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.

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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.

Mixing by fluid flow is important in a variety of situations
in nature and technology. One effect fluid motion can have is to
strongly enhance diffusion. The extent of diffusion enhancement depends
on the properties of the flow. I will give an overview of the area, and
will discuss a sharp criterion describing a class of incompressible
flows that are especially effective mixers. The criterion uses spectral
properties of the dynamical system associated with the flow, and is
derived from a general result on decay rates for dissipative semigroups

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 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.

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.

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.

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

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.

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.

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.