Temporal decay of strong solutions for generalized Newtonian fluids with variable power-law index

We are interested in studying the decay properties for solutions of a system of nonlinear partial differential equations (PDEs) modelling the rheological response of electrorheological fluids. The electrorheological fluid is a viscous fluid with a special property: When it is disposed to an electro-magnetic field, the viscosity exhibits a significant change. For instance, there exist some electrorheological fluids whose viscosity changes by a factor of $1000$ as a response to the application of an electric field within 1ms. These days, some useful electrorheological fluids were found with the quality and potential for a wide range of scientific and industrial applications, including for example, clutches, shock absorbers and actuators. In this paper, we consider the reduced model for the incompressible electrorheological fluids, which consists of the following system of PDEs:

where the extra stress tensor is of the form

Indeed, the exponent $p(\cdot)$ depends on the magnitude of the electric field $|\boldsymbol{E}|$, which is the solution of quasi-static Maxwell’s equations. However, since the equations for $|\boldsymbol{E}|$ decouple from (1.1)-(1.2), we can consider $p(\cdot)$ as a given function and restrict ourselves to the study of the equations (1.1)-(1.3). In the above system of equations, $\boldsymbol{u}:Q_{T}\rightarrow\mathbb{R}^{d}$, $p:Q_{T}\rightarrow\mathbb{R}$, denote the velocity field and pressure respectively, and $\boldsymbol{D}\boldsymbol{u}$ is the symmetric velocity gradient, i.e. $\boldsymbol{D}\boldsymbol{u}=\frac{1}{2}(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^{T})$. Here we prescribe the initial condition

Such electrorheological models were studied in [17, 18], where the detailed description of the model is presented and existence theory and numerical approximation are developed.

In the present paper, we shall investigate the decay properties of the model (1.1)-(1.3). Regarding this matter, for the Navier–Stokes equations, there are a large number of contributions; see, for example, [19, 20] where the Fourier splitting method was developed, and [9, 8, 24, 21, 22, 15] with the references therein for related results such as upper and lower bounds on the decay rate for various norms and classes of initial data. For the magnetohydrodynamics equations where the Navier–Stokes equations are coupled with Maxwell’s equations, see [12, 23, 1, 6]. On the other hand, for the non-Newtonian fluid flow model, the algebraic $L^{2}$ decay of solutions was examined in [5, 14, 4], and the similar results for the non-Newtonian fluids combined with Maxwell’s equations were studied in [10, 11].

To the best of our knowledge, there is no result for the decay properties of solutions to generalized Newtonian fluids with a variable power-law index. In this paper, we shall study the $L^{2}$ decay rates for the $L^{2}$-norm and $H^{1}$-seminorm of strong solution to the electrorheological fluids.

In this section, we first introduce some notations and preliminaries which will be used throughout the paper. For two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$, $\boldsymbol{a}\cdot\boldsymbol{b}$ denotes the scalar product, and $C$ denotes a generic positive constant, which may differ at each appearance. For $1\leq p\leq\infty$ and $k\in\mathbb{N}$, we mean by $W^{k,p}(\mathbb{R}^{d})$ the standard Sobolev space and we denote $H^{k}(\mathbb{R}^{d})=W^{k,2}(\mathbb{R}^{d})$. Furthermore, for simplicity, we write $\|\cdot\|_{p}=\|\cdot\|_{L^{p}(\mathbb{R}^{d})}$ and $\|\cdot\|_{k,p}=\|\cdot\|_{W^{k,p}(\mathbb{R}^{d})}$. We also recall Korn’s inequality (see, for instance, Lemma 2.7 in [16]).

Next, since we deal with the variable power-law index, we introduce the variable-exponent Lebesgue space. We denote a set of all measurable functions $p:\Omega\rightarrow[1,\infty)$ by $\mathcal{P}(\Omega)$ and we call a function $p\in\mathcal{P}(\Omega)$ a variable exponent. Let $\Omega\subset\mathbb{R}^{d}$ be open and $p\in\mathcal{P}(\Omega)$. We define the bounds for $p$ as $p^{-}={\rm{ess}}\,\inf_{x\in\Omega}p(x)$ and $p^{+}={\rm{ess}}\,\sup_{x\in\Omega}p(x).$ For each measurable functions $f:\Omega\rightarrow\mathbb{R}$, we define the modular $|f|_{p(\cdot)}$ by

Then we define the variable-exponent Lebesgue spaces with the corresponding Luxembourg norms:

If $1<p^{-}\leq p^{+}<\infty$, it is straightforward to show that the variable-exponent space $L^{p(\cdot)}(\Omega)$ is a reflexive, separable Banach space. Throughout this paper, we shall assume that $1<p^{-}\leq p^{+}<\infty$.

Next we introduce the minimum regularity of the exponent $p(\cdot)$, which guarantees the validity of various results from the theory of classical Lebesgue space: log-Hölder continuity.

Note that, since $p\mapsto\frac{1}{p}$ is bilipschitz form $[p^{-},p^{+}]$ to $\left[\frac{1}{p^{+}},\frac{1}{p^{-}}\right]$, if $p\in\mathcal{P}(\Omega)$ and $p^{+}<\infty$, we can show that $p\in\mathcal{P}^{\rm{log}}(\Omega)$ if and only if $p$ is globally Hölder continuous. For further information, see [2] as an extensive source of information for the variable-exponent spaces.

Next, we introduce the notation

and define the following energies:

In this paper, for the sake of simplicity, we consider the case of three space dimensions. Note that the proof for the general case follows in a similar manner. The existence of strong solutions of the model (1.1)-(1.3) can be found in some literatures. For example, in [3], it was shown that the local strong solutions exist under the condition $\frac{7}{5}<p^{-}\leq p^{+}\leq 2$ in a bounded domain with periodic boundary conditions. On the other hand, for the case of the whole space $\mathbb{R}^{3}$, in [16] the authors considered the model with a constant exponent $p$, and proved the existence of global-in-time strong solutions provided that $p\geq\frac{11}{5}$. For the variable-exponent case, we can show the existence of strong solutions in $\mathbb{R}^{3}$ to the equations (1.1), (1.2) with

where the exponent $p(\cdot)$ only depends on the spatial variable $x\in\mathbb{R}^{3}$. Indeed, if we assume $p\in W^{1,\infty}(\mathbb{R}^{3})\cap\mathcal{P}^{\rm{log}}(\mathbb{R}^{3})$ with $p^{-}\geq\frac{11}{5}$, by following the argument presented in [16] we can show in a straightforward manner that the global strong solutions of (1.1), (1.2) and (2.3) exist. Furthermore, based on the method presented in [13], we can also show the existence of global strong solutions to (1.1), (1.2) and (2.3) provided that $p\in W^{1,\infty}(\mathbb{R}^{3})\cap\mathcal{P}^{\rm{log}}(\mathbb{R}^{3})$ and $p^{-}>\frac{5}{3}$, under the smallness assumption of the initial data. The existence of global strong solutions of the model (1.1)-(1.3) in $\mathbb{R}^{3}$ where $p=p(t,x)$ depends on both time and space variables is still open.

In this paper, we consider the aforementioned global-in-time strong solutions of the model (1.1), (1.2) and (2.3) with the condition $p^{-}\geq\frac{11}{5}$. Note however, that our proofs of Theorem 2.4 and Theorem 2.5 also work for the space-time-dependent power-law index, and hence we can extend the current results to the case of the equations (1.1)-(1.3) if the existence of corresponding strong solutions for $p=p(t,x)$ is guaranteed. Here strong solutions of the model (1.1), (1.2) and (2.3) mean that $\boldsymbol{u}\in L^{\infty}((0,T);H^{1}(\mathbb{R}^{3})^{3})\cap L^{2}((0,T);H^{2}(\mathbb{R}^{3})^{3})$, $|\nabla\boldsymbol{u}|\in L^{p(\cdot)}(Q_{T})\cap L^{\infty}((0,T);L^{p(\cdot)}(\mathbb{R}^{3}))$ and $\partial_{t}\boldsymbol{u}\in L^{2}(Q_{T})^{3}$ with the following energy inequalities:

Now we are ready to state our main theorems. Note here that the condition $p\in\mathcal{P}^{\rm{log}}(\mathbb{R}^{3})\cap W^{1,\infty}(\mathbb{R}^{3})$ is required for the existence of strong solutions, and not for the proof of Theorem 2.4, and hence will be omitted in the statement of the theorem. On the other hand, the condition $p\in W^{1,\infty}(\mathbb{R}^{3})$ is needed for the proof of Theorem 2.5.

In this section, we aim to prove Theorem 2.4. Note that the following lemma also holds for the case $p=p(t,x)$ and the proof for this general case is exactly same as the proof below by adding $t$ to the exponent. We first define $\boldsymbol{G}(\boldsymbol{D}\boldsymbol{u})\mathrel{\mathop{:}}=\left((\overline{\boldsymbol{D}}\boldsymbol{u})^{p(x)-2}-1\right)\boldsymbol{D}\boldsymbol{u}$ and rewrite the equations (1.1) as

Furthermore, we need the following estimate, where $\widehat{f}$ denotes the Fourier transformation of $f$.

We begin with the following a priori estimate.