PDEs Seminar (Spring 2005)

 Organizers: Ronghua Pan and Andrzej Swiech


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


No Talk.


Speaker: Ronghua Pan, Georgia Tech

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

Abstract: We study the local well-poseness and singularity formation of smooth solutions for the
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.*****



Speaker:   Prof. Diaraf SECK, Universit Cheikh Anta Diop (FASEG) , Dakar SENEGAL (Host: Evans Harrell)

Title:  Bernoulli free  boundary problems.

Abstract: We study  the existence and uniqueness results  of the  Bernoulli free
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.



Speaker:   Xu-Yuan Chen, Georgia Tech

Title:  A Uniqueness Theorem for Nonlinear Reaction Diffusion Equations

Abstract:  It is well known that the Cauchy problem of the heat equation
$u_t=\Delta u$ 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.



Speaker:  Vladimir Oliker, Emory University

Title: Some nonlinear problems in geometry and optics leading to
          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.


Speaker:  Shaoqiang  Tang,  CalTech

Title: Low order regularizations for dynamic phase trnasitions
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.



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.


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.


SpeakerWen Shen, Penn State Univ.

Title Non-cooperative and semi-cooperative differential games

: 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.




SpeakerTong 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.


 Spring break (no talk!)



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.




Speaker:  Konstantina Trivisa, University of Maryland 

TitleOn a Multidimensional Model for the Dynamic Combustion of Compresssible Reacting gases

Abstract: In this talk  a multidimensional model will be introduced  for
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.



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.




Speaker:  Tao Luo, Georgetown University

Title:  Blow up of BV- norms  for Non-smooth Measure preserving Transport

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.




Speaker:  Qingbo Huang,  Wright State University, (Host Andrzej Swiech)

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)





Please contact me to volunteer to talk or recommend speakers.