## Seminars and Colloquia by Series

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 $L^2$.
Series: PDE Seminar
Tuesday, February 12, 2013 - 15:05 , Location: Skiles 006 , Miles Wheeler , Brown University , Organizer: Zhiwu Lin
We provide the first construction of exact solitary waves of large amplitude with an arbitrary distribution of vorticity.  Small amplitude solutions have been constructed by Hur and later by Groves and Wahlen using a KdV scaling. We use continuation to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. This generalizes the classical result of Amick and Toland.
Series: PDE Seminar
Tuesday, February 5, 2013 - 15:05 , Location: Skiles 006 , Yannick Sire , Universite Paul Cezanne d'Aix-Marseille III , Organizer: Zhiwu Lin
I will describe a joint work with Vincent Millot (Paris 7) where we investigate the singular limit of a fractional GL equation towards the so-called boundary harmonic maps.
Series: PDE Seminar
Tuesday, January 29, 2013 - 15:00 , Location: Skiles 006 , Prof. Yachu Li , Shanghai Jiao Tong University , , Organizer: Ronghua Pan
We study the Dirichlet and Neumann type initial-boundary value problems for strongly degenerate parabolic-hyperbolic equations. We suggest the notions of entropy solutions for these problems and establish the uniqueness of entropy solutions. The existence of entropy solutions is also discussed（joint work with Yuxi Hu and Qin Wang).
Series: PDE Seminar
Tuesday, January 8, 2013 - 15:05 , Location: Skiles 006 , Fausto Gozzi , LUISS University, Rome, Italy , Organizer: Zhiwu Lin
In this talk we first present some applied examples (coming from Economics and Finance) of Optimal Control Problems for Dynamical Systems with Delay (deterministic and stochastic). To treat such problems with the so called Dynamic Programming Approach one has to study a class of infinite dimensional HJB equations for which the existing theory does not apply due to their specific features (presence of state constraints, presence of first order differential operators in the state equation, possible unboundedness of the control operator). We will present some results on the existence of regular solutions for such equations and on existence of optimal control in feedback form.
Series: PDE Seminar
Tuesday, November 20, 2012 - 15:05 , Location: Skiles 006 , Rafael de la Llave , Georgia Tech , Organizer: Zhiwu Lin
We prove an a-posteriori KAM theorem which applies to some ill-posed Hamiltonian equations. We show that given an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, there is a true solution nearby. Furthermore, the solution is "whiskered" in the sense that it has stable and unstable directions. We do not assume that the equation defines an evolution equation. Some examples are the Boussinesq equation (and system) and the elliptic equations in cylindrical domains. This is joint work with Y. Sire. Related work with E. Fontich and Y. Sire.
Series: PDE Seminar
Tuesday, November 13, 2012 - 15:05 , Location: Skiles 006 , Ming Chen , University of Pittsburgh , Organizer: Zhiwu Lin
We prove via explicitly constructed initial data that solutionsto the gravity-capillary wave system in R^3 representing a 2d air-waterinterface immediately fail to be C^3 with respect to the initial data ifthe initial (h_0, \psi_0) \in H^{s + 1/2} \times H^s for s<3, where h isthe free surface and \psi is the velocity potential.