Nonlinear Analysis and PDEs
April 16 - April 17, UT Austin

All people are invited to attend.
However, for organizational reasons, please fill out the registration form at the link below.

Participant Inscription

The conference will take place in two different buildings on Friday and on Saturday:
Friday 16th will be in the ACES building, room 6.304.
Saturday 17th will be in the RLM building, room 6.104.


Here is a picture of the speakers together with the organizer:

Speakers and Abstracts

Luis Caffarelli: Non-local porous medium type equations

We will discuss several properties of the porous media equation, when the pressure is a (integral) potential of the density, instead of a power of the pressure: regularity, compact expanding support of the density, and related properties.

Nicola Gigli: On the Heat flow on metric measure spaces

I will discuss the well posedness of the definition of Heat flow in metric measure spaces with Ricci curvature bounded from below, as gradient flow of the Entropy w.r.t. the quadratic Wasserstein distance. In particular, uniqueness will be addressed. I will also present some open problems in this setting.

Fanghua Lin: Flows of liquid crystals and incompressible viscoelastic fluids

The study of flows of liquid crystals plays an important role in the understanding of many other incompressible viscoelastic fluids. The equations describing these Non-Newtonian fluids are deduced from the similar reasonings in physics and, they have shared much of the mathematical properties as well. In this lecture I shall explain some recent work concerning model equations, and to describe some connections.

Ludovic Rifford: Generic regularity of weak KAM solutions and Mañé conjecture

We will discuss the regularity properties of viscosity solutions of Hamilton-Jacobi equations with or without Dirichlet conditions and explain why it is related to the so-called Mañé Conjecture in Aubry-Mather theory.

Luis Silvestre: On the well posedness of the integro-differential Bellman equation.

We prove an interior regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. It is the fractional order version of the theorem of Evans and Krylov about the regularity of solutions to concave uniformly elliptic partial differential equations.

Panagiotis Souganidis: Homogenization and enhancement for the G-equation in periodic media

I will present homogenization results about the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, it is shown that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic Lfirst-order (spatio-temporal homogeneous) level set equation. Moreover there is a rate of convergence and, under certain conditions, the averaging enhances the velocity of the underlying front. At scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulf shape associated with the effective Hamiltonian. This is joint work with J. Nolen and P. Cardaliaguet.

Alexis Vasseur: Relative entropy method applied to the stability of shocks for systems of conservation laws

We develop a theory based on relative entropy to show stability and uniqueness of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of big amplitude), among bounded entropic weak solutions having an additional strong trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, the strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.