| Abstract: |
| In this talk, I will present the boundary regularity theory and the global well-posedness of regular classical solutions on the dynamic Prandtl equations.
To begin, we first establish the up-to-boundary regularity theory for the dynamic Prandtl system. The main obstacle lies in the lack of an explicit expression for the fundamental solution of a certain ultra-parabolic operator in the half-space. Our key strategies in overcoming this regularity issue include identifying the collaboration mechanism between diffusion and transport for a simplified operator, and then combining Fourier analysis, enhanced dissipation theory, and iterative methods to establish a series of hypoelliptic estimates for linear and quasilinear equations. By combining the established boundary regularity with the presented local theory, we also prove the global--in--$t,x$ well-posedness of regular classical solutions for the Prandtl system. This is joint work with Prof. Hao Jia (University of Minnesota) and Prof. Zhen Lei (Fudan University). |
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