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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.

Series: PDE Seminar

In this talk, we shall study the stability of the Prandtl boundary layer
equations in three space variables. First, we obtain a well-posedness
result of the three-dimensional Prandtl equations under some constraint
on its flow structure. It reveals that the classical Burgers equation
plays an important role in determining this type of flow with special
structure, that avoids the appearance of the complicated secondary flow
in the three-dimensional Prandtl boundary layers. Second, we give an
instability criterion for the Prandtl equations in three space
variables. Both of linear and nonlinear stability are considered. This
criterion shows that the monotonic shear flow is linearly stable for the
three dimensional Prandtl equations if and only if the tangential
velocity field direction is invariant with respect to the normal
variable, which is an exact complement to the above well-posedness
result for a special flow. This is a joint work with Chengjie Liu and
Tong Yang.

Series: PDE Seminar

Find the abstract at the link: http://people.math.gatech.edu/~gchen73/research/GA_Tech.pdf

Series: PDE Seminar

We survey some recent results by the speaker, Jason Metcalfe and Daniel
Tataru for small data local well-posedness of quasilinear Schrödinger
equations. In addition, we will discuss some applications recently
explored with Jianfeng Lu and recent progress towards the large data
short time problem. Along the way, we will attempt to motivate analysis
of the problem with connections to problems from Density Functional
Theory.