Abstract: We consider the T^4 cubic NLS which is energy-critical. We study the unconditional uniqueness of solution to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method, and does not require the existence of solutions in Strichartz type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates, which fail on T^4. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel-Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the H^1 unconditional uniqueness for the NLS under the infinite hierarchy framework. This work establishes a unified scheme to prove H^1 uniqueness for the R^3/R^4/T^3/T^4 energy-critical Gross-Pitaevskii hierarchies and thus the corresponding NLS.
Abstract: The tools from machine learning (ML) offer exciting new prospects in scientific computing in general, and the numerical solution of PDEs in particular. In this talk, I will discuss some recent results in this context, by focusing on problems with a variational formulation, and discussing three main ingredients of their solution by ML: (i) approximation quality, i.e. how accurate the representation of the solution by a neural network can in principle be; (ii) optimization, i.e. how effective are the methods to train the parameter of the network; and (iii) generalization error, i.e. how much data is necessary to obtain a solution accurate also outside this data set. In particular I will show that this third aspect typically requires using importance sampling methods for data acquisition. These results will be illustrated on eigenvalue problems in high dimension. This is joint work with Grant Rotskoff.
The seminar will be available on zoom at the link https://umd.zoom.us/j/98038768667?pwd=L3lrQ0lZVmQyM1ZQLzA0dkp3OUJXUT09
Abstract: In this talk, we are interested in studying linear stability properties of perturbations around the Couette flow with constant density for an isentropic compressible fluid.Â In the inviscid case, we show a generic Lyapunov type instability for the density and the irrotational part of the velocity, namely their L^2 norm grows asÂ t^(1/2). Instead, the solenoidal component of the velocity experience inviscid damping, meaning that it decays to zero even in the absence of viscosity.Â For a viscous compressible fluid, we show that the perturbations decay exponentially fast on a faster time-scale with respect to the standard diffusive one, a phenomenon also called enhanced dissipation. This is a joint work with P. Antonelli and P. Marcati.Â Â
Abstract: In the recent years, several major results have been obtained in the problem of finding a constant Q-curvature metric in a given conformal class in dimensions bigger than 5. This talk will cover new results concerning existence and multiplicity of such metrics. I will first present a rather general geometric approach to prove existence and multiplicity of regular metrics, giving several explicit examples. Then I will move to the case of singular metrics, i.e. complete metrics with constant Q-curvature outside of a closed set. This requires to develop several tools to handle 4th order equations (but applicable actually to higher order ones). I will also provide some explicit examples of such metrics and investigate their multiplicity. I will state open problems as well.
Abstract: In this talk I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).