Abstract
In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity.
Highlights
1 Introduction The Navier–Stokes/Allen–Cahn system, which is a combination of the compressible Navier–Stokes equations with an Allen–Cahn phase field description, is considered in this paper
Ding et al [6] proved the existence and uniqueness of global strong solutions to (1)–(4) with free boundary conditions and with the lower bound of the initial density
The following result means that the strong solution obtained by Theorem 1 is a classical solution provided that the initial data (ρ0, u0, χ0) satisfies some additional conditions
Summary
The Navier–Stokes/Allen–Cahn system, which is a combination of the compressible Navier–Stokes equations with an Allen–Cahn phase field description, is considered in this paper. Ding et al [6] proved the existence and uniqueness of global strong solutions to (1)–(4) with free boundary conditions and with the lower bound of the initial density. Considerable progress has been made to the compressible Navier–Stokes/ Allen–Cahn system, one of the natural questions is whether one could obtain the global classical solutions without any small assumption on the initial data or perturbations, where the time t could tend to +∞? The following result means that the strong solution obtained by Theorem 1 is a classical solution provided that the initial data (ρ0, u0, χ0) satisfies some additional conditions. There exists a small time T0 > 0 depending only on (ρ0, u0, χ0) such that the initial boundary value problem (1)–(7) admits a unique classical solution (ρ, u, χ) satisfying that Substituting (25) and (26) into (23), and integrating the resultant inequality over (0, t), one has t1 ν(ρ)u2x dx +
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