Global existence and optimal decay rates for a generic non--conservative compressible two--fluid model

Abstract

We investigate global existence and optimal decay rates of a generic non-conservative compressible two–fluid model with general constant viscosities and capillary coefficients, and our main purpose is three–fold: First, for any integer $\ell \ge 3$, we show that the densities and velocities converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and the k($\in [1, \ell ]$)–order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, which are the same as ones of the compressible Navier–Stokes–Korteweg system. This can be regarded as non-straightforward generalization from the compressible Navier–Stokes–Korteweg system to the two–fluid model. Compared to the compressible Navier–Stokes–Korteweg system, many new mathematical challenges occur since the corresponding model is non-conservative, and its nonlinear structure is very terrible, and the corresponding linear system cannot be diagonalizable. One of key observations here is that to tackle with the strongly coupling terms, we will introduce the linear combination of the fraction densities ($\beta ^+\alpha ^+\rho ^++\beta ^-\alpha ^-\rho ^-$), and explore its good regularity, which is particularly better than ones of two fraction densities ($\alpha ^\pm \rho ^\pm$) themselves. Second, the linear combination of the fraction densities ($\beta ^+\alpha ^+\rho ^++\beta ^-\alpha ^-\rho ^-$) converges to its corresponding equilibrium state at the $L^2$ rate $(1+t)^{-\frac{3}{4}}$, and its k($\in [1, \ell ]$)–order spatial derivative converges to zero at the $L^2$ rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$, but the fraction densities ($\alpha ^\pm \rho ^\pm$) themselves converge to their corresponding equilibrium states at the $L^2$ rate $(1+t)^{-\frac{1}{4}}$, and the k($\in [1, \ell ]$)–order spatial derivatives of them converge to zero at the $L^2$ rate $(1+t)^{-\frac{1}{4}-\frac{k}{2}}$, which are slower than ones of their linear combination ($\beta ^+\alpha ^+\rho ^++\beta ^-\alpha ^-\rho ^-$) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well–chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two–fluid model.

Appendix A: Analytic Tools

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

Lemma A.1

Let $0\le i, j\le k$, then we have

$$\begin{aligned} \left\| \nabla ^i f\right\| _{L^p}\lesssim \left\| \nabla ^jf\right\| _{L^q}^{1-a}\left\| \nabla ^k f\right\| _{L^r}^a \end{aligned}$$

where a satisfies

$$\begin{aligned} \frac{i}{3}-\frac{1}{p}=\left( \frac{j}{3} -\frac{1}{q}\right) (1-a)+\left( \frac{k}{3}-\frac{1}{r}\right) a. \end{aligned}$$

Especially, while $p=q=r=2$, we have

$$\begin{aligned} \left\| \nabla ^if\right\| _{L^2}\lesssim \left\| \nabla ^jf\right\| _{L^2}^\frac{k-i}{k-j}\left\| \nabla ^kf\right\| _{L^2}^\frac{i-j}{k-j}. \end{aligned}$$

Proof

This is a special case of [24, pp. 125, THEOREM]. $\square$

Next, to estimate the $L^p$–norm of the spatial derivatives of the product of two functions, we shall recall the following estimate:

Lemma A.2

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

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

Proof

See [19]. $\square$

Finally, the following two lemmas concern the estimate for the low–frequency part and the high–frequency part of f.

Lemma A.3

If $f\in L^r({\mathbb {R}}^3)$ for any $2\le r\le \infty$, then we have

$$\begin{aligned} \Vert f^l\Vert _{L^r}+\Vert f^h\Vert _{L^r}\lesssim \Vert f\Vert _{L^r}. \end{aligned}$$

Proof

For $2\le r\le \infty$, by virtue of Young’s inequality for convolutions, for the low frequency, it holds that

$$\begin{aligned} \Vert f^l\Vert _{L^r}\lesssim \Vert {\mathfrak {F}}^{-1}\phi \Vert _{L^1}\Vert f\Vert _{L^r}\lesssim \Vert f\Vert _{L^r}, \end{aligned}$$

and hence

$$\begin{aligned} \Vert f^h\Vert _{L^r}\lesssim \Vert f\Vert _{L^r}+\Vert f^l\Vert _{L^r}\lesssim \Vert f\Vert _{L^r}. \end{aligned}$$

$\square$

Lemma A.4

Let $f\in H^k({\mathbb {R}}^3)$ for any integer $k\ge 2$. Then there exists a positive constant $C_0$ such that

$$\begin{aligned} \Vert \nabla ^j f^h\Vert _{L^2}\le C_0 \Vert \nabla ^{j+1}f\Vert _{L^2}, \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla ^{j+1} f^l\Vert _{L^2}\le C_0\Vert \nabla ^{j}f\Vert _{L^2}, \end{aligned}$$

for any $0\le j\le k-1$.

Proof

This lemma can be shown directly by the definitions of the low–frequency and high–frequency of f and the Plancherel theorem, and thus we omit the details. $\square$