Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum

Abstract

In this paper, we study the initial-boundary value problem of the 1D compressible Navier--Stokes/Cahn--Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem. No any initial compatibility conditions are required via time weighted techniques, which leads to a loss of regularity near the initial time. Therefore, the uniqueness of solutions obtained in this paper is even more challenging. To address this issue, we establish refined growth estimates and singular-in-time weighted energy estimates that induce a Gronwall-type structure, which ultimately allows us to close the uniqueness proof in Eulerian coordinates without passing to Lagrangian coordinates.