The PDE seminar is held on Fridays at 3:00 in Kassar 105unless otherwise specified (*). Coffee and cookiesare 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.
| September | | On the Hilbert Expansion of the Boltzmann equations |
| September | | Models for transport in porous media Abstract: In this talk, I will discuss general models used to model flux of fluids in porous media: two and three phase isothermal flows, non-isothermal flows with phase change, combustion problems. These models have important applications, such as, oil recovery, improved oil recovery, soil and groundwater remediation, CO2 sequestration. I will present the PDEs and we I will discuss some analytic, semi-analytic and numerical techniques used for obtaining the solutions of these PDE. I will present also a brief discussion of open questions in different problems. |
| September | | (Joint PDE/LCDS seminar, 4:15 pm, B&H 161) Large Deviations for a Large Class of 1-D Markov Processes and Applications to Reaction Diffusion Equations |
| October | | Quadruple Junction Solutions in the Entire Three Dimensional Space In this talk, I will discuss the quadruple junction solutions in the entire three dimensional space to a vector-valued Allen-Cahn equation which models multiple phase separation. The solution is the basic profile of the local structure near a quadruple junction in three dimensional crystalline material under the generalized Allen-Cahn model, and is the three dimensional counterpart of triple junction solution which is two dimensional. I will start with one dimensional heteroclinic solutions, and describe how we can construct higher dimensional solutions from the lower dimensional ones, and explain the complications and difficulties in constructing such a solution in three dimensions. |
| October | | Lagrangian Dynamics on an infinite-dimensional torus Abstract |
| October | | Strichartz estimates for the water-wave problem with surface tensionIn recent work with V.M. Hur and G. Staffilani, we study dispersive properties of the water-wave problem with surface tension. Based on a formulation of Ambrose-Masmoudi, we formulate the problem as a nonlinear dispersive equation weakly coupled to a nonlinear transport equation. We show that the solution to the dispersive equation satisfies mixed time-space $L^p L^q$ estimates commonly referred to as Strichartz estimates. That such estimates hold is a direct reflection of the regularizing effect of surface tension. |
| October | | (Joint PDE/LCDS seminar, special time, room to be announced) Title TBA |
| October | | The Analysis of the Boltzmann Collision Operator In the study of the qualitative aspects of the solutions of the Boltzmann equation in kinetic theory, an invariable step is the use of weighted L^p (convolution) estimates for the collision operator. This applied analysis talk is intended to revisit this topic and show how an essential tool in harmonic analysis, namely, a radial symmetrization lemma, can be used to simplify the existent proofs and provide new results in this direction, giving a better understanding of the constants governing these inequalities. Despite the technicalities of the subject I intent to keep the presentation as simple and accessible as possible. This is a joint program with R. J. Alonso (Rice) and I. M. Gamba (Texas - Austin). |
| November | | (Joint PDE/LCDS seminar, 4:30pm, B&H 155) Landau damping: relaxation without dissipation |
| November | | New singular solutions of the biharmonic NLS Abstract |
| November | | Dynamics of dilute block copolymer mixtures Block copolymers are macromolecules that can form variety of microstructures as a result of incomplete phase separation. For this reason, they are natural candidates for controlled nanoscale self-assembly and possess novel material properties. This talk focuses on the mathematical issues surrounding density functional models of dilute diblock copolymer mixtures and their related gradient flows. Isolated structures emerge in the subcritical regime that resemble amphiphilic bilayers and micelles. Existence and stability properties of these solutions are discussed, including a surprising secondary bifurcation that results in self-replication phenomena. Well above the subcritical regime, there is a Ostwald-ripening-type process which is inhibited by long range interactions. The dynamics can be reduced by a series of approximations to interacting particle systems and coarse-grained statistical descriptions which characterize the large scale behavior. |
| November | | Kinetic models for the sedimentation of rigid-rods and connections to the Keller-Segel model |
| November | |
| December | | Regularity analysis for systems of reaction-diffusion pdesWe are interested in proving the regularity of solutions to some systems of reaction--diffusion equations, which is actually a consequence of global boundedness. Our approach is inspired by De Giorgi's method for elliptic regularity with rough coefficients. The proof uses the specific structure of the system to be considered and is not a mere adaptation of scalar techniques; in particular the entropy dissipation of the system plays a crucial role in the analysis. |
| December | | Pseudo-Riemannian Calibrated Geometry and Optimal TransportationOn a manifold M, there is a naturally occurring pseudo-Riemannian metric and K\"ahler form on the product M x M, which is determined by the distance function M. The graph of the solution to the optimal transportation problem for given smooth densities on M is a calibrated maximal Lagrangian submanifold in M x M, with respect to a conformal metric on M x M. The graph of the optimal map is special Lagrangian in the sense of Hitchin. This variational characterization of optimal transportation is different from the traditional one. The calibrations which detect these special Lagrangians are pseudo-Riemannian analogues of the special Lagrangian calibrations for Calabi-Yau manifolds. Like in the Calabi-Yau case, the moduli space of such submanifolds is itself a manifold of dimension b_1(M). On the sphere, a parabolic flow related to mean curvature flow converges to the maximal submanifold. This is joint work with Kim and McCann, and also with Kim and Streets. |