Johns Hopkins
Mathematical  Physics Seminar

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, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time integrable rate $1/(t(\log t)^{1/2})$. We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an $N$-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. This is joint work with Igor Rodnianski."

Title: From Instability to Singularity Formation in Incompressible Fluids

Abstract: 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 particle transport, leveraging the instability seen in the classical Taylor-Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. This is joint work with Tarek Elgindi (Duke University).

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 an asymptotic stability result for the exterior of Schwarzschild as a solution to the Einstein-massless Vlasov system, assuming spherical symmetry. These results follow via concentration estimates on suitable stable manifolds in phase space. 

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 on the nonlinear problem.  

Title: Well-Posedness of the 3D Peskin Problem

Abstract: 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 consider general nonlinear elastic laws, which is called the fully nonlinear 3D Peskin problem.   The main result of this work proves that the evolution problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.  This is a joint work with Eduardo García-Juárez, Po-Chun Kuo, and Yoichiro Mori.