PDE Seminar Fall 2022

Fridays, 3:30 — 4:20pm, Stevenson Center 1431 (in-person) or via Zoom (online). When possible, in-person talks will be live streamed.

Date: Friday, September 2, 2022.

  • Speaker: Brian Luczak, Vanderbilt University.
  • Title: Continuous harmonic functions on a ball that are not in Hs for s>1/2.
  • Abstract: We show that there are harmonic functions on a ball 𝔹n of ℝn, n ≥ 2, that are continuous up to the boundary (and even Hölder continuous) but not in the Sobolev space Hs(𝔹n) for any s sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension n ≥ 2, we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.
  • Zoom link: https://vanderbilt.zoom.us/j/92794509994

Date: Friday, October 7, 2022.

  • Speaker: Dongxiao Yu, University of Bonn.
  • Title: Nontrivial global solutions to some quasilinear wave equations in three space dimensions
  • Abstract: In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John’s counterexample ☐u=u_tu_{tt} and the 3D compressible Euler equations without vorticity for t ≥ 0.
  • Zoom link: https://vanderbilt.zoom.us/j/97526207493

Date: Friday, October 21, 2022.

  • Speaker: Ian Tice, Carnegie Mellon University.
  • Title: Traveling wave solutions to the free boundary Navier-Stokes equations.
  • Abstract: Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.

Date: Friday, November 18, 2022.

  • Speaker: Dejan Gajic, Leipzig University.
  • Title: Late-time tails for geometric wave equations with inverse-square potentials
  • Abstract: I will introduce a new method for obtaining the precise late-time asymptotic profile of solutions to geometric wave equations with inverse-square potentials on asymptotically flat spacetimes. This setting serves as a convenient toy model for understanding novel dynamical properties in the context of Einstein’s equations of general relativity that arise in a variety of situations, e.g. when considering the gravitational properties of electromagnetically charged matter, when describing dynamical, rapidly rotating black holes and when considering higher, odd, spacetime dimensions.
  • Zoom link: https://vanderbilt.zoom.us/j/99209740964

Date: Friday, December 2, 2022.

  • Speaker: Leonardo Abbrescia, Vanderbilt University.
  • Title: Classical Developments of Compressible Fluid Flow
  • Abstract: The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.

    There is a monumental gap in our understanding of compressible fluids in the physical 3D setting compared to the 1D case. This is due in large to the (provable) inaccessibility of the technical PDE tools used in 1D when quantifying the dynamics in 3D. Nevertheless, Christodoulou’s 2007 celebrated breakthrough on shock singularities for the Euler equation has sparked a dramatic wave of results and ideas in multiple space dimensions that have the potential to make the first meaningful dent in the global-in-time theory of compressible fluids. Roughly, shocks are a form of singularity where the fluid solution remains regular but certain first derivatives blow up.

    In this talk I will discuss the recent culmination of the wave of results initiated by Christodoulou: my work on the maximal classical development (MCD) for compressible fluids, joint with J. Speck. Roughly speaking, the MCD describes the largest region of spacetime where the Euler equations admit a classical solution. For an open set of smooth data, my work reveals the intimate relationship between shock singularity formation and the full structure of the MCD. This fully solves the 162 year old open problem of extending Riemann’s historic 1D result to 3D without symmetry assumptions. In addition to the mathematical contribution, the geo-analytic information of the MCD is precisely the correct “initial data” needed to physically describe the fluid “past” the initial shock singularity in a weak sense. I will also briefly discuss the countless open problems in the field, all of which can be viewed as “building blocks” which will shine the first lights onto the outstanding global-in-time open problem of fluids.

Date: Friday, December 10, 2022.

  • Speaker: Sifan Yu, Vanderbilt University.
  • Title: Characteristic initial value problem for 3D compressible Euler equations.
  • Abstract: We consider the characteristic initial value problem for 3D compressible Euler equations (“CharIVP” for short), which is a Cauchy problem whose initial data set is given in a pair of transversally intersecting characteristic initial hypersurface. The resolution of this problem will provide the first solution to a multi-speed characteristic initial value problem in the context of compressible fluid mechanics in multiple spatial dimensions. The setup of “CharIVP” is expected to be useful for studying the long-time dynamics of solutions. We use the geometric framework from [Speck, 19], where the Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. We determine the initial data, which consists of a pair of transversally intersecting null hypersurfaces embedded in Cartesian spacetime and constrained fluid data prescribed on such hypersurfaces. Then we prove a well-posedness result in a “characteristic diamond” region bounded below by initial null hypersurfaces, and propagate regularity with the help of the integral identities derived in [Abbrescia-Speck, 20]. This is a joint work with Jared Speck.