The traveling wave problem for the shallow water equations with drag

The shallow water equations describe the effective dynamics of an incompressible fluid in the regime in which the horizontal scale is much larger than the vertical scale. These equations may be obtained from the free-boundary incompressible Navier-Stokes system via a formal asymptotic expansion in a parameter that measures this vertical-to-horizontal scale, and as such can model a number of interesting physical phenomena present in the NS system. In this talk we focus on the SW system obtained from NS with Navier slip conditions imposed at the fluid bottom, which results in a useful and interesting drag term in the SW system. We will discuss the construction of traveling wave solutions for the SW system as well as the limits obtained by sending the viscosity and surface tension parameters to zero. This is joint work with Noah Stevenson.