PDE Seminar Calendar

The PDE seminar is held on Fridays at 3:00 in Kassar 105 unless otherwise specified (*). Coffee and cookies are usually served after the talks.

In Spring 2012, the seminar is co-organized by Walter Strauss and Hongjie Dong.

For more information, or if you would like to give a talk, please send email to Hongjie Dong at hdong_at_dam_dot_brown_dot_edu.

Spring 2012

January
27	Gerard-Varet David, Paris 7 Homogenization and Boundary Layer I will present a joint work with Nader Masmoudi, on the homogenization of elliptic systems with Dirichlet boundary condition, when the coefficients of both the system and the boundary datum are \epsilon-periodic. We show that, as \epsilon\to 0, the solutions converge in L2 with a power rate in \epsilon, and identify the homogenized limit system. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. Our analysis answers a longstanding open problem, raised by Bensoussan, Lions and Papanicolaou.
February
3	Jared Speck, MIT The Global Stability of the Minkowski Spacetime Solution to the Einstein-Nonlinear Electromagnetic System in Wave Coordinates, (special time: 2:00-2:50 pm) The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of general relativity to a model of nonlinear electromagnetic fields. In this talk, I will discuss the family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable from a sufficiently regular Lagrangian, ii) they reduce to the linear Maxwell model in the weak-field limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a proof of the global nonlinear stability of the 1 + 3--dimensional Minkowski spacetime solution to the coupled system. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in a wave coordinate gauge. The analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. The analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic potentials.
February
3	Mahir Hadzic, MIT The classical Stefan problem and the vanishing surface tension limit We develop a new unified framework for the treatment of well-posedness for the Stefan problem with and without surface tension. This approach yields new estimates for the regularity of the moving surface in the absence of surface tension, which allows us to prove that solutions of the Stefan problem with positive surface tension converge to solutions of the Stefan problem without surface tension. Our techniques rely on a fluid-mechanics inspired approach which, in a suitable sense, combines the Eulerian and the Lagrangian viewpoint. This is joint work with S. Shkoller.
February
10	Canceled
February
17	Xueke Pu, Brown University KdV limit of Euler-Poisson system
February
24	Yoshiaki Teramoto, Setsunan University Navier-Stokes flow down a vertical flat wall We consider the motion of a viscous incompressible fluid flow down a vertical flat plane under the effect of gravity. We formulate the problem for downward periodic disturbances from the laminar steady flow as an evolution equation in a function space. Under certain assumptions, we show global-in-time existence of solutions of the initial value problem for small disturbances. It is crucial that the operator arising in the linearized problem has compact resolvent operators and generates an analytic semigroup in some function space.
March
2	Yuxi Zheng, Yeshiva University Hyperbolic systems of wave equations in nematic liquid crystals In modelling liquid crystals, Leslie and Erickson utilize both hyperbolic and parabolic terms along with the core Oseen-Frank potential energy. We focus on the hyperbolic aspect of the motion. We illustrate through asymptotic models the singularity formation of solutions from smooth initial data, the various concepts of weak solutions, and the existence and stability theory for the solutions. Toward the end of the talk, we present our latest work of the global existence of solutions to its Cauchy problem for the system modeling a type of nematic liquid crystal that has equal splay and twist coefficients. We use the three-component director variable, rather than the two-component spherical angle variable, in our equations to avoid a persistent shortcoming in previous works.
March
9	Matania Ben-Artzi, Hebrew University From Maxwell to Dirac: spectral theory of first-order systems The class of strongly propagative systems contains many of the well-known physical systems, such as the Maxwell equations, crystal optics, elasticity, Dirac equation (with and without mass)...The spectral structure of these (nonelliptic) operators is studied by identifying the derivative of the spectral measure ("density of states" in the physical language) as a trace operator on the "slowness surfaces". As an application, a "limiting absorption principle" is established, as well as global spacetime estimates. (Based on joint work with Tomio Umeda, Hyogo University, Japan).
March
16	Miles Wheeler, Brown University Large-amplitude solitary water waves with vorticity We consider the solitary water wave problem with vorticity. Small amplitude solutions have been constructed by Hur and later by Groves and Wahlижn. We use degree theory to prove a continuation result, constructing a global connected set of solutions. We also discuss some properties of this connected set.
March
23	Pierre Raphael, Universite Paul Sabatier The flow near the ground state for the mass critical gKdV equation Abstract
March
30	Spring break
April
6	Vlad Vicol, University of Chicago Shape dependent maximum principles and applications We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with Peter Constantin.
April
13 (special time: 2pm)	Zhen Lei, Fudan University and Caltech Convection and Incompressible Euler Navier-Stokes Equations In this talk I will first report a Liouville type Theorem for the axi-symmetric Navier-Stokes equations. The proof relies on the extra conservation law which is obtained by using the convection term. In the second part, I will talk about a 3D model equation for Navier-Stokes equations. This 3D model is obtained by ignoring the convection terms in a reformulated axi-symmetric Navier-Stokes equations. The 3D model possesses the properties that the energy law is satisfied, it is incompressible, the nonlinearity is nonlocal, etc., which are similar to those for the Naiver-Stokes equations. Consequently, in most cases, whatever you can prove for the Navier-Stokes equations, you may also have similar results for our 3D model. Then I will report that the 3D inviscid model can develop finite time singularities starting from smooth initial data with finite energy. The above two parts suggest that convection terms may play a positive role for the global regularity theory of the incompressible Euler and Navier-Stokes equations. At last, I will report a very recent result of constructing a family of large solutions for the 3D Navier-Stokes equations which is obtained by making use of the convection to break the scaling. Those are a series work joint with a number of collaborators: professor Thomas Y. Hou from Caltech, professor Congming Li from University of Colorado, professor Fang-hua Lin from Courant Institute, professor Qi S. Zhang from University of California at Riverside and professor Yi Zhou from Fudan university.
April
13	Xiaodong Yan, University of Connecticut A liouville theorem for higher order elliptic system We study positive solutions to higher order elliptic system (-\Delta)^m u=v^p, (-\Delta)^m v=u^q in R^n. We shall prove that there are no positive radial solution (u,v) for 1/(p+1)+1/(q+1).1-2m/n. We also prove that if p>1, q>1 and max(2m(p+1)/(pq-1), 2m(q+1)/(pq-1)>=n-2m, there are no positive solutions.
April
20
April
27	Yanyan Li, Rutgers University Some analytic aspects of conformally invariant fully nonlinear equations We will discuss some work on conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville type theorems, Harnack inequalities, and Bocher type theorems.
May
4