Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

Abstract

In this paper, we study the uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in the half-space R + 3 . We consider the Navier-slip boundary condition for velocity and the Dirichlet boundary condition for the gradient of density. By establishing the conormal energy estimates, we prove that there exists a unique strong solution of the model in a finite time interval [ 0 , T 0 ] , where T 0 is independent of the capillary and viscosity coefficients, and the solution is uniformly bounded in a conormal Sobolev space. Based on the aforementioned uniform estimates, we further show that there exists a constant 0 < T 1 ⩽ T 0 , such that the solutions of this model converge to the solution of the inhomogeneous incompressible Euler equations with the rates of convergence in L ∞ ( 0 , T 1 ; L 2 ( R + 3 ) ) and L ∞ ( 0 , T 1 ; H 1 ( R + 3 ) ) , as the capillary and viscosity coefficients tend to zero simultaneously.