Georgia Institute of Technology College of Sciences

This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.

A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.

In this talk we concentrate on wave maps – a geometric nonlinear wave equation – and we discuss the existence and stability of self-similar solutions, as in all higher dimensions they appear to drive the generic blowup behavior. We outline a novel framework for studying global-in-space stability of such solutions; we then men-tion some long-awaited results that we thereby obtained, and, finally, we discuss the new mathematical challenges that our approach generates.

We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University)

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao.