Nonlinear Partial Differential Equations

Date: March 3, 2015

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Vu Hoang, Rice University

Title: Nonlinear PDEs seminar

Abstract:

Title: Singularities for a nonlocal version of Burgers' equation

Abstract:
I consider a non-local, nonlinear transport equation in one space dimension (without dissipation). Here, the velocity field is given by a integral operator with weakly singular kernel. The goal is to prove the existence of blowup in finite time, starting from, say, continuously differentiable initial data. Previous, known blowup proofs rely on studying a Lyapunov functional and on the use of clever identities and/or clever inequalities. A drawback of this approach is that it gives little information on the nature of the singularity. In this talk, I present ongoing work whose chief goal is to gain a deeper understanding of the singularity formation for these equations, and maybe providing a more systematic approach. We construct blowup solutions such that the gradient of blows up at one point only, the solution otherwise retaining its initial regularity. (joint work with A. Kiselev, M. Radosz and X. Xu)