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.

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 2009

January
16	Seick Kim, Yonsei University, Korea Harnack's inequality for second order elliptic operators In this talk, I will review Harnack inequalities for second order elliptic operators in the Euclidean setting and their extension to the Riemannian manifolds. Harnack inequalities play a central role in the theory of elliptic and parabolic differential equations, many of their applications, and to analysis at large. Their role in analysis began with the classical Liouville theorem for holomorphic functions, and has continued in producing refined covering theorems, as well as underscoring the importance of BMO and the De Giorgi classes. These inequalities are essential in Perron's method of solving the Dirichlet problem for the Laplacian. This method is at the root of the modern theory of viscosity solutions for fully nonlinear equations. Thus coming from a rich tradition the Harnack estimate is a key tool in modern developments of analysis and PDEs.
January
23	Francis Filbet,Universitat of Paris, Lyon 1, France Approximation of PDE models for chemotaxis A finite volume method is presented to discretize the Patlak-Keller-Segel (PKS) modelling chemosensitive movements. First, we prove existence and uniqueness of a numerical solution to the proposed scheme. Then, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation converges to the solution to the PKS system when the initial mass is lower than this threshold. Numerical simulations are performed to verify accuracy and the properties of the scheme. Finally, in the last section we investigate blow-up of the solution for large mass.
January
30	Samuel Walsh, Brown University Steady Stratified Capillary-Gravity Waves We will discuss two-dimensional traveling stratified water waves propagating over an impermeable flat bed and with a free surface. The wave's motion is assumed to be driven by surface tension on the upper boundary and a gravitational force acting on the body of the fluid. Such waves are commonly seen to form when, for example, a wind blows over a quiescent body of water. We shall present some new results on the existence of global continua of classical solutions of this type. In the process, we shall also answer some open questions for the constant density case.
February
20	Jerome Vetois, University of Cergy Bubble tree decompositions for critical anisotropic equations We describe the asymptotic behavior in energy space of Palais-Smale sequences for an anisotropic problem on a domain in the Euclidian space. This description is well-known in the isotropic case. In the general case, we emphasize the crucial role played by the geometry of the domain.
February
27	Charis Tsikkou, Brown University Hyperbolic Conservation Laws With Large Initial Data. Is The Cauchy Problem Well-Posed?
March
6	Xinan Ma, USTC and IAS Curvature estimates for the level sets of three harmonic function and minimal graph From the H.Lewy theorem and Gleason-Wolff counterexample, we know the three dimension harmonic function has very special geometry properties. In this talk,we give the sharp Gaussian curvature and principal curvature estimates of the level sets of three harmonic function and minimal graph on convex ring. In the end, we also state a constant rank theorem on the second fundamental form of the convex level sets for a class fully nonlinear elliptic equations.
March
13	Shuanglin Shao, IAS Profile decomposition for Airy equation and applications in critical gKdV This talk is divided into two parts. In the first part, I will discuss how to construct a linear profile decomposition for the Airy equation; in the second part, I will discuss several applications.
March
20	Camillo De Lellis, Universitat Zürich h–Principle and fluid dynamics Abstract
April
10	Robert Strain, Princeton University and U Penn Global existence and Newtonian limit for the Relativistic Boltzmann Equation near Vacuum with some short range interactions We discuss recent work to prove global existence of solutions to the Cauchy Problem for the Relativistic Boltzmann equation with near Vacuum data. Our proof generalizes the work of Glassey (2006) to some restricted bounded cross sections which need not decay at infinity. Our estimates are independent of the speed of light globally in time. We will discuss applications to the Newtonian limit, c \to \infty if time permits.
April
17	Christopher Larsen, Worcester Polytechnic Institute Fracture evolution and locality The main goal of recent mathematical work on quasi-static fracture triggered by the Francfort-Marigo approach is to turn Griffith's criterion for crack growth into a model for predicting crack paths. The main drawback of the Francfort-Marigo model is its reliance on global minimization, which in particular results in violations of Griffith's criterion. I will discuss some efforts towards creating a variational model based on local minimization that also predicts crack paths, and why a fully satisfactory (variational) approach seems impossible. Time permitting, I will also discuss some recent work on dynamics.
April
17 (4:15-5:15pm)	Emmanuel Hebey, Universite de Cergy-Pontoise Stability for Einstein-scalar field Lichnerowicz equations We discus existence and stability for Einstein-scalar field Lichnerowicz equations in the inhomogeneous context of compact Riemannian manifolds.
April
24 (at 12pm, 37 Manning 102)	Daniel Spirn, University of Minnesota Dynamics of concentrations in mixed flows
April
24 (at 4pm)	Ovidiu Savin, Columbia University On parabolic Monge-Ampere equations We discuss geometric properties of parabolic Monge-Ampere equations of the type $u_t=f(x,t)(\det D^2u)^p$, with $f$ measurable function $0 < \lambda \le f \le \Lambda$. This is a joint work with P. Daskalopoulos.
April
28 (at 3pm, B&H 166)	Dong Li, IAS TBA
May
1	Natasa Pavlovic, UT Austin The quintic NLS as the mean field limit of a Boson gas with three-body interactions In this talk we will discuss the dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit where the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a limiting Gross-Pitaevskii (GP) hierarchy for which we prove existence and uniqueness of solutions. For factorized initial data, the solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schr\"odinger equation. Time permitting, we will discuss the new approach to look at the the Cauchy problem for focusing and defocusing GP hierarchy. In that line of work we consider the GP hierarchy in dimensions d\geq 1, for cubic, quintic, focusing and defocusing interactions. We prove local in time existence and uniqueness of solutions in generalized Sobolev spaces of sequences of marginal density matrices. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in d=3. For defocusing interactions, we prove the existence and uniqueness of solutions globally in time in certain cases. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. These results hold without the assumption of factorized initial conditions. This is joint work with Thomas Chen.
May
8	Mahir Hadzic, Brown University Stability in the Stefan problem with surface tension