We study the global well-posedness of magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to (1.1)-(1.4). Then we derive the existence of a uniform attractor for (1.1)-(1.4).
In this paper we consider the global well-posedness of compressible magnetohydrodynamic system in $\R^d$ with $d\geq2$, in the framework of the critical Besov spaces. We can show that if the initial data, the shear viscosity and the magnetic diffusion coefficient are small comparing with the volume viscosity, then compressible magnetohydrodynamic system has a unique global solution.
We prove the global well-posedness of the Cauchy problem to the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical Besov spaces, which generalize the result in [10]. Meanwhile , we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established.
A global existence result is established for a free boundary problem of planar magnetohydrodynamic fluid flows with radiation and large initial data. Particularly, it is novelty to embrace the constant transport coefficient. As a by-product, the free boundary is shown to expand outward at an algebraic rate from above in time.
In this paper we study the initial-boundary value problem for the magnetohydrodynamic system in three dimensional exterior domain. We show an existence theorem of global in time strong solution for small initial data and we also show its asymptotic behavior when time goes to infinity.
The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic (MHD) model in two dimensional space. Based on Agmon, Douglis and Nirenberg's estimates for the stationary Stokes equation and the Solonnikov's theorem of $L^p$-$L^q$-estimates for the evolution Stokes equation, it is shown that the mixed-type MHD equations exist a global strong solution.
In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
We prove the global-in-time existence of H^2 solutions of the equations of compressible magnetohydrodynamics with zero magnetic resistivity in three space dimensions. Initial data are taken to be small in H^2 modulo a constant state and initial densities are positive and essentially bounded. The present work generalizes the results obtained by Kawashima.
The existence of global-in-time classical solutions to the Cauchy problem of incompressible Magnetohydrodynamic flows with zero magnetic diffusivity is considered in two dimensions. The linearization of equations is a degenerated parabolic-hyperbolic system. The solution is constructed as a small perturbation of a constant background in critical spaces. The deformation gradient has been introduced to decouple the subtle coupling between the flow and the magnetic field. The $L^1$ dissipation of the velocity is obtained.
We study the two-dimensional generalized magnetohydrodynamics system with generalized dissipation and diffusion in terms of fractional Laplacians. It is known that the classical magnetohydrodynamics system with full Laplacians in both dissipation and diffusion terms admits a unique global strong solution pair. Making use of the special structure of the system in the two-dimensional case, we show in particular that the solution pair remains smooth when we have zero dissipation but only magnetic diffusion with its power of the fractional Laplacian $β> \frac{3}{2}$.
In this article, the Cauchy problem of three-dimensional (3-D) incompressible magnetohydrodynamic system was investigated. If the initial $\mathcal{M}^{1,1}$ norms of the vorticity $ω$ and the current density $j$ are both sufficiently small, then some uniform estimates with respect to time for the coupling terms between the fluid and the magnetic field can be established, which lead to a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments.
In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - ν(-\triangle)^α u b_t + u \cdot \nabla b & = & b \cdot \nabla u - κ(-\triangle)^β b \end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslantα< 1 / 2,\,\, β\geqslant 1, \,\,3α+ 2β>3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $α=0$ and $β>\frac 32$.
We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole space R^3 , in the case where there are no diffusivity in the vertical direction and only a small diffusivity in the horizontal direction (of size $ε$$α$ with 0 \textless{} $α$ $\le$ $α$0, for some $α$0 \textgreater{} 0). We prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution globally exists in time for large initial data, when the rotation is fast enough.
This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient $κ(θ)$ satisfies \begin{equation*} C_{1}(1+θ^q)\leq κ(θ)\leq C_2(1+θ^q) \end{equation*} for some constants $q>0$, and $C_1,C_2>0$.
We discover cancellations upon $H^{2}(\mathbb{R}^{n})$-estimate of the Hall term for $n \in \{2,3\}$. As its consequence, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of only horizontal components of velocity and magnetic fields. Second, we prove the global regularity of the $2\frac{1}{2}$-dimensional electron magnetohydrodynamics system with magnetic diffusion $(-Δ)^{\frac{3}{2}} (b_{1}, b_{2}, 0) + (-Δ)^α (0, 0, b_{3})$ for $α> \frac{1}{2}$. Lastly, we extend this result to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics sys…
We are concerned with the global existence of finite energy weak solutions to 3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in a general smooth bounded domain. The perfectly conducting wall boundary condition is imposed on the magnetic field. Due to the degeneracy of Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies in the bounded function space and having a positive lower bound. Particularly, these bounds are preserved by the density transport equation which helps yield a satisfying compactness argument of the magnetic field.
In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $H^3(\mathbb{R}^3)$. Our result may also be considered as the extension of work of Lei-Lin-Zhou from the second-order semi-linear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.
Global weak solutions and incompressible limit of two-dimensional isentropic compressible magnetohydrodynamic equations with ripped density and large initial data
In this paper, we establish the global existence to the three-dimensional incompressible Hall-MHD equations for a class of large initial data, whose $L^{\infty}$ norms can be arbitrarily large. In addition , we give an example to show that such a large initial value does exist. Our idea is splitting the generalized heat equations from Hall-MHD system to generate a small quantity for large time $t$.
We study the two-dimensional generalized magnetohydrodynamics system with dissipation and diffusion in terms of fractional Laplacians. In particular, we show that in case the diffusion term has the power $β= 1$, in contrast to the previous result of $α\geq \frac{1}{2}$, we show that $α> \frac{1}{3}$ suffices in order for the solution pair to remain smooth for all time.
In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced "transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.
The existence of global-in-time classical solutions to the Cauchy problem of compressible magnetohydrodynamic flows with zero magnetic diffusivity is considered in two dimensions. The linear structure is a degenerated hyperbolic-parabolic system. The solution is constructed as a small perturbation of a constant background in critical spaces. The deformation gradient is introduced to decouple the subtle coupling between the flow and the magnetic field. The $L^1$ dissipation for the velocity is obtained, and the $L^2$ dissipations for the density and the magnetic field are also achieved.
Global weak solutions and incompressible limit to the isentropic compressible magnetohydrodynamic equations in 2D bounded domains with ripped density and large initial data
In this paper, the Cauchy's problem for fractional MHD system with the Hall and ion-slip effects is considered. By exploring the structure of semilinear and quasilinear terms, we prove the global existence of solutions for a class of large initial data. Both the velocity and magnetic fields could be arbitrarily large in $H^3(\mathbb{R}^3)$.
In one-dimensional unbounded domains, we consider the equations of a planar compressible magnetohydrodynamic (MHD) flow with constant viscosity and heat conductivity. More precisely, we prove the global existence of strong solutions to the MHD equations with large initial data satisfying the same conditions as those of Kazhikhov's theory in bounded domains (Kazhikhov 1987 Boundary Value Problems for Equations of Mathematical Physics (Krasnoyarsk)). In particular, our result generalizes the Kazhikhov's theory for the initial boundary value problem in bounded domains to the unbounded case.
We address the compressible magnetohydrodynamics (MHD) equations in $\mathbb{R}^3$ and establish a blow-up criterion for the local strong solutions in terms of the density only. Namely, if the density is away from vacuum ($ρ= 0$) and the concentration of mass ($ρ=\infty$), then a local strong solution can be continued globally in time. The results generalise and strengthen the previous ones in the sense that there is no magnetic field present in the criterion and the assumption on the pressure is significantly relaxed. The proof is based on some new a priori estimates for three-dimensional com…
Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum
The full compressible magnetohydrodynamic system in three-dimensional exterior domains is investigated. For the initial-boundary-value problem of this system with slip boundary condition for the velocity, adiabatic one for the temperature, and perfect one for the magnetic field, the global existence and uniqueness of classical solutions is established, under the condition that the initial data are of small energy but possibly large oscillations. In particular,the initial density and temperature are both allowed to vanish. Moreover, the large-time behavior of the classical solutions is also obt…
In this paper, we consider two Approximate Deconvolution Magnetohydrodynamics models which are related to Large Eddy Simulation. We first study existence and uniqueness of solutions in the double viscous case. Then, we study existence and uniqueness of solutions of the Approximate Deconvolution MHD model with magnetic diffusivity, but without kinematic viscosity. In each case, we give the optimal value of regularizations where we can prove global existence and uniqueness of the solutions. The second model includes the Approximate Deconvolution Euler Model as a particular case. Finally, an asym…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-Δ)^α$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $α$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $α=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global r…
This paper studies the Cauchy problem of the incompressible magnetohydrodynamic systems with or without viscosity $ν$. Under the assumption that the initial velocity field and the displacement of the initial magnetic field from a non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all $ν\geq 0$ and all space dimension $n \geq 2$. Such a result holds true uniformly in nonnegative viscosity parameter. The proof is based on the inherent strong null structure of the systems which was first introduced for incompressi…
In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corrresponding result in \cite{Chae-Degond-Liu}, the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estim…
In this paper, we obtain the global-in-$x$ Sobolev stability of Prandtl layer expansions for 2-D steady incompressible MHD flows with shear outer ideal MHD flows $(1,0,σ,0)$ ($σ\geq 0$) on a moving plate. It is worth noticing that in the Sobolev sense, the degeneracy of the tangential magnetic field in our result is allowed, which is much different from both the unsteady case in [25,26,27] and the steady case in [3,4].
This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-Δ)^β$. We establish the global regularity for the case when $β>1$. This result significantly improves previous work which requires $β>\frac32$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when $β=1$.
We consider the motion of an inviscid compressible fluid under the mutual interactions with magnetic field. We show that the initial value problem is ill--posed in the class of weak solutions for a large class of physically admissible data. We also consider the same problem for inviscid heat--conductive fluid and show the same result under certain restrictions imposed on the magnetic field. The main tool is the method of convex integration adapted to the Euler system with `variable coefficients'.
By observing a new relation between the magnetic pressure and the hydrodynamic pressure, global existence of classical solution to the full perfect MHD equations with large data is established, in particular including the case when all the viscosity, heat conductivity and diffusivity coefficients are constant. This can be viewed as an analog of the classical work by Kazhikhov-Shelukhin for the Navier-Stokes equations to the MHD equations. In addition, the vanishing shear viscosity limit is proved.
We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D flows for the Hall-MHD system, and prove the global exist…
In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations contain vacuum states. Moreover, the large-time…
The authors study the Cauchy problem of the magnetohydrodynamic equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For two spatial dimensions, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) provided the smooth initial data are of small total energy, and obtain some a priori decay with rates (in large time) for the pressure, the spatial gradient of both the velocity field and the magnetic field. Moreover, for three spatial dimensions case, some similar decay …
In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $(γ-1)^{\frac{1}{6}}E_{0}^{\frac{1}{2}}$, together with the $\|H_{0}\|_{L^{2}}$, is suitably small, a result on the existence of global classical solutions is obtained. It should be pointed out that the initial energy $E_{0}$ except the $L^{2}$- norm of $H_{0}$ can be large as $γ$ goes to 1, and that throughout the proof of the theorem in the present paper, we make no restriction upon the initial data $(ρ_{0},u_{0})$. Our result improves the one established by Li-Xu…
We revisit the 3D Cauchy problem of compressible heat-conducting magnetohydrodynamic equations with vacuum as far field density. By delicate energy method, we derive global existence and uniqueness of strong solutions provided that $(\|ρ_0\|_{L^\infty}+1)\big[\|ρ_0\|_{L^3}+ \|ρ_0\|_{L^\infty}+1)^2\big(\|\sqrt{ρ_0}u_0\|_{L^2}^2 +\|b_0\|_{L^2}^2\big)\big]\big[\|\nabla u_0\|_{L^2}^2+(\|ρ_0\|_{L^\infty}+1)\big(\|\sqrt{ρ_0}E_0\|_{L^2}^2+\|\nabla b_0\|_{L^2}^2\big)\big]$ is properly small. In particular, the smallness condition is independent of any norms of the initial data. This work improves our …
This paper establishes the global existence and uniqueness of smooth solutions to the two-dimensional compressible magnetohydrodynamic system when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also obtained. These results are achieved through a new approach of diagonalizing a system of coupled linearized equations. The standard method of diagonalization via the eigenvalues and eigenvectors of the matrix symbol is very difficult to implement here. This new process allows us to obtain an integral rep…
We deal with the barotropic compressible magnetohydrodynamic equations in three-dimensional (3D) bounded domain with slip boundary condition and vacuum. By a series of a priori estimates, especially the boundary estimates, we prove the global well-posedness of classical solution and the exponential decay rate to the initial-boundary-value problem of this system for the regular initial data with small energy but possibly large oscillations. The initial density of such a classical solution is allowed to contain vacuum states. Moreover, it is also shown that the oscillation of the density will gr…
We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{ü}-Xu-Zhong for the inhomogeneous case to the ful…
We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov (1987)], we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the …
Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criterion for Navier-Stokes equations and magnetohydrodynamics system have caught much attention recently, the Hall term has presented much difficulty. In this manuscript we discover a certain cancellation within the Hall term and obtain various new regularity criterion: first, in terms of a gradient of only the third componen…
We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.
Global well-posedness and exponential decay of 2D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum
This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Magnetohydrodynamic (MHD) equations with vacuum as far field density. We establish the global existence and uniqueness of strong solutions to the 2D Cauchy problem on the whole space $\mathbb{R}^2$, provided that the initial density and the initial magnetic decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the gradients of …
We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions.
The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained…
We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $ \Bbb R^2$. We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations.
We prove the existence of a mild solution to the three dimensional incompressible stochastic magnetohydrodynamic equations in the whole space with the initial data which belong to the Sobolev spaces.
The global regularity for the incompressible magnetohydrodynamic equations (MHD) in three dimensions is a long standing open problem of fluid dynamics and PDE theory. The Navier-Stokes equations can be viewed as a special case of MHD with a constant magnetic field, whose global regularity problem is known as a Clay Millennium Prize Problem. In this article, we prove the global regularity of axially symmetric solutions to the ideal MHD in three dimensions for a family of non-trivial magnetic fields. The proofs are based on the special structures of MHD and can of course also applied to the resi…
This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.
In this brief note we study the $n$-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.
