, A., Statistical theories of turbulence, (Advances in Mathematical Sciences and Applications, (2003), Gakkotosho Tokyo)
[37] Ladyzhenskaya, O. A., The mathematical theory of viscous incompressible flow, (1969), Gordon and Breach New York · Zbl 0184.52603
[38] Ladyzhenskaya, O. A., New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, (Boundary Value Problems of Mathematical Physics V, (1970), American Mathematical Society Providence, RI)
[39] Malek, J.; Necas, J.; Novotny, A., Measure-valued solutions and asymptotic behavior of a multipolar model of a boundary layer, Czechoslovak Math. J., 42, 3, 549-576, (1992) · Zbl 0774.76008
[40] Malek, J.; Necas, J.; Rokyta, M.; Ruzicka, M., Weak and measure-valued solutions to evolutionary pdes, Appl. Math. Math. Comput., vol. 13, (1996), Chapman & Hall London · Zbl 0851.35002
[41] Mikulevicius, R.; Rozovskii, B. L., Stochastic Navier-Stokes equations and turbulent flows, SIAM J. Math. Anal., 35, 5, 1250-1310, (2004) · Zbl 1062.60061
[42] Necas, J.; Novotny, A.; Silhavy, M., Global solution to the compressible isothermal multipolar fluids, J. Math. Anal. Appl., 162, 223-242, (1991) · Zbl 0757.35060
[43] Necas, J.; Silhavy, M., Multipolar viscous fluids, Quart. Appl. Math., XLIX, 2, 247-266, (1991) · Zbl 0732.76003
[44] Pardoux, E., Equations aux dérivées partielles stochastiques monotones, (1975), Université Paris-Sud, Thèse de Doctorat · Zbl 0363.60041
[45] Peszat, S.; Zabczyk, J., Stochastic partial differential equations with levy noise. an evolution equation approach, Encyclopedia Math. Appl., vol. 113, (2007), Cambridge University Press · Zbl 1205.60122
[46] Razafimandimby, P. A., On stochastic models describing the motions of randomly forced linear viscoelastic fluids, J. Inequal. Appl., 2010, (2010), 27 pages · Zbl 1394.76011
[47] Razafimandimby, P. A.; Sango, M., Weak solutions of a stochastic model for two-dimensional second grade fluids, Bound. Value Probl., 2010, (2010), 47 pages · Zbl 1188.35207
[48] Razafimandimby, P. A.; Sango, M., Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids, C. R. Math. Acad. Sci. Paris, 348, 13-14, 787-790, (2010) · Zbl 1202.60103
[49] Razafimandimby, P. A.; Sango, M., Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behaviour, Nonlinear Anal. Theory Methods Appl., 75, 11, 4251-4270, (2012) · Zbl 1258.35214
[50] Razafimandimby, P. A.; Sango, M., On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids, Appl. Anal., 91, 12, 2217-2233, (2012) · Zbl 1256.60022
[51] Robinson, J. C., Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge Texts Appl. Math., (2001), Cambridge University Press Cambridge · Zbl 0980.35001
[52] Rüdiger, B., Stochastic integration with respect to compensated Poisson random measure on separable Banach spaces, Stoch. Stoch. Rep., 76, 3, 213-242, (2004) · Zbl 1052.60045
[53] Sango, M., Magnetohydrodynamic turbulent flows: existence results, Phys. D Nonlinear Phenom., 239, 12, 912-923, (2010) · Zbl 1193.76162
[54] Sango, M., Density dependent stochastic Navier-Stokes equations with non Lipschitz random forcing, Rev. Math. Phys., 22, 6, 669-697, (2010) · Zbl 1194.35521
[55] Taniguchi, T., The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by levy processes, J. Math. Anal. Appl., 385, 2, 634-654, (2012) · Zbl 1233.60041
[56] Temam, R., Navier-Stokes equations, (1979), North-Holland · Zbl 0454.35073
[57] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., vol. 68, (1988), Springer-Verlag New York · Zbl 0662.35001
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