## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, October 1, 2013 - 15:05 , Location: Skiles 006 , , Georgia Tech , , Organizer:
It is well known that solutions of compressible Euler equations in general form discontinuities (shock waves) in finite time even when the initial data is $C^\infty$ smooth. The lack of regularity makes the system hard to resolve. When the initial data have large amplitude, the well-posedness of the full Euler equations is still wide open even in one space dimenssion. In this talk, we discuss some recent progress on large data solutions for the compressible Euler equations in one space dimension. The talk includes joint works with Alberto Bressan, Helge Kristian Jenssen, Robin Young and Qingtian Zhang.
Series: PDE Seminar
Tuesday, September 3, 2013 - 15:05 , Location: Skiles 005 , Renato Calleja , UNAM, Mexico , Organizer:
I will present  a method for constructing periodic or quasi-periodic solutions for forced strongly dissipative systems. Our method applies to the varactor equation in electronic engineering and to the forced non-linear wave equation with a strong damping term proportional to the wave velocity. The strong damping leads to very few small divisors which allows to prove the existence by using a fixed point contraction theorem. The method also leads to efficient numerics. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.
Series: PDE Seminar
Tuesday, August 20, 2013 - 15:00 , Location: 006 , Cristel Chandre , Center for Theoretical Physics, Univ. Aix-Marseille , Organizer: Chongchun Zeng
Dirac'stheory of constrained Hamiltonian systems allows for reductions of the dynamics in a Hamiltonian framework. Starting from an appropriate set of constraints on the dynamics, Dirac'stheory provides a bracket for the reduced dynamics. After a brief introduction of Dirac'stheory, I will illustrate the approach on ideal magnetohydrodynamics and Vlasov-Maxwell equations. Finally I will discuss the conditions under which the Dirac bracket can be constructed and is a Poisson bracket.
Series: PDE Seminar
Tuesday, April 23, 2013 - 15:05 , Location: Skiles 006 , Jin Feng , University of Kansas , Organizer: Zhiwu Lin
Using metric derivative and local Lipschitz constant, we define action integral and Hamiltonian operator for a class of optimal control problem on curves in metric spaces. Main requirement on the space is a geodesic property (or more generally, length space property). Examples of such space includes space of probability measures in R^d, general Banach spaces, among others. A well-posedness theory is developed for first order Hamilton-Jacobi equation in this context. The main motivation for considering the above problem comes from variational formulation of compressible Euler type equations. Value function of the variation problem is described through a Hamilton-Jacobi equation in space of probability measures. Through the use of geometric tangent cone and other properties of mass transportation theory, we illustrate how the current approach uniquely describes the problem (and also why previous approaches missed). This is joint work with Luigi Ambrosio at Scuola Normale Superiore di Pisa.
Series: PDE Seminar
Friday, April 19, 2013 - 14:00 , Location: Skiles 006 , Prof. Alexandaer Kiselev , Univrsity of Wisconsin,-Madison , , Organizer: Ronghua Pan
We discuss a system of two equations involving two diffusing densities, one of which is chemotactic on the other, and absorbing reaction. The problem is motivated by modeling of coral life cycle and in particular breeding process, but the set up is relevant to many other situations in biology and ecology. The models built on diffusion and advection alone seem to dramatically under predict the success rate in coral reproduction. We show that presence of chemotaxis can significantly increase reproduction rates. On mathematical level, the first step in understanding the problem involves derivation of sharp estimates on rate of convergence to bound state for Fokker-Planck equation with logarithmic potential in two dimensions.
Series: PDE Seminar
Tuesday, April 16, 2013 - 15:05 , Location: Skiles 006 , Chen, Zhang , Shandong University , Organizer: Zhiwu Lin
In this talk, globally modified non-autonomous 3D Navier-Stokes equations with memory and perturbations of additive noise will be discussed. Through providing theorem on the global well-posedness of the weak and strong solutions for the specific Navier-Stokes equations, random dynamical system (continuous cocycle) is established, which is associated with the above stochastic differential equations. Moreover, theoretical results show that the established random dynamical system possesses a unique compact random attractor in the space of C_H, which is periodic under certain conditions and upper semicontinuous with respect to noise intensity parameter.
Series: PDE Seminar
Tuesday, April 9, 2013 - 15:05 , Location: Skiles 006 , Xu, Xiang , Carnegie Mellon University , Organizer: Zhiwu Lin
In the Landau-de Gennes theory to describe nematic liquid crystals, there exists a cubic term in the elastic energy, which is unusual but is used to recover the corresponding part of the classical Oseen-Frank energy. And the cost is that with its appearance the current elastic energy becomes unbounded from below. One way to deal with this unboundedness problem is to replace the bulk potential defined as in with a potential that is finite if and only if $Q$ is physical such that its eigenvalues are between -1/3 and 2/3. The main aim of our talk is to understand what can be preserved out of the physical relevance of the energy if one does not use a somewhat ad-hoc potential, but keeps the more common potential. In this case one cannot expect to obtain anything meaningful in a static theory, but one can attempt to see what a dynamical theory can predict.
Series: PDE Seminar
Tuesday, April 2, 2013 - 15:05 , Location: Skiles 006 , Mahir Hadzic , MIT , Organizer: Zhiwu Lin
We study small perturbations of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant on a spatially periodic background. These solutions model a quiet fluid in a spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our result extends the stability results of Rodnianski and Speck for the Euler-Einstein system with positive cosmological constant to the case of dust (i.e. a pressureless fluid). The main difficulty that we overcome is the degenerate nature of the dust model that loses one degree of differentiability with respect to the Euler case. To resolve it, we commute the equations with a well-chosen differential operator and develop a new family of elliptic estimates that complement the energy estimates. This is joint work with J. Speck.
Series: PDE Seminar
Tuesday, February 26, 2013 - 15:05 , Location: Skiles 006 , Xu, Ming , Ji'Nan University, Guangzhou, China , Organizer: Zhiwu Lin
In the report, we give an introduction on our previous work mainly on elliptic operators and its related function spaces. Firstly we give the problem and its root, secondly we state the difficulties in such problems, at last we give some details about some of our recent work related to it.
Series: PDE Seminar
Tuesday, February 19, 2013 - 15:00 , Location: Skiles 006 , Prof. Hermano Frid , IMPA, Rio De Janeiro, Braizil , , Organizer: Ronghua Pan
We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem whichgeneralizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential  inclusion, for a real-valued function $u(x,t)$, $$0\in \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)+\nabla_x\cdot \nabla_\eta\psi(x/\ve,x,t,u,\nabla u) - f(x/\ve,x,t, u),$$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where  $\Psi(z,x,\cdot)$ is strictly convex and $\psi(z,x,t,u,\cdot)$ is a $C^1$ convex function, both with quadratic growth,satisfying some additional technical hypotheses. As functions of the oscillatory variable, $\Psi(\cdot,x,u),\psi(\cdot,x,t,u,\eta)$ and  $f(\cdot,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $\AA$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to usual $L^2$ convergence in the Cartesian product $\Om\X\Pi$, where $\Pi$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra.   To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with uniformly bounded  sequences in $L^2$.