Seminar

Abstract: We introduce and analyse a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier-Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of stresses, and the Beavers-Joseph-Saffman law.

The QQ systems are systems of polynomial equations that arise in various geometric settings, including the enumerative geometry of Nakajima varieties and elements of the (deformed)  geometric Langlands correspondence. These equations are related to the integrable models of spin chain type, linked to quantum groups and Yangians. Specifically, the solutions to the QQ-system equations characterize the spectrum of these integrable models via the so-called Bethe ansatz equations.

Abstract: In this talk, we will study the Cauchy problem for the Euler equations in vorticity form. With the initial data in L^1∩L^∞, the uniqueness of the solution is guaranteed by Yudovich's classic result. However, it is a long-standing open question if it is possible to extend the result to the L^p scale. We will disprove it by constructing a one-parameter family of solutions with the same initial data and external force.