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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.

Series: PDE Seminar

A class of kinetic models for the collective self-organization of
agents is presented. Results on the global existence of weak solutions as
well as a hydrodynamic limit will be discussed. The main tools employed in
the analysis are the velocity averaging lemma and the relative entropy
method. This is joint work with T. Karper and A. Mellet.

Series: PDE Seminar

The existence of large BV (total variation) solution for compressible Euler equations in one space dimension is a major open problem in the hyperbolic conservation laws, where the small BV existence was first established by James Glimm in his celebrated paper in 1964. In this talk, I will discuss the recent progress toward this longstanding open problem joint with my collaborators. The singularity (shock) formation and behaviors of large data solutions will also be discussed.

Series: PDE Seminar

Abstract: the theory of weak turbulence has been put forward by appliedmathematicians to describe the asymptotic behavior of NLS set on a compactdomain - as well as many other infinite dimensional Hamiltonian systems.It is believed to be valid in a statistical sense, in the weaklynonlinear, infinite volume limit. I will present how these limits can betaken rigorously, and give rise to new equations.

Series: PDE Seminar

For a $C^{1,1}$-uniformly elliptic matrix $A$, let $H(x,p)=$ be the corresponding
Hamiltonian function. Consider the Aronsson equation associated with $H$:
$$(H(x,Du))x H_p(x,Du)=0.$$
In this talk, I will indicate everywhere differentiability of any viscosity solution of the above Aronsson's equation.
This extends an important theorem by Evans and Smart on the infinity harmonic functions (i.e. $A$
is the identity matrix).

Series: PDE Seminar

Shock waves are idealizations of steep spatial gradients of
physical quantities such as pressure and density in a gas,
or stress in an elastic solid. In this talk, I outline the mathematics
of shock waves for nonlinear partial differential equations
that are simple models of physical systems. I will focus on
non-classical shocks and smooth waves that they approximate. Of particular interest are comparisons between nonlinear traveling
waves influenced strongly by dissipative effects such as viscosity or
surface tension, and spreading waves generated by the balance between dispersion and
nonlinearity, when the nonlinearity is non-convex.