Andrea Bertozzi, "Processing images with nonlinear PDEs"
Abstract: We illustrate how PDEs can arise naturally in the analysis and processing of images. We review some of the now classical methods in this area and show some novel new methods in which physics-based models play a role. Some specific problems include denoising, segmentation, and image "inpainting" which involves filling being occluding objects.
Jerry Bona, "Initial-boundary-value problems for nonlinear waves"
Abstract: We discuss recent work on well posedness and long time asymptotics to initial boundary value problems for equations for long waves.
James Colliander, "On rough blowup solutions of L2 critical NLS"
Abstract: The goal of the work reported upon in this talk is to understand the L2 critical nonlinear Schrödinger (NLS) blowup dynamics for rough initial data. Much of the known theory for NLS blowup relies upon energy conservation and is thus restricted to H1 solutions, even though the NLS evolution problem is well-posed in L2. After surveying some of the known theory of blowup, the talk will describe recent work which provides some insights into the blowup phenomena for Hs solutions for certain s < 1.
Georg Dolzmann, "A 2D compressible membrane theory as a Gamma-limit of a nonlinear elasticity model for incompressible membranes in 3D"
Abstract: We derive a two-dimensional compressible elasticity model for thin elastic sheets as a Gamma-limit of a fully three-dimensional incompressible theory. The energy density of the reduced problem is obtained in two steps: first one optimizes locally over out-of-plane deformations, then one passes to the quasiconvex envelope of the resulting energy density. This work extends the results by LeDret and Raoult on smooth and finite-valued energies to the case incompressible materials. The main difficulty in this extension is the construction of a recovery sequence which satisfies the nonlinear constraint of incompressibility pointwise everywhere.
**This is joint work with Sergio Conti.
Marius Mitrea, "Sharp estimates for Green potentials and a solution of the Chang-Krantz-Stein conjecture"
Absract: One of the most classical applications of the Calderon-Zygmund theory is the fact that, in the whole Euclidean space, any two partial derivatives on the (harmonic) Newtonian potential give rise to a bounded operator on Lp, 1<p<∞. This has been further extended to values p<1 by Fefferman-Stein, granted that the Lebesgue scale is replaced by Hardy spaces. In the present talk I will consider similar issues when the Euclidean space is replaced by a bounded Lipschitz domain D. One typical result is as follows. If G is the Green operator for the Dirichlet Laplacian associated with D, then two derivatives on G may fail to map Lp(D) into itself for any p>1, though this operator always maps the Hardy space Hp(D) into itself for p<1, sufficiently close to 1.
Daniel Spirn, "Dynamics of superfluid vortices"
Abstract: The Ginzburg-Landau equations provide a model for the motion of superfluids. One feature of such fluids is the formation of regions, called vortices, where the fluid is no longer a superfluid. I will discuss recent progress in the asymptotic description of these vortices in both static and dispersive cases.
Konstantina Trivisa, "On a multidimensional model for the dynamic combustion of compressible reacting fluids."
Abstract: In this work we present a multidimensional model for the dynamic combustion of compressible reacting fluids formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl and P.L. Lions.
Gunther Uhlmann, "Travel time tomography"
Abstract: Travel time tomography consists in determining the index of refraction or sound speed of a medium by measuring the travel times of waves going through the medium. In differential geometry this is known as the the boundary rigidity problem. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points of the boundary of a compact Riemannian manifold with boundary. The inverse boundary problem consists in determining the Riemannian metric from the boundary distance function. We will describe some recent progress on this problem.