Day 4 (June 13)

 Elasticity is important in continuum mechanics with a wide range of applications and is challenging in analysis. In this talk we shall first review some basic mathematical results and then discuss some special elastic effects in elastic fluids. The first elastic effect is the stabilizing effect of elasticity on the vortex sheets in compressible elastic flows. Some recent results on linear and nonlinear stability of compressible vortex sheets will be presented. The second effect is on the vanishing viscosity process of compressible viscoelastic flows on the half plane under the no-slip boundary condition. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. Our results show that the deformation tensor can prevent the formation of strong boundary layers. The talk is based on the recent results joint with several collaborators.

 The supercooled Stefan problem describes freezing of supercooled water. In contrast to the Stefan problem that describes melting of ice, the supercooled problem exhibits unstable bahaviour which makes the usual PDE methods break down. We will discuss some recent progress which employs a stochastic version of optimal transport, involving optimal stopping times of the Brownian motion.

 This talk concerns the dynamics of nonisentropic compressible Euler equations in multidimensions in a physical vacuum. The emphasize is on the Hadamardstyle local well-posedness within lowregularity weighted Sobolev spaces, demonstrating existence, uniqueness, and continuous dependence on initial data, and a sharp a priori energy estimates and continuation criteria. This talk is based on the joint work with Dr. Sicheng Liu in Macau University.

 Classical variational approach of maximizing the kinetic energy with various constraints provides vortex stability in several special cases, but in general this approach fails when the vorticity is concentrated at several points ("multi-vortex") in the fluid domain. This is simply because such configurations are not local kinetic energy maximizers, even when we restrict the admissible class using all the other coercive coserved quantities of fluid motion. In this talk, we present several results on the stability of multi-vortex solutions, obtained by combining variational approaches with dynamical bootstrapping schemes. We focus on the case of multiple Lamb dipoles weakly interacting with each other. This is based on joint works with Ken Abe, Kyudong Choi, and Yao Yao.

 In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation.