Asymptotic behavior of traveling wave solutions of the equations for the flow of a fluid with small viscosity and capillarity

Abstract

We study the oscillations of the traveling wave solutions of \[ { v t = − p ( u ) x + ϵ v x x − δ u x x x u t = v x , \left \{ {_{{v_t} = - p{{\left ( u \right )}_x} + \epsilon {v_{xx}} - \delta {u_{xxx}}}^{{u_t} = {v_x},}} \right . \] for small ϵ \epsilon and δ \delta . These solutions give information about the structure of the shock layers in fluids with small viscosity and capillarity. We conclude that the traveling wave has oscillations with increasing amplitude when ϵ \epsilon and δ \delta approach zero such that δ ≠ O ( ϵ 2 ) \delta \ne O\left ( {{\epsilon ^2}} \right ) . When δ = o ( ϵ 2 ) \delta = o\left ( {{\epsilon ^2}} \right ) , if there are oscillations, their amplitude decreases to zero as ϵ \epsilon and δ \delta approach zero. When δ = ϵ 2 \delta = {\epsilon ^2} the shape of the traveling wave is independent of the magnitude of ϵ \epsilon and δ \delta .