Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

Abstract

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in R 3 . We establish the global well-posedness of the MHD system if the initial data satisfiesfor some small c > 0 which only depends on ρ, ρ. As a byproduct, we also get the decay estimate of the solution.

Local well‐posedness to the 2D Cauchy problem of nonhomogeneous heat‐conducting Navier–Stokes and magnetohydrodynamic equations with vacuum at infinity

“…In the absence of vacuum, that is, the initial density has a positive lower bound, the local existence of strong solutions in some Besov spaces was established by Abidi and Paicu 2 and later extended by Chen et al 3 to be the global one. Meanwhile, Chen et al 4 showed global well‐posedness to the three‐dimensional (3D) Cauchy problem for discontinuous initial density. In the presence of vacuum, that is, in the case that the initial density vanishes in some region, there has been a considerable number of researches on the nonhomogeneous fluid equations since the works of Lions 5 and Choe and Kim, 6 where the global‐in‐time weak solutions and local strong solutions to the nonhomogeneous Navier–Stokes equations were obtained, respectively.…”

Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations

“…Chen et al [6] obtained the local existence of strong solutions to the 3D Cauchy problem, while Huang and Wang [12] investigated the global existence of strong solution with general large data and vacuum in 2D bounded domains provided that the compatibility condition (4) holds. Recently, by virtue of time-weighted estimates and the delicate spatial-weighted estimates, Lü et al [19] showed the global existence of strong solutions to the 2D Cauchy problem without using the compatibility condition (4). At the same time, Song [22] studied the local wellposedness of strong solutions with density-dependent viscosity, which extended the main result of [6].…”

“…Later, Chen et al [5] proved a global solution for the initial data belonging to critical Besov spaces, see also [3] for related improvement. Meanwhile, Chen et al [4] showed global well-posedness to the 3D Cauchy problem for discontinuous initial density. On the other hand, in the presence of vacuum, motivated by [7], under the following compatibility condition…”

Global solvability to the 3D incompressible magneto-micropolar system with vacuum

“…In the absence of vacuum, Abidi-Paicu [1] established the local and global (with small initial data) existence of strong solutions in the framework of Besov spaces. Chen et al [7] proved a global solution for the global well-posedness to the 3D Cauchy problem for the bounded density. In the presence of vacuum, Chen et al [8] obtained the unique local strong solutions to the 3D Cauchy problem with general initial data and some compatibility condition.…”