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 2010

February
5	Douglas Wright, Drexel University (Talk canceled)
February
12	Laszlo Erdoes, Mathematisches Institut der LMU Theresienstr TBA
February
19	Douglas Wright, Drexel University Interaction manifolds in reaction diffusion systems We consider a general planar reaction diffusion equation which wehypothesize has a localized traveling wave solution. Underassumptions which are no stronger than those needed to prove the stability of asingle pulse, we prove that the PDE has solutions which are roughlythe linear superposition of two pulses, so long as they move alongtrajectories which are not parallel. In particular we prove that ifthe initial data for the equation is close to the sum of two separatedpulses, then the solution converges exponentially fast to such asuperposition so long as the distance between the two pulses remainssufficiently large.
March
5	Eric Carlen, Rutgers University (Talk canceled)
March
12	Distinguished lectures by Andrea Bertozzi (special time and location) TBA
March
19	Vedran Sohinger, MIT Bounds on the growth of high Sobolev norms of solutions to 1D Nonlinear Schrodinger Equations In this talk, we study the growth of Sobolev norms of solutions to 1D Nonlinear Schrodinger Equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate of the low-to-high frequency cascade. We present a frequency decomposition method which allows us to obtain polynomial bounds in the case of the 1D Hartree equation with sufficiently regular convolution potential, and which allows us to bound the growth of fractional Sobolev norms of the Cubic NLS on the real line.
March
26	Kotaro Tsugawa, Nagoya University Local well-posedness for quadratic nonlinear Schr\"odinger equations We consider the Cauchy problem for Schr\"odinger equations with quadratic nonlinearities in one space dimension. We prove sharp local well-posedness and ill-posedness results by using a modification of Bourgain's norm. As an application, we also prove a local well-posedness result for the ``good'' Boussinesq equation. This is joint work with Nobu Kishimoto.
April
9	Wen Shen, PSU Recent results on a model for granular flow In this talk we present some recent results on the mathematical aspect of a model for granular flow.The model was proposed by Hadeler & Kuttler (1999, "Granular Matter"). In one space dimension, this model can be written as a 2x2 hyperbolic system of balance laws, in which the unknowns are the slope of the standing layer and the thickness of the moving layer.If the slope does not change sign, under suitable assumptions we prove the global smooth solutions and global existence of large BV solutions.At the slow erosion limit, i.e., as the height of the moving layer tends to 0, we show that the limiting behavior of the mountain profile provides an entropy solution to a scalar integro-differential conservation law.At the end of the talk, we discuss some further ideas and some work in progress.Part of these results are joint work with Debora Amadori from L'Aquila, Italy.
April
16	Sophie Chen, IAS Conformal deformation to scalar flat metrics with constant mean curvature on the boundary We study the problem of finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. This problem was proposed by Escobar as a generalization of Riemann mapping theorem. The problem turns out to be solving an elliptic equation with (Neumann type condition) critical nonlinearity on the boundary. Here, the critical exponent is in the sense of trace Sobolev inequality. In this talk, we present a proof which solves most cases left of the problem and reduces the remaining case to positive mass theorem.
April
23	Nestor Guillen, University of Texas (Asutin) Instantaneous regularization phenomena for the Stefan problem with Gibbs-Thomson law The two-phase Stefan problem is expected to exhibit waiting time phenomena, in other words, an initial Lipschitz free boundary might remain just Lipschitz for some positive time. On the other hand, adding a Gibbs-Thomson correction term to the model is believed to stabilize the interface and make it more regular. I will present a partial result in this direction, namely, I will show that weak solutions whose free boundaries are Lipschitz in space and time are actually C^{2,\alpha} in space and become so instantaneously. This will follow first from observing that the De Giorgi-Moser-Nash theory for parabolic equations works even when one has a singular right hand side and secondly from the regularity theorem of almost-minimal boundaries of Almgren, De Giorgi and Tamanini.
April
30	Jerome Vetois, Nice University Stability of sign-changing solutions for critical elliptic equations On a Riemannian manifold, we describe the asymptotic behavior of families of sign-changing solutions of elliptic equations with asymptotically critical nonlinearities. Such families are said to be stable if any family of solutions, bounded in the energy space, is in fact bounded in C^0. We present stability and instability results. In particular, we emphasize the threshold role played by the geometric Yamabe potential. As an application of our stability results, we derive multiplicity results of sign-changing solutions.
May
7 (Special time: 2pm)	Panagiotis Souganidis, University of Chicago Fronts in oscillatory media
May
14	Gigliola Staffilani, MIT Periodic Derivative NLS: almost sure global existence and invariance of its weighted Wiener measure In this talk I will present a recent work in collaboration with A. Nahmod, H. Oh and L. Ray-Bellet in which we prove that the periodic derivative nonlinear Schr\"odinger equation in one dimension is globally well-posed for initial data in the support of its associated weighted Wiener measure which we suitably construct. In particular, almost surely global well-posedness is proved in a certain Banach space of initial data that scales like $H^{\frac{1}{2}-\epsilon}(\T),$ for small $\epsilon >0$. We also show the invariance of this measure