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 that the RT instability still occurs when the strength of base magnetic field is small or the base magnetic field is horizontal with proper large horizontal period cell. Moreover, by analyzing the stability condition in magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in vertical direction can be inhibited by the stabilizing effect of pressure, which also plays an important role in the mathematical proof for stability of the magnetic RT problem. In addition, we will extend the results in magnetic RT problem to the (compressible) viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than the one of magnetic fields.
This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We are concerned with the Rayleigh-Taylor instability when the upper fluid is heavier than…
We consider the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. After constructing Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, we turn to an analysis of the equations obtained from linearizing around such a steady state. By a natural variational approach, we construct normal mode solutions that grow exponentially in time with rate like $e^{t \sqrt{\absξ}}$, where $ξ$ is the spatial frequency of the normal mod…
We study the equations obtained from linearizing the compressible Navier-Stokes equations around a steady-state profile with a heavier fluid lying above a lighter fluid along a planar interface, i.e. a Rayleigh-Taylor instability. We consider the equations with or without surface tension, with the viscosity allowed to depend on the density, and in both periodic and non-periodic settings. In the presence of viscosity there is no natural variational framework for constructing growing mode solutions to the linearized problem. We develop a general method of studying a family of modified variationa…
We consider the motion of a viscous compressible and heat conducting fluid confined in the gap between two rotating cylinders (Taylor-Couette flow). The temperature of the cylinders is fixed but not necessarily constant. We show that the problem admits a time--periodic solution as soon as the ratio of the angular velocities of the two cylinders is a rational number.
In this paper, we investigate the Rayleigh-Taylor instability problem for two compressible, immiscible, inviscid flows rotating with an constant angular velocity, and evolving with a free interface in the presence of a uniform gravitational field. First we construct the Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, then we turn to an analysis of the equations obtained from linearization around such a steady state. In the presence of uniform rotation, there is no natural variational framework for constructing growing mode soluti…
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…
We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time exist…
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 …
We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma--vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh--Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash--Moser iteration scheme. Our result is uniform in the light sp…
Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleig…
This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit de…
We consider 3D free-boundary compressible elastodynamic system under the Rayleigh-Taylor sign condition. It describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor satisfies the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin [84] by Nash-Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach and also establish the incompressible limit. We apply the tangential smoothing method to define the approximation system. The ke…
Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces
We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh-Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving \emph{a priori} estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a ne…
We are concerned with a model of ideal compressible isentropic two-fluid magnetohydrodynamics (MHD). Introducing an entropy-like function, we reduce the equations of two-fluid MHD to a symmetric form which looks like the classical MHD system written in the nonconservative form in terms of the pressure, the velocity, the magnetic field and the entropy. This gives a number of instant results. In particular, we conclude that all compressive extreme shock waves exist locally in time in the limit of weak magnetic field. We write down a condition sufficient for the local-in-time existence of current…
We study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flo…
We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for 2D planar flows provided that the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a nat…
The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction.Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be sign…
We establish a variational framework for nonlinear instabilities in a setting of the ideal magnetohydrodynamic (MHD) equations. We apply a variational method to various kind of smooth steady states which are shown to be nonlinearly unstable for both incompressible and compressible ideal MHD equations. Destabilizing effect of compressibility is justified as well as stabilizing effect of magnetic field lines arising in MHD dynamics, which distinguishes from the Rayleigh-Taylor instability in the absence of magnetic field lines.
We consider the plasma-vacuum interface problem in a classical statement when in the plasma region the flow is governed by the equations of ideal compressible magnetohydrodynamics, while in the vacuum region the magnetic field obeys the div-curl system of pre-Maxwell dynamics. The local-in-time existence and uniqueness of the solution to this problem in suitable anisotropic Sobolev spaces was proved in [P. Secchi, Y. Trakhinin, Nonlinearity 27 (2014), 105-169], provided that at each point of the initial interface the plasma density is strictly positive and the magnetic fields on either side of…
