Well-posedness, analyticity and time decay of the 3D fractional magneto-hydrodynamics equations in critical Fourier-Besov-Morrey spaces with variable exponent

Springer Science and Business Media LLC - Tập 8 - Trang 723-742 - 2022
Achraf Azanzal1, Chakir Allalou1, Said Melliani1
1Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Beni Mellal, Morocco

Tóm tắt

In this paper, we establish the global well-posedness and analyticity of the 3D fractional magnetohydrodynamics equations in the critical Fourier-Besov-Morrey spaces with variable exponent, which can be seen as a meaningful complement to the corresponding results of the magnetohydrodynamics equations in usual Fourier-Besov-Morrey spaces. Furthermore, we get time decay rate estimate of mild solutions.

Tài liệu tham khảo

Abbassi, A., Allalou, C., Oulha, Y.: Well-posedness and stability for the viscous primitive equations of geophysics in critical Fourier-Besov-Morrey spaces. In: The International Congress of the Moroccan Society of Applied Mathematics, pp. 123–140. Springer, Cham (2019) Abidin, M.Z., Chen, J.: Global well-posedness and analyticity of generalized porous medium quation in Fourier-Besov-Morrey spaces with variable exponent. Mathematics 9, 498 (2021) Abidin, M.Z., Chen, J.: Global well-posedness for fractional Navier-Stokes equations in variable exponent Fourier-Besov-Morrey spaces. Acta Math. Sci. 41, 164–176 (2021) Almeida, A., Caetano, A.: Variable exponent Besov-Morrey spaces. J. Fourier Anal. Appl. 26, 1–420 (2020) Almeida, A., Hasanov, J., Stefan, J.: Maximal and potential operators in variable exponent Morrey spaces. Georg. Math. J. 15, 195–208 (2008) Azanzal, A., Abbassi, A., Allalou, C.: Existence of solutions for the Debye-Hückel system with low regularity initial data in critical Fourier-Besov-Morrey spaces. Nonlinear Dyn. Syst. Theory 21, 367–380 (2021) Azanzal, A., Allalou, C., Melliani, S.: Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces. J. Elliptic Parabol. Equ. (2021). https://doi.org/10.1007/s41808-021-00140-x Azanzal, A., Allalou, C., Abbassi, A.: Well-posedness and analyticity for generalized Navier-Stokes equations in critical Fourier-Besov-Morrey spaces. J. Nonlinear Funct. Anal. 2021, 24 (2021) Azanzal, A., Abbassi, A., Allalou, C.: On the Cauchy problem for the fractional drift-diffusion system in critical Fourier-Besov-Morrey spaces. Int. J. Optim. Appl. 1, 28 (2021) Bahouri, H.: The Littlewood-Paley theory: a common thread of many works in nonlinear analysis. Eur. Math. Soc. Newsl. 11, 15–23 (2019) Cannone, M.: A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoam. 13, 515–541 (1997) Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972) El Baraka, A., Toumlilin, M.: Global well-posedness for fractional Navier-Stokes equations in critical Fourier-Besov-Morrey spaces. Moroccan J. Pure Appl. Anal. 3, 1–14 (2017) El Baraka, A., Toumlilin, M.: Global well-posedness and decay results for 3D generalized magneto-hydrodynamic equations in critical Fourier-Besov-Morrey spaces. Electron. J. Differ. Equ. 65, 1–20 (2017) El Baraka, A., Toumlilin, M.: Well-posedness and stability for the generalized incompressible magneto-hydrodynamic equations in critical Fourier-Besov-Morrey spaces. Acta Math. Sci. 39, 1551–1567 (2019) El Baraka, A., Toumlilin, M.: The uniform global well-posedness and the stability of the 3D generalized magnetohydrodynamic equations with the coriolis force. Commun. Optim. Theory 2019, 12 (2019) Fu, J., Xu, J.: Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents. J. Math. Anal. Appl. 381, 280–298 (2011) Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Stanford Univ. Calif. 16(4), 269–315 (1964). https://doi.org/10.1007/bf00276188 He, C., Huang, X., Wang, Y.: On some new global existence results for 3D magnetohydrodynamic equations. Nonlinearity 27, 343 (2014) Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet. Math. Nachr. 4, 213–231 (1950) Kato, T.: Strong \(L^{p}\)-solutions of the Navier-Stokes equation in \(\mathbb{R}^{m}\) with applications to weak solutions. Math. Z. 187, 471–480 (1984) Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Spectr. Theory Differ. Equ. 448, 25–70 (1995) Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157, 22–35 (2001) Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. CRC Press, London (2002) Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) Lin, F., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. J. Differ. Equ. 259, 5440–5485 (2015) Lin, F., Zhang, P.: Global small solutions to an MHD-type system: the three-dimensional case. Commun. Pure Appl. Math. 67, 531–580 (2014) Liu, Q., Zhao, J.: Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces. J. Math. Anal. Appl. 420, 1301–1315 (2014) Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer Science and Business Media, Berlin (2000) Toumlilin, M.: Global well-posedness and analyticity for generalized porous medium equation in critical Fourier-Besov-Morrey spaces. Open J. Math. Anal. 3, 71–80 (2019) Wang, W.: Global well-posedness and analyticity for the 3D fractional magneto-hydrodynamics equations in variable Fourier-Besov spaces. Z. Angew. Math. Phys. 70, 1–16 (2019) Wang, Y., Wang, K.: Global well-posedness of the three dimensional magneto-hydrodynamics equations. Nonlinear Anal. Real World Appl. 17, 245–251 (2014) Ye, Z.: Global well-posedness and decay results to 3D generalized viscous magneto-hydrodynamic equations. Ann. di Mat. 195, 1111–1121 (2016) Zhou, X., Xiao, W.: Algebra properties in Fourier-Besov spaces and their applications. J. Funct. Spaces (2018). https://doi.org/10.1155/2018/3629179