Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 03/05/2010
17:00 		Aaron N. K. Yip (Purdue) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
A model of crystal growth with corner regularization
We investigate a dynamic model of two dimensional crystal growth described by a forward-backward parabolic equation. The ill-posed region of the equation describes the motion of corners on the surface. We analyze a fourth order regularized version of this equation and show that the dynamical behavior of the regularized corner can be described by a traveling wave solution. The speed of the wave is found by rigorous asymptotic analysis. The interaction between multiple corners will also be presented together with numerical simulations. This is joint work in progress with Fang Wan.
Mon, 10/05/2010
17:00 		Robert Pego (Carnegie Mellon University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Spectral stability for solitary water waves 
I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.
Mon, 17/05/2010
17:00 		Martin Fuchs (Universität des Saarlandes) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Recent regularity results for variational problems and for nonlinear elliptic systems
Mon, 24/05/2010
17:00 		Varga kalantarov (Koç University) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
CANCELED
Mon, 31/05/2010
17:00 		James Glimm (SUNY at Stony Brook) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Mathematical, Numerical and Physical Principles for Turbulent Mixing
Numerical approximation of fluid equations are reviewed. We identify numerical mass diffusion as a characteristic problem in most simulation codes. This fact is illustrated by an analysis of fluid mixing flows. In these flows, numerical mass diffusion has the effect of over regularizing the solution. Simple mathematical theories explain this difficulty. A number of startling conclusions have recently been observed, related to numerical mass diffusion. For a flow accelerated by multiple shock waves, we observe an interface between the two fluids proportional to Delta x-1, that is occupying a constant fraction of the available mesh degrees of freedom. This result suggests
  • (a) nonconvergence for the unregularized mathematical problem or
  • (b) nonuniqueness of the limit if it exists, or
  • (c) limiting solutions only in the very weak form of a space time dependent probability distribution.
The cure for the pathology (a), (b) is a regularized solution, in other words inclusion of all physical regularizing effects, such as viscosity and physical mass diffusion. We do not regard (c) as a pathology, but an inherent feature of the equations. In other words, the amount and type of regularization of an unstable flow is of central importance. Too much regularization, with a numerical origin, is bad, and too little, with respect to the physics, is also bad. For systems of equations, the balance of regularization between the distinct equations in the system is of central importance. At the level of numerical modeling, the implication from this insight is to compute solutions of the Navier-Stokes, not the Euler equations. Resolution requirements for realistic problems make this solution impractical in most cases. Thus subgrid transport processes must be modeled, and for this we use dynamic models of the turbulence modeling community. In the process we combine and extend ideas of the capturing community (sharp interfaces or numerically steep gradients) with conventional turbulence models, usually applied to problems relatively smooth at a grid level. The numerical strategy is verified with a careful study of a 2D Richtmyer-Meshkov unstable turbulent mixing problem. We obtain converged solutions for such molecular level mixing quantities as a chemical reaction rate. The strategy is validated (comparison to laboratory experiments) through the study of 3D Rayleigh-Taylor unstable flows.
Mon, 07/06/2010
17:00 		Mikhail Feldman (University of Wisconsin at Madison) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Shock Reflection and Free Boundary Problems
In this talk we describe some recent work on shock reflection problems for the potential flow equation. We will start with discussion of shock reflection phenomena. Then we will describe the results on existence, structure and regularity of global solutions to regular shock reflection. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the sonic arc). We will discuss techniques to handle such free boundary problems and degenerate elliptic equations. This talk is based on joint works with Gui-Qiang Chen, and with Myoungjean Ba
Mon, 14/06/2010
17:00 		Panagiotis Souganidis (University of Chicago) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Random homogenization: theory and applications

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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