Past OxPDE Lunchtime Seminar

20 June 2014
12:00
Abstract
We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.
  • OxPDE Lunchtime Seminar
13 June 2014
12:00
Prof. Mikhail Feldman
Abstract
We discuss shock reflection problem for compressible gas dynamics, various patterns of reflected shocks, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.
  • OxPDE Lunchtime Seminar
13 June 2014
10:30
Prof. Suncica Canic
Abstract
Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids. In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same. In such problems, the geometric nonlinearities of the fluid-structure interface and the significant exchange in the energy between a moving fluid and a structure require sophisticated ideas for the study of their solutions. In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with different mechanical characteristics. No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers. In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions for a class of problems describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid, giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.} A stable, modular, loosely coupled scheme will be presented, and an existence proof showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed. Our results reveal a new physical regularizing mechanism in FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution. All theoretical results will be illustrated with numerical examples. This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).
  • OxPDE Lunchtime Seminar
30 May 2014
12:00
Dr. Weijun Xu
Abstract
We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.
  • OxPDE Lunchtime Seminar
23 May 2014
12:00
Dr. Antonio Segatti
Abstract
In this talk, I will introduce and analyse an elastic surface energy recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. As it will be clear, the topology and the geometry of the surface will play a fundamental role in understanding the behavior of thin films of liquid crystals. In particular, our results regards the existence of minimizers, the existence of the gradient flow of the energy and, in the case of an axisymmetric toroidal particle, a detailed characterization of global and local minimizers. This last item is supplemented with numerical experiments. This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).
  • OxPDE Lunchtime Seminar
9 May 2014
12:00
Abstract
In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.
  • OxPDE Lunchtime Seminar
9 May 2014
11:00
Prof. Ya-Guang Wang
Abstract
We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.
  • OxPDE Lunchtime Seminar

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