# [Seminar] Geometric PDE and Applied Analysis Seminar

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Geometric PDE and Applied Analysis Seminar (February 9, 2024)

**Talk 1: 14:00-15:00**

Speaker: Prof. Yoshiyuki Kagei (Tokyo Institute of Technology)

Title: Eckhaus instability of the compressible Taylor vortices

Abstract: This talk is concerned with the bifurcation and stability of the compressible Taylor vortex. Consider the compressible Navier-Stokes equations in a domain between two concentric infinite cylinders. If the outer cylinder is at rest and the inner one rotates with sufficiently small angular velocity, a laminar flow, called the Couette flow, is stable. When the angular velocity of the inner cylinder increases, beyond a certain value of the angular velocity, the Couette flow becomes unstable and a vortex pattern, called the Taylor vortex, bifurcates and is observed stably. This phenomenon is mathematically formulated as a bifurcation and stability problem. In this talk, the compressible Taylor vortices are shown to bifurcate near the criticality for the incompressible problem when the Mach number is sufficiently small. The localized stability of the compressible Taylor vortices is considered under axisymmetric perturbations and it is shown that the Eckhaus instability of compressible Taylor vortices occurs as in the case of the incompressible ones.

**Talk 2: 15:00-16:00**

Speaker: Prof. Ryo Takada (University of Tokyo)

Title: Large time behavior of global solutions to the rotating Navier-Stokes equations

Abstract: We consider the large time behavior of global solutions for the initial value problem of the Navier-Stokes equations with the Coriolis force in the three-dimensional whole space. We establish the \(L^p\) temporal decay estimates with the dispersion effect of the Coriolis force for global solutions. Moreover, we show the large time asymptotics of global solutions behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution. This talk is based on the joint work with Takanari Egashira (Kyushu University).

**Talk 3: 16:15-17:15**

Speaker: Prof. Lorenzo Cavallina (Tohoku University)

Title: A characterization of radial symmetry for composite media by overdetermined level sets

Abstract: In this talk, we introduce the concept of ``overdetermined level set" (that is, a set where both a given function and the absolute value of its gradient are constant). As a corollary of J.Serrin's celebrated symmetry theorem for overdetermined elliptic problems, we know that the ball is the only domain such that the solution to some elliptic problem for the Dirichlet Laplacian, called the torsion problem, admits at least one overdetermined level set. Moving beyond the classical case, we study how this symmetry result generalizes to a multi-phase setting (that is, when the Laplacian is replaced with an elliptic operator in divergence form with piece-wise constant coefficients that take finitely many values). Notice that, in a multi-phase setting, various types of overdeterminations are possible, depending on the number and relative position of the overdetermined level sets. In this talk, we give a complete characterizion in the two-phase setting by means of overdetermined level sets. The content of this talk is based on a joint work with Giorgio Poggesi (Univ. of Western Australia).

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