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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.

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.