A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.Comment: Final version, to appear in Nonlinearit

Ferroelectric Properties of Aurivillius Phase Bi5CrxFe1-xTi3O15 Films

Aurivillius phase Bi5CrxFe1-xTi3O15 (BCxF1-xTO) thin films are prepared by the chemical solution deposition method and the effect of Cr content on the structural, optical bandgap, electric transport, and dielectric/ferroelectric properties of BCxF1-xTO thin films are investigated in detail. X-ray diffraction analysis shows that all of BCxF1-xTO films are a complete solid solution and maintain Aurivillius structure. The replacement of Fe3+ with smaller Cr3+ decreases the overall lattice volume and gradually increases the bandgap of BCxF1-xTO thin film. The random mixing of Cr and Fe in BCxF1-xTO decreases the long-range lattice distortion in a-b plane and the smallest a/b ratio is found at BC0.5F0.5TO. Cr-doping also changes the gran shape from the spheres to plates and BC0.5F0.5TO consists of only plate-like grains. This indicates that a decrease in the lattice distortion promotes the grain growth along a-b plane and facilitates the appearance of the inherent crystal shape of Aurivillius phase. Ferroelectric properties of BCxF1-xTO films are examined by measuring P-E hysteresis loops. Cr-doping increases saturated polarization (PS) and decreases coercive field (EC). When 50 atomic % of Cr is doped, PS and EC of BC0.5F0.5TO is 35 μC/cm2 and 125 kV/cm, respectively. This is due to the fact that the decrease in the long-range distortion in a- b plane promotes the alignment of ferroelectric dipoles under electric field. The frequency dependent dielectric properties at different temperatures and the leakage current show that Cr doping increases the carrier concentration and the space charge polarization. However, the plate- like shape grains of Cr-rich BCxF1-xTO films suppress the transport of carriers from grains to grains and prevents a dramatic increase in the leakage current. The results of this study provide a design rule to control the ferroelectric of Aurivillius phase BCxF1-xTO thin films by modifying the composition and lattice distortion

Local Well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics

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 control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.Comment: Typos correcte

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

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. A local existence and uniqueness result was recently proved by Trakhinin and Wang [64] by using Nash-Moser iteration. However, that result loses regularity going from data to solution. In this paper, we show that the Nash-Moser iteration scheme in [64] can be improved such that the local-in-time smooth solution exists and is unique when the initial data is smooth and satisfies the compatibility condition up to infinite order. Second, 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. As a result, we can prove the uniqueness and the continuous dependence on initial data provided the local existence, and a continuation criterion for smooth solution. Finally, we extend the local well-posedness theorem to the case of initial data only satisfying compatibility conditions up to finite order, assuming these can be approximated by data satisfying infinitely many compatibility conditions.Comment: 61 pages. Add an existence theorem for smooth solutions, a continuation criterion and the construction of initial data satisfying the compatibility conditions up to infinite order. Final version, accepted by Arch. Rational Mech. Ana

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

We consider 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension and it is not assumed to be irrotational. We prove the local well-posedness by introducing carefully-designed approximate equations which are asymptotically consistent with the a priori energy estimates. The energy estimates yield no regularity loss and are uniform in Mach number. Also, they are uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds initially. We can thus simultaneously obtain incompressible and vanishing-surface-tension limits. The method developed in this paper is a unified and robust hyperbolic approach to free-boundary problems in compressible Euler equations. It can be applied to some important complex fluid models as it relies on neither parabolic regularization nor irrotational assumption. This paper joined with our previous works [46,47] rigorously proves the local well-posedness and the incompressible limit for a compressible gravity water wave with or without surface tension.Comment: 63 page

Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

We prove the incompressible limit of compressible ideal magnetohydrodynamic (MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space $H^m~(m\geq 2)$, while the incompressible problem is still well-posed in $H^m$. The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight $\varepsilon^2$. Thus, the energy functional should be defined by using the anisotropic Sobolev space $H_*^{2m}$. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is parallel to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD such as current-vortex sheets.Comment: 28 page

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

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 [17,32,16]. Our proof is based on the combination of the techniques developed in our previous works [17,32,16], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.Comment: 35 pages. Final version, accepted by Nonlinearity. We extend the result to the case of a general diffeomorphis