We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to small viscosity coefficients. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in $L^\infty$ sense. It shows that the transverse magnetic field can prevent the boundary layers from occurring in some physical regime.
In this paper, we consider a two-dimensional non-resistive magnetohydrodynamic model, taking the fluctuation of absolute temperature into account. Combining the method of weak convergence developed by Lions [20], Feireisl et al. [7, 8] from compressible Navier-Stokes(- Fourier) system and the new technique of variable reduction proposed by Vasseur et al. [26] and refined by Novotny et al. [22] from compressible two-fluid models, weak solutions are shown to exist globally in time with finite energy initial data. The result is the first one on global solvability to full compressible, viscous, no…
In this paper, we prove the a priori estimates in Sobolev spaces for the free-boundary compressible inviscid magnetohydrodynamics equations with magnetic diffusion under the Rayleigh-Taylor physical sign condition. Our energy estimates are uniform in the sound speed. As a result, we can prove the convergence of solutions of the free-boundary compressible resistive MHD equations to the solution of the free-boundary incompressible resistive MHD equations, i.e., the incompressible limit. The key observation is that the magnetic diffusion together with elliptic estimates directly controls the Lore…
In this paper, we investigate the Cauchy problem of the compressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density. We prove that the 2D Cauchy problem has a unique local strong solution provided the initial density and magnetic field decay not too slow at infinity. Furthermore, if the initial data satisfies some additional regularity and compatibility conditions, the strong solution becomes a classical one. Additionally, we establish a blowup criterion for the 2D compressible non-resistive MHD depending solely on the density and magnetic fields.
In this paper, we consider the three-dimensional full compressible viscous non-resistive MHD system. Global well-posedness is proved for an initial-boundary value problem around a strong background magnetic field. It is also shown that the unique solution converges to the steady state at an almost exponential rate as time tends to infinity. The proof is based on the celebrated two-tier energy method, due to Guo and Tice [\emph{Arch. Ration. Mech. Anal.}, 207 (2013), pp. 459--531; \emph{Anal. PDE.}, 6 (2013), pp. 287--369], reformulated in Lagrangian coordinates. The obtained result may be view…
In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $ρ$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,ρ)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L…
We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common …
In general, the resistivity is inversely proportional to the electrical conductivity, and is usually taken to be zero when the conducting fluid is of extremely high conductivity (e.g., ideal conductors). In this paper, we first establish the global well-posedness of strong solution to an initial-boundary value problem of the one-dimensional compressible, viscous, heat-conductive, non-resistive MHD equations with general heat-conductivity coefficient and large data. Then, the non-resistive limit is justified and the convergence rates are obtained, provided the heat-conductivity satisfies some g…
We investigate the Cauchy problem to the compressible planar non-resistive magnetohydrodynamic equations with zero heat conduction. The global existence of strong solutions to such a model has been established by Li and Li (J. Differential Equations 316: 136--157, 2022). However, to our best knowledge, so far there is no result on the behavior of the entropy near the vacuum region for this model. The main novelty of this paper is to give a positive response to this problem. More precisely, by a series of a priori estimates, especially the singular type estimates, we show that the boundedness o…
This paper solves the global well-posedness and stability problem on a special $2\frac12$-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in $L^p$-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion, and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced …
In this paper, we consider the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity, but the initial vacuum can be permitted inside the region. By deriving a priori $ν$ (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for both the compressible resistive MHD equations and non-resistive MHD equations are also established, respectively.
This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in a periodic domain. We present a systematic approach to establishing the global existence of smooth solutions when the initial data is close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. When there is no magnetic diffusion, the magnetic field and the density are governed by forced transport equations and the problem considered here is difficult. This paper implements several key observations and ideas to maximize the enhanced dissipat…
We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ>1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-εestimates to the regular solutions. At the second level, we show global existence of…
We investigate the initial-boundary value problem for one-dimensional compressible, heat-conductive, non-resistive MHD equations of viscous, ideal polytropic fluids in the Lagrangian coordinates. The existence and Lipschitz continuous dependence on the initial data of global weak solutions are established. Uniqueness of weak solutions follows as a direct consequence of stability.
Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；Department of Mathematics, University of Notre Dame（圣母大学数学系）
We consider an initial-boundary value problem for the one-dimensional equations of compressible isentropic viscous and non-resistive magnetohydrodynamic flows. The global well-posedness of strong solutions with general large data is established. Moreover, the vanishing resistivity limit is justified and the thickness of magnetic boundary layers is analyzed. The proofs of these results are based on a full use of the so-called "effective viscous flux", the material derivative and the structure of the equations.
We study the initial-boundary value problem for 1D compressible MHD equations of viscous non-resistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are constructed by approximation of global regular solutions, the existence of which has recently been obtained by Jiang and Zhang in [17]. Uniqueness of weak solutions is also proved as a consequence of Lipschitz continuous dependence on the initial data. Furthermore, long time behavior for global solutions is investigated. Specifically, based on the uniform-in-time boun…
This paper concerns the viscous and non-resistive MHD systems which govern the motion of electrically conducting fluids interacting with magnetic fields. We consider an initial-boundary value problem for both compressible and (nonhomogeneous and homogeneous) incompressible fluids in an infinite flat layer. We prove the global well-posedness of the systems around a uniform magnetic field which is vertical to the layer. Moreover, the solution converges to the steady state at an almost exponential rate as time goes to infinity. Our proof relies on a two-tier energy method for the reformulated sys…
Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；Department of Mathematics, University of Notre Dame（圣母大学数学系）；School of Mathematics and Statistics, Guangdong University of Technology（广东工业大学数学与统计学院）
