Abstract
The three-dimensional incompressible magnetohydrodynamic equations with stochastic external forces are considered. First the existence and uniqueness of local strong solution to the stochastic magnetohydrodynamic equations are proved when the external forces satisfy some conditions. The proof is based on the contraction mapping principle, stopping time and stochastic estimates. The strong solution is a weak solution for the fluid variables with a given complete probability space and a given Brownian motion. Then, the global existence of strong solutions in probability is established if the initial data are sufficiently small, and the noise is multiplicative and non-degenerate.
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16 July 2019
The authors have retracted this article [1] because the article shows [2]. All authors agree to this retraction.
16 July 2019
The authors have retracted this article [1] because the article shows [2]. All authors agree to this retraction.
References
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, New York (2011)
Barbu, V., Da Prato, G.: Existence and ergodicity for the 2D stochastic MHD equations. Preprint di Matematica
Bensoussan, A., Temam, R.: Équations stochastiques du type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)
Breckner, H.: Galerkin approximation and the strong solution of the Navier–Stokes equation. J. Appl. Math. Stoch. Anal. 13(3), 239–259 (2000)
Brzeźniak, Z., Motyl, E.: Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains. J. Differ. Equ. 254, 1627–1685 (2013)
Capínski, M., Gatarek, D.: Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension. J. Funct. Anal. 126(1), 26–35 (1994)
Capínski, M., Peszat, S.: Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations. Nonlinear Differ. Equ. Appl. 4, 185–200 (1997)
Capínski, M., Peszat, S.: On the existence of a solution to stochastic Navier-Stokes equations. Nonlinear Anal. 44, 141–177 (2001)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, London (1992)
Evans, L.C.: An Introduction to Stochastic Differential Equations. American Mathematical Society, Providence (2013)
Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102(3), 367–391 (1995)
Flandoli, F., Maslowski, B.: Ergodicity of the 2D Navier–Stokes equations under random perturbations. Commun. Math. Phys. 171, 119–141 (1995)
Glatt-Holtz, N., Ziane, M.: Strong pathwise solutions of the stochastic Navier–Stokes system. Adv. Differ. Equ. 14(5–6), 567–600 (2009)
Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, Berlin (2009)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stchastic Calculus. Springer, Berlin (1991)
Kim, J.U.: Strong solutions of the stochastic Navier–Stokes equations in \(\mathbb{R}^3\). Indiana Univ. Math. J. 59, 1417–1450 (2010)
Kuo, H.H.: Introduction to Stochastic Integration. Springer, Berlin (2006)
Menaldi, J.L., Sritharan, S.S.: Stochastic 2-D Navier–Stokes Equation. Appl. Math. Optim. 46, 31–53 (2002)
Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)
Mikulevicius, R., Rozovskii, B.L.: Global \(L^2\)-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33(1), 137–176 (2005)
Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Sango, M.: Magnetohydrodynamic turbulent flows: existence results. Phys. D 239, 912–923 (2010)
Sundar, P.: Stochastic magnetohydrodynamic system perturbed by general noise. Commun. Stoch. Anal. 4(2), 253–269 (2010)
Taniguchi, T.: The existence of energy solutions to 2-dimensional non-Lipschtz stochastic Navier–Stokes equations in unbounded domains. J. Differ. Equ. 251, 3329–3362 (2011)
Taylor, M.E.: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Cambridge Univ Press, London (2000)
Acknowledgments
D. Wang’s research was supported in part by the National Science Foundation under Grant DMS-1312800. Z. Tan and H. Wang’s research were supported in part by the National Natural Science Foundation of China-NSAF (No. 11271305).
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The authors have retracted this article [1] because the article shows significant overlap with a previously published article by Jong Uhn Kim [2]. All authors agree to this retraction.
[1] Tan, Z., Wang, D. & Wang, H. Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations. Math. Ann. (2016) 365: 1219. https://doi.org/10.1007/s00208-015-1296-7
[2] Kim JU. Strong Solutions of the Stochastic Navier-Stokes Equations in R 3. Indiana University mathematics journal. (2010) Jan 1:1853-86.
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Tan, Z., Wang, D. & Wang, H. RETRACTED ARTICLE: Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations. Math. Ann. 365, 1219–1256 (2016). https://doi.org/10.1007/s00208-015-1296-7
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DOI: https://doi.org/10.1007/s00208-015-1296-7