In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.
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 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.
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…
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.
In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitab…
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…
In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^{m}(\mathbb{R}^3)$ for any $m>\frac{7}{2}$. Namely, either the system is locally ill-posed in $H^{m}(\mathbb{R}^3)$, or it is locally well-posed, but there exists an initial data in $H^{m}(\mathbb{R}^3)$, for which the $H^{m}(\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$. In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and $B_0(x)=b_{0θ}(r,z)e_θ$, and reduce the system to the axisymmet…
We prove the existence of both local and global smooth solutions to the Cauchy problem in $\R^3$ for the incompressible magnetohydrodynamics (MHD) system. We also prove that the solution to the incompressible MHD system can be obtained as the incompressible limit of the corresponding compressible system. We apply methods in extracting weak dissipative structure which were suggested by Lei-Liu-Zhou.
Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space
Shanghai Jiao Tong University（上海交通大学）；University of Notre Dame（圣母大学）
For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with $γ$-law gas equation of state, $γ\ge 1$, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes. In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompre…
We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, we prove that the weak solutions constructed by Beekie-Buckmaster-Vicol [2] for the ideal MHD can be obtained as a strong vanishing viscosity and resistivity limit of a sequence of weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishin…
Existence of local-in-time unique solution and loss of smoothness of full Magnet-Hydro-Dynamics system (MHD) is considered for periodic initial data. The result is proven using Fujita-Kato's method in $\ell^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwells's equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.
In this paper, we prove that as the viscosity and resistivity go to zero, the solution of the Cauchy problem for the incompressible MHD equations converges to the solution of the ideal MHD equations in the same topology with the initial data. Our proof mainly depends on the method introduced by the paper \cite{G.L.Y} and the constructions of the incompressible MHD equations.
In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic converge…
The current paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of \cite{AKH,Hassainia,hz0}. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied.
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.
In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the weighted Hölder space, where the closeness is independent of the dissipation coefficients.
As a continuation of \cite{LXY}, the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a $L^\infty$ estimate on the error by multi-scale analysis.
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.
We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to o…
This paper is concerned with the vanishing viscosity and magnetic resistivity limit for the two-dimensional steady incompressible MHD system on the half plane with no-slip boundary condition on velocity field and perfectly conducting wall condition on magnetic field. We prove the nonlinear stability of shear flows of Prandtl type with nondegenerate tangential magnetic field, but without any positivity or monotonicity assumption on the velocity field. It is in sharp contrast to the steady Navier-Stokes equations and reflects the stabilization effect of magnetic field. Unlike the unsteady MHD sy…
We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in $\Bbb R^n$, $n=2,3$, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in $\Bbb R^n$. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the beh…
On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics
Institute of Mathematics of the Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；College of Sciences Yangzhou University（扬州大学理学院）；Hohai University（河海大学）
We investigate the stability and instability of the magnetic Rayleigh--Bénard problem with zero resistivity. An stability criterion is established, under which the magnetic Bénard problem is stable. The proof mainly is based on a three-layers energy method and an idea of magnetic inhibition mechanism. The stable result first mathematically verifies Chandrasekhar's assertion in 1955 that the thermal instability can be inhibited by strong magnetic field in magnetohydrodynamic (MHD) fluid with zero resistivity (based on a linearized steady magnetic Bénard equations). In addition, we also provide …
In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $(u_0,B_0)\in H^{\frac{1}{2}+σ}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $σ\in (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $σ$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31-65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Mor…
This paper is devoted to the analysis of a singular perturbation problem for a $2$-D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non…
We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity…
In \cite{ChCa}, Califano and Chiuderi conjectured that the energy of incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimension provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state $(e_3,0).$ In particular, we prove that for such data, 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field…
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…
In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy…
We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^γ_tW^{s,p}_x$, where the exponents $(s,γ,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Ladyženskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regular…
In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. The free boundary problem for MHD is an important problem not only for mathematical fluid dynamics but also some application to the field of engineering In fact, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. In this paper, an electromagnetic field generated by some currents in an external domain keeps an…
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）
We consider the plasma-vacuum interface problem in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field. The lower plasma region is governed by the incompressible inviscid and resistive MHD, the upper vacuum region is governed by the pre-Maxwell equations, and the effect of surface tension is taken into account on the free interface. The global well-posedness of the problem, supplemented with physical boundary conditions, around the equilibrium is established, and the solution is shown to decay to the equilibrium almost exponentially. Our results reveal the strong …
We reformulate in Lagrangian coordinates the two-phase free boundary problem for the equations of Magnetohydrodynamics in a infinite slab, which is incompressible, viscous and of zero resistivity, as one for the Navier-Stokes equations with a force term induced by the fluid flow map. We study the stabilized effect of the magnetic field for the linearized equations around the steady-state solution by assuming that the upper fluid is heavier than the lower fluid, $i. e.$, the linear Rayleigh-Taylor instability. We identity the critical magnetic number $|B|_c$ by a variational problem. For the ca…
