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Series: PDE Seminar

We prove an a-posteriori KAM theorem which applies to some ill-posed Hamiltonian equations. We show that given an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, there is a true solution nearby. Furthermore, the solution is "whiskered" in the sense that it has stable and unstable directions. We do not assume that the equation defines an evolution equation. Some examples are the Boussinesq equation (and system) and the elliptic equations in cylindrical domains. This is joint work with Y. Sire. Related work with E. Fontich and Y. Sire.

Series: PDE Seminar

We prove via explicitly constructed initial data that solutionsto the gravity-capillary wave system in R^3 representing a 2d air-waterinterface immediately fail to be C^3 with respect to the initial data ifthe initial (h_0, \psi_0) \in H^{s + 1/2} \times H^s for s<3, where h isthe free surface and \psi is the velocity potential.

Series: PDE Seminar

From its physical origin, the viscosity and heat conductivity in
compressible fluids depend on absolute temperature through power laws.
The mathematical theory on the well-posedness and regularity on this setting
is widely open. I will report some recent progress made on this direction,
with emphasis on the lower bound of temperature, and global existence of
solutions in one or multiple dimensions. The relation between thermodynamics
laws and Navier-Stokes equations will also be discussed. This talk is based
on joint works with Weizhe Zhang.

Series: PDE Seminar

Active scalars appear in many problems of fluid dynamics. The most
common examples of active scalar equations are 2D Euler, Burgers, and
2D surface quasi-geostrophic (SQG) equations. Many questions about
regularity and properties of solutions of these equations remain open.
I will discuss the recently introduced
idea of nonlocal maximum principle, which helped prove global
regularity of
solutions to the critical SQG equation. I will describe some further
recent developments on regularity and blowup of solutions to active
scalar equations.

Series: PDE Seminar

The existence of self-similar blow-up for the viscous incompressible
fluids was a classical question settled in the seminal of works of
Necas, et al and Tsai in the 90'. The corresponding scenario for the
inviscid Euler equations has not received as much attention, yet it
appears in many numerical simulations, for example those based on vortex
filament models of Kida's high symmetry flows. The case of a
homogeneous self-similar profile is especially interesting due to its
relevance to other theoretical questions such the Onsager conjecture or
existence of Landau type solutions. In this talk we give an account of
recent studies demonstrating that a self-similar blow-up is
unsustainable the Euler system under various weak decay assumptions on
the profile. We will also talk about general energetics of the Euler
system that, in part, is responsible for such exclusion results.

Series: PDE Seminar

The talk focuses on positive equilibrium (i.e. time-independent)solutionsto mathematical models for the dynamics of populations structured by ageand spatial position. This leads to the study of quasilinear parabolicequations with nonlocal and possibly nonlinear initial conditions. Weshallsee in an abstract functional analytic framework how bifurcationtechniquesmay be combined with optimal parabolic regularity theory to establishtheexistence of positive solutions. As an application of these results wegivea description of the geometry of coexistence states in a two-parameterpredator-prey model.

Series: PDE Seminar

In this talk we will give a very elementary explanation of
issues associated with the unique global solvability of three
dimensional Navier-Stokes equation and then discuss various
modifications of the classical system for which the unique solvability
is resolved. We then discuss some of the fascinating issues associated
with the stochastic Navier-Stokes equations such as Gaussian & Levy
Noise, large deviations and invariant measures.

Series: PDE Seminar

We shall describe our recent work on the extension of sharp
Hardy-Littlewood-Sobolev inequality, including the reversed HLS
inequality with negative exponents. The background and motivation will
be given. The related integral curvature equations may be discussed if
time permits.

Series: PDE Seminar

Time-averages are common observables in analysis of experimental data
and numerical simulations of physical systems. We describe a
PDE-theoretical framework for studying time-averages of dynamical
systems that evolve in both fast and slow scales. Patterns arise upon
time-averaging, which in turn affects long term dynamics via nonlinear
coupling. We apply this framework to geophysical fluid dynamics in
spherical and bounded domains subject to strong Coriolis force and/or
Lorentz force.

Series: PDE Seminar

In this talk, we consider the Cauchy problem of a modified two-component Camassa-Holm shallow water system. We first establish local well-possedness of the Cauchy problem of the system. Then we present several blow-up results of strong solutions to the system. Moreover, we show the existence of global weak solutions to the system. Finally, we address global conservative solutions to the system. This talk is based on several joint works with C. Guan, K. H. Karlsen, K. Yan and W. Tan.