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 Will Feldman.
Many kinetic equations have the corresponding fluid limits. In the zero limit of the Knudsen number, one derives the Euler equation out of the Boltzmann equation and the heat equation out of the neutron transport equation. While there are good numerical solvers for both equations, it is not well-understood when the two regimes co-exist. In this talk, we model the layer between the fluid and the kinetic using a half-space equation, study its well-posedness, design a numerical solver, and utilize it to couple the two sets of equations that govern separate domains.
I will describe the vanishing discount problem for fully nonlinear, possibly degenerate, elliptic PDEs and show general convergence results on this problem under mild assumptions. The vanishing discount problem is a prototype of numerous asymptotic problems for fully nonlinear PDEs. In this problem, since the limiting equation (often called ergodic/cell problem) is not monotone and has many (viscosity) solutions in general, proving convergence is a challenging task. I will also explain a new representation formula for solutions along the way. This is a joint work with Hitoshi Ishii and Hiroyoshi Mitake.
We prove strong gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flattens and quantifiably circularizes, losing its initial structure. (joint work with Jean-Michel Roquejoffre)
We will discuss the homogenization of periodic oscillating Dirichlet boundary data problems in general domains for second order elliptic equations. These problems are connected with the study of boundary layers in fluid mechanics and with the study of higher order asymptotic expansions in classical homogenization theory. The talk will be aimed at a general audience. Due to a singular behavior of the averaging near boundary points with rational normal direction the regularity of the limiting problem has so far not been well understand. We will explain some progress on this issue which displays a sharp contrast between the case of linear and nonlinear equations. In the linear case we establish continuity estimates, while in the nonlinear case we show that discontinuity is a generic phenomenon. This talk is based on joint work with Inwon Kim.
Equations with small scales abound in physics and applied science. When the coefficients vary on microscopic scales, we expect the local fluctuations to average out under certain assumptions and the coefficients have some equivalent homogeneity on large scales -- homogenization occurs. In this talk, I will try to explain some probabilistic/analytic approaches we use to obtain the first order random fluctuations in stochastic homogenization. It turns out that in high dimensions, a formal two-scale expansion only leads to the correct "local" fluctuation, but not the "global" one.