Fall 2006 Midwest PDE Seminar

 September 22-24, 2006

Friday, September 22, Chaired by Tong Li

2:00-Coffee and pick up conference materials in Room 114 MacLean Hall.

3:00- Dongsheng Li (Xian Jiaotong): Elliptic equations on convex domains with nonhomogeneous Dirichlet boundary conditions

4:00- Hailiang Liu (Iowa State): Critical Thresholds in Nonlinear Balance Laws

Saturday, September  23, Chaired by Juan Gatica

8:00 -Coffee and pick up conference materials in Muhly Lounge, 3 MacLean Hall.

8:30 - Qingbo Huang (Wright State): Alexandrov type inequalities and the regularity of  weak solutions of the reflector problem
9:30- Xiaodong  Yan: Singular set for critical points of polyconvex functionals from nonlinear elasticity

10:30-Coffee Break  Muhly Lounge
11:00- Gui-Qiang Chen (Northwestern): On Nonlinear Partial Differential Equations of Mixed Type.

12:00  Lunch at Muhly Lounge

Afternoon Session: Chaired by Gerhard Strohmer
2:00 - M. Feldman (Wisconsin): Existence and regularity of solutions to shock reflection problem.

3:00- Coffee break in Muhly Lounge, 3 Maclean Hall. .
3:30 - Ronghua Pan (Georgia Tech): Large time behavior and BV estimates on p-system with damping.

4:30- Fengbo Hang (Princeton): An integral equation in conformal geometry

7:30- Dessert Party at Tong and Lihe's house.

Sunday, September 24 , Chaired by Yi Li

8:30-- Coffee in Muhly Lounge, 3 Maclean Hall.

9:00 -- Markus Keel (Minnesota): Energy Transfer in a Nonlinear Hamiltonian PDE.

10:00- Wen Shen (Penn State): Optimal Tracing of Viscous Shocks in Solutions of Conservation Laws.

11:00- Gary Lieberman (Iowa State): On elliptic equations with strongly singular lower order terms.

Abstracts

Elliptic equations on convex domains with nonhomogeneous Dirichlet boundary conditions.

Dongsheng Li,  Xi'an Jiao Tong University

Abstract: In this paper, we will study the differentiability on the boundary of solutions of elliptic non-divergence differential equations on convex domains. The results are divided into two cases: (i) at the boundary points where the blow-up of the domain is not the half space, if the boundary function is differentiable then the solution is differentiable;(ii) at the boundary points where the blow-up of the domain is the half space, the differentiability of the solution needs an extra Dini condition for the boundary function. Counterexample is given to show that our results are optimal. This is a joint work with Lihe Wang.

Critical Thresholds in Nonlinear Balance Laws

Hailiang Liu, Iowa State University


We study the questions of global regularity vs. finite time breakdown in nonlinear balance laws, with momentum equation $u_t+u\cdot\nabla_x u=\nabla_x F$ governing the velocity field, which shows up in different contexts dictated by different modeling of $F$'s. To address these questions, we propose the notion Critical Threshold (CT), where a conditional finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold. With the standard energy method one studies the growth of the velocity gradient Du. Our approach is based on spectral dynamics, tracing the eigenvalues, eig(Du), which determine the boundaries of CT surfaces in configuration space.

We demonstrate the CT phenomena with several prototype models. We begin with 1D models, including Euler-Poisson equations with different forces, and a relaxation system arising in traffic flows. We then move on to n-dimensional models, including restricted Euler equations, restricted Euler-Poisson equations and a rotational Euler system. We show how the CT phenomenon is associated with different mechanisms. Our study of these models reveals the critical dependence of the CT phenomenon on initial spectral gaps.

Alexandrov type inequalities and the regularity of weak solutions of the reflector problem

Qingbo Huang, Wright State University

We will discuss geometric inequalities of Alexandrov type for
the reflector antenna problem and their application in
establishing results on strict antennas and C1 regularity
of weak solutions. This is a joint work with Luis Caffarelli
and Cristian Gutierrez.
 

Singular set for critical points of polyconvex functionals from nonlinear elasticity

Xiaodong Yan, Michigan State University

Partial regularity is proved for Lipschitzian critical points of
polyconvex functionals motivated by nonlinear elasticity provided
the Lipschitz norm is small enough. In particular, the singular set for a
Lipschitzian critical point has Haudorff dimension strictly less than n
when Lipschitz norm of u is small enough. Moreover, it is shown that the singular set of a Lipschitzian global minimizer has Hausdorff dimension strictly less than n.

On Nonlinear Partial Differential Equations of Mixed Type

 Gui-Qiang Chen, Northwestern University

Abstract: In this talk we will discuss some aspects of recent
developments in the study of nonlinear partial differential
equations of mixed type, including nonlinear partial
differential equations of mixed parabolic-hyperbolic type and
mixed elliptic-hyperbolic type, as well as more complicated
nonlinear systems of mixed-composite type.
Examples include nonlinear degenerate diffusion-convection
equations, the transonic flow equation, and steady or self-similar
Euler equations for compressible fluids.
Further trends and open problems in this direction will also be
addressed. This talk will be mainly based on joint works with
Jun Chen, Mikhail Feldman, and Benoit Perthame respectively.

Existence and regularity of solutions to shock reflection problem

Mikhail Feldman, University of Wisconsin-Madison

We show existence of global solutions to shock reflection by large-angle
wedges for potential flow (joint work with G.-Q. Chen). We reduce the shock
reflection problem to a free boundary problem for a nonlinear elliptic
equation, with ellipticity degenerate near a part of the boundary (the sonic
line), and solve this problem by an iteration procedure. We also study optimal
regularity of solutions near the sonic line.


Large time behavior and BV estimates on p-system with damping

Ronghua Pan, Georgia Tech

We establish the uniform BV estimates on p-system
with damping for generic BV data. Previous results are valid
for isothermal ideal gas or perturbation near constant states.
Essential difficulty occurs when one considers the BV data with
different endstates due to diffusion. Recently, we are able to
carry out the uniform BV estimate through fractional step Glimm
scheme with the help of entropy analysis.
The large time behavior of weak solutions with sharp decay rates
was obtained as a by-product. This is a joint work with C. M. Dafermos.


An integral equation in conformal geometry

Fengbo Hang, Princeton University

Motivated by the study of isoperimetric inequalities, we introduce a conformal invariant integral equation on manifolds with boundary and discuss regularity issues, Liouville type theorems and a possible way toward existence results (joint work with X. Wang and X. Yan).

Energy Transfer in a Nonlinear Hamiltonian PDE

Markus Keel, University of Minnesota

We'll discuss recent work with J. Colliander, G. Staffilani, H. Takaoka, and T. Tao. The particular
result is motivated by the much harder goal of understanding how energy can be exchanged between the
different modes of an infinite dimensional Hamiltonian system.

The energy of our non-integrable PDE is conserved, but one can ask
whether it's possible for smoother norms to grow in time.  Such
growth provides at least some quantitative measure of how energy
might cascade from lower modes to arbitrarily high modes of
the solution.  Our result gives an example of such behavior in  an autonomous nonlinear Schr\"odinger equation (the defocusing, cubic equation on the torus).



Optimal Tracing of Viscous Shocks in Solutions of Conservation Laws

Wen Shen, Penn State University

We study qualitatively a scalar conservation law with viscosity:
$$ u_t+f(u)_x=u_{xx}\,.$$
We consider the problem of identifying the location of viscous
shocks, thus obtaining an optimal finite dimensional description
of solutions to the viscous conservation law.
We introduce a nonlinear functional whose minimizers yield the
viscous traveling profiles which ``optimally fit'' the given solution.
We prove that, outside an initial time interval and away from times
of shock interactions, our functional remains very small, i.e. the
solution can be accurately represented by a finite number of
viscous traveling waves.




On elliptic equations with strongly singular lower order terms

Gary Lieberman, Iowa State University

The properties of solutions of elliptic equations with bounded lower order terms are well-known, but when these terms become unbounded near the boundary of the domain, results are not so familiar. In this talk, we examine two basic concerns:
(1) The existence of viscosity solutions when the lower order terms blow up like the reciprocal of distance to the boundary.
(2) Global smoothness of solutions when the lower terms blow up but satisfy appropriate sign conditions.
Issue (2) was studied in the late 70's and early 80's but some strong, unnecessary additional hypotheses were added.


 

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