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

• DOI: 10.1088/1361-6544/ad1b8c
• Publication Date: 2024-01-22T06:27:53Z
• Authors: Fucai Li; Shuxing Zhang; Zhipeng Zhang
• Keywords: 0101 mathematics; 01 natural sciences

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.