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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 $L^2$.

Series: PDE Seminar

We provide the first construction of exact solitary waves of large
amplitude with an arbitrary distribution of vorticity. Small amplitude
solutions have been constructed by Hur and later by Groves and Wahlen
using a KdV scaling. We use continuation to construct a global connected
set of symmetric solitary waves of elevation, whose profiles decrease
monotonically on either side of a central crest. This generalizes the
classical result of Amick and Toland.

Series: PDE Seminar

I will describe a joint work with Vincent Millot (Paris 7) where we
investigate the singular limit of a fractional GL equation towards the
so-called boundary harmonic maps.

Series: PDE Seminar

We study the Dirichlet and Neumann type initial-boundary value problems for strongly degenerate parabolic-hyperbolic equations. We suggest the notions of entropy solutions for these problems and establish the uniqueness of entropy solutions. The existence of entropy solutions is also discussed（joint work with Yuxi Hu and Qin Wang).

Series: PDE Seminar

In this talk we first present some applied examples (coming from
Economics and Finance) of
Optimal Control Problems for Dynamical Systems with Delay (deterministic
and stochastic).
To treat such problems with the so called Dynamic Programming Approach
one has to study a class of infinite dimensional HJB equations for which
the existing theory does not apply
due to their specific features (presence of state constraints, presence
of first order differential operators in the state equation, possible
unboundedness of the control operator).
We will present some results on the existence of regular solutions for
such equations and on existence of optimal control in feedback form.