Apr
24
3:00pm 3:00pm

The One Dimensional Problem for a Fractional Elliptic System

By Kelei Wang, Wuhan University

In this talk I will discuss an elliptic system involving fractional Laplacians. This system is used to model the "phase separation" phenomena with anomalous diffusion. We are mainly concerned with the uniqueness of one dimensional profiles. It turns out that some trivial facts in the classical Laplacian case can become rather serious in this fractional setting. To attack this problem we need various tools: Yau's gradient estimate and hyperbolic geometry, Almgren's monotonicity formula and blowing down analysis, the method of moving plane and so on.

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Apr
24
1:30pm 1:30pm

Classifications of Solutions to Some Nonlocal Operators

By Mouhammed Moustapha Fall, African Institute for Mathematical Sciences (AIMS) in Senegal

In the first part of the talk, we will discuss classifications of distributional solutions to general elliptic equations driven by nonlocal operators generated by a stable Lévy process (not necessarily symmetric). In the second part of the talk, we will show how to transform a nonlocal equation to a local one and thus. The argument is based on a kind of 'factorization' property of local operators by many nonlocal operators. This allow us to obtain further classifications of some distributional solutions with possibly variable coefficients. We will present several applications. Joint work with Tobias Weth.

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Apr
24
11:00am11:00am

Holder Continuity Smoothing for Hypoelliptic Kinetic Equations

By Alexis Vasseur, The University of Texas at Austin

We consider generalized Fokker-Planck equations with rough elliptic coefficients. Typically, the parabolic regularization takes effect in the velocity space only. We show that the Liouville transport operator extends the regularity to the space variable as well. Precisely, we show that any solution with integrable and bounded initial data, becomes Holder continuous for positive times. The proof is based on the De Giorgi method for elliptic operators. The method is extended to the hypoelliptic case, using the averaging lemmas. This is a joint work with Francois Golse.

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Apr
24
9:30am 9:30am

A PDE System Modeling Biological Network Formation

By Peter Markowich, King Abdullah University of Science and Technology

We present a PDE model for biological (fluid) network formation. The model consists of a Poisson equation for the fluid pressure (based on Darcy’s law) coupled to a reaction diffusion equation for the network conductance vector. We discuss existence&uniqueness issues, bifurcations of stationary solutions and diffusion-caused network formation by a Turing-type instability.

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