报告摘要：A fundamental problem in the theory of compressible fluid is to provide a precise description of shock formation and development from smooth initial data. In this talk, I will present recent work in collaboration with Buckmaster, Drivas, and Shkoller on the shock development problem for the 2D compressible Euler under azimuthal symmetry. The work represents the first full description of shock development; in particular, in addition to describing the shock front, we give the first detailed description of the phenomena of weak discontinuities conjectured by Landau and Lifschitz over half a century ago. We prove that along the slowest surface (relative to the shock front), all fluid variables except the entropy have C^{1,1/2} one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, the density and entropy form C^{1,1/2} one-sided cusps, while the pressure and normal velocity remain C^2; as such, we term this surface a weak contact discontinuity. These detailed results are new even for the 1D Euler system.

报告人简介：Vlad is currently a professor at Courant Institute for Mathematical Sciences. He is well-known for a series of breakthrough work in singularity formation for compressible fluids, Onsager conjecture, convex integration, inviscid limit problem and so on. He got the Clay Research Award, MCA Prize, Alfred P. Sloan Research Fellowship in Mathematics. He is also an invited speaker at Hadamard Lectures, IHES, Paris, France.