Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth

1.   Introduction

Traditionally, the Navier-Stokes equations have received quite a bit of attention. Recently, attention to the behavior of fluids with various viscosities has been increasing dramatically. It is because we can find such fluids everywhere. For example, water, yogurt, lubricants, sand in water, ink, gum solutions, nail polish, ketchup, molasses, ice, paint, custard, paper pulp, even blood in our body. The behavior of many of them can be described in the power law model. In that sense, we are interested in the power law model, which is a generalized Navier-Stokes system.

As mentioned in [16], electrorheological fluids are viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. The motion is governed by a system of partial differential equations consisting of electric field $E$, polarization, density $\rho$, velocity $u$, pressure $\pi$, and deviatoric stress tensor $S$. Refer to [11] for the detail description. In [11] the viscosity is the form of power law model, and the power $p$ depends on the electric field $p(|E|^2)$.

In this article we consider the following stationary system related with non-Newtonian fluids:

where $g\in L^{\infty}(\Omega)$ is a given exterior force, and $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $n = 2, 3$. Here, $D$ denotes the symmetric part of $\nabla$, given by

In addition, $S(x, D(u))$ denotes the deviatoric stress. Then $T : = S- \pi I$ is called the full shear stress, such that $-{\rm{div}}\, T$ represents the sum of the internal forces due to friction, which depends mostly on the material of the fluid.

The continuous deviatoric stress tensor $S:\Omega\times\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}$ is assumed to be $C^{1}$-regular in the gradient variable $z$ for every $z\in \mathbb{R}^{n}\setminus \{0\}$, with $S_{z}(\cdot)$ being ${\rm{Carathéodory}}$ regular and satisfying the following nonstandard growth, monotonicity and continuity assumptions:

for every $z\in \mathbb{R}^{n}\setminus \{0\}$, $z_{1}, z_{2}$, $\xi\in\mathbb{R}^{n}$ and $x, x_{1}, x_{2}\in\Omega$, where $0 <\nu\leq L$ and $\mu\ge 0$ are fixed numbers. The variable exponent function $p(\cdot):\Omega\to[0, +\infty)$ is continuous with modulus of continuity $\omega:[0, \infty)\to [0, \infty)$ satisfying

where $\gamma_{1}>3/2$ if $n = 2$ and $\gamma_{1}>9/5$ if $n = 3$. We assume that the variable exponent $p(\cdot)$ is Lipschitz continuous, i.e.

for some constant $c_p>0$. For simplicity, we set

for some fixed $B_{4R}(x_{0})\subset\subset \Omega$.

Here, we denote the variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$, by the set of all measurable functions $f:\Omega \to \mathbb{R}^{n}$ satisfying

and the space $W^{1, p(\cdot)}_{0, \sigma}(\Omega)$ is the set of all divergence free $f\in W^{1, 1}_{0}(\Omega)$ with $\|f\|_{L^{p(\cdot)}(\Omega)} + \|\nabla f\|_{L^{p(\cdot)}(\Omega)} <\infty$. For more details, we refer to [6,9]. We say that $u \in W^{1, p(\cdot)}_{loc}(\Omega)^{n}$ is a weak solution of (1), if $u$ satisfies

where $u\otimes v: = \{u^{i}v^{j}\}_{i, j}$ is a tensor product of $u$ and $v$ in $\mathbb{R}^{n}$.

In [13], the existence of weak solutions was provided for constant $p > \frac{2n}{n+2}$. In [14,15], the existence of strong solution of (1) was proved when $p$ is constant. In [14], the solution belongs to $C^{1, \alpha}$ for $p>\frac{3}{2}$ when $n = 2$ (up to boundary), and for $p>\frac{6}{5}$ when $n = 3$ (interior regularity). In [15], for $\frac{3n}{n+2} <p <2$, it is shown that $(1+|D(u)|)^{\frac{p-2}{2}}\nabla D(u)\in L^2_{loc}(\Omega)^{n\times n}$, and $u\in W^{2, t}_{loc}(\Omega)^{n}$ for all $1\le t <2$ when $n = 2$, and $u\in W^{2, \frac{3p}{1+p}}_{loc}(\Omega)^n$ when $n = 3$. In [4], for $S(z) = (1+|z|^{p-2})z$ under slip or no-slip boundary conditions a regularity is provided for shear thickening fluid $p>2$.

In case of anisotropic dissipative potential $f$, where $S = \nabla f$ and

with exponents $1 <p_1\le p_2 <\infty$ and $2\le p_2 <p_1\frac{n+2}{n}$, the existence of weak solutions is given in [3]. For the anisotropic fluid, it is shown in [5] that $u\in W^{1, p_{1}}_{0}(\Omega)^{n}\cap W^{2, s}_{loc}(\Omega)^{n}$ for some $s>1$ for $p_1>6/5$ when $n = 2$, and $p_1>9/5$ when $n = 3$, and $p_2 <p_1\frac{n+1}{n}$. Also in that article, there is a good introduction about isotropic and anisotropic cases, and the power depending on $x$, $p(x)$.

For a variable $p(x)$ depending on $E$, in [16] it is proved that a weak solution of (1) exists and it has the second weak derivative for $ \gamma_1 > \dfrac{3n}{n+2}$ and $\mu >0$. Nonetheless, it is not shown there that the second weak derivative of a weak solution belongs to Lebesgue space with variable exponent. It is just shown that a weak solution belongs to $W^{2, \gamma_1}(\Omega)^{n}$.

The lower bound of $p(\cdot)$, $\gamma_1$, that a weak solution of (1) exists is decreased to $\frac{2n}{n+2}$ in [10]. But if $\gamma_1 > \frac{2n}{n+2}$, we cannot say that $u \otimes u : D(\varphi) \in L^1$ for all $u, \varphi \in W^{1, p(\cdot)}_0(\Omega)^{n}$. In this case, $u$ is not able to be taken as a test function, hence we just consider $ \gamma_1 > \frac{3n}{n+2}$.

On the other hand, in [7] it was proved that a solution of $p(x)$-Laplace equation has weak second derivative in $L^2$. In that paper, $\mu = 0$ is allowed, but the model is not related with fluid directly. The existence of strong solution of (1) for $S(x, D(u)) = (\mu+|D(u)|)^{p(x)-2}D(u)$ with $\mu>0$ was proved in [8] for $\gamma_1>2$. And the same result was proved in [11] for $\gamma_1>\frac{9}{5}$.

In this paper, we handle non-Newtonian problem (1) where $\mu$ is allowed to be $0$ and $p(\cdot)$ is Lipschitz continuous. The result of this paper is the following theorem.

Theorem 1.1. Let $\mu\in [0, \infty)$. We assume that (2) are satisfied and $p(\cdot)$ is Lipschitz continuous function with (3). Let $u$ be a weak solution of (1) with (2). Then $u$ has the following properties:

and for all $1\leq t < 2$,

2.   Proof of main theorem

To simplify the notation, the letter $c$ will always denote any positive constant, which may vary throughout the paper. Moreover, we denote $p' = \frac{p}{p-1}$ as the Hölder conjugate exponent of $p$ and $p^* = \dfrac{n p}{n-p}$ as the Sobolev exponent of $p$ for every $p \in (1, n)$.

We recall a useful lemma about higher integrability from [1].

Lemma 2.1. Let $u \in W^{1, p(\cdot)}_{loc}(\Omega)^n$ be a weak solution of (1) and assume that $S$ fulfills conditions $(2)$ with Lipschitz continuous variable exponent, $p(\cdot)$. Then there exist constants $c, \sigma >0$, both depending on $n, \gamma_1, \gamma_2, c_p$ such that if $B_{2R} \subset \subset \Omega$, then

Here, ${{\rlap{-} \smallint }_{B_{R}}} f \, dx$ means the average of $f\in L^{1}(B_{R})$ over $B_R$. We take a constant $R_{0}\leq 1$ such that

and assume $0 <R\leq R_{0}$, throughout this paper. This assumption will be frequently used in the proof, for instance (11) and (12).

Remark 1. Although the authors only considered the case $\mu>0$ in [1], the statement is still valid for $\mu = 0$, since the proof is also available for $\mu = 0$. In this paper, we can remove the dependence of $c$ on $\gamma_{2}$ since $\gamma_{2}\leq 2$.

Since $\lim_{t \to 0^+} t^\epsilon \log t = 0$ and $\lim_{t \to +\infty} \tfrac{\log t}{t^\epsilon} = 0$ for any $\epsilon >0$, we have the following useful estimate.

Lemma 2.2. There exists a constant $c(\epsilon)>0$ depending only on $\epsilon$ such that

for all $t>0$.

Our proof is mainly based on that of [15]. We divide our proof of main theorem in three steps. At first, we derive estimation related to finite difference of $D(u)$. After then, applying fractional Sobolev embedding theorem, we can show that $u$ is locally bounded. Finally, we prove that $u$ has second derivatives using difference quotient method.

Step 1. Estimation of $(|D(u)(x+\lambda {e_k})|^{2}+|D(u)(x)|^{2}+\mu^{2})^{\frac{p(x)-2}{2}}|{\Delta _{\rm{\lambda }}} D(u)|^{2}$.

From the modulus of continuity, there exists a radius $R_{0}>0$ such that $\omega(8R_{0})\leq \frac{\sigma}{16}$. We use the following weak formulation

for all $\varphi \in W^{1, p(x)}_{0}(\Omega)^n$, where $\pi \in L^{p(\cdot)'}(\Omega)$. Let $\eta\in C^{\infty}_{c}(B_{2R}(x_{0}))$, where $B_{2R}(x_{0})\subset \Omega$ such that $\eta\equiv 1$ in $B_{R}(x_{0})$, $|\nabla\eta|\leq \dfrac2R$ and $|\nabla^{2}\eta|\leq \dfrac{4}{R^{2}}$ for some $0 <R\leq R_{0}$. Now we choose a test function $\varphi = \Delta_{-\lambda, k}(\eta^{2}\Delta_{\lambda, k}u)$, where $\Delta_{\lambda, k}f(x) = f(x+\lambda e_{k})-f(x)$. For simplicity, we will denote ${\Delta _{\rm{\lambda }}}: = \Delta_{\lambda, k}$ for $\lambda \in [-R/4, R/4]$ and $B_{2R}: = B_{2R}(x_{0})$. By simple computation, we see

The ellipticity condition in $(2)_2$ implies that

where $O_{1}^{+}: = \{x\in B_{2R} : |Du|(x+\lambda e_{k}) + |Du|(x) + \mu > 0\}$.

Note that the set $O_{1}^{+} = B_{2R}$ whenever $\mu>0$. Indeed, we here introduced the set $O_{1}^{+}$ to prove Theorem 1.1 for $\mu = 0$, in a simple and unified way with respect to $\mu\in [0, \infty)$.

Combining (7)-(9), we obtain

For simplicity, we set $\nu = 1$ since it does not make any trouble for the proof.

Estimation of $I_{1}$. To estimate $I_{1}$, we introduce a measurable set $O_{2}^{+}$:

We use the continuity assumption in $(2)_3$ to see

Therefore, Lemma 2.2 and (6) reveals

Estimation of $I_{2}$. Noting that (6) implies

and

we discover

Note that

Applying (14) and Korn's inequality, we have

And the first integral term in the last line is estimated as follows:

Combining (13), (15) and (16), we obtain

Estimation of $I_3$ will be postponed to Step 2.

Estimation of $I_{4}$. Recalling Lemma 2.1 and the De Rham theory in [10], we see $\pi\in L^{q(\cdot)}(\Omega)$ for $q(x) = (1+\sigma)p'(x)$. Observe that

by the assumption $\omega(4R_{0}) <\dfrac{\sigma}{16}$. Via a similar estimating procedure of $I_{2}$, we compute

In light of Korn's inequality, it holds that

Combining (16), (17) and (18), we obtain

Estimation of $I_{5}$. Korn's inequality yields

Consequently, it follows from combining the estimates of $I_{1}, \, I_2, \, I_4 \textrm{ and }\, I_{5}$, and (10) that

Step 2. Boundedness of $u$.

In this step, we shall prove that $u$ is locally bounded. Note that Lemma 2.1 implies that $u$ is locally Hölder continuous by Morrey embedding in case of $n = 2$ and $\gamma_{1} = 2$. So it is not necessary to consider this case. We recall $I_3$ and apply integration by parts formula for finite difference to see

We now assume that

Next we estimate $I_{3, 1}$

Defining $\theta: = \frac{s(n+2)-3n}{2n}$, we see $0 <\theta <1$ and

where $s'$ is the Hölder conjugate exponent of $s$ and $s^{*}$ is the Sobolev exponent of $s$. It then follows from interpolation that

and

On the other hand, integration by parts formula for finite difference reveals

Merging up (20) and (21), we obtain

Finally, since $p(\cdot)$ is Lipschitz continuous, (19) and (22) give the boundedness of $I_0$:

where the constant $c$ depends on $\|u\|_{W^{1, s}(B_{3R})^n}$ and $\|\pi\|_{L^{p_{1}'}(B_{2R})}$. Set

Then $\gamma_{1} \leq \hat{s} <2$ and $\frac{s(\hat{s}-2)}{\hat{s}} = \gamma_{1}-2$. Thus, H$\ddot{\mathit{\boldsymbol{o}}}$lder's inequality and (23) reveal

This implies

and then we obtain by Nikolskii's embedding theorem in [2] that

We set

then it follows that

Hence (25) can be written as

Here, we have used Korn's inequality. According to Sobolev embedding theorem and the fact that $\sigma(s)\leq s^{*}$, we see

and so, by covering, we also have

By bootstrap argument, we can conclude that

where $n < \sigma(n) = \frac{2n}{3-\gamma_{1}}$. By Morrey embedding theorem, $u$ is locally Hölder continuous, and so $u$ is locally bounded.

Step 3. Completing the proof of Theorem 1.1.

For $1 <s <\frac{2n}{3-\gamma_{1}}$, we estimate $I_{3}$ again

We select $s = 4-\gamma_{1}$. Then $2\leq s < \frac{2n}{3-\gamma_{1}}$ and

By following the calculations in (24), we also have

Applying (27) and Young's inequality to (26), we find

Combining (19), (28) and the assumption $p(x) \leq 2$, we obtain

where

We now divide (27) by $|\lambda|$ to deduce

for all $\lambda \in \left(-\frac{R}{4}, \frac{R}{4}\right)$, where we have used Korn's inequality and (19).

By difference quotient method (See [12,Section 5.8.2]), there exists the weak derivative of $\nabla u$ such that

For the time being, we suppose that

for all $q \in [1, \infty)$. Then we see that (30) deduce

Since there exists a function $w_k \in L^2(B_R)$ by (29) such that

we find $\left(1+2|D(u)(x)|^2 \right)^{\frac{p(x)-2}{4}} \left|\frac{\partial}{\partial x_k}\nabla u \right| = w_k$.

Verification of (31). To do this, we write $h(x) = \frac 12 +|D(u)(x)|^2$ and $h_\lambda(x) = \frac 12 +|D(u)(x+\lambda {e_k})|^2$, and compute

And we estimate

Hence we have

Since the integral on the right-hand side above inequality converges to 0 as $\lambda \to 0$, (31) is valid, and (4) is proved.

To verify (5), we put $V = \left(1+|D(u)|^2 \right)^{\frac{p(x)}{4}}$. Then

Using (4), the higher integrability of $D(u)$ and Lemma 2.2, we know $V \in W^{1, 2}_{loc}(\Omega)$. Therefore,

Note that following inequality holds by Young's inequality:

for any $p \in [1, 2], t \in [1, 2) \textrm{ and } |b| >0.$ For the case $n = 2$ with $t\in [1, 2)$, we estimate

Then, we immediately deduce $(5)_{1}$ from (32) and Korn's inequality. One can prove $(5)_{2}$ in a similar way. This completes our proof.

Acknowledgments

Bae was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2015R1D1A1A01057976 and 2016K2A9A2A06005080) and Youn was supported by NRF (2015R1A4A1041675). The authors would like to thank referees for valuable comments and suggestions. The authors would like to express their sincere gratitudes Jihoon Ok for helpful discussions.