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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.

Series: PDE Seminar

In the Landau-de Gennes theory to describe nematic liquid crystals,
there
exists a cubic term in the elastic energy, which is unusual but is used to
recover
the corresponding part of the classical Oseen-Frank energy. And the cost
is that
with its appearance the current elastic energy becomes unbounded from below.
One way to deal with this unboundedness problem is to replace the bulk
potential
defined as in with a potential that is finite if and only if $Q$ is
physical such
that its eigenvalues are between -1/3 and 2/3.
The main aim of our talk is to understand what can be preserved out
of the
physical relevance of the energy if one does not use a somewhat ad-hoc
potential,
but keeps the more common potential. In this case one cannot expect to
obtain anything
meaningful in a static theory, but one can attempt to see what a dynamical
theory can
predict.

Series: PDE Seminar

We study small perturbations of the well-known
Friedman-Lemaitre-Robertson-Walker (FLRW) solutions to the dust-Einstein
system with a positive cosmological constant on a spatially periodic
background. These solutions model a quiet fluid in a spacetime undergoing
accelerated expansion. We show that the FLRW solutions are nonlinearly
globally future-stable under small perturbations of their initial data. Our
result extends the stability results of Rodnianski and Speck for the
Euler-Einstein system with positive cosmological constant to the case of
dust (i.e. a pressureless fluid). The main difficulty that we overcome is
the degenerate nature of the dust model that loses one degree of
differentiability with respect to the Euler case. To resolve it, we commute
the equations with a well-chosen differential operator and develop a new
family of elliptic estimates that complement the energy estimates. This is
joint work with J. Speck.

Series: PDE Seminar

In the report, we give an
introduction on our previous work mainly on elliptic operators and
its related function spaces. Firstly we give the problem and its
root, secondly we state the difficulties in such problems, at last we
give some details about some of our recent work related to it.

Series: PDE Seminar

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem whichgeneralizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$0\in \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)+\nabla_x\cdot \nabla_\eta\psi(x/\ve,x,t,u,\nabla u) - f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where $\Psi(z,x,\cdot)$ is strictly convex and $\psi(z,x,t,u,\cdot)$ is a $C^1$ convex function, both with quadratic growth,satisfying some additional technical hypotheses. As functions of the oscillatory variable, $\Psi(\cdot,x,u),\psi(\cdot,x,t,u,\eta)$ and $f(\cdot,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $\AA$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to usual $L^2$ convergence in the Cartesian product $\Om\X\Pi$, where $\Pi$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with uniformly bounded sequences in $L^2$.