Gui-Qiang Chen

Title: Singular Equations, Entropy, and Conservation Laws

Abstract: In this talk we will discuss some relations between singular equations and entropy for nonlinear conservation laws and related equations, and apply these relations to solving some important nonlinear partial differential equations with degeneracy.

M. G. Grillakis

Title: On the global existence and regularity of Nonlinear Schrodinger Equations

Abstract: I will discus certain apriori Energy estimates and how they can be used to show global existence for a radially symmetric critical Schrodinger equation. This is a different proof of a recent result by J. Bourgain.

Sze-Bi Hsu

Title: CHAOTIC VIBRATIONS OF THE ONE-DIMENSIONAL WAVE EQUATION DUE TO SELF-EXCITATION BOUNDARY CONDITION

Abstract: In this talk we present the joint work with Goong Chen and J. Zhou on the chaotic vibrating string. Consider the one-dimensional linear wave equation $w_{tt}- w_{xx} = 0$ on the unit interval $x \in (0, 1)$ with the boundary conditions: at the left end $x=0$, the string is fixed while at the right end $x=1$, a nonlinear boundary condition $w_{x} = \alpha w_{t} - \beta w^{3}_{t}, \ \ \alpha , \beta > 0 $ takes effect. The nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system and use the method of characteristic to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of $u_{n+1} = F(u_{n})$ where $F$ is the nonlinear reflection relation. We say the $PDE$ system is chaotic if the mapping $F$ is chaotic as an interval map. We shall study behavior of iterates of the map $u_{n+1} = F (u_{n})$ in two cases namely, controlled hysteresis and natural hysteresis and also each with energy injection at the left end $x=0$.

Bob Jerrard

Title: Functions of boounded higher variation

Abstract: We define a class of functions that is a natural generalization of the space BV, and we establish certain properties of these functions. In particular we prove a generalization of the classical theorem of De Giorgi on the rectifiability of sets of finite perimeter.

Congming Li

Title: Prescribed Gaussian or Scalar Curvatures on n-spheres

Abstract: Given a function on the standard n-sphere, can it be realized as the scalar curvature of some pointwise conformal metric? This is equivalent to the solvability of a nonlinear elliptic equation. It involves many difficulties arised in the study of nonlinear PDEs. Here, under the well-known `flatness conditions', we give a necessary and sufficient conditions for the solvability of the problem in the class of rotatinally symmetric functions.

Tong Li

Title: Mathematical Modelling of the Traffic Flows

Abstract: We establish mathematical theory of various micro-macro traffic flow models. In particular, for a 'higher-order' continuum model of traffic flows, we establish the global existence of solutions and their convergence to a solution of the equilibrium equation as the relaxation time goes to zero. The main innovation of the model is that it addresses the anisotropic feature of traffic flows. As a direct consequence, the hyperbolic system with relaxation is degenerate in the sense that the subcharacteristic conditions are not satisfied in the strict sense. We also study the instabilities in the traffic flow. The instabilities in the traffic flow are the direct results of 'car following' process which is based on a cycle of stimulus and delayed response. The process resembles a feedback control process in which oscillations may occur. It turns out that the model describing the instabilities is a hyperbolic system with relaxation which violates the subcharacteristic conditions or satisfies the so-called supercharacteristic conditions. In fact, the weakly nonlinear asymptotic limit of such a system is the Burger's equation with a source term enforces growth. The balance of the growth with the nonlinear decay of the convection yields asymptotically discontinuous periodic waves similar to roll waves the shallow water waves.

Chang-Shou Lin

Title: The spherical Harnack inequality for singular solutions of scalar curvature equation

Abstract: In this talk, we prove the spherical Harnack inequality $$\max_{|x|=r}u\leq c \ \min_{|x|=r}u ,$$ for a singular solution of \begin{equation}{\label{eq:1}} \Delta u+K(x)u^{\frac{n+2}{n-2}}=0\ \ \ \mbox{for}\ 0<|x|\leq 2 , \end{equation} where $K$ satisfies $$c_1 \ |x|^{\alpha -1}\leq |\bigtriangledown K(x)|\leq c_2 \ |x|^{\alpha -1} $$ for some $\alpha$.\\ We also discuss the theorem of removable singularity for equation (\ref{eq:1}) and the asymptotical symmetry of solutions, and some applications to entire solutions of the scalar curvature equation.

Fanghua Lin

Title: Sobolev mappings,density problems and defect measures

Abstract:

Tai-Ping Liu

Title: Wave Tracing Technique for Conservation Laws

Abstract: To study the interacting of dominating shock, expansion, or other waves with neighboring flows, it is often necessary to accurately trace these waves. In order to do this, a new approach of pointwise estimates is developed in recent years. We will illustrate the basic ideas, such as the construction of Green functions, conservation formulation, time-asymptotic wave patterns and nonlinear coupling for various examples including the numerical shocks, combustions and the resolution of discontinuity for viscous conservation laws.

Frank Pacard

Title: Construction of singular solutions for some class of semilinear elliptic equations

Abstract: The singular Yamabe problem can be stated as follows : Let $\Sigma \subset S^n$ be a closed subset of $S^n$, $n\geq 3$. Does there exist a complete metric $g$ defined in $S^n \setminus \Sigma$ such that $g$ is conformal to the standard metric in $S^n$, $S^n \setminus \Sigma$ equipped with $g$ is complete and has constant scalar curvature $R >0$ ? We report on some recent progress about this problem in the case where $\Sigma$ is constituted by finitely many distinct closed submanifolds of $S^n$ and also when $\Sigma$ is constituted by finitely many points.

D. Phillips

Title: Phase Transitions in Superconductors and Liquid Crystals

Abstract:

Friedmar Schulz

Title: Unique Continuation

Abstract:

Eitan TADMOR

Title: Approximate solutions to the incompressible Euler equations with no concentrations

Abstract: We present a sharp local condition for the lack of concentration (and hence -- the $L^2$ convergence of) a sequence of approximate solution to the incompressible Euler equations. We apply this characterization to greatly simplify known existence results for 2D flows in the full plane (--- with special emphasize on rearrangement invariant regularity spaces), and obtain new existence results of solutions without energy concentrations in any number of spatial dimensions. Our results identify the 'critical' regularity which prevent concentration, regularity which is quantified in terms of Lebesgue, Lorentz, Orlicz and Morrey spaces. In particular, the strong convergence criterion cast in terms of circulation logarithmic decay rates due to DiPerna \& Majda is simplified (--- removing the weak control of the vorticity at infinity) and extended (--- to any number of space dimensions). Our approach relies on using a generalized Div-Curl Lemma to replace the role that elliptic regularity theory has played previously in this problem.

Terry Tao

Title: Global well-posedness and blowup for the wave map equation in $R^{1+1}$

Abstract: A wave map is a function $\phi$ from Minkowski space $\R^{n+1}$ to a target Riemannian manifold $M$ which minimizes the functional $\int |\nabla \phi|^2 - |\phi_t|^2$. Wave maps evolve according to a non-linear wave equation. It is known in all dimensions that the wave map equation is locally well-posed in the subcritical Sobolev spaces $H^s$, $s > n/2$, but the critical case of $\dot H^{n/2}$ is still open. This conjecture is especially interesting in the case $n=2$ since the critical norm is the energy norm. In particular, in the case $n=1$ we have global well-posedness for $H^s$, $s \geq 1$ by energy conservation. In work of Mark Keel and the author, we extend the global well-posedness results to $s > 3/4$ by a variant of a technique of Bourgain, combined with some conservation laws of Pohlmeyer in the one-dimensional case. We also show that one has ill-posedness and blowup in the $\dot H^{1/2}$ norm in this case.

Hwai-chiuan Wang

Title: Semilinear Elliptic Equations in Unbounded Domains

Abstract: Let $\Omega \subset \Bbb{R}^{N}$ be the upper half strip with a hole$.$ In this paper we show: $(1)$ there exists a positive higher energy solution of semilinear elliptic equations in $\Omega $; $(2)$ we describe the dynamic systems of solutions of equation $(1)$ in various $\Omega ;$ $(3)$Let $% \Omega _{0}=\Omega _{1}\cup \Omega _{2},$ where $\Omega _{1}\cap \Omega _{2}$ is bounded, and $\alpha _{i}=\alpha (\Omega _{i})$ the index of $J$ in $% \Omega _{i}$ for $i=0,1,2.$ $J$ satisfies the $(PS)_{\alpha _{0}}$- condition if and only if the inequality $\alpha _{0}<\min \{\alpha _{1},\alpha _{2}\}$ holds; $(4)$there exists $s_{0}>0$ such that $\alpha (D_{s}^{r})$ admits a minimizer if $s>s_{0}$, but $\alpha (D_{s}^{r})$ does not admit any minimizer if $s < s_{0},$ where $D_{s}^{r}$ is an interior flask domain.

Title: Fourier restriction estimates for cones

Abstract: I will discuss recent progress on the so-called cone restriction problem. The main result is as follows: Let $f$ and $g$ be $L^2$ densities on the light cone in $d+1$ dimensions, with compact support not including the vertex and with disjoint conical support. Then $\widehat{fd\sigma}\widehat{gd\sigma}\in L^p$ for $p>1+\frac{2}{d+1}$. This sort of improvement on the Strichartz inequality (which implies the above estimate with $p=1+\frac{2}{d-1}$) has been previously considered by several authors - Bourgain, Klainerman-Macedon and Tao-Vargas. The point of our result is that the range of exponents is almost sharp (the \lq\lq almost" is because we do not consider the endpoint). We will also discuss corollaries including an $L^p\rightarrow L^p$ restriction theorem for the cone in $3+1$ dimensions and some related results concerning the X-ray transform.

P. Yang

Title: The Chern-Gauss-Bonnet integral and the Paneitz equation

Abstract: I will talk about the fourth order equation defining the Q-curvature invariant. In a joint work with Chang and Qing we extend the Cohn-Vossen inequality for the Gauss-Bonnet integral of complete surfaces to the setting of 4-dimensional conformally flat manifold. In particular, for domains in R^4, we prove a finiteness result for the topology of the domain in terms of a finiteness assumption on the Q-curvature integral. This result also yields a compactification criteria for conformally flat 4-manifolds.

Yuxi Zheng

Title: A Nonlinear Variational Wave Equation

Abstract: We study a nonlinear wave equation derived through the variational principle on a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. It has been known for the equation that smooth initial data may develop singularities in finite time, a sequence of weak solutions may develop concentrations, while oscillations may persist. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For monotone wave speed functions in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data mollification and step (a) for not-as-smooth initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is Young measure theory and related techniques.