Georgia Institute of Technology College of Sciences

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.

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.

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.