Title: Stability for Vlasov and Hartree equations Abstract: I focus on presenting the survival threshold of wave numbers found in mean field equations such as Vlasov and Hartree equations, which completely characterizes the large time behavior of linearized equations near non-trivial backgrounds into plasma oscillations, Landau damping, and phase mixing regime. I shall also mention progress […]
Title: Well-Posedness of the 3D Peskin ProblemAbstract: This work introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and we show that these equations admit a boundary integral reduction. This provides an evolution equation for the elastic interface. We also […]
Title: Singularity formation in incompressible fluids. Abstract: I will discuss a recent example of singularity formation in the 3d Euler equation. This is based on work joint with Federico Pasqualotto.
Title: The stability of irrotational shocks and the Landau law of decay. Abstract: We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, […]
Title: From Instability to Singularity Formation in Incompressible FluidsAbstract: In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for the 3d incompressible Euler equations. I will then outline a new mechanism that allows us to overcome the effects of […]
Title: On linear and non-linear stability of collisionless systems on black hole exteriors. Abstract: I will present upcoming linear and non-linear stability results for collisionless systems on spherically symmetric black holes. On the one hand, I will discuss the decay properties of massive Vlasov fields on the exterior of Schwarzschild spacetime. On the other hand, I will discuss […]