Analytic Stability and Singularities for Kelvin Helmholtz, Rayleigh Taylor, Problems Comparison with the Stability of Water Waves Problems
Analytical and Stochastic Fluid Dynamics
October 12, 2005 02:45 PM to 03:30 PM
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Abstract: |
Rayleigh Taylor and Kelvin Helmholtz are equations describing interfaces in fluids. Such interfaces are known by experiments and numerical simulations to be highly unstable. On the other hand a series of recent of results (Sijie Wu, Gilles Lebeau and Vladimir Kamotski) show that these solutions whenever they exist with a minimal regularity are solutions of nonlinear elliptic equations. Therefore they are analytic. This shows for instance that any solution with small regularity (say in C1+∞ but not in C∞ ) at some time will evolve in very singular systems. In the present lecture I will discuss these issues and compare the situation corresponding to Kelvin Helmholtz, Rayleigh Taylor and water waves equations. Conclusions are as follow The equation for water waves describes the stability of the phenomena as long as the surface of the water (which may be a multivalued function) do not self intersect. From this point of view Rayleigh Taylor and Kelvin Helmholtz problem are very similar but very different from water waves. When a singularity appears say a cusp in vorticity as predicted by the numerical computations of Moore and Orzag and proven on one example by Caflisch and Orellana the solution can in no way be extended by a curve of the same regularity what will appear will be a very weak solution in the sense of Delors or a curve which will not be arc-chord in the sense of Guy David and may be in agreement with the numerical simulations of Krasny. Therefore mathematical proof of the existence of weak enough solutions concentrated on such curves is a challenging open problem. |
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Lecture #12208
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