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Register a new userA priori Estimates for the Free-Boundary problem of Compressible Resistive MHD Equations and Incompressible Limit
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Junyan Zhang
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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 Lorentz force, magnetic field and pressure wave simultaneously.
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In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.
A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view. Some estimates on the second fundamental form and velocity of the free surface are also obtained.
We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $eta$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.
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 [85] 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 key observation is that the structure of the wave equation of pressure together with Christodoulou-Lindblad elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations in spite of a potential loss of derivative in the source term. We first establish the nonlinear energy estimate without loss of regularity for the free-boundary compressible elastodynamic system. The energy estimate is also uniform in sound speed which yields the incompressible limit. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in boundary energy and analyze higher order wave equations as in the previous works of compressible Euler equations (cf. Lindblad-Luo [60] and Luo [62]) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in [60,62] can still be recovered for a slightly compressible elastic medium by further delicate analysis which is completely different from Euler equations.
In this paper, we address the space-time decay properties for strong solutions to the incompressible viscous resistive Hall-MHD equations. We obtained the same space-time decay rates as those of the heat equation. Based on the temporal decay results in cite{cs}, we find that one can obtain weighted estimates of the magnetic field $B$ by direct weighted energy estimate, and then by regarding the magnetic convection term as a forcing term in the velocity equations, we can obtain the weighted estimates for the vorticity, which yields the corresponding estimates for the velocity field. The higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in cite{k01}. It should be emphasized that the the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight. The same arguments also yield the sharp space-time decay rates for strong solutions to the usual MHD equations.
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