On the Cauchy problem of 3D incompressible Navier-Stokes-Cahn-Hilliard system

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions to the 3D incompressible Navier-Stokes-Cahn-Hilliard system, establishing local and global well-posedness under certain conditions and deriving decay rates.

Contribution

It provides new results on the well-posedness and decay rates for the 3D Navier-Stokes-Cahn-Hilliard system, including global solutions with small initial data.

Findings

Local well-posedness established via Banach fixed point theorem

Global well-posedness obtained under small initial data in rac12-norm

Optimal decay rates for higher-order derivatives derived

Abstract

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for 3D incompressible Navier-Stokes-Cahn-Hilliard equations. First, using Banach fixed point theorem, we establish the local well-posedness of solutions. Second, assuming $\|(u_0,\nabla\phi_0)\|_{\dot{H}^\frac12}$ is sufficiently small, we obtain the global well-posedness of solutions. Moreover, the optimal decay rates of the higher-order spatial derivatives of the solution are also obtained.