Past Partial Differential Equations Seminar

30 April 2018
16:00
Connor Mooney
Abstract

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

  • Partial Differential Equations Seminar
23 April 2018
16:00
Myoungjean Bae
Abstract

I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component. This talk is based on a joint work with S. Weng (Wuhan University, China).
 

  • Partial Differential Equations Seminar
4 April 2018
15:00
Abstract

The uniformly degenerate elliptic equation is a special class of degenerate elliptic equations. It appears frequently in many important geometric problems. For example, the Beltrami-Laplace operator on conformally compact manifolds is uniformly degenerate elliptic, and the minimal surface equation in the hyperbolic space is also uniformly degenerate elliptic. In this talk, we discuss the global regularity for this class of equations in the classical Holder spaces. We also discuss some applications.

  • Partial Differential Equations Seminar
5 March 2018
16:00
Alberto Bressan
Abstract

A well known result by Schaeffer (1973) shows that generic solutions to a scalar conservation law are piecewise smooth, containing a finite family of shock curves.

In this direction, it is of interest to find other classes of nonlinear hyperbolic equations where nearly all solutions (in a Baire category sense) are piecewise smooth, and classify their singularities.

The talk will mainly focus on conservative solutions to the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$. For an open dense set of $C^3$ initial data, it is proved that the conservative solution is piecewise smooth in the $t - x$ plane, while the gradient $u_x$ can blow up along  finitely  many characteristic curves. The analysis relies on a variable transformation which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.   

A detailed description of the solution profile can be given, in a neighborhood of every singular point and every singular curve.

Some results on structurally stable singularities have been obtained  also for dissipative solutions, of the above wave equation. Recent progress on the Burgers-Hilbert equation, and related open problems, will also be discussed.

These results are in collaboration with Geng Chen, Tao Huang, Fang Yu, and Tianyou Zhang.

  • Partial Differential Equations Seminar

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