1. Introduction

In this paper, we study Vlasov equation coupled with a shear thickening fluid (see e.g. [Reference Ha, Kim and Kim7, Reference Mucha, Peszek and Pokorny10])

(1.1)\begin{equation} \left\{\begin{array}{rcl} \partial_t f + v\cdot \nabla_x f+\nabla\cdot [(u-v)f] & = & 0,\\ \partial_t u - \nabla \cdot S(Du)+ (u\cdot\nabla_x)u + \nabla_x\pi & = & -\int_{{{\mathbb{R}}}^{3}} (u-v) f\,{\rm d}v,\\ \nabla \cdot u & = & 0. \end{array} \right. \end{equation}

Here $u:{{\mathbb {R}}}^{3}\times (0, T)\rightarrow {{\mathbb {R}}}^{3}$ is the flow velocity vector, $b:{{\mathbb {R}}}^{3}\times (0, T)\rightarrow {{\mathbb {R}}}^{3}$ is the magnetic vector, $\pi :{{\mathbb {R}}}^{3}\times (0, T)\rightarrow {{\mathbb {R}}}$ is the total pressure and $Du$ is the symmetric part of the velocity gradient, i.e.

\[ Du = D_{ij}u:= \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right), \quad i,j=1,2,3. \]

To motivate the conditions on the stress tensor $S$, we recall the following examples of constitutive laws

(1.2)\begin{equation} S(Du) = (\mu_0+\mu_1|Du|^{p-2})Du \end{equation}

where $\mu _0\geq 0$ and $\mu _1>0$ are constants (see e.g. [Reference Astarita and Marrucci1, Reference Wilkinson12]). The system is completed by the initial data:

(1.3)\begin{equation} u|_{t=0}=u_0,\quad \nabla_x\cdot u_0=0,\quad f|_{t=0}=f_0. \end{equation}

We recall some known results for the Newtonian case. Hamdache [Reference Hamdache8] proved the global existence of weak solutions to the time-dependent Stokes system coupled with the Vlasov equation in a bounded domain. Later, existence of weak solution was extended to the Vlasov–Navier–Stokes system by Boudin et al. [Reference Boudin, Desvillettes, Grandmont and Moussa3] in a periodic domain (refer also to [Reference Goudon, Jabin and Vasseur4, Reference Goudon, Jabin and Vasseur5] for hydrodynamic limit problems). When the fluid is inviscid, Baranger and Desvillettes established the local existence of solutions to the compressible Vlasov–Euler equations [Reference Baranger and Desvillettes2].

On the other hand, recently, Mucha et al. in [Reference Mucha, Peszek and Pokorny10] considered (1.1) and $S(Du)=(1+|Du|^{p-2})Du$ with $p\geq \frac {11}{5}$ for the case of a periodic domain, and they established the existence of solutions $(f,u)$ for a large initial data (see also [Reference Ha, Kim and Kim7]). Moreover, a divergence free vector $u$ and nonnegative function $f$ satisfy

\begin{equation*} \begin{aligned} u & \in L^{\infty}(0,T;\dot{W}^{1,p}_{div}(\mathbb{T}^{3}))\cap C([0,T];L^{2}_{div}(\mathbb{T}^{3}))\cap L^{2}(0,T;W^{2,2}(\mathbb{T}^{3}))\cap \\ & \quad\cap L^{\infty}(0,T;W^{1,2}({{\mathbb{R}}}^{3})) \cap L^{p}(0,T;\dot W^{1,3p}(\mathbb{T}^{3})),\quad \partial_t u\in L^{2}(0,T;L^{2}(\mathbb{T}^{3})),\\ f & \in L^{\infty}((0,T)\times\mathbb{T}^{3}\times{{\mathbb{R}}}^{3})\cap L^{\infty}(0,T;L^{1}(\mathbb{T}^{3}\times{{\mathbb{R}}}^{3})),\quad M_2 f\in L^{\infty}([0,T]). \end{aligned} \end{equation*}

Here, for non-negative and integrable functions $f$, we denote

\[ M_\alpha f(t):=\int_{\mathbb{T}^{3}\times{{\mathbb{R}}}^{3}}|v|^{\alpha} f(t,x,v)\,{\rm d}x\,{\rm d}v, \quad M_0f = \|f\|_{L^{1}} = 1. \]

Recently, Han-Kwan [Reference Han-Kwan9] showed the large time behaviour of small data solutions to the Vlasov–Navier–Stokes system, that is $p=2$ in (1.1) on ${{\mathbb {R}}}^{3} \times {{\mathbb {R}}}^{3}$. More speaking, he proved that for all $t \geq 0$ and $\alpha \in (0,3/2)$

(1.4)\begin{equation} E(t) \leq \frac{\varphi_\alpha(E(0))}{(1+t)^{\alpha}},\quad E(t):=\frac 12\left(\int_{{{\mathbb{R}}}^{3}\times{{\mathbb{R}}}^{3}} |v|^{2}f\,{\rm d}x\,{\rm d}v + \int_{{{\mathbb{R}}}^{3}} |u|^{2} \,{\rm d}x\right). \end{equation}

In this direction, we focus that Han-Kwan'a result extends to (1.1)–(1.3). Precisely, the solutions $(f,u)$ for Vlasov equation coupled with a shear thickening fluid (1.1)–(1.3) with a large initial data on ${{\mathbb {R}}}^{3} \times {{\mathbb {R}}}^{3}$ have the decay rate (1.4) under some assumptions. Unlike Han's results, we cannot use characteristic method (see [Reference Han-Kwan9, § 3] for the fluid equations, and thus, we give additional conditions. Our analysis is based on Han's approach, however to deal with the non-Newtonian case, we focus on the control of the stress tensor. Now, we present the main result of the paper.

Theorem 1.1 Let $p\geq \frac {11}{5}$ and

(1.5)\begin{equation} \|\rho_f\|_{L^{\infty}((0,T)\times {{\mathbb{R}}}^{3})}<\infty, \quad \rho_f(t,x):=\int_{{{\mathbb{R}}}^{3}}f(t,x,v)\,dv. \end{equation}

Suppose that the initial data $(f_0,u_0)$ satisfy

1. (i) $0\leq f_0\in (L^{1}\cap L^{\infty })({{\mathbb {R}}}^{3}\times {{\mathbb {R}}}^{3})$, $\textrm {supp}f_0(x,\cdot )\subset B(R)$ for some $R>0$ and a.a. $x\in {{\mathbb {R}}}^{3}$, where $B(R)$ is a ball centred at $0$ with radius $R$,

2. (ii) $u_0\in (L^{p}\cap W^{1,2}_{div})({{\mathbb {R}}}^{3})$ with $1\leq p<2$.

Then, there exists a constant $C$ (independent of $t$) such that global strong solutions $(f,u)$ to (1.1)–(1.3) satisfies

(1.6)\begin{equation} E(t) \leq \frac{C}{(1+t)^{\alpha}}, \quad \forall t \geq 0,\quad \alpha \in (0,3/2). \end{equation}

Remark 1.2 The result in [Reference Mucha, Peszek and Pokorny10] is also hold on the whole space. If the density function $f$ is ignored, the system (1.1) becomes to the Navier–Stokes equation of non-Newtonian type:

(1.7)\begin{equation} \partial_t u - \nabla \cdot (1+|Du|^{p-2})Du+ (u\cdot\nabla x)u + \nabla \pi=0. \end{equation}

For this equations, we knew that the solution for (1.7) satisfies the optimal temporal decay rate, that is, $\|u(t)\|_{L^{2}}\leq C(1+t)^{-{3}/{4}}$ for $p\geq 11/5$, which is the same result to Navier–Stokes equations (see also [Reference Guo and Zhu6, Reference Nečasová and Penel11]).