Meets Tuesdays at 4:30 pm in Skiles 255 unless otherwise indicated

01/11/05

No Talk.01/18/05

Title: **Relativistic Euler Equations in
(3+1)-Dimensional Spacetime**

relativisitc Euler equations in (3+1)-dimensional spacetime. The local well-poseness is established

via a construction of a convex entropy if the initial data is in a sub-luminous region away from

vacuum. However, the classical solutions are proved to blow up in finite time for any

non-trivial finite initial energy or for infinite initial energy with large radial momentum.

**** This is a joint work with Joel Smoller at University of Michigan.*****

01/25/05

Speaker: Prof. Diaraf SECK, Universit Cheikh Anta Diop (FASEG) , Dakar SENEGAL (Host: Evans Harrell)Title: Bernoulli free boundary
problems.

boundary problems(the exterior and interior cases ) for the p-Laplace

operator (1<p<\infty). Using the shape optimization theory, the derivative

with respect to the domain, we prove existence and uniqueness results and

monotony results. And we show the existence of the free boundary problems.

In the interior case, it is known that there is not always an existence

result reult. We show an isoperiperimetric inequality. That is the

optimal estimation for the upper bound of the Bernoulli constant.

02/01/05

Speaker: Xu-Yuan Chen, Georgia Tech

Title: A Uniqueness Theorem for Nonlinear
Reaction Diffusion Equations

$u_t=\Delta u$ has nontrivial classical solutions with zero initial

data. The uniqueness for the heat equation only holds under some growth

conditions on the solutions at space infinity. On the contrary, we will

show that for a class of nonlinear reaction diffusion equations, the

uniqueness of solutions to the Cauchy problem holds without any growth

conditions. Our examples include $u_t=\Delta u+u-u^3$. The existence of

solutions with singular initial data will also be discussed.

02/08/05

Speaker: Vladimir Oliker, Emory University

Monge-Ampere equations

Abstract: Many problems in geometry concerning existence of a closed

hypersurface in Euclidean space with a prescribed curvature function require an

investigation of a second order PDE of Monge-Ampere type. Similarly,

the corresponding PDE's are of Monge-Ampere type in several

classes of problems in optics which require determination of a convex

hypersurface which for a given energy source will redirect and

redistribute that energy in a prespecified manner. In my talk I intend to survey

several such problems and describe geometric ideas (some of which go

back to Minkowski) which allow to solve these equations (in weak sense)

by purely geometric means. If time permits, I will also explain the connection

of such problems to the Monge-Kantorovich theory.

02/15/05

Speaker: Shaoqiang Tang, CalTech

Abstract: In the last a few decades, extensive explorations have been made on

stationary phase transitions, e.g. theory of remormalization group. When dynamics

is concerned, major difficulty comes from instabilities. Before the presence of a

better approach from the perspective of physics, we aim at an attack on this

challenging issue at phenomenological level.

We shall investigate possible stabilizations, to substantiate our

understanding of nonlinear interactions among instability and dissipative

mechanisms. In particular, we shall propose a category of discrete BGK models

for regularization. Suliciu model and Jin-Xin relaxation model are special cases.

For Suliciu model, theoretical we obtain stability results under tri-linear structural

relation. We further demonstrate numerically that low order dissipation mechanisms

is capable to stablize phase transition systems. Moreover, this approach applies

to high space dimensions. With a relaxation model, numerical simulations

produces interesting patterns. This may shed insight into further

investigations on dynamic phase transitions, as well as related physical systems.

02/22/05

Speaker: Guiqiang Chen, Northwestern University

Title: Free Boundary Problems and
Multidimensional Transonic Shocks

Abstract: In this talk, we will first discuss
some natural connections

between multidimensional
transonic shock waves and free boundary

problems for the Euler equations
for compressible fluid flow.

Then we will present some new
approaches developed recently

for solving such free boudary
problems through some concrete

examples and address further
applications in fluid dynamics.

The examples and further
applications especially include

the existence and stability of
multidimensional

transonic shocks in steady
compressible flow in the whole

space $R^n, n\ge 3,$ and past an
infinite de Laval nozzle under

the perturbation of the nozzle
boundary.

The nonlinear stability of
multidimensional shocks in steady

Euler flow past an infinite
curved wedge or cone

under the $BV$ perturbation of
the obstacle and the nonlinear

stability of supersonic vortex
sheets in steady

Euler flow under the $BV$
perturbation of the boundaries will

also be addressed.

03/01/05

Speaker:
Hailiang Liu, Iowa State University

Title: Wave breaking in a class of nonlocal
dispersive wave equations

Abstract: The Korteweg de Vries (KdV)
equation is well known as an approximation model

for small amplitude and long waves in different physical contexts,

but wave breaking phenomena related to short wavelengths are not
captured

in. We introduce a class of nonlocal dispersive wave equations

which incorporate physics of short wavelength scales. The model is

identified by the renormalization of an infinite dispersive differential

operator and the number of associated conservation laws. Several
well-known

models are thus rediscovered. Wave breaking criteria are obtained

for several typical models including the Burgers-Poisson system
and the

Camassa-Holm equation.

03/08/05

Speaker: Wen Shen,
Penn State Univ.

Abstract: In this talk I will present some recent results we

have on differential games. For the n-person

non-cooperative games in one space dimension, we

consider the Nash equilibrium solutions.

When the system of Hamilton--Jacobi equations for

the value functions is strictly hyperbolic, we show

that the weak solution of a corresponding system of

conservation laws determines an n-tuple of feedback

strategies. These yield a Nash equilibrium solution

to the noncooperative differential game.

However, in the multi-dimensional cases, the system of

Hamilton-Jacobi equations is generically ill-posed.

In an effort of obtaining meaningful stable solutions,

we propose an alternative ``semi-cooperative'' pair of

strategies for the two players, seeking a Pareto optimum

instead of a Nash equilibrium. In this case, we prove

that the corresponding Hamiltonian system for the value

functions is always weakly hyperbolic.

This is a joint work with Alberto Bressan.

03/15/05

Speaker: Tong Li, University of Iowa.

Title: Nonlinear Dynamics of Traffic Jams

Abstract: A class of
traffic flow models that capture the nonlinear dynamics of

traffic jams are proposed. The self-organized oscillatory behavior and
chaotic transitions

in traffic systems are identified and formulated. The results can
explain the appearance

of a phantom traffic jams observed in real traffic flow.

There is a qualitative agreement when the analytical results are
compared

with the empirical findings for freeway traffic and with previous

numerical simulations.

03/22/05

Spring break (no talk!)

03/29/05

Speaker: Feimin Huang, Chinese Academy of
Sciences and Courant Institute

Title: Contact Discontinuity for Gas Motions

Abstract: The contact discontinuity is one of
the basic wave patterns in gas

motions. The stability of contact discontinuities with general
perturbations

is a long standing open problem. One of the reasons

is that contact discontinuities are linearly degenerate waves in the

nonlinear settings, like the Navier-Stokes equations and the Boltzmann

equation. The nonlinear diffusion waves generated by the

perturbations in sound-wave families couple and interact with the
contact

discontinuity and then cause analytic difficulties. Another reason is
that

in contrast to the basic nonlinear waves, shock waves and rarefaction
waves,

for which the corresponding characteristic speeds are strictly
monotone,

the characteristic speed is constant across a contact discontinuity,

and the derivative of contact wave decays slower than the one for
rarefaction wave.

Here, we succeed in obtaining the time asymptotic stability of a damped
contact wave pattern

with an convergence rate for the Navier-Stokes equations and the
Boltzmann equation

in a uniform way. One of the key observations is that even though the
energy

estimate involving the lower order may grow

at the rate $(1+t)^{\frac 12}$, it can be compensated by the
decay in the

energy estimate for derivatives which is of the order of $(1+t)^{-\frac

12}$. Thus, these reciprocal order of decay

rates for the time evolution of the perturbation are essential to close
the

priori estimate containing the uniform bounds of the $L^\infty$ norm on
the

lower order estimate and then it gives the decay of the solution to the

contact wave pattern.

04/05/05

Speaker: Konstantina
Trivisa, University of Maryland

Title: On a Multidimensional Model for the
Dynamic Combustion of Compresssible Reacting gases

the dynamic combustion of compressible reacting gases formulated by the

Navier Stokes equations in Euler coordinates. For the chemical model

we consider a one way irreversible chemical reaction governed by the

Arrhenius kinetics. The existence of globally defined weak solutions of the

Navier-Stokes equations for compressible reacting fluids is established by using weak

convergence methods, compactness and interpolation arguments in the spirit

of Feireisl and P.L. Lions. In addition, conditions on the initial data will be introduced yielding

blow up of smooth solutions to the Navier-Stokes and Euler equations

for compressible reacting gases.

04/12/05

Speaker: Guozhen Lu, Wayne State University (Host Andrzej Swiech)Title:Subelliptic convexity and fully nonlinear PDEs on the Heisenberg group

Abstract: In this talk, we review some results in recent years on convexity in the

subelliptic setting, and properties of convex functions, and fully nonlinear

subelliptic equations on the Heisenberg group or more general settings.

04/19/05

Speaker: Tao Luo, Georgetown University

Abstract: In this talk, I will first review some results on the transport

equations with non-smooth coefficients of Diperna-Lions,

Colombini-Lerner, Ambriosio, Depaul and Columbini-Luo-Rauch. Then

I will sketch a proof of the Blow up of BV-norms when the

coefficients are not Lipshitzean. This is a joint work with F.

Columbini and J. Rauch.

04/26/05

Speaker: Qingbo Huang, Wright State University, (Host Andrzej Swiech)Title: On the Alexandrov type inequalities for reflector problem

Abstract: The Alexandrov inequality and the interior gradient

estimate are important in the study of the Monge-Ampere equation.

However, it turns out that establishing the inequalities of these

types in the setting of the reflector problem is much more difficult

than that for the Monge-Ampere equation.

In this talk, we will discuss some recent joint work

with Caffarelli and Gutierrez on this problem.

05/03/05 (finals week, probably no talk)

Speaker:

Title:Abstract:

Please contact me to volunteer to talk or recommend speakers.