Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model

Abstract

We investigate a generic compressible two-fluid model with common pressure ($P^+=P^-$) in $\mathbb {R}^3$. Under some smallness assumptions, Evje–Wang–Wen [Arch Rational Mech Anal 221:1285–1316, 2016] obtained the global solution and its optimal decay rate for the 3D compressible two-fluid model with unequal pressures $P^+\ne P^-$. More precisely, the capillary pressure $f(\alpha ^-\rho ^-)=P^+-P^-\ne 0$ is taken into account, and is assumed to be a strictly decreasing function near the equilibrium. As indicated by Evje–Wang–Wen, this assumption played a key role in their analysis and appeared to have an essential stabilization effect on the model. However, global well–posedness of the 3D compressible two-fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we prove global existence and large time behavior of classical solutions to the 3D compressible two-fluid model with common pressure. To the best of our knowledge, we establish the first result on the global existence of classical solutions to the 3D compressible two-fluid model with common pressure and without capillary effects. The method relies upon careful analysis of the linearized system, exploitation of the algebraic structure of the nonlinear system, and the introduction of an auxiliary velocity $v=(2\mu ^++\lambda ^+)u^+-(2\mu ^-+\lambda ^-)u^-$ which plays the role of the effective viscous flux (since in this system $P^+= P^-$) in the single phase case: such velocity has better regularity than phase velocities $u^\pm$.

Analytic tools

We recall the Sobolev interpolation of the Gagliardo–Nirenberg inequality.

Lemma A.1

(Gagliardo–Nirenberg interpolation, general case). Let $1 \le$ $p, q \le \infty , m \in \mathbb {N}, k \in \mathbb {N}_0$, with $0 \le k<m$, and let a, r be such that

$$\begin{aligned} 0 \le a \le 1-k / m \end{aligned}$$

and

$$\begin{aligned} (1-a)\left( \frac{1}{p}-\frac{m-k}{3}\right) +a\left( \frac{1}{q}+\frac{k}{3}\right) =\frac{1}{r} \in (-\infty , 1]. \end{aligned}$$

Then there exists a constant $c=c(m, p, q, a, k)>0$ such that

$$\begin{aligned} \left\| \nabla ^k f\right\| _{L^r} \le c\Vert f\Vert _{L^q}^a\left\| \nabla ^m f\right\| _{L^p}^{1-a} \end{aligned}$$

(A.1)

for every $f \in L^q\left( \mathbb {R}^3\right) \cap \dot{W}^{m, p}\left( \mathbb {R}^3\right)$(where the homogeneous Sobolev space $\dot{W}^{m, p}\left( \mathbb {R}^3\right)$ is the space of all functions $f\in L_{loc}^1(\mathbb {R})^3$ whose $\alpha$-th weak derivative $\partial ^\alpha f$ belongs to $L^p(\mathbb {R}^3)$ with $|\alpha |=m$), with the following exceptional cases:

(i) If $k=0, m p<3$, and $q=\infty$, we assume that f vanishes at infinity.

(ii) If $1<p<\infty$ and $m-k-3 / p$ is a nonnegative integer, then (A.1) only holds for $0<a \le 1-k / m$.

Proof

See [12, pp. 403, Theorem 12.87] and [16, pp. 125, Theorem]. $\square$

Lemma A.2

([6, 10, 13]). For any integer $k\ge 1$, we have

$$\begin{aligned} \left\| \nabla ^k(fg)\right\| _{L^p} \lesssim \left\| f\right\| _{L^{p_1}}\left\| \nabla ^kg\right\| _{L^{p_2}} +\left\| \nabla ^kf\right\| _{L^{p_3}}\left\| g\right\| _{L^{p_4}}, \end{aligned}$$

and

$$\begin{aligned} \left\| \nabla ^k(fg)-f\nabla ^kg\right\| _{L^p} \lesssim \left\| \nabla f\right\| _{L^{p_1}}\left\| \nabla ^{k-1}g\right\| _{L^{p_2}} +\left\| \nabla ^kf\right\| _{L^{p_3}}\left\| g\right\| _{L^{p_4}}, \end{aligned}$$

where $p, p_1, p_{2}, p_{3}, p_{4} \in [1, \infty ]$ and

$$\begin{aligned} \frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}. \end{aligned}$$