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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.

Series: PDE Seminar

In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the
Hamilton-Jacobi type.
The method can be applied in different settings. One of these settings
concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$
where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p)
is $C^2$, strictly convex superlinear in p.
Let D be a compact smooth domain with boundary $\partial D$ contained in
$T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then
this is also the case on the interior of D.
There are several variants of this result in different settings.
To make the result accessible to the layman, we will explain the method
on the function distance to a closed subset of an Euclidean space. This
example contains all the ideas of the general case.

Series: PDE Seminar

We consider the focusing 3D quantum many-body dynamic which
models a dilute bose gas strongly confined in two spatial directions.
We assume that the microscopic pair interaction is focusing and
matches the Gross-Pitaevskii scaling condition. We carefully examine
the effects of the fine interplay between the strength of the
confining potential and the number of particles on the 3D N-body
dynamic. We overcome the difficulties generated by the attractive
interaction in 3D and establish new focusing energy estimates. We
study the corresponding BBGKY hierarchy which contains a diverging
coefficient as the strength of the confining potential tends to
infinity. We prove that the limiting structure of the density matrices
counterbalances this diverging coefficient. We establish the
convergence of the BBGKY sequence and hence the propagation of chaos
for the focusing quantum many-body system. We derive rigorously the 1D
focusing cubic NLS as the mean-field limit of this 3D focusing quantum
many-body dynamic and obtain the exact 3D to 1D coupling constant.

Series: PDE Seminar

Vortices arise in many problems in condensed matter physics, including superconductivity, superfluids, and Bose-Einstein condensates. I will discuss some results on the behavior of two of these systems when there are asymptotically large numbers of vortices. The methods involve suitable renormalization of the energies both at the vortex cores and at infinity, along with a renormalization of the vortex density function.

Series: PDE Seminar

We identify sufficient conditions on initial data to ensure the existence of a unique strong solution to the Cauchy problem for the Compressible Navier-Stokes equations with degenerate viscosities and vacuum (such as viscous Saint-Venants model in $\mathbb{R}^2$). This is a recent work joint with Yachun Li and Ronghua Pan.