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Series: PDE Seminar

In 1966 V. Arnold observed that solutions to the Euler equations of incompressible fluids can be
viewed as geodesics of the kinetic energy metric on the group of volume-preserving diffeomorphisms.
This introduced Riemannian geometric methods into the study of ideal fluids. I will first review this approach
and then describe results on the structure of singularities of the associated exponential map and (time premitting)
related recent developments.

Series: PDE Seminar

This talk gives a blowup criteria to the incompressible
Navier-Stokes equations in BMO^{-s} on the whole space R^3, which implies
the well-known BKM criteria
and Serrin criteria. Using the result, we can get the norm of
|u(t)|_{\dot{H}^{\frac{1}{2}}} is decreasing function. Our result can
obtained by the compensated compactness and Hardy space result of [6] as
well as [7].

Series: PDE Seminar

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension.We obtain this result by deriving global $W^{1,p}$-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation techniquetogether with a new two-parameter scaling argument.The talk is based on my joint work with Luan Hoang (Texas Tech University) and Truyen Nguyen (University of Akron)

Series: PDE Seminar

In this talk I will present a new notion of Ricci curvature that applies
to finite Markov chains and weighted graphs. It is defined using tools
from optimal transport in terms of convexity properties of the Boltzmann
entropy functional on the space of probability measures over the graph.
I will also discuss consequences of lower curvature bounds in terms of
functional inequalities. E.g. we will see that a positive lower bound
implies a modified logarithmic Sobolev inequality.
This is joint work with Jan Maas.

Series: PDE Seminar

It is well known that solutions of compressible Euler equations in general form discontinuities (shock waves) in finite time even when the initial data is $C^\infty$ smooth. The lack of regularity makes the system hard to resolve. When the initial data have large amplitude, the well-posedness of the full Euler equations is still wide open even in one space dimenssion. In this talk, we discuss some recent progress on large data solutions
for the compressible Euler equations in one space dimension. The talk includes joint works with Alberto Bressan, Helge Kristian Jenssen, Robin Young and Qingtian Zhang.

Series: PDE Seminar

I will present a method for constructing periodic or
quasi-periodic solutions for forced strongly dissipative systems. Our
method applies to the varactor equation in electronic engineering and to
the forced non-linear wave equation with a strong damping term
proportional to the wave velocity. The strong damping leads
to very few small divisors which allows to prove the existence by using a
fixed point contraction theorem. The method also leads to efficient
numerics. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.

Series: PDE Seminar

Dirac'stheory of constrained Hamiltonian systems allows for reductions
of the dynamics in a Hamiltonian framework. Starting from an appropriate
set of constraints on the dynamics, Dirac'stheory provides a bracket
for the reduced dynamics. After a brief introduction of Dirac'stheory, I
will illustrate the approach on ideal magnetohydrodynamics and
Vlasov-Maxwell equations. Finally I will discuss the conditions under
which the Dirac bracket can be constructed and is a Poisson bracket.

Series: PDE Seminar

Using metric derivative and local Lipschitz constant, we define action
integral and Hamiltonian operator for a class of optimal control problem
on curves in metric spaces. Main requirement on the space is a geodesic
property (or more generally, length space property). Examples of such
space includes space of probability measures in R^d, general Banach
spaces, among others. A well-posedness theory is developed for first
order Hamilton-Jacobi equation in this context.
The main motivation for considering the above problem comes from
variational formulation of compressible Euler type equations. Value
function of the variation problem is described through a Hamilton-Jacobi
equation in space of probability measures. Through the use of geometric
tangent cone and other properties of mass transportation theory, we
illustrate how the current approach uniquely describes the problem (and
also why previous approaches missed).
This is joint work with Luigi Ambrosio at Scuola Normale Superiore di Pisa.

Series: PDE Seminar

We discuss a system of two equations involving two diffusing
densities, one of which is chemotactic on the other, and absorbing
reaction. The problem is motivated by modeling of coral life cycle and
in particular breeding process, but the set up is relevant to many
other situations in biology and ecology. The models built on diffusion
and advection alone seem to dramatically under predict the success
rate in coral reproduction. We show that presence of chemotaxis can
significantly increase reproduction rates. On mathematical level, the
first step
in understanding the problem involves derivation of sharp estimates on
rate of convergence to bound state for Fokker-Planck equation with
logarithmic potential in two dimensions.

Series: PDE Seminar

In this talk, globally modified non-autonomous 3D
Navier-Stokes equations with memory and perturbations of additive
noise will be discussed. Through providing theorem on the global
well-posedness of the weak and strong solutions for the specific
Navier-Stokes equations, random dynamical system (continuous
cocycle) is established, which is associated with the above
stochastic differential equations. Moreover, theoretical results
show that the established random dynamical system possesses a unique
compact random attractor in the space of C_H, which is periodic
under certain conditions and upper semicontinuous with respect to
noise intensity parameter.