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Series: PDE Seminar

In fluid dynamics, one of the most classical issues is to understand the dynamics of viscous fluid flows past solid bodies (e.g., aircrafts, ships, etc...), especially in the regime of very high Reynolds numbers (or small viscosity). Boundary layers are typically formed in a thin layer near the boundary. In this talk, I shall present various ill-posedness results on the classical Prandtl equation, and discuss the relevance of boundary-layer expansions and the vanishing viscosity limit problem of the Navier-Stokes equations. I will also discuss viscosity effects in destabilizing stable inviscid flows.

Series: PDE Seminar

We establish Gevrey class regularity of solutions to dissipative equations. The main tools are the Kato-Ponce inequality for Gevrey estimates in Sobolev spaces and the Gevrey estimates in Besov spaces using the paraproduct decomposition. As an application, we obtain temporal decay of solutions for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations.

Series: PDE Seminar

I will describe results of global existence and scattering for water waves (inviscid, irrotational), in the case of small data. I will examine two physical settings: gravity, but no capillarity; or capillarity, but no gravity. The proofs rely on the space-time resonance method, which I will briefly present. This is joint work with Nader Masmoudi and Jalal Shatah.

Series: PDE Seminar

I will discuss a natural elliptic obstacle problem that arises in the study of the Abelian sandpile. The Abelian sandpile is a deterministic growth model from statistical physics which produces beautiful fractal-like images. In recent joint work with Wesley Pegden, we characterize the continuum limit of the sandpile processusing PDE techniques. In follow up work with Lionel Levine and Wesley Pegden, we partially describe the fractal structure of the stable sandpiles via a careful analysis of the limiting obstacle problem.

Series: PDE Seminar

We consider the stationary nonlinear Schrodinger equation when the potential changes sign and may vanish at infinity. We prove that there exists a sign-changing ground state and the so called energy doubling property for sign-changing solutions does not hold. Furthermore, we find that the ground state energy is not equal to the infimum of energy functional over the Nehari manifold. These phenomena are quite different from the case of positive potential.

Series: PDE Seminar

In this talk, I will show recent results on the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle for $L^p$-viscosity solutions of fully nonlinear, uniformly elliptic partial differential equations with unbounded inhomogeneous terms and coefficients. I will also discuss some cases when the PDE has superlinear terms in the first derivatives. This is a series of joint works with Andrzej Swiech.

Series: PDE Seminar

An important question in geometry and analysis is to know when two $k$-forms $f$ and $g$ are equivalent. The problem is therefore to find a map $\varphi$ such that $\varphi^*(g) =f$. We will mostly discuss the symplectic case $k=2$ and the case of volume forms$k=n$. We will give some results on the more difficult case where $3\leq k\leq n-2$, the case $k=n-1$ will also be considered.

Series: PDE Seminar

The basic problem faced in geophysical fluid dynamics isthat a mathematical description based only on fundamental physicalprinciples, the so-called the ``Primitive Equations'', is oftenprohibitively expensive computationally, and hard to studyanalytically. In this talk I will survey the main obstacles inproving the global regularity for the three-dimensionalNavier-Stokes equations and their geophysical counterparts. Eventhough the Primitive Equations look as if they are more difficult tostudy analytically than the three-dimensional Navier-Stokesequations I will show in this talk that they have a unique global(in time) regular solution for all initial data.Inspired by this work I will also provide a new globalregularity criterion for the three-dimensional Navier-Stokesequations involving the pressure.This is a joint work with Chongsheng Cao.

Series: PDE Seminar

We consider Riemann problems for the compressible Euler system in aerodynamics in two space dimensions. The solutionsinvolve shock waves, hyperbolic and elliptic regions. There are also regions which we call semi-hyperbolic. We have shownbefore the existence of such solutions, and now we show regularity of the boundaries of such regions.

Series: PDE Seminar

The incompressible Navier-Stokes equations provide an adequate
physical model of a variety of physical phenomena. However, when the
fluid speeds are not too low, the equations possess very complicated
solutions making both mathematical theory and numerical work
challenging. If time is discretized by treating the inertial term
explicitly, each time step of the solver is a linear boundary value
problem. We show how to solve this linear boundary value problem using
Green's functions, assuming the channel and plane Couette geometries.
The advantage of using Green's functions is that numerical derivatives
are replaced by numerical integrals. However, the mere use of Green's
functions does not result in a good solver. Numerical derivatives can
come in through the nonlinear inertial term or the incompressibility
constraint, even if the linear boundary value problem is tackled using
Green's functions. In addition, the boundary value problem will be
singularly perturbed at high Reynolds numbers. We show how to eliminate
all numerical derivatives in the wall-normal direction and to cast the
integrals into a form that is robust in the singularly perturbed limit.
[This talk is based on joint work with Tobasco].