A Liouville Type Theorem For The Density Dependent Navier-stokes Equations And The Applications Of Related Skills In The Study Of The Wave Profile Of Water Waves

In this paper, we study the density dependent incompressible Navier-Stokesequations and wave profiles of steady water waves. For the density dependentincompressible Navier-Stokes equations, we proved a Liouville theorem, showingthat non-trivial solutions satisfying certain boundedness condition do not exist.We use the technique of Chae used in proving the Liouville theorem for theincompressible Euler equations to study the profiles of steady water waves. Forsolitary waves on rotational and irrotational flows, we derive the relations betweenthe L1and L2norms of the wave profiles. We do the same for periodic waves.From these relations, we can tell in certain circumstances whether the flow underthe water wave is rotational or irrotational.