Meets Tuesdays at 4:30 pm in Skiles 255 unless
otherwise indicated
01/11/05
No Talk.
01/18/05
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.*****
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.
02/01/05
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.
02/08/05
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.
02/15/05
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.
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.
Title: Non-cooperative and
semi-cooperative differential games
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
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.
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
Title: Blow up of BV- norms for
Non-smooth Measure preserving Transport
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.