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Register a new userFree Boundary Problems: The Forefront of Current and Future Developments
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This is an introduction to The Theme Issue on Free Boundary Problems and Related Topics, which consists of 14 survey/review articles on the topics, of Philosophical Transactions of the Royal Society A: Physical, Mathematical and Engineering Sciences, 373, no. 2050, The Royal Society, 2015.
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For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,alpha}$ regularity, as well-known in the scalar case cite{AC,C2}. While in cite{MTV2} the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of cite{D}. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in cite{DR, DSS}.
We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more generally for minimizers of the Alt-Phillips functional.
The semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove existence of solutions of the incompressible semi-geostrophic equations in a fully three-dimensional domain with a free upper boundary condition. We show that, using methods similar to those introduced in the pioneering work of Benamou and Brenier, who analysed the same system but with a rigid boundary condition, we can prove the existence of solutions for the incompressible free boundary problem. The proof is based on optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures given by Ambrosio and Gangbo. We also show how these techniques can be modified to yield the same result also for the compressible version of the system.
In this paper, we investigate the convergence rates of inviscid limits for the free-boundary problems of the incompressible magnetohydrodynamics (MHD) with or without surface tension in $mathbb{R}^3$, where the magnetic field is identically constant on the surface and outside of the domain. First, we establish the vorticity, the normal derivatives and the regularity structure of the solutions, and develop a priori co-norm estimates including time derivatives by the vorticity system. Second, we obtain two independent sufficient conditions for the existence of strong vorticity layers: (I) the limit of the difference between the initial MHD vorticity of velocity or magnetic field and that of the ideal MHD equations is nonzero. (II) The cross product of tangential projection on the free surface of the ideal MHD strain tensor of velocity or magnetic field with the normal vector of the free surface is nonzero. Otherwise, the vorticity layer is weak. Third, we prove high order convergence rates of tangential derivatives and the first order normal derivative in standard Sobolev space, where the convergence rates depend on the ideal MHD boundary value.
The Oxygen Depletion problem is an implicit free boundary value problem. The dynamics allow topological changes in the free boundary. We show several mathematical formulations of this model from the literature and give a new formulation based on a gradient flow with constraint. All formulations are shown to be equivalent. We explore the possibilities for the numerical approximation of the problem that arise from the different formulations. We show a convergence result for an approximation based on the gradient flow with constraint formulation that applies to the general dynamics including topological changes. More general (vector, higher order) implicit free boundary value problems are discussed. Several open problems are described.
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