Abstract: We give the first mathematical construction of traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations. Our proof is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers.
Abstract: We will discuss the long-time dynamics of the derivative nonlinear Schr\"odinger equation. For small, localized initial data, where no solitons arise, we prove dispersive estimates globally in time.  Under the same assumptions, we further prove modified scattering and asymptotic completeness. To the best of our knowledge, this is the first result to achieve an asymptotic completeness theory in a quasilinear setting. Our approach combines the method of testing by wave packets of Ifrim and Tataru, a bootstrap argument, and the Klainerman–Sobolev vector field method.
Abstract: Kinetic equations model systems, such as a gas, where particles move through space according to a velocity that is diffusing (due to, say, collisions with other particles). The presence of spatial boundaries in these models causes technical issues because they are first order in the spatial variable and therefore cannot be defined everywhere on the boundary. In this talk, I will present $L^1-L^\infty$ estimates that yield sharp bounds on the behavior at the spatial boundary. The main estimate is a kinetic version of the Nash inequality. This is a joint work with Giacomo Lucertini and Weinan Wang.
Abstract: In this talk, we first explore image resolution and ill-posed-ness of inverse scattering problems. In particular, we would like to discuss how certain properties of the inclusion might enhance (global or local) resolution in imaging. Examples include inclusions with high contrast or with points on interface surface having high extrinsic curvature.    Along this line, we also discuss concentration of certain resonance modes. First, we explore concentration of plasmon resonance around boundary points of high curvature. With further symmetry, we can have a more precise description of the concentration phenomenon via an investigation of a related quantum integrable system. We then look into the transmission eigenvalue problem, and describe the concentration of almost-transmission eigenfunctions along the interface surface with Weyl’s Law. We explore how these almost-transmission eigenfunctions define an index function that help reconstruct the inclusion.    These are joint works with Habib Ammari (ETH Zurich), Hungyu Liu (CityU of HK), Mahesh Sunkula (UCR), Jun Zou (CUHK).Â
Abstract: In this talk, we study the large-time behavior of the pressureless Euler system with nonlocal alignment and interaction forces. For general interaction potentials and communication weights, we obtain quantitative convergence of classical solutions. In 1D, $(\lambda,\Lambda)$-convex potentials yield exponential decay for bounded weights and sharp algebraic rates for weakly singular ones. For the Coulomb-quadratic potential, we prove exponential convergence with bounded weights and polynomial bounds with singular ones. In multi-dimensions, $(\lambda,\Lambda)$-convex potentials give exponential or improved algebraic decay depending on the weight. In all cases, the density converges (up to translation) to the interaction energy minimizer, while velocities align to a constant. Our results highlight that convergence rates are determined solely by the local behavior of the communication kernel: bounded weights produce exponential decay, while weakly singular ones yield algebraic rates.
Abstract: Mean curvature flow, the gradient flow of the area functional, is the most natural geometric heat flow for embedded hypersurfaces. Being non linear, the flow develops singularities, at which it stops being smooth. One fundamental, often delicate, question for such non linear flows is that of backwards uniqueness. In this talk I will discuss recent backwards uniqueness results, obtained jointly with Josh Daniels-Holgate, which can address some singularities. I will also compare these results to (commonly more robust) forward uniqueness results, and also to the situation in  other equations.