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

Noah Stevenson (Princeton University)
Thursday, January 30, 2025
4:00-5:00 PM
4088 East Hall Map
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.
Building: East Hall
Event Type: Workshop / Seminar
Tags: Applied Mathematics, Mathematics
Source: Happening @ Michigan from Differential Equations Seminar - Department of Mathematics, Department of Mathematics