Speaker: Charles Collot (Courant Institute, NYU)
Time: 15:00-17:00, August 5,7,9
Room: Room 1418, Management building, East Compus of USTC
Many partial differential equations display particular phenomena at the heart of which lie special solutions of the equation. Indeed, in some asymptotic configurations, one expects the solution to be well described by, for example, stationary states, traveling waves, backward or forward self-similar solutions. An open problem for many equations is to prove such a decomposition of the solution (soliton resolution), to study these special solutions, and to understand the phenomena they are responsible for. This course will focus on the singularity formation problem for the semi-linear heat equation and the Prandtl's system, and will aim at describing precisely some blow-up solutions. The first one is a famous model equation for which the theory has been well developed since the work of Fujita in the sixties, while the second is a fluid mechanics problem that has been intensively studied by physicists since the eighties, but that is still poorly understood at a mathematical level. In both problems, smooth solutions may not exist for all times: some become singular in finite time. This course aims at explaining this phenomenon, presenting basic and key notions, and focusing on the recent work of the author on energy supercritical phenomena, anisotropic singularities, and shock formation in fluid mechanics (with Merle, Rapha\el, Szeftel, and Ghoul, Ibrahim, Masmoudi).
We shall first go over the Cauchy theory for nonlinear parabolic equations and quasilinear hyperbolic transport equations, to emphasize some key properties as regularising effects or the geometry of characteristics. Then, a key notion, that of self-similarity, will be explained both from a physical and mathematical perspective. It accounts for a simplification of the dynamics of the equation in certain regimes, especially during singularity formation. We will then show the existence of backward self-similar solutions, and prove that they appear as universal blow-up profiles in some configurations. The core of the course will be the stabilisation of truly or degenerate self-similar singular solutions. First, the linearized problem, modulation techniques, and topological tools will be presented for the stability of supercritical self-similar blow-up for the heat equation. Then, the full nonlinear analysis will be done for degenerate self-similar solutions to a reduced equation linked to the Prandtl's system. We will finish the course by the description of anisotropic singularities (with different behaviors in different space directions) for a Burgers equation with transverse viscosity. The last course will be devoted to a more general presentation of recent and related results in singularity formation for parabolic and dispersive equations, as well as for fluid problems.
