PDE Seminar Fall 2018

Fridays, 4:10pm, Stevenson Center 1307

Date: Friday, November 9, 2018.

  • Speaker: Chenyun Luo, Department of Mathematics, Vanderbilt University.
  • Title: Local well-posedness for the motion of a compressible, self-gravitating liquid droplet with free surface boundary.
  • Abstract: We prove the local well-posedness for the compressible free-boundary Euler equations under the influence of the self-gravity. We construct the local (in time) solution by tangentially smoothed Euler equations whose lifespan Tε depends on the smoothing coefficient ε and then obtain a positive lower bound for Tε by establishing an energy estimate that is uniform in ε. Our approach does not require the classical artificial viscosity approximation scheme introduced by Coutand and Shkoller.

Date: Friday, November 16, 2018.

  • Speaker: Wenqing Hu, Department of Mathematics, Missouri University of Science and Technology.
  • Title: On 2d incompressible Euler equations with partial damping and some related model problems.
  • Abstract: The statistical approach led R. H. Kraichnan to conjecture the existence of energy and enstrophy transfers between different Fourier modes in 2d Navier-Stokes turbulence [Phys. Fluids 10 (1967), no. 7, 1417–1423]. The present work is motivated by an interesting question: what if some of the Fourier modes are removed or added? In the first part of this talk, we illustrate that damping of spatial frequencies of Fourier modes can have an effect on properties of the solutions, hence in turn on the transfer process of 2d turbulence, as the modes are associated with vorticity gradients generated at small scales in the turbulence. This part is based on joint work with Vladimir Sverak and Tarek Elgindi.In the second part of this talk we introduce a finite-dimensional model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states, and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. Our model problem is used to illustrate some geometric features related to 2-dimensional Euler equations with partial damping. This part is based on a recent work of the speaker.

Date: Friday, November 30, 2018.

  • Speaker: Willie Wong, Department of Mathematics, Michigan State University.
  • Title: Stability with modified scattering of almost plane-wave solutions to the membrane equations.
  • Abstract: I will present a recent result with my student L. Abbrescia, where we prove the global nonlinear stability of planar travelling wave solutions to the membrane equations in three and higher spatial dimensions. The membrane equations exhibit good null structure, and hence this result is complementary to the earlier result of Speck, Holzegel, Luk, and the speaker showing nonlinear stability of plane-symmetric blow-up for genuinely nonlinear quasilinear wave equations. An interesting facet of our proof is that the structure of the perturbed equations forces us to close the estimates with modified rates of scattering. This can be interpreted as a leakage of energy from the infinite-energy background travelling wave to the perturbation.

Date: Friday, December 7, 2018.

  • Speaker: Jinping Zhuge, Department of Mathematics, University of Kentucky.
  • Title: Recent progress on boundary layer problems in periodic homogenization.
  • Abstract: This talk is concerned with periodic homogenization of linear elliptic
    equations in divergence form with oscillating Dirichlet data or Neumann data of rst
    order. For example, the Dirichlet problem with oscillating data reads as follows:-div(A(x/ε) ∇ uε) in Ω ,
    uε(x)=f(x,x/ε) on ∂Ω

    where A(y) and f(x, y) are 1-periodic in y, and ε > 0 is tiny. Recently, it has been known that, if Ω is uniformly convex (and smooth), the above equation homogenizes (convergence in some sense) to

    -div(Â ∇ u0) in Ω ,
    u0(x)=f̄(x) on ∂Ω

    where  is the homogenized coecient matrix (constant) and f̄ is the homogenized boundary data. In this talk, I will present the recent progress on this problem, including the sharp convergence rates and the regularity of the homogenized boundary
    data.