# Seminars & Events for Analysis of Fluids and Related Topics

##### Global well-posedness for the 2D Muskat problem

The Muskat problem was originally introduced by Muskat in order to model the interface between water and oil in tar sands. In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media. In this talk I will prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the initial data is monotonic or has slope strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Naverov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.

##### Some recent results on wave turbulence and quantum kinetics theories.

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales.

Despite the fact that wave fields in nature are enormous diverse; to describe the processes of random wave interactions, there is a common mathematical concept; the wave kinetic equations.

After the production of the first Bose-Einstein Condensates (BECs), there has been an explosion of physics research on the kinetic theory associated to BECs and their thermal clouds.

In this talk, we will summarize our recent progress on this topic.

##### Well-posedness for Stochastic Continuity Equations with Rough Coefficients.

According to the theory of Diperna/ Lions, the continuity equation associated to a Sobolev (or BV) vector field with bounded divergence has a unique weak solution in L^p. Under the addition of white in time stochastic perturbations to the characteristics of the continuity equation, it is known that uniqueness can be obtained under a relaxation of the regularity conditions and the requirement of bounded divergence. In this talk, we will consider the general stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients and discuss conditions under which the equation has a unique solution.

##### The optimal design of wall-bounded heat transport

Flowing a fluid is a familiar and efficient way to cool: fans cool electronics, water cools nuclear reactors, and the atmosphere cools the Earth. In this talk, we discuss a class of problems from fluid dynamics concerning the design of incompressible wall-bounded flows achieving optimal rates of heat transport for a given flow intensity budget. Guided by a perhaps unexpected connection between this optimal design problem and various “energy-driven pattern formation” problems from materials science, we construct flows achieving nearly optimal rates of heat transport in their scaling with respect to the intensity budget. The resulting flows share striking similarities with self-similar elastic wrinkling patterns, such as can be seen in the shape of a hanging drape or nearby the edge of a torn plastic sheet.

##### Solutions after blowup in ODEs and PDEs: spontaneous stochasticity

We discuss the extension of solutions beyond a finite blowup time, i.e., the time at which the system ceases to be Lipschitz continuous. For larger times solutions are defined first by using a (physically motivated) regularization of equations and then taking the limit of a vanishing regularization parameter. We report on several generic situations when such a limit leads to stochastic solutions defining uniquely a probability to choose one or the other (non-unique) path. Moreover, such solutions appear to be independent of the details of regularization procedure, thus, following from the properties of original ideal system alone. In this talk, we provide some rigorous results for systems of ODEs with singularities by using the methods of dynamical system theory applied to renormalized equations.

##### Self-similar structure of caustics and shock formation

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We use this insight to study shock formation in the dKP equation, as well as shocks in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

##### Well-posedness problems for the magneto-hydrodynamics models

We will talk about some recent results on the well-posedness problems in Sobolev spaces for the magneto-hydrodynamics with and without Hall effect, i.e., the Hall MHD and classical MHD models. One of the purposes of the work is to search the optimal Sobolev space of well-posedness for the two models. Another purpose is to understand the nonlinear Hall term $\nabla\times((\nabla\times b)\times b)$ in the Hall MHD, which appears more singular than $u\cdot\nabla u$ in the NSE, but with special geometry.