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Series: PDE Seminar

Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a result such as a comparison theorem which can deal with only measure theoretic norms (e.g. L-n and L-infinity) of the right hand side of the equation has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We will discuss this estimate in the context of fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])

Series: PDE Seminar

In large part, the waves that we observe in the open ocean are created by wind blowing over the water. The precise nature of this process occurs has been intensely studied, but is still not understood very well at a mathematically rigorous level. In this talk, we side-step that issue, somewhat, and consider the steady problem. That is, we prove the existence of small-amplitude traveling waves in a two phase air-water system that can be viewed as the eventual product of wind generation. This is joint work with Oliver Buhler and Jalal Shatah.

Series: PDE Seminar

This talk will be an amalgamation of aspects of scientific computing - development, verification, and interpretation - with application to the Vlasov-Poisson (VP) system, an important nonlinear partial differential equation containing the essential difficulties of collisionless kinetic theories. I will describe our development of a discontinuous Galerkin (DG) algorithm, its verification via convergence studies and comparison to known Vlasov results, and our interpretation of computational results in terms of dynamical systems ideas. The DG method was invented for solving a neutron transport model, successfully adapted to fluid motion including shock propagation, applied to the Boltzmann equation, and developed in the general context of conservation laws, and elliptic and parabolic equations. Our development for the VP system required the simultaneous approximation of the hyperbolic Vlasov equation with the elliptic Poisson equation, which created new challenges. I will briefy discuss advantages of the method and describe our error estimates and recurrence calculations for polynomial bases. Then, I will show results from a collection of benchmark computations of electron plasma dynamics, including i) convergence studies of high resolution linear and nonlinear Landau damping with a comparison to theoretical parameter dependencies ii) the nonlinear two-stream instability integrated out to (weak) saturation into an apparently stable equilibrium (BGK) state with detailed modeling of this state, and iii) an electric field driven (dynamically accessible) example that appears to saturate into various periodic solutions. I will interpret such final states, in analogy to finite-dimensional Hamiltonian theory, as Moser-Weinstein periodic orbits, and suggest a possible variational path for proof of their existence. Finally, I will comment briefly on recent progress on extensions to the Maxwell-Vlasov system, including estimates and computational results.

Series: PDE Seminar

We will prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton-Jacobi equations on a general metric space. Then, we will present some consequences: in particular the equivalence of the log-Sobolev inequality and the hypercontractivity property of theHamilton-Jacobi "semi-group", (and if time allows) that Talagrand’s transport-entropy inequalities in metric space are characterizedin terms of log-Sobolev inequalities restricted to the class of c-convex functions (based on a paper in collaboration with N. Gozlan and P.M. Samson).

Series: PDE Seminar

The stability of a liquid drop of prescribed volume hanging from a circular cylindrical tube in a gravity field has been a problem of continuing interest. This problem was treated variationally in the late '70s by Henry Wente who showed there was a continuous family indexed by increasing volume which terminated in a final unstable equilibrium due to one or the other of two specific geometric mechanisms. I will describe a similar problem arising in mathematical biology for drops at the bottom of a rectangular tube and explain, among other things, how the associated instability occurs through exactly three physical mechanisms.

Series: PDE Seminar

This talk considers a scalar conservation (balance) law equation with random (martingale measure) source term. A new notion of entropic solution is introduced as the underlying calculus for change of variable needs to be changed into Ito's calculus. This is due to irregularities in the trajectory of particles caused by randomness. In the new notion, entropy production has additional terms. We discuss ways to handle such term so that a uniqueness theory can still be established. Additionally, stochastic generalizations of compensated compactness will be given. This was a joint work with David Nualart. It appeared in Journal of Functional Analysis, Vol 255, Issue 2, 2008, pages 313-373.

Series: PDE Seminar

We review the theory of the (extended) divergence-measure fields providing an up to date account of its basic results established by Chen and Frid (1999, 2002), as well as the more recent important contributions by Silhavy (2008, 2009). We include a discussion on some pairings that are important in connection with the definition of normal trace for divergence-measure fields. We also review its application to the uniqueness of Riemann solutions to the Euler equations in gas dynamics, as given by Chen and Frid (2002). While reviewing the theory, we simplify a number of proofs allowing an almost self-contained exposition.

Series: PDE Seminar

We prove an analog of the Caffarelli-Kohn-Nirenberg theorem for weak solutions of a system of PDE that model a viscoelastic fluid in the presence of an energy damping mechanism. The system was recently introduced in a method of establishing the global in time existence of weak solutions of the well known Oldroyd model, which remains an open problem.

Series: PDE Seminar

Fundamental issues such as the global regularity problem concerning the surface quasi-geostrophic (SQG) and related equations have attracted a lot of attention recently. Significant progress has been made in the last few years. This talk summarizes some current results on the critical and supercritical SQG equations and presents very recent work on the generalized SQG equations. These generalized equations are active scalar equations with the velocity fields determined by the scalars through general Fourier multiplier operators. The SQG equation is a special case of these general models and it corresponds to the Riesz transform. We obtain global regularity for equations with velocity fields logarithmically singular than the 2D Euler and local regularity for equations with velocity fields more singular than those corresponding to the Riesz transform. The results are from recent papers in collaboration with D. Chae and P. Constantin, and with D. Chae, P. Constantin, D. Cordoba and F. Gancedo.

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.