## Past Seminars- Nonlinear PDEs

• Yi Zhang
Tue Jan 16, 2018
3:00 pm
Given a planar infinity harmonic function u, for each $\alpha>0$ we show a quantitative $W^{1,\,2}_{\loc}$-estimate of $|Du|^{\alpha}$, which is sharp when $\alpha\to 0$.  As a consequence we obtain an $L^p$-Liouville property for infinity harmonic functions in the whole plane
• Gautam Iyer
Thu Jan 11, 2018
4:00 pm
Consider a diffusive passive scalar advected by a two dimensional incompressible flow. If the flow is cellular (i.e.\ has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. We show that on intermediate time scales, the effective behaviour...
• Albert Fathi
Tue Dec 5, 2017
3:00 pm
This is a joint work with Piermarco Cannarsa and Wei Cheng.  We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.  The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property...
• Dejan Slepcev
Tue Oct 17, 2017
3:00 pm
The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov,  asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and...
• Zaher Hani
Tue Oct 10, 2017
3:00 pm
We consider the nonlinear Schroedinger equation posed on a large box of characteristic size $L$, and ask about its effective dynamics for very long time scales. After pointing out some “more or less” trivial time scales along which the effective dynamics can be easily described, we start inspecting some much longer time scales where we...
• Hiroyoshi Mitake
Thu Sep 21, 2017
3:00 pm
In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness...
• Vera Mikyoung Hur
Tue Jun 6, 2017
3:00 pm
In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some...