Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary

Abstract: We study traveling wave solutions to the free boundary problem associated to a generalized Navier-Stokes Fourier system, which models a viscous, incompressible, heat-conducting fluid. The fluid is assumed to occupy a horizontally infinite strip-like domain with flat rigid bottom and moving upper surface. The fluid is acted upon by gravity as well as external sources of bulk force and boundary stress and an external heat source. Additionally, we allow for temperature-dependent viscosity and capillary coefficients, the latter of which gives rise to Marangoni stresses on the free surface. We develop a small data well-posedness theory in Sobolev spaces that shows that if the sources of force, stress, and heat are small, then there exists a unique solution depending continuously on these data.