Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients

TL;DR

This paper proves the global existence and uniqueness of solutions for a complex fluid system with variable properties, using advanced regularity techniques and assuming initial conditions are close to equilibrium.

Contribution

It establishes the global well-posedness of the inhomogeneous incompressible Navier-Stokes-Korteweg system with variable coefficients, extending previous results to more general capillary and viscosity terms.

Findings

Global existence and uniqueness of solutions proved

Solutions exist for initial data close to equilibrium in critical spaces

Relies on maximal regularity of heat, Stokes, and Lamé equations

Abstract

We consider the global well-posedness of the inhomogeneous incompressible Navier-Stokes-Korteweg system with a general capillary term. Based on the maximal regularity property, we obtain the global existence and uniqueness of solutions to the incompressible Navier-Stokes-Korteweg system with variable viscosity and capillary terms. By assuming the initial density $\rho_0$ is close to a positive constant, additionally, the initial velocity $u_0$ and the initial density $\nabla \rho_0$ are small in critical space $\dot B^{-1+d/p}_{p,1}(\mathbb R^{d})$ $(1<p<d).$ This work relies on the maximal regularity property of the heat equation, of the Stokes equation, and of the Lam\'e equation.