The purpose of this presentation is to introduce a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.
University of Cambridge - Texas A&M University - Weizmann Institute of Science
Friday, October 30, 2020 - 3:00pm to 4:00pm
In this talk I will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness and the effect of physical boundaries on the potential formation of singularity. In addition, I will present a blow-up criterion based on a class of inviscid regularization for these equations.
We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group SO(3)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.
In recent years, there has been significant interest in continuum models of crystal surface evolution and facet formation. However, in the most physically relevant case, when the free energy of the surface is the total variation energy, even existence of solutions to the continuum PDE is unknown. Furthermore, attempts at developing a robust numerical method for simulating solutions suffer from significant stiffness, preventing numerical study of the equation’s behavior on fine spatial grids. In this talk, I will describe a new approach to simulating solutions of the crystal surface evolution equation based on combining the formal gradient flow structure of this equation with modern operator splitting techniques. This is based on joint work with Jian-Guo Liu, Jianfeng Lu, Jeremy L. Marzuola, and Li Wang.
I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by Harrison-Pugh and by Almgren-David. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.
In this talk, I present two PDE models of a chemical reaction, and I show that they are two different sides of the same coin: namely, the solutions of one PDE converge to the solutions of the other. The proof of this fact is surprisingly elementary (but not easy), because it just requires some integration by parts. This is based on joint work with Lawrence C. Evans. No chemistry background required.
We will show optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is joint work with Jingang Xiong.
We study the long-time asymptotic behavior of the linearized Euler and nonlinear Navier-Stokes equations close to Couette flow. As a main result we show that suitable forcing breaks asymptotic stability results at the level of the vorticity, but that solutions never the less exhibit convergence of the velocity field. Thus, here linear inviscid damping persists despite instability of the vorticity equations.