Global mild solutions to three-dimensional magnetohydrodynamic equations in Morrey spaces. (English) Zbl 1508.35081
Summary: In this paper, we consider the Cauchy problem of three-dimensional incompressible magnetohydrodynamic equations. Some uniform estimates with respect to time for the coupling terms between the fluid and the magnetic field will be presented, under the condition that the initial \(\mathcal{M}^{1 , 1}\) norms of the vorticity and the current density are both sufficiently small. By the above estimates, we can obtain a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments. The asymptotic behaviours of the solutions are also obtained.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L60 | First-order nonlinear hyperbolic equations |
Keywords:
three-dimensional incompressible magnetohydrodynamic equations; Morrey spaces; global-in-time well-posedness; mild solutions; asymptotic behavioursReferences:
| [1] | de Almeida, M. F.; Ferreira, L. C.F., On the Navier-Stokes equations in the half-space with initial and boundary rough data in Morrey spaces, J. Differ. Equ., 254, 1548-1570 (2013) · Zbl 1255.76022 |
| [2] | de Almeida, M. F.; Ferreira, L. C.F.; Lima, L. S.M., Uniform global well-posedness of the Navier-Stokes-Coriolis system in a new critical space, Math. Z., 287, 735-750 (2017) · Zbl 1379.35220 |
| [3] | Ben-Artzi, M.; Croisille, J.; Fishelov, D., Navier-Stokes Equations in Planar Domains (2013), World Scientific · Zbl 1273.35002 |
| [4] | Bradshaw, Z.; Tsai, T., Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations, Commun. Partial Differ. Equ., 45, 9, 1168-1201 (2020) · Zbl 1448.35360 |
| [5] | Cannone, M., A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13, 515-541 (1997) · Zbl 0897.35061 |
| [6] | Cannone, M., Ondelettes, Paraproduits et Navier-Stokes (1995), Diderot Editeur: Diderot Editeur Paris · Zbl 1049.35517 |
| [7] | Carpio, A., Asymptotic behavior for the vorticity equations in dimensions two and three, Commun. Partial Differ. Equ., 19, 5-6, 827-872 (1994) · Zbl 0816.35108 |
| [8] | Cottet, G., Équations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci., Sér. 1 Math., 303, 105-108 (1986) · Zbl 0606.35065 |
| [9] | Cottet, G.; Soler, J., Three-dimensional Navier-Stokes equations for singular filament initial data, J. Differ. Equ., 74, 234-253 (1988) · Zbl 0675.76072 |
| [10] | Cowling, T. G., Magnetohydrodynamics, (Interscience Tracts on Physics and Astronomy, vol. 4 (1957), Interscience Publishers, Inc./Interscience Publishers, Ltd.: Interscience Publishers, Inc./Interscience Publishers, Ltd. New York/London) · Zbl 0133.24001 |
| [11] | Duvaut, G.; Lions, J. L., Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46, 241-279 (1972) · Zbl 0264.73027 |
| [12] | Federbush, P., Navier and Stokes meet the wavelet, Commun. Math. Phys., 155, 219-248 (1993) · Zbl 0795.35080 |
| [13] | Ferreira, L. C.F., On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier-Stokes equations, J. Math. Pures Appl., 105, 228-247 (2016) · Zbl 1334.35204 |
| [14] | Ferreira, L. C.F.; Villamizar-Roa, E. J., Exponentially-stable steady flow and asymptotic behavior for the magnetohydrodynamic equations, Commun. Math., Sci.9, 499-516 (2011) · Zbl 1246.76151 |
| [15] | Fujita, H.; Kato, T., On the Navier-Stokes initial value problem, I, Arch. Ration. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 |
| [16] | Gallagher, I.; Gallay, T., Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., 332, 287-327 (2005) · Zbl 1096.35102 |
| [17] | Gallay, T.; Wayne, C. Eugene, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbf{R}^2\), Arch. Ration. Mech. Anal., 163, 3, 209-258 (2002) · Zbl 1042.37058 |
| [18] | Gallay, T.; Wayne, C. Eugene, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Commun. Math. Phys., 255, 1, 97-129 (2005) · Zbl 1139.35084 |
| [19] | Gallay, T.; Maekawa, Y., Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity, Anal. PDE, 6, 4, 973-991 (2013) · Zbl 1350.35144 |
| [20] | Giga, M. H.; Giga, Y.; Saal, J., Asymptotic behavior of solutions and self-similar solutions, (Nonlinear Partial Differential Equations. Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, vol. 79 (2010), Birkhäuser Boston, Ltd.: Birkhäuser Boston, Ltd. Boston, MA) · Zbl 1215.35001 |
| [21] | Giga, Y.; Kambe, T., Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Commun. Math. Phys., 117, 4, 549-568 (1988) · Zbl 0661.76018 |
| [22] | Giga, Y.; Miyakawa, T., Navier-Stokes flow in \(\mathbb{R}^3\) with measures as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ., 14, 577-618 (1989) · Zbl 0681.35072 |
| [23] | Giga, Y.; Miyakawa, T.; Osada, H., Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal., 104, 223-250 (1988) · Zbl 0666.76052 |
| [24] | Konieczny, P.; Yoneda, T., On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differ. Equ., 250, 3859-3873 (2011) · Zbl 1211.35218 |
| [25] | Hishida, T.; Schonbek, M. E., Stability of time-dependent Navier-Stokes flow and algebraic energy decay, Indiana Univ. Math. J., 65, 4, 1307-1346 (2016) · Zbl 1354.35067 |
| [26] | Iftimie, D.; Karch, G.; Lacave, C., Asymptotics of solutions to the Navier-Stokes system in exterior domains, J. Lond. Math. Soc. (2), 90, 3, 785-806 (2014) · Zbl 1304.35493 |
| [27] | Jia, H.; Šverák, V., Minimal \(L^3\)-initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45, 3, 1448-1459 (2013) · Zbl 1294.35065 |
| [28] | Jia, H.; Šverák, V., Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions, Invent. Math., 196, 1, 233-265 (2014) · Zbl 1301.35089 |
| [29] | Karch, G.; Pilarczyk, D.; Schonbek, M. E., \( L^2\)-asymptotic stability of singular solutions to the Navier-Stokes system of equations in \(\mathbb{R}^3\), J. Math. Pures Appl. (9), 108, 1, 14-40 (2017) · Zbl 1368.35207 |
| [30] | 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) · Zbl 0545.35073 |
| [31] | Kato, T., Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Bras. Mat. (N. S.), 22, 127-155 (1992) · Zbl 0781.35052 |
| [32] | Kato, T., The Navier-Stokes equation for an incompressible fluid in \(\mathbb{R}^2\) with a measure as the initial vorticity, Differ. Integral Equ., 7, 949-966 (1994) · Zbl 0826.35094 |
| [33] | Koch, H.; Tataru, D., Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 22-35 (2001) · Zbl 0972.35084 |
| [34] | Kozono, H., Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations, J. Funct. Anal., 176, 2, 153-197 (2000) · Zbl 0970.35106 |
| [35] | Kozono, H.; Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Commun. Partial Differ. Equ., 19, 959-1014 (1994) · Zbl 0803.35068 |
| [36] | Landau, L.; Lifchitz, E., Physique théorique (“Landau-Lifshits”). Tome 8 (1990), “Mir”: “Mir” Moscow, Électrodynamique des milieux continus. [Electrodynamics of continuous media], Second Russian edition revised by Lifchitz [Lifshits] and L. Pitayevski [L.P. Pitaevskiĭ], Translated from the second Russian edition by Anne Sokova · Zbl 0764.73001 |
| [37] | Lei, Z.; Lin, F., Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64, 1297-1304 (2011) · Zbl 1225.35165 |
| [38] | Peetre, J., On the theory of \(\mathfrak{L}_{p , \lambda}\) spaces, J. Funct. Anal., 4, 71-87 (1969) · Zbl 0175.42602 |
| [39] | Planchon, F., Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(\mathbb{R}^3\), Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 13, 319-336 (1996) · Zbl 0865.35101 |
| [40] | Sermange, M.; Temam, R., Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36, 635-664 (1983) · Zbl 0524.76099 |
| [41] | Taylor, M. E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Commun. Partial Differ. Equ., 17, 1407-1456 (1992) · Zbl 0771.35047 |
| [42] | Wu, J., Quasi-geostrophic-type equations with initial data in Morrey spaces, Nonlinearity, 10, 1409-1420 (1997) · Zbl 0906.35078 |
| [43] | Yamazaki, M., The Navier-Stokes equations in the weak-Ln space with time-dependent external force, Math. Ann., 317, 635-675 (2000) · Zbl 0965.35118 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
