Blow-up criterion and the global existence of strong/classical solutions to Navier–Stokes/Allen–Cahn system

Abstract

In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions can be improved to classical solutions. Precisely, when the viscosity only depends on the density, we prove the global existence and uniqueness of the Navier–Stokes/Allen–Cahn equations.

Appendix

In the proof of local existence, it is easy to show that $\rho ,u\in L^\infty (Q_{T^{**}})$($T^{**}>0$), $A^{-1}\le \rho \le A$ (for some constant $A>1$), which leads to the following lemma.

Lemma 4.6

Suppose that $\rho ,u\in L^\infty (Q_{T^{**}})$, $A^{-1}\le \rho \le A$, $\chi$ is a smooth solution to (1.1)$_3$–(1.1)$_4$ with $\chi _0\in [0,1]$ and $\chi |_{x=0,1}=0$. Then, it holds that for any $t\in [0,T^{**})$

$$\begin{aligned} \chi (x,t)\in [0,1]. \end{aligned}$$

Proof

Equations (1.1)$_3$ and (1.1)$_4$ can be rewritten as

$$\begin{aligned} \rho ^2 \chi _t+ \rho ^2u \cdot \chi _x-\chi _{xx}=-\rho \chi (\chi ^2-1). \end{aligned}$$

(4.22)

Then, we test (4.22) by $\chi$ to obtain

$$\begin{aligned} \rho ^2 \partial _t (\chi ^2-1)- (\chi ^2-1)_{xx}+\rho ^2 u\cdot (\chi ^2-1)_x+2\rho (\chi ^2-1)=-2\rho (\chi ^2-1)^2-2\chi _x^2\le 0, \end{aligned}$$

which with the help of the maximum principle yields

$$\begin{aligned} \chi ^2-1\le 0. \end{aligned}$$

(4.23)

Considering a new function $Y(x,t)=e^{At}\chi (x,t)$, one can deduce from (4.22) that

$$\begin{aligned} \rho ^2Y_t+\rho ^2uY_x-Y_{xx}+\left[ A+\rho (\chi ^2-1)\right] Y=0, \end{aligned}$$

which by the maximum principle implies

$$\begin{aligned} \chi \ge 0. \end{aligned}$$

It together with (4.23) completes the proof of Lemma 4.6. $\square$