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Title:
Formation and development of singularities for the compressible Euler equations
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Abstract:
We give a complete description of the formation and development of singularities for the compressible Euler equations in two space dimensions, under azimuthal symmetry. Our proof applies mutatis mutandis in the drastically simpler situations of one-dimensional flows, or multi-dimensional flows with radial symmetry. We prove that for smooth and generic initial data with azimuthal symmetry, the 2D compressible Euler equations yield a local in time smooth solution, which in finite time forms a first gradient singularity, the so-called pre-shock. We then show that a discontinuous entropy producing shock wave instantaneously develops from the pre-shock. Simultaneous to the development of the shock, two other characteristic surfaces of higher-order cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau and Lifshitz [17, Chapter IX, ยง96], who conjectured their existence. We prove that along the characteristic surface moving with the fluid, a weak contact discontinuity is formed, while along the slowest surface in the problem, a weak rarefaction wave emerges. The constructed solution is the unique solution of the Euler equations in the class of entropy-producing weak solutions with azimuthal symmetry and with regularity determined by the fact that it arises from a generic pre-shock. The talk is based on joint work with T. Buckmaster, T. Drivas, and S. Shkoller.
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