**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.

Reactive Processes in Inhomogeneous Media

We study fine details of spreading of reactive processes (e.g., combustion) in multi-dimensional inhomogeneous media. One typically observes a transition from one equilibrium (e.g., unburned fuel) to another (e.g., burned fuel) to happen on short spatial as well as temporal scales. We demonstrate that this phenomenon also occurs in one of the simplest models for reactive processes, reaction-diffusion equations with ignition reaction functions (as well as with some monostable, bistable, and mixed reaction functions, in a slightly weaker form), under very general hypotheses. Specifically, we show that in up to three spatial dimensions, the width (both in space and time) of the zone where reaction occurs stays uniformly bounded in time for some fairly general classes of initial data, and this bound even becomes independent of the initial datum as well as the reaction function, after an initial time interval. Such results have recently been obtained in one spatial dimension but were unknown in higher dimensions. As one indication of the added difficulties, we also show that three dimensions is indeed the borderline case, and the result is false for general inhomogeneous media in four and more dimensions.

The boundary Harnack principle for second order elliptic equations in John and uniform domains.

Consider second order uniformly elliptic equations \(Lu=a_{ij}D_{ij}u+b_iD_iu=0\) in a bounded domain in \(R^n\). We discuss direct proofs of the boundary Harnack principle for the ratios of positive solutions in John domains, and the Holder regularity near the boundary in uniform domains. This is joint work with Hyejin Kim.

Defects and local profiles: Hamiltonâ€“Jacobi equations

Defects and local profiles: Quasilinear equations

Defects and local profiles: Fully nonlinear equations

Inviscid limits for the stochastic Navier-Stokes equation

We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow. We prove that the limiting inviscid invariant measures are supported on bounded vorticity solutions of the 2D Euler equations. This is joint work with N. Glatt-Holtz and V. Sverak.

Quantitative stochastic homogenization of viscous Hamilton-Jacobi equations

In this joint work with S. Armstrong, we prove estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence.

Finite time extinction for stochastic sign fast diffusion and self-organized criticality

We will first shortly review the informal derivation of a continuum limit for the Bak-Tang-Wiesenfeld model of self-organized criticality. This will lead to the stochastic sign fast diffusion equation. A key property of models exhibiting self-organized criticality is the relaxation of supercritical states into critical ones in finite time. However, it has remained an open question for several years whether the continuum limit - the stochastic sign fast diffusion - satisfies this relaxation in finite time. We will present a proof of this

De Giorgi iteration for stochastic partial differential equations

We apply De Giorgi iteration method to study linear stochastic partial differential equations \[\partial_t u = div (A\nabla u) + S \dot{W}_t\] with measurable coefficients. We are interested in the moment estimates of the solution \(u=u(t, x;\omega)\) to such an equation. Under mild structure conditions, we will give an estimate on the tail probability of \(u\), which, in particular, implies that all the moments of \(\|u(t)\|_{\infty}\) are finite. The main innovation of our work lies in the method of handling the contribution of the white-noise term \(S\dot{W}_t\). Rather than treating the white-noise term as one object, we analyze its contribution in each scale. We show that \(S\dot{W}_t\) has a small probability to contribute in a large scale. Our method provides more precise control on the contribution of the white-noise term and relies less on the linear structure of the equation. Joint work with Elton P. Hsu and Zhenan Wang.

Long time behavior of the Navier-Stokes and related equations

I will review resent results on the long time behavior of equations arising in fluid dynamics, such as the 3D Navier-Stokes and critical surface quasi-geostrophic equations.

For questions, contact Tianling Jin at: tj