Tuesday, September 1, 2009 - 15:00 , Location: Skiles 255 , Lincoln Chayes , UCLA , Organizer: Zhiwu Lin
This seminar concerns the analysis of a PDE, invented by J.M. Lasry and P.L. Lions whose applications need not concern us. Notwithstanding, the focus of the application is the behavior of a free boundary in a diffusion equation which has dynamically evolving, non--standard sources. Global existence and uniqueness are established for this system. The work to be discussed represents a collaborative effort with Maria del Mar Gonzalez, Maria Pia Gualdani and Inwon Kim.
Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flowTuesday, August 25, 2009 - 15:05 , Location: Skiles 255 , David Hoff , Indiana University, Bloomington , Organizer:
We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories. I'm going to integrate into the talk something like a "pretalk" in an attempt to motivate the more technical material and to make things accessible to a general analysis audience.
Tuesday, May 5, 2009 - 15:00 , Location: Skiles 255 , Giuseppe Savare , Università degli Studi di Pavia, Italy , Organizer:
Some interesting nonlinear fourth-order parabolic equations, including the "thin-film" equation with linear mobility and the quantum drift-diffusion equation, can be seen as gradient flows of first-order integral functionals in the Wasserstein space of probability measures. We will present some general tools of the metric-variational approach to gradient flows which are useful to study this kind of equations and their asymptotic behavior. (Joint works in collaboration with U.Gianazza, R.J. McCann, D. Matthes, G. Toscani)
Tuesday, April 14, 2009 - 15:05 , Location: Skiles 255 , Joy Ko , Brown University, Providence , Organizer:
I will talk about the highlights of a collaborative and multidisciplinary program investigating qualitative features of steady water waves with vorticity in two dimensions. Computational and analytical results together with data from the oceanographic community have resulted in strong evidence that key qualitative features such as amplitude, depth, streamline shape and pressure profile can be fundamentally affected by the presence of vorticity. Systematic studies of constant vorticity and shear vorticity functions will be presented.
Global Weak Solutions for an Incompressible Charged Fluid with Multi-Scale Couplings - Initial-Boundary Value ProblemTuesday, April 7, 2009 - 15:05 , Location: Skiles 255 , Joseph Jerome , Northwestern University, Evanston , Organizer:
The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was investigated by the speaker in [Transport Theory Statist. Phys. 31 (2002), 333-366], where a local existence-uniqueness theory was demonstrated, based upon Kato's framework for examining evolution equations. In this talk, the existence of a global distribution solution is proved to hold for the model, in the case of the initial-boundary value problem. Connection of the above analysis to significant applications is discussed. The solution obtained is quite rudimentary, and further progress would be expected in resolving issues of regularity.
Tuesday, March 31, 2009 - 15:05 , Location: Skiles 255 , Kris Jenssen , Penn State University, College Station , Organizer:
We study the problem of constructing systems of hyperbolic conservation laws with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a (typically overdetermined) system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed with techniques from exterior differential systems (Cartan-Kahler theory). The cases of 2x2- and 3x3-systems can be treated in detail, and explicit examples show that already the 3x3-case is fairly complex. We also analyze general rich systems. We also characterize conservative systems with the same eigencurves as compressible gas dynamics. This is joint work with Irina Kogan (North Carolina State University).
Tuesday, March 24, 2009 - 15:05 , Location: Skiles 255 , Thomas Sideris , University of California, Santa Barbara , Organizer:
We will give an overview of results on the global existence of solutions to the initial value problem for nonlinear elastic and viscoelastic materials in 3d without boundary. Materials will be assumed to be isotropic, but both compressible and incompressible cases will be discussed. In the compressible case, a key null condition must be imposed to control nonlinear interactions of pressure waves. This necessary assumption is consistent with the physical model. Initial conditions are small perturbations of a stress free reference state. Existence is proven using a fixed point argument which combines energy estimates and with some new dispersive estimates.
Friday, March 13, 2009 - 16:05 , Location: Skiles 255 , Eitan Tadmor , University of Maryland, College Park , Organizer:
We discuss the global regularity vs. finite time breakdown in Eulerian dynamics, driven by different models of nonlinear forcing. Finite time breakdown depends on whether the initial configuration crosses intrinsic, O(1) critical thresholds (CT). Our approach is based on spectral dynamics, tracing the eigenvalues of the velocity gradient which determine the boundaries of CT surfaces in configuration space. We demonstrate this critical threshold phenomena with several n-dimensional prototype models. For n=2 we show that when rotational forcing dominates the pressure, it prolongs the life-span of sub-critical 2D shallow-water solutions. We present a stability theory for vanishing viscosity solutions of the 2D nonlinear "pressureless" convection. We revisit the 3D restricted Euler and Euler-Poisson equations, and obtain a surprising global existence result for a large set of sub-critical initial data in the 4D case.
Tuesday, March 10, 2009 - 15:05 , Location: Skiles 255 , Hans Knüpfer , Courant Institute, New York , Organizer:
We consider the the following fourth order degenerate parabolic equation h_t + (hh_xxx)_x = 0. The equation arises in the lubrication approximation regime, describing the spreading of a thin film liquid with height profile h >= 0 on a plate. We consider the equation as free boundary problem, defined on its positivity set. We address existence and regularity of classical solutions in weighted Hölder and Sobolev spaces.
Tuesday, March 3, 2009 - 15:05 , Location: Skiles 255 , Marianne Korten , Kansas State University, Manhattan , Organizer:
In this talk I will describe recent work with C. N. Moore about the two-phase Stefan problem with a degenerate zone. We start with local solutions (no reference to initial or boundary data) and then obtain intrinsic energy estimates, that will in turn lead to the continuity of the temperature. We then show existence and uniqueness of solutions with signed measures as data. The uniqueness problem with signed measure data has been open for some 30 years for any degenerate parabolic equation.