**The seminar meets regularly on Wednesdays** at **4pm** in **Eckhart 202**. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Tianling Jin.

Oct 7

Quasiperiodic homogenization.

Oct 16

Stochastic homogenization of a conductivity problem in a domain with heterogeneous and random interfaces.

We consider a linear stationary conductivity problem in a domain which contains a large number of conductivity resistant interfaces of small length scale. On the microscopic level, this conductivity problem is equipped with jump conditions across the interfaces; more precisely, the normal flux is continuous across the interfaces but the potential field undergoes a jump. We present a rigorous derivation of an effective (homogenized) macroscopic conductivity model that captures the large-scale behavior of the aforementioned problem. We adopt the stationary ergodic setting of Blanc, Le Bris and Lions; that is, the random interfaces are obtained as the image of a periodic interface structure under a random diffeomorphism. Such a homogenization theory finds applications in certain spectroscopic imaging techniques. This is a joint work with H. Ammari, J. Garnier and L. Giovangigli.

On integro-differential equations involving Levy measures with small (and directionally dependent) support

Elliptic integro-differential operators have been well studied in probability for many years due to their importance as the generators of a general class of Markov processes. Recently they have also drawn significant attention from the PDE community, especially in studying nonlinear problems involving integro-differential operators. Important and interesting features of the solutions arise when the Levy measures corresponding to the operators are allowed to have only a small support set. We will discuss why these types of Levy measures are natural, how solutions in this setting deviate from their second order counter parts, and mention some recent results on Holder regularity.

Lecture 1: The Navier-Stokes Equations

Monday October 28, 4 - 5 PM. Ryerson 251.

I will present a selection of results and open problems concerning the incompressible Navier-Stokes and related equations.

Lecture 2: Regularity, absence of anomalous dissipation and long time behavior in 2D hydrodynamic models

Tuesday, October 29, 4:30 - 5:30 PM. Eckhart 202.

The models discussed include SQG (the surface quasi-geostrophic equation) and variants, and the 2D damped and driven Euler equations.

Lecture 3: Fractionally dissipated, stochastically forced 2D Euler equations

Wednesday, October 30, 4 - 5 PM. Eckhart 202.

I will describe a proof of unique ergodicity for this system.

Scale-invariant solutions of Navier Stokes equation and implications to Leray-Hopf solutions

In this talk, I will first discuss the existence of scale invariant solutions to Navier Stokes equation, with arbitrary \(-1\) homogeneous initial data. As these solutions are not small, linearized analysis seem to suggest non-trivial bifurcations, which imply non-uniqueness. Under a plausible spectral condition, we show rigorously that these bifurcations do occur; moreover by appropriately localizing such solutions, we show non-uniqueness of Leray-Hopf solutions, for initial data which are compact, and smooth away from origin, with a singularity at origin of the order \(O(\frac{1}{|x|})\), which will be sharp. This is joint work with V.Sverak

Stochastic differential equations with Sobolev regular coefficients

We briefly review some recent and less recent results on existence and uniqueness of solutions to parabolic equations with Sobolev regular coefficients. We next explore in details the consequences of these results on the theory of stochastic differential equations with non regular coefficients and related issues. The talk presents works with and by PL Lions.

Correctors for boundary layers

In this talk we will focus on a corrector problem arising in the homogenization of elliptic systems with oscillating coefficients and boundary data, so-called boundary layers. The corrector is a solution of an elliptic system in a half-space. While both the coefficients and the boundary data are periodic functions, the boundary forces a non trivial, non periodic behavior of the solution. We will discuss recent results on the existence and asymptotic behavior of the boundary layer corrector.

For questions, contact Tianling Jin at: tj