Georgia Institute of Technology College of Sciences

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension.We obtain this result by deriving global $W^{1,p}$-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion.

We will start by describing some general features of quasilinear dispersive and wave equations. In particular we will discuss a few important aspects related to the question of global regularity for such equations. We will then consider the water waves system for the evolution of a perfect fluid with a free boundary. In 2 spatial dimensions, under the influence of gravity, we prove the existence of global irrotational solutions for suitably small and regular initial data. We also prove that the asymptotic behavior of solutions as time goes to infinity is

In this talk I will present a new notion of Ricci curvature that applies to finite Markov chains and weighted graphs. It is defined using tools from optimal transport in terms of convexity properties of the Boltzmann entropy functional on the space of probability measures over the graph. I will also discuss consequences of lower curvature bounds in terms of functional inequalities. E.g. we will see that a positive lower bound implies a modified logarithmic Sobolev inequality. This is joint work with Jan Maas.

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

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

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

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.

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

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.