Well-posedness of the 3D Stochastic Generalized Rotating Magnetohydrodynamics Equations

Yıl 2022, Cilt: 6 Sayı: 4, 513 - 527, 30.12.2022

Mohamed Toumlilin , Muhammad Zain Al-abidin

Öz

In this paper we treat the 3D stochastic incompressible generalized
rotating magnetohydrodynamics equations. By using littlewood-Paley
decomposition and Itô integral, we establish the global well-posedness result for small initial data $(u_{0}, b_{0})$ belonging in the critical Fourier-Besov-Morrey spaces
$\mathcal{F\dot{N}}_{2,\lambda,q}^{\frac{5}{2}-2 \alpha +\frac{\lambda}{2}}(\mathbb{R}^{3})$. In addition, the proof of local existence is also founded on a priori estimates of the stochastic parabolic equation and the iterative contraction method.

Anahtar Kelimeler

Stochastic magnetohydrodynamics equation, well-posedness, Fourier_Besov_Morrey spaces

Kaynakça

• [1] Z.I. Ali, Stochastic generalized magnetohydrodynamics equations: well-posedness, Appl. Anal., 98.13 (2019), 2464-2485.
• [2] J.Y. Chemin, et al., Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations, Oxford Lecture Series in Mathematics and its Applications 32, Oxford University Press, Oxford, 2006.
• [3] M. F. De Almeida, L. C. F. Ferreira, L. S. M. Lima, Uniform global well-posedness of the Navier-Stokes-Coriolis system in a new critical space, Math. Z. 287 (2017), 735-750.
• [4] A. El Baraka, M. Toumlilin, Uniform well-posedness and stability for fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces, Open J. Math. Anal. 3.1 (2019), 70-89.
• [5] A. El Baraka, M. Toumlilin, The uniform global well-posedness and the stability of the 3D generalized magnetohydrodynamic equations with the Coriolis force, Communications in Optimization Theory, Vol. 2019 (2019), Article ID 12, 1-15.
• [6] L.C. Ferreira, L.S. Lima; Self-similar solutions for active scalar equations in Fourier-BesovMorrey spaces, Monatsh. Math. 175.4 (2014), 491-509.
• [7] M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z. 265 (2010), 481-491.
• [8] M. Sango, T. A.Tegegn, Harmonic analysis tools for stochastic magnetohydrodynamics equations in Besov spaces, International Journal of Modern Physics B, 30.28n29 (2016), 1640025.
• [9] W. Wang, Global existence and analyticity of mild solutions for the stochastic Navier–Stokes–Coriolis equations in Besov spaces, Nonlinear Analysis: Real World Applications, 52 (2020), 103048.
• [10] W. Wang, G. Wu, Global Well-posedness of the 3D Generalized Rotating Magnetohydrodynamics Equations, Acta Mathematica Sinica-English Series, 34.6(2018), 992-1000.
• [11] W.H. Wang, G. Wu, Global Mild Solution of Stochastic Generalized Navier–Stokes Equations with Coriolis Force, Acta Mathematica Sinica, English Series 34.11 (2018), 1635-1647.
• [12] W. H. Wang, G. Wu, Global well-posedness of the 3D generalized rotating magnetohydrodynamics equations, Acta Mathematica Sinica, English Ser. 34(2018), 992-1000.
• [13] W. Wang, G. Wu, Global mild solution of the generalized Navier-Stokes equations with the Coriolis force, Appl. Math. Lett. 76 (2018), 181-186.
• [14] W. Xiao, X. Zhou, On the Generalized Porous Medium Equation in Fourier-Besov Spaces, J. Math. Study. 53 (2020), 316-328.