We study a general convergence theory for the analysis of numerical solutions to the magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (= 2; 3). First, we introduce the concept of dissipative weak solutions and prove the weak-strong uniqueness property for dissipative weak solutions, meaning a dissipative weak solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the dissipative weak solution as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax-equivalence theorem. Finally, as an application of this theory, we show the convergence analysis for two numerical methods.
In this paper we are concerned with the global well-posedness for the compressible MHD equations with large data. We show that if the shear viscosity $μ$ is a positive constant and the bulk viscosity $λ$ is the power function of the density, that is, $λ(ρ)=ρ^β$ with $β>3$, then the two dimensional compressible MHD system with the periodic boundary conditions on the torus $\mathbb{T}^2$ have a unique global classical solution $(ρ, u,H)$. In this work we extended the results about compressible Navier-Stokes equations in \cite{Karzhikhov} to compressible MHD equations by applying several new tech…
In this paper, we investigate the three dimensional stationary compressible Navier-Stokes equations, and obtain Liouville type theorems if a smooth solution $(ρ, \mathbf{u})$ satisfies some suitable conditions. In particular, our results improve and generalize the corresponding result.
In this paper, we transform compressible MHD system written in Euler coordinate to Lagrange coordinate in critical Besov space. Then we construct unique local solutions for compressible MHD system. Our results improve the range of Lebesgue exponent in Besov space from $[2, N)$ to $[2, 2N)$ with $N$ stands for dimension. In addition, we give a lower bound for the maximal existence time which is important for our construction of global solutions. Based on the local solution, we obtain a unique global solution with high oscillating initial velocity and density by using effective viscous flux and …
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.
In this note we consider the ideal compressible magneto-hydrodynamics (MHD) equations in a special two dimensional setting. We show that there exist particular initial data for which one obtains infinitely many entropy-conserving weak solutions by using the convex integration technique. Our result is also true for the isentropic case.
In this paper, we study the global well-posedness of classical solutions to the 2D compressible MHD equations with large initial data and vacuum. With the assumption $μ=const.$ and $λ=ρ^β,~β>1$ (Vaǐgant-Kazhikhov Model) for the viscosity coefficients, the global existence and uniqueness of classical solutions to the initial value problem is established on the torus $\mathbb{T}^2$ and the whole space $\mathbb{R}^2$ (with vacuum or non-vacuum far fields). These results generalize the previous ones for the Vaǐgant-Kazhikhov model of compressible Navier-Stokes.
An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with N…
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 study the barotropic compressible magnetohydrodynamic equations with the shear viscosity being a positive constant and the bulk one being proportional to a power of the density in a general two-dimensional bounded simply connected domain. For initial density allowed to vanish, we prove that the initial-boundary-value problem of 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data provided the shear viscosity is a positive constant and bulk one is $λ=ρ^β$ with $β>4/3$. As we known, this is the first result …
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 establish new $L^p$ gradient estimates of the solutions in order to discuss Cauchy problem for the full compressible magnetohydrodynamic(MHD) systems in $\mathrm{R}^3$. We use the "$\rm{div}-\rm{curl}$" decomposition technique (see \cite{{HJR},{MR}}) and new modified effective viscous flux and vorticity to calculate "$\Vert\nabla \mathbf{u}\Vert_{L^3}$" and "$\Vert\nabla \mathbb{H}\Vert_{L^3}$".As a result, we obtain global well-posedness for the solution with the initial data being in a class of space with lower regularity, while the energy of which should be suitably small.
In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the…
Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier--Stokes (cNS) and magnetohydrodynamics (MHD) equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in $H^s$ for data in $H^s$. We accomplish this by a novel energy estimate carried out in Lagrangian…
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…
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 are concerned with optimal decay rates for higher order spatial derivatives of classical solutions to the full compressible MHD equations in three dimensional whole space. If the initial perturbation are small in $H^3$-norm and bounded in $L^q(q\in \left[1, \frac{6}{5}\right))$-norm, we apply the Fourier splitting method by Schonbek[Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for the second order spatial derivatives of solutions and the third order spatial derivatives of magnetic field in $L^2$-norm. These results improve the work of Pu and Guo [Z. …
In this paper, we are concerned with global existence and optimal decay rates of solutions for the three-dimensional compressible Hall-MHD equations. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch.Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for hi…
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 …
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at vanishing shear viscosity, both the boundary-layer thickness and the boundary-layer solution are discussed. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are ba…
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…
For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work \cite{GMWZ6}, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in term…
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…
In this paper, firstly, we prove the global well-posedness of three dimensional compressible magnetohydrodynamics equations for some classes of large initial data, which may have large oscillation for the density and large energy for the velocity and magnetic field. Secondly, we prove the optimal time decay for the compressible magnetohydrodynamics equations with low regularity assumptions about the initial data. Especially, we can obtain the optimal $L^{2}$ time decay rate when the initial data small in the critical Besov space (no small condition in space $H^{N/2+1}$). When we calculate the …
We establish the convergence of an approximation scheme to a model for aurora type phenomena. The latter, mathematically, means a system describing the short wave-long wave (SW-LW) interactions for compressible magnetohydrodynamic (MHD) flows, introduced in a previous work, which presents short waves, governed by a nonlinear Schrödinger (NLS) equation based on the Lagrangian coordinates of the fluid, and long waves, governed by the compressible MHD system. The NLS equation and the compressible MHD system are also explicitly coupled by an interaction potential in the NLS equation and an interac…
In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-σ_1}$ with $σ_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{σ_1}{2}}$ for $t\rightarrow+\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usua…
Strong solutions to the initial-boundary-value problem of compressible MHD equations with degenerate viscosities and far field vacuum in 3D exterior domains
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. To capture the behavior of the solutions at small shear viscosity, we also discuss the boundary-layer thickness and the boundary-layer solution. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (unif…
In this paper, the 3-D compressible MHD equations with initial vacuum or infinity electric conductivity is considered. We prove that the $L^\infty$ norms of the deformation tensor $D(u)$ and the absolute temperature $θ$ control the possible blow-up (see [5][18][20]) for strong solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ and $θ$ as the critical time approaches. The viscosity coefficients are only restricted by the physic…
Global uniform regularity for the 3D compressible MHD equations near a background magnetic field
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Department of Mathematics, University of Notre Dame（圣母大学数学系）
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 mainly focus on optimal time decay estimation for large-solution about compressible magnetohydrodynamic equations in 3D whole space, provided that $(σ_{0}-1,u_{0},M_{0})\in L^1\cap H^2$. In [2](Chen et al.,2019), they proved time decay estimation of $\|(σ-1,u,M)\|_{H^1}$ being $(1+t)^{-\frac{3}{4}}$. Based on it, we obtained that of $\|\nabla(σ-1,u,M)\|_{H^1}$ being $(1+t)^{-\frac{5}{4}}$ in [24]. Therefore, we are committed to improving that of $\|\nabla^2 (σ-1,u,M)\|_{L^2}$ in this paper. Thanks to the method adopted in [25] (Wang and Wen, 2021), we get the optimal time decay esti…
We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small $L^2$-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-St…
Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$.…
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 establish criteria of stability and instability for the stratified compressible magnetic Rayleigh--Taylor (RT) problem. More precisely, if under the stability condition $Ξ<1$, we show the existence of unique solution with algebraic decay in time for the (compressible) magnetic RT problem with proper initial data in Lagrangian coordinates. The stability result presents that sufficiently large vertical (base) magnetic field can inhibit the development of RT instability. On the other hand, if $Ξ>1$, there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This shows …
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…
