Georgia Institute of Technology College of Sciences

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])

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.

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.

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.