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Register a new userZero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics
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We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh-Taylor sign condition holds true initially. This result is a continuation of the authors previous works [13,27,12]. Our proof is based on the combination of the techniques developed in our previous works [13,27,12], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.
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We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by $kappa>0$ whose solution converges to that of the MHD equations as $kappato 0$. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.
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 study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.
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.
We consider 3D free-boundary compressible ideal magnetohydrodynamic (MHD) system under the Rayleigh-Taylor sign condition. It describes the motion of a free-surface perfect conducting fluid in an electro-magnetic field. The local well-posedness was recently proved by Trakhinin and Wang [66] by using Nash-Moser iteration. In this paper, we prove the a priori estimates without loss of regularity for the free-boundary compressible MHD system in Lagrangian coordinates in anisotropic Sobolev space, with more regularity tangential to the boundary than in the normal direction. It is based on modified Alinhac good unknowns, which take into account the covariance under the change of coordinates to avoid the derivative loss; full utilization of the cancellation structures of MHD system, to turn normal derivatives into tangential ones; and delicate analysis in anisotropic Sobolev spaces. Our method is also completely applicable to compressible Euler equations and thus yields an alternative estimate for compressible Euler equations without the analysis of div-curl decomposition or the wave equation in Lindblad-Luo [42], that do not work for compressible MHD. To the best of our knowledge, we establish the first result on the energy estimates without loss of regularity for the free-boundary problem of compressible ideal MHD.
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