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.
We shall discuss geometric approaches and some new tools designed for the study of regularity issues in the theory of elliptic and parabolic PDEs.
We provide a new proof of the Holder continuity of bounded solutions to Hamilton-Jacobi equations with rough coefficients, first showed by Cardaliaguet using probability methods, and by Cardaliaguet and Silvestre using explicit super and sub solutions. Our proof uses the De Giorgi method first applied to regularity for elliptic equations with rough coefficients. The method allows to obtain a sort of Harnack inequality. This is a preliminary work. Further applications, especially to homogenization, remain to be investigated. This is a joint work with Chi Hin Chan.
In this talk, we will start with the classical Li-Yau type differential Harnack estimate, and then discuss some recent work on various nonlinear PDEs, and their applications.
In this talk I will describe a full asymptotic expansion for tempered solutions of the wave equation on Lorentzian spacetimes endowed with an end structure modeled on the radial compactification of Minkowski space. Solutions of the wave equation on such spaces have two main asymptotic regimes: along the light cones and interior to the light cone. The rates of decay seen in the expansion (a classical object) can be expressed in terms of a purely quantum object, namely, the resonances of a related elliptic problem on an asymptotically hyperbolic manifold. In particular, even on Minkowski space, these methods give a new understanding of the Klainerman—Sobolev estimates. This is joint work with Andras Vasy and Jared Wunsch.
We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by a linear evolution and in general enstrophy is lost in the weak limit. Joint work with Nader Masmoudi.
I will present a recent result on the Dirichlet boundary value problem for parabolic equations in time-varying domains. The equations are in either divergence or non-divergence form with boundary blowup low-order coefficients. The domains satisfy a very weak exterior measure condition. The proof is based on the growth lemma.
Lecture 1: Partial Differential Equations and weak solutions
Lecture 2: On some interesting classes of solutions of incompressible Euler's equation and their models.
Lecture 3: On long-time behavior of solutions of Hamiltonian PDEs
We will begin by presenting some new instability and ill-posedness results for the incompressible Euler equations. These ill-posedness results are based upon some special exact solutions to the 3D Euler equations which bring out both the non-locality and the non-linear nature of the vortex stretching term. We will then discuss a general ill-posedness result for a large class of equations arising in hydrodynamics. This is a joint work with Nader Masmoudi.