On the existence of unique global-in-time solutions and temporal decay rates of solutions to some non-Newtonian incompressible fluids

Abstract

In this paper, we deal with a class of incompressible non-Newtonian fluids. We first give some conditions to the viscous part of the stress tensor to set our model. We then show that there exists a unique regular solution globally in time if $u_{0}\in L^{2}\cap \dot{B}^{1}_{\infty ,1}$ and is sufficiently small in $\dot{B}^{1}_{\infty ,1}$. We finally derive temporal decay rates of the solution which are consistent with the decay rates of the linear part of our model.

Appendix

1.1 $W^{2,q}(\mathbb {R}^{3})\subset \dot{B}^{1}_{\infty ,1}(\mathbb {R}^{3})$ when $q>3$

Let $f\in W^{2,q}$. By (2.2),

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{1}_{\infty ,1}}\le C\sum ^{0}_{j=-\infty }2^{j(1+\frac{3}{q})} \left\| \Delta _{j}f\right\| _{L^{q}} + C\sum ^{\infty }_{j=1}2^{j(1+\frac{3}{q})} \left\| \Delta _{j}f\right\| _{L^{q}}=\text {I+II}. \end{aligned}$$

Since $1+\frac{3}{q}>0$, we have $\text {I}\le C \Vert f\Vert _{L^{q}}$. And since $q>3$,

$$\begin{aligned} \text {II}=C\sum ^{\infty }_{j=1}2^{j(-1+\frac{3}{q})}2^{2j} \left\| \Delta _{j}f\right\| _{L^{q}}\le C\Vert f\Vert _{\dot{B}^{2}_{q,\infty }} \le C\Vert f\Vert _{\dot{B}^{2}_{q,q}}. \end{aligned}$$

We note $\dot{W}^{2,q}=\dot{F}^{2}_{q,2}$, where $\dot{F}^{s}_{p,q}$ is the homogeneous Triebel–Lizorkin space. Since $q>3$, we have $\dot{F}^{2}_{q,2}\subset \dot{B}^{s}_{q,q}$. For the definition of the Triebel–Lizorkin space and the embedding property, see [13].

1.2 Proof of (3.12)

We bound $\Vert f\Vert _{\dot{B}^{1}_{\infty ,1}}$ as

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{\dot{B}^{1}_{\infty ,1}}&\le C \sum ^{N}_{j=-\infty } 2^{\frac{5}{2}j}\left\| \Delta _{j}f\right\| _{L^{2}} +\sum ^{\infty }_{j=N+1} 2^{-\frac{j}{2}}2^{\frac{3j}{2}}\left\| \Delta _{j}f\right\| _{L^{\infty }} \le C \left\| f\right\| _{L^{2}}\sum ^{N}_{j=-\infty } 2^{\frac{5}{2}j}+C2^{-\frac{N}{2}} \Vert f\Vert _{\dot{B}^{\frac{3}{2}}_{\infty ,1}} \\&\le C2^{\frac{5}{2}N}\left\| f\right\| _{L^{2}}+C2^{-\frac{N}{2}} \Vert f\Vert _{\dot{B}^{\frac{3}{2}}_{\infty ,1}}. \end{aligned} \end{aligned}$$

By choosing N such that $2^{\frac{5}{2}N}\left\| u\right\| _{L^{2}}\simeq 2^{-\frac{N}{2}} \Vert u\Vert _{\dot{B}^{\frac{3}{2}}_{\infty ,1}}$, we obtain

$$\begin{aligned} \left\| f\right\| _{\dot{B}^{1}_{\infty ,1}}\le C \left\| f\right\| ^{\frac{1}{6}}_{L^{2}}\left\| f\right\| ^{\frac{5}{6}}_{\dot{B}^{\frac{3}{2}}_{\infty ,1}} \end{aligned}$$

which implies (3.12).