Time:9:00-10:00, June 5 - 7 2024

Venue:E5-329, Yungu Campus

Zoom ID:952 2949 3423

Passcode:004072

Speaker:Diego Cordoba Gazolaz, Instituto de Ciencias Matemáticas

Title:Finite Time Singularities in Incompressible Fluids

Abstract:This lecture series delves into the phenomenon of finite time singularities in incompressible fluids. We will explore the existence and characteristics of these singularities through several physical scenarios, governed by different sets of equations. The focus will be on the incompressible Euler equations, the ncompressible porous media equation, and the generalized quasi-geostrophic equation.

1st Lecture9:00-10:00, Wednesday, June 5 2024

Instant Blow-Up for a Family of 2D Incompressible Active Scalars: In this part, we will construct classical solutions that lose regularity instantaneously. We will examine a family of 2D incompressible active scalar equations to understand the mechanisms behind this immediate breakdown of regularity.

2nd Lecture9:00-10:00, Thursday, June 6 2024

Finite Time Singularities for the 3D Euler Equations and the Hypodissipative Navier-Stokes Equations: This section will focus on constructing classical solutions within the local well-posedness regime that experience a blow-up in finite time. We will present two distinct scenarios to illustrate the development of singularities in 3D.

3rd Lecture9:00-10:00, Friday, June 7 2024

Finite Time Singularities for the Incompressible Porous Media Equation: Here, we will analyze the Muskat problem, which models the dynamics of the interface between two incompressible, immiscible fluids with different constant densities. We will show that there exist analytic initial data in the stable regime for the Muskat problem that evolve into the unstable regime, eventually leading to a breakdown of the solution.