We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.
- You are here:
- Home
We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation.
The basic model of an isolated self-gravitating gaseous star is given by the gravitational Euler-Poisson system. For any value of the adiabatic index strictly between 1 and 4/3 we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface in the Lagrangian coordinates. The leading order singular behaviour is driven by collapsing dust solutions. This is a joint work with Yan Guo (Brown) and Juhi Jang (USC).
We study whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main results are the following:
(1) On the one hand, we are able to show that for any non-negative smooth stationary vorticity that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation.
Abstract: In this talk, we consider the Cauchy problem of the N-dimensional incompressible viscoelastic fluids with N ≥ 2. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group e∓itΛ. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.
Collective behavior can be seen in many animal species, such as flocking birds, herding mammals, and swarming bacteria. In the continuum limit, these phenomena can be modeled by nonlocal PDEs. In this talk, after discussing some PDE models for collective dynamics, I will focus on the analysis of the Keller-Segel equation, which models bacterial chemotaxis.
The main goal of the thesis is to study integro-differential equations. Integro-differential equations arise naturally in the study of stochastic processes with jumps. These types of processes are of particular interest in finance, physics and ecology. In the thesis, we study uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n.
Abstract: In this talk, I will present the uniqueness of conservative
solutions to Camassa-Holm and two-component Camassa-Holm equations.
Generic regularity and singular behavior of those solutions are also
studied in detail. If time permitting, I will also mention the recent
result on wellposedness of cubic Camassa-Holm equations.
The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity.
Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice.
Pages
