Global Well-Posedness and Decay Rates for the Three-Dimensional Compressible Phan-Thein–Tanner Model

Abstract

In this paper, we are concerned with the global well-posedness and decay rates of strong solutions for the three-dimensional compressible Phan-Thein–Tanner model. We prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in $H^3$-framework. Moreover, if the initial data belong to $L^{1}$, the convergence rates of the higher-order spatial derivatives of the solution are obtained by combining the decay estimates for the linearized equations and the Fourier splitting method.

Appendix A. Analytic Tools

We will extensively use the Sobolev interpolation of the Gagliardo-Nirenberg inequality.

Lemma 5.1

Let $0\le m,\alpha \le l$, then we have

$$\begin{aligned} \Vert \nabla ^{\alpha }f\Vert _{L^{p}}\lesssim \Vert \nabla ^{m}f\Vert _{L^{q}}^{1-\theta }\Vert \nabla ^{l}f\Vert _{L^{r}}^{\theta }. \end{aligned}$$

(5.1)

where $0\le \theta \le 1$ and $\alpha$ satisfies

$$\begin{aligned} \frac{\alpha }{3}-\frac{1}{p}=\left( \frac{m}{3}-\frac{1}{q}\right) (1-\theta )+\left( \frac{l}{3}-\frac{1}{r}\right) \theta . \end{aligned}$$

(5.2)

Here when $p=\infty$ we require that $0<\theta <1$.

Proof

This can be found in [30, p. 125, Theorem]. $\square$

We recall the following commutator estimate:

Lemma 5.2

Let $m\ge 1$ be an integer and define the commutator

$$\begin{aligned}{}[\nabla ^{m},f]g=\nabla ^{m}(fg)-f\nabla ^{m}g, \end{aligned}$$

then we have

$$\begin{aligned} \Vert [\nabla ^{m},f]g\Vert _{L^{p}}\lesssim \Vert \nabla f\Vert _{L^{p_{1}}}\Vert \nabla ^{m-1}g\Vert _{L^{p_{2}}}+\Vert \nabla ^{m}f\Vert _{L^{p_{3}}}\Vert g\Vert _{L^{p_{4}}}. \end{aligned}$$

(5.3)

and for $m\ge 0$

$$\begin{aligned} \Vert \nabla ^{m}(fg)\Vert _{L^{p}}\lesssim \Vert f\Vert _{L^{p_{1}}}\Vert \nabla ^{m}g\Vert _{L^{p_{2}}}+\Vert \nabla ^{m}f\Vert _{L^{p_{3}}}\Vert g\Vert _{L^{p_{4}}}. \end{aligned}$$

(5.4)

where $p, p_{2}, p_{3}\in (1, \infty )$ and $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}.$

Proof

For $p=p_{2}=p_{3}=2$, it can be proved by using Lemma 5.1. For the general cases, one may refer to [19, Lemma 3.1]. $\square$

We should now recall the following elementary but useful inequality.

Lemma 5.3

Assume that $\Vert \varrho \Vert _{L^{\infty }}\le 1$ and $p>1$. Let $g(\varrho )$ be a smooth function of $\varrho$ with bounded derivatives of any order, then for any integer $m\ge 1$, we have

$$\begin{aligned} \Vert \nabla ^{m}g(\varrho )\Vert _{L^{p}}\lesssim \Vert \nabla ^{m}\varrho \Vert _{L^{p}}. \end{aligned}$$

(5.5)

Proof

The proof is similar to the proof of Lemma A.2 in [38] and is omitted here. $\square$

Lemma 5.4

[6] If $r_1>1$ and $r_2\in [0, r_1]$, then it holds that

$$\begin{aligned} \int _{0}^{t} (1+t-s)^{-r_1}(1+s)^{-r_2}ds\le C(r_1,r_2)(1+t)^{-r_2}. \end{aligned}$$

(5.6)

Lemma 5.5

[1] Let $f\in H^2(\mathbb {R}^{3})$. Then we have

$$\begin{aligned}&\Vert f\Vert _{L^{\infty }}\lesssim \Vert \nabla f\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla f\Vert _{H^{1}}^{\frac{1}{2}} \lesssim \Vert \nabla f\Vert _{H^{1}},\\&\Vert f\Vert _{L^{6}}\lesssim \Vert \nabla f\Vert _{L^{2}},\\&\Vert f\Vert _{L^{q}}\lesssim \Vert f\Vert _{H^{1}},\ 2\le q\le 6. \end{aligned}$$