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Series: PDE Seminar

In this talk, we will present some results on global classical solution to the two-dimensional compressible Navier-Stokes equations with density-dependent of viscosity, which is the shear viscosity is a positive constant and the bulk viscosity is of the type $\r^\b$ with $\b>\frac43$. This model was first studied by Kazhikhov and Vaigant who proved the global well-posedness of the classical solution in periodic case with $\b> 3$ and the initial data is away from vacuum. Here we consider the Cauchy problem and the initial data may be large and vacuum is permmited. Weighted stimates are applied to prove the main results.

Series: PDE Seminar

Abstract is available at: http://people.math.gatech.edu/~gchen73/seminar/April15.pdf

Series: PDE Seminar

We consider some recent models from stochastic or optimal control
involving a very large number of agents. The goal is to derive mean
field limits when the number of agents increases to infinity. This
presents some new unique difficulties; the corresponding master equation
is a non linear Hamilton-Jacobi equation for instance instead of the
linear transport equations that are more typical in the usual mean field
limits. We can nevertheless pass to the limit by looking at the problem
from an optimization point of view and by using an appropriate kinetic
formulation. This is a joint work with S. Mischler, E. Sere, D. Talay.

Series: PDE Seminar

This is a special PDE seminar in Skiles 005. In this talk, we will introduce the compactness framework for approximate solutions to sonic-subsonic flows governed by the irrotational steady compressible Euler equations in arbitrary dimension. After that, similar results will be presented for the isentropic case. As a direct application, we establish several existence theorems for multidimensional sonic-subsonic Euler flows. Also, we will show the recent progress on the incompressible limits.

Series: PDE Seminar

The Euler-Maxwell system describes the interaction between a
compressible fluid of electrons over a background of fixed ions and the
self-consistent electromagnetic field created by the motion.We
show that small irrotational perturbations of a constant equilibrium
lead to solutions which remain globally smooth and return to
equilibrium. This is in sharp contrast with the case of neutral fluids
where shock creation happens even for very nice initial data.Mathematically,
this is a quasilinear dispersive system and we show a small data-global
solution result. The main challenge comes from the low dimension which
leads to slow decay and from the fact that the nonlinearity has some
badly resonant interactions which force a correction to the linear
decay. This is joint work with Yu Deng and Alex Ionescu.

Series: PDE Seminar

Cubic focusing and defocusing Nonlinear Schroedinger Equations admit
spatially (and temporally) periodic standing wave solutions given
explicitly by elliptic functions. A natural question to ask is: are they
stable in some sense (spectrally/linearly, orbitally,
asymptotically,...), against some class of perturbations (same-period,
multiple-period, general...)? Recent efforts have slightly enlarged our
understanding of such issues. I'll give a short survey, and describe an
elementary proof of the linear stability of some of these waves. Partly
joint work in progress with S. Le Coz and T.-P. Tsai.

Series: PDE Seminar

In this talk we examine the cubic nonlinear wave and
Schrodinger equations. In three dimensions, each of these equations is
H^{1/2} critical. It has been showed that such equations are well-posed and scattering when the H^{1/2} norm is bounded, however, there is
no known quantity that controls the H^{1/2} norm. In this talk we use
the I-method to prove global well posedness for data in H^{s}, s >
1/2.

Series: PDE Seminar

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy.
The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Series: PDE Seminar

Motivated by the theory of hydrodynamic turbulence, L. Onsager
conjectured in 1949 that solutions to the incompressible Euler equations
with Holder regularity less than 1/3 may fail to conserve energy. C.
De Lellis and L. Székelyhidi, Jr. have pioneered an approach to
constructing such irregular flows based on an iteration scheme known as
convex integration. This approach involves correcting “approximate
solutions" by adding rapid oscillations, which are designed to reduce
the error term in solving the equation. In this talk, I will discuss an
improved convex integration framework, which yields solutions with
Holder regularity 1/5- as well as other recent results.

Series: PDE Seminar

For the water waves system we have shown the formation in finite time of
splash and splat singularities. A splash singularity is when the
interface remain smooth but self-intersects at a point and a splat
singularity is when it self-intersects along an arc. In this talk I will
discuss new results on stationary splash singularities for water waves
and in the case of a parabolic system a splash can also develop but not a
splat singularity.