Abstract: In this talk, we will investigate the propagation of Lipschitz regularity by solutions to various nonlinear, nonlocal parabolic equations. We will locally analyze models such as the Michelson-Sivashinsky equation, incompressible Navier-Stokes system, and advection diffusion problems that include the dissipative SQG equation. Depending on the model, we will either show global well-posedness, derive new regularity criteria, or provide different proofs to and generalize previously obtained results. In particular, we will show that for abstract drift-diffusion problems, it is possible to break certain, supercritical Holder-type barriers and get regularity, which is rather surprising.
Abstract: In this talk, we consider nematic liquid crystal flow (NLCF) and Landau-Lifshitz-Gilbert equation (LLG) in \R^2. (NLCF) is a system strongly coupling the nonhomogeneous incompressible Navier-Stokes equation and the transported harmonic map heat flow, while (LLG) serves as the basic evolution equation for the spin fields in the continuum theory of ferromagnetism. We will investigate how parabolic gluing method can be developed and applied to construct finite time blow-up solutions. This talk is based on joint works with Chen-Chih Lai, Fanghua Lin, Changyou Wang, Juncheng Wei and Qidi Zhang.
Abstract: Nonlinear scalar field theories on the line such as the phi^4 model or the sine-Gordon model feature soliton solutions called kinks. They are expected to form the building blocks of the long-time dynamics for these models. In this talk I will survey recent progress on the asymptotic stability problem for kinks and I will present a recent result (joint work with W. Schlag) on the asymptotic stability of the sine-Gordon kink under odd perturbations.
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