Events for 11/19/2024 from all calendars

Student/Postdoc Working Geometry Seminar

Time: 1:00PM - 2:00PM

Location: BLOC 302 (tent)

Speaker: Derek Wu, Texas A&M

Title: GIN X (note special time and place)

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Noah Stevenson, Princeton

Title: On the traveling and stationary wave problems for a system of viscous shallow water equations

Abstract: There is a rich history of the study of traveling wave solutions to the free boundary Euler equations - otherwise known as the water wave problem; nevertheless, it is only within the last six years that the analogous study of traveling wave solutions to free boundary fluids with viscosity - such as the incompressible Navier-Stokes equations - has commenced. Even more recently, this traveling wave study has expanded to a larger family of dissipative fluid systems including compressible flows, fluids obeying Darcy’s law, and the shallow water equations. This talk focuses specifically on the shallow water equation’s version of the water wave problem. The shallow water equations, which are derived from the free boundary Navier-Stokes equations with Navier slip boundary conditions via a rescaling, asymptotic expansion, and depth-averaging procedure, are both mathematically and computationally important. For these equations we shall discuss three recent results: (1) the traveling wave problem and the limits of vanishing viscosity and capillarity, (2) the existence of families of two-dimensional roll wave solutions, and (3) the stationary wave problem with variable bathymetry and the nature of large solutions. These theorems are unified by the fact the main engine of their proofs is the implicit function theorem, although each result utilizes a different manifestation. These are a Nash-Moser variant to handle derivative loss, a multiparameter bifurcation theorem, and an analytic global implicit function theorem.