Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

Abstract

In this paper, we study the degenerate parabolic system

$$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$

where $X=\{X_{1},\ldots,X_{m} \}$ is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients $a_{ij}^{\alpha \beta }$ are measurable functions and their skew-symmetric part can be unbounded. After proving the $L^{2}$ estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.

1 Introduction

Let $\{X_{1},\dots,X_{q}\}$ be a system of smooth real vector fields in a neighborhood Ω̃ of some bounded domain $\varOmega \subset \mathbb{R}^{n}$ $(n\geq q)$, satisfying Hörmander’s rank condition up to the order s and free up to the order s. The main purpose of this paper is to study higher integrability for weak solutions to nondiagonal quasilinear degenerate parabolic system

$$ u_{t}^{i} + X_{\alpha }^{\ast } \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$

(1.1)

where $i,j=1,2,\ldots,N$; $\alpha,\beta =1,2,\ldots,q$; $z=(x,t) \in Q_{T}=\varOmega \times (0,T)$; $X_{\alpha }^{\ast }=-X_{\alpha }+c _{\alpha }$ $(c_{\alpha }=\sum_{k = 1}^{n} b_{\alpha k}(x)\frac{ \partial }{\partial x_{k}}\in C^{\infty })$ is the transposed vector field of $X_{\alpha }$. The assumptions on functions $g_{i}$, $f_{i}^{\alpha }$ and the coefficients will be specified later.

A function $u \in W_{2}^{1,1}(Q_{T}, \mathbb{R}^{N})$ is called a weak solution to (1.1) if

$$ \iint _{Q_{T}} \bigl[ {u_{t}^{i}{\psi ^{i}} + a_{ij}^{\alpha \beta } {X_{\alpha }} {\psi ^{i}} {X_{\beta }} {u^{j}}} \bigr] \,dz = \iint _{Q_{T}} \bigl[ {{g_{i}} {\psi ^{i}} + f_{i}^{\alpha }{X_{\alpha }} {\psi ^{i}}} \bigr] \,dz, $$

for all $\psi \in C_{0}^{\infty }({Q_{T}},\mathbb{R}^{N})$.

In the Euclidean space, regularity to elliptic and parabolic equations and systems has been studied by many authors (see [1,2,3,4,5,6,7,8] and the references therein). Giaquinta in [5] proved the reverse Hölder estimates for weak solutions to diagonal elliptic systems with Hölder continuous coefficients and obtained the higher integrability of weak solutions. Giaquinta and Struwe in [2] treated partial regularity for weak solutions to diagonal quasilinear parabolic systems with the natural growth conditions and got Hölder continuity. Wiegner in [9] derived Hölder continuity of weak solutions to nondiagonal elliptic systems with VMO coefficients and natural growth conditions. Recently, Földes and Phan [10] got the higher integrability for gradients of weak solutions to a linear elliptic equation having the skew-symmetric part of coefficients unbounded.

Based on Hömander’s fundamental work [11], there has been tremendous work on degenerate PDEs arising from non-commuting vector fields; see, for example, [12,13,14,15,16,17,18,19,20,21]. Di Fazio and Fanciullo in [14] obtained gradient estimates for weak solutions to linear diagonal elliptic systems with bounded VMO coefficients. Dong and Niu [17] got the higher $L^{p}$ estimates for the gradient of weak solutions to nondiagonal quasilinear degenerate elliptic systems. In [16, 18], Dong and her collaborators studied Morrey and Hölder regularity for weak solutions to diagonal and nondiagonal parabolic systems with bounded VMO coefficients.

However, as far as we know, there is no relevant research about quasilinear degenerate parabolic systems with skew-symmetric coefficients. In this paper, we try to generalize the results in [10] to quasilinear degenerate parabolic systems constructed by Hörmander’s vector fields. The aim of this paper is to get the higher integrability for weak solutions to (1.1). In order to state our results, we make the following hypotheses:

(H1) The coefficients $a_{ij}^{\alpha \beta }(z) = A_{ij}^{\alpha \beta }(z) + B_{ij}^{\alpha \beta }(z)$, where $A_{ij}^{\alpha \beta }$ are symmetric ($A_{ij}^{\alpha \beta } = A_{ji}^{\alpha \beta }$), bounded, and satisfy the uniform ellipticity condition that for some $\varLambda >0$,

$$ A_{ij}^{\alpha \beta }(z){\xi _{i}^{\alpha }} {\xi _{j}^{\beta }} \geq \varLambda { \vert \xi \vert ^{2}},\qquad \bigl\vert {A_{ij}^{\alpha \beta }(z)} \bigr\vert \leq {\varLambda ^{ - 1}}, \quad {\text{a.e. }} z\in Q_{T}, \forall \xi \in \mathbb{R}^{qN}; $$

$B_{ij}^{\alpha \beta }(z)$ are skew-symmetric ($B_{ij}^{\alpha \beta } = - B_{ji}^{\alpha \beta }$) and belong to BMO space (therefore they can be unbounded).

(H2) For any $(z,u,\xi )\in Q_{T} \times \mathbb{R}^{N}\times \mathbb{R}^{qN}$,

$$\begin{aligned} &\bigl\vert {{g_{i}}(z,u,\xi )} \bigr\vert \le {g^{i}}(z) + L{ \vert \xi \vert ^{{\gamma _{0}}}}, \\ &\bigl\vert {f_{i}^{\alpha }(z,u,\xi )} \bigr\vert \le g_{i}^{\alpha }(z) + L \vert \xi \vert , \end{aligned}$$

where $1\leq \gamma _{0}<1/q_{0}$, $q_{0}=\frac{Q+2}{Q + 4}$, L is a positive constant satisfying $L<\varLambda$, and

$$ {g^{i}}(z) \in L^{pq_{0}}(Q_{T}),\qquad g_{i}^{\alpha }\in {L^{p}}(Q_{T}),\quad p \geq 2. $$

Here Q is the homogeneous dimension relative to Ω, and in the sequel we set $\tilde{g}=(g^{i})$, $\tilde{\tilde{g}}=(g_{i}^{\alpha })$, $\tilde{q}=2q_{0}$.

Now we state our main result.

Theorem 1.1

Suppose that (H1) and (H2) hold. Let $u \in W_{2}^{1,1}( {Q_{T}},\mathbb{R}^{N})$ be a weak solutions to (1.1), then there exists a constant $\varepsilon _{0}>$ such that for any $p \in [2,2 +\tilde{q}{\varepsilon _{0}})$, we have $Xu \in L_{ \mathrm{loc}}^{p} (Q_{T},\mathbb{R}^{N})$, and for every $Q_{T}' \subset \subset Q_{T}$, there exists a constant $C>0$ such that

$$ \Vert Xu \Vert _{L^{p}(Q_{T}')}\leq C \bigl( \Vert \tilde{g} \Vert _{L^{pq_{0}}(Q_{T})} ^{q_{0}} + \Vert \tilde{\tilde{g}} \Vert _{L^{p}(Q_{T})} \bigr). $$

(1.2)

The main difficulty in the proof is establishing the reverse Hölder inequality for gradients of weak solutions. We first establish the $L^{2}$ estimates of weak solutions by constructing suitable test functions. Then the reverse Hölder inequality of gradients is obtained by the $L^{2}$ estimates and the Gehring lemma on a metric measure space.

The paper is organized as follows. In Sect. 2, we introduce some concepts and results related to Hörmander’s vector fields that will be used in our proof. Section 3 is devoted to establishing the reverse Hölder inequality for gradients of weak solutions to (1.1) and giving the proof of Theorem 1.1.

2 Preliminaries

Let

$$ X_{\alpha }=\sum_{k=1}^{n}b_{\alpha k} \frac{\partial }{\partial x_{k}},\quad b_{\alpha k}\in C^{\infty }, \alpha =1,2,\dots ,q $$

be a family of vector fields in a neighborhood Ω̃ of some bounded domain $\varOmega \subset \mathbb{R}^{n}$. For a multiindex $\alpha =(i_{1},\ldots, i_{k})$, denote by $X_{\beta }=[X_{i_{1}},[X _{i_{2}},\ldots,[X_{i_{k-1}},X_{i_{k}}]]\ldots ]$ the commutator of vector fields $X_{1},\dots,X_{q}$ with length $k=|\beta |$. We say that the vector fields $X_{1},\ldots,X_{q}$ satisfy Hörmander’s condition up to the order s (see [11]) provided there exists $s>0$ such that $\{X_{\beta }\}_{|\beta |\leq s}$ span the tangent space at each point in $\mathbb{R}^{n}$.

We denote by $Xu=(X_{1} u,\dots,X_{q}u)$ the gradient of u with respect to the system $X=\{X_{1},\ldots,X_{q}\}$ and hence

$$ \bigl\vert Xu(x) \bigr\vert = \Biggl(\sum_{\alpha =1}^{q} \bigl\vert X_{\alpha }u(x) \bigr\vert ^{2} \Biggr) ^{\frac{1}{2}}. $$

An absolutely continuous curve $\gamma:[a,b]\to \tilde{\varOmega }$ is said to be admissible for the family X, if there exist functions $c_{\alpha }(t)$, $a\leq t\leq b$, satisfying

$$ \sum_{\alpha =1}^{q} c_{\alpha }(t)^{2} \leq 1 \quad\text{{and}}\quad \gamma '(t)=\sum _{\alpha =1}^{q}{c_{\alpha }}(t)X_{\alpha } \bigl( \gamma (t)\bigr), \quad\text{a.e. } t\in [a,b]. $$

The Carnot–Carathéodory distance induced by X is defined by

$$ d_{X}(x,y)=\inf \bigl\{ T>0:\text{there is an admissible curve } \gamma, \gamma (0)=x, \gamma (T)=y\bigr\} . $$

Then $d_{X}$ is a local metric on Ω̃. The metric ball is denoted by

$$ B_{R}(x)=B(x,R)=\bigl\{ y\in \varOmega:d_{X}(x,y)< R \bigr\} . $$

If one does not need to consider the center of the ball, then we also write $B_{R}$ instead of $B(x,R)$.

It is well known that the doubling property for metric balls holds true (see [22, 23]): there exist positive constants $R_{d}>0$ and $C_{d}\geq 1$ such that for any $x\in \varOmega$ and $0<2R\le R_{d}$,

$$ \bigl\vert B(x,2R) \bigr\vert \leq C_{d} \bigl\vert B(x,R) \bigr\vert . $$

Here, $|B(x,R)|$ denotes the Lebesgue measure of $B(x,R)$. The number $Q=\log _{2} C_{d}$ is called the homogeneous dimension relative to Ω. Clearly, $Q\geq n$. From the doubling property, we can see that

$$ \vert B_{tR} \vert \geq Ct^{Q} \vert B_{R} \vert , \quad\forall R \leq R_{d}, t \in (0,1), $$

where $C=C_{d}^{-2}$. In particular, if the vector fields $X_{1}, \dots,X_{q}$ are free up to the order s, there exist two positive constants $C_{1}$ and $C_{2}$ such that ([24])

$$ C_{1}R^{Q}\leq \bigl\vert B(x,R) \bigr\vert \leq C_{2}R^{Q}. $$

For ${z_{0}}=(x_{0},t_{0})\in {Q_{T}}\subset \mathbb{R}^{n + 1}$, the parabolic cylinder with vertex at $z_{0}$ is defined by

$$ {Q_{R}}({z_{0}})={B_{R}}({x_{0}}) \times \biggl({t_{0}- \frac{R^{2}}{2}},{t_{0}+ \frac{R^{2}}{2}} \biggr]. $$

Let $I_{R} (t_{0}) = ({t_{0}-\frac{R^{2}}{2}},{t_{0}+\frac{R ^{2}}{2}} ]$, and the parabolic boundary of $Q_{R}({z_{0}})$ be denoted by

$$ \partial _{p}{Q_{R}}(z_{0}) = \biggl( { \partial {B_{R}}({x_{0}}) \times \biggl({t_{0}- \frac{R^{2}}{2}},{t_{0}+\frac{R^{2}}{2}} \biggr]} \biggr) \cup \biggl( {B_{R}}(x_{0}) \times \biggl\{ t_{0} - \frac{R ^{2}}{2} \biggr\} \biggr). $$

For any $(x,t),(y,s) \in {Q_{T}}$, the parabolic distance in $Q_{T}$ is defined by

$$ d_{p} \bigl((x,t),(y,s)\bigr) = \sqrt{d_{X} (x,y)^{2} + \vert {t - s} \vert }, $$

and the parabolic ball is defined by

$$ B_{p}({z_{0}},R) = \bigl\{ {(x,t) \in {Q_{T}}:{d_{p}} \bigl( {( {x_{0}},{t_{0}}),(x,t)} \bigr) < R} \bigr\} . $$

To simplify the notations, in the sequel, $Q_{R} (z_{0})$, $B_{R} (x _{0})$, and $I_{R} (t_{0})$ are written as $Q_{R}$, $B_{R}$, and $I_{R}$, respectively. Furthermore, if E is a Lebesgue measurable set with Lebesgue measure $|E|$, we set to be the integral average of u on E.

We define the parabolic Sobolev space by

$$ W_{p}^{1,1} (Q_{T}) = \bigl\{ u\in L^{p} (Q_{T}):X_{\alpha }u, \partial _{t} u \in L^{p} (Q_{T}), \alpha =1,2, \ldots,q \bigr\} , $$

with the norm

$$ \Vert u \Vert _{W_{p}^{1,1}(Q_{T})}= \Vert u \Vert _{L^{p}(Q_{T})}+ \Vert \partial _{t}u \Vert _{L ^{p}(Q_{T})}+\sum _{\alpha =1}^{q} \Vert X_{\alpha }u \Vert _{L^{p}(Q_{T})}. $$

For any $f\in L_{\mathrm{loc}}^{1}({Q_{T}})$, if

$$\begin{aligned} \Vert f \Vert _{\mathrm{BMO}} &=\sup_{z_{0}\in Q_{T},\rho >0} \frac{1}{ \vert Q _{T}\cap Q_{\rho }({z_{0}}) \vert } \iint _{Q_{T}\cap Q_{\rho }({z_{0}})} \vert f-f_{ Q_{T}\cap Q_{\rho }({z_{0}})} \vert \,dz\\ &< \infty , \end{aligned}$$

we say that $f\in {\mathrm{BMO}}(Q_{T})$ (i.e., f has bounded mean oscillation).

Lemma 2.1

(Sobolev inequality, see [12, 23])

For every compact set $K\subset \varOmega$, there exist constants $C>0$ and $\bar{R}>0$ such that for any metric ball $B=B(x_{0},R)$ with $x_{0}\in K$ and $0< R\leq \bar{R}$, it holds that for any $f\in C^{ \infty }(\overline{B_{R}})$,

where is the integral average of f on $B_{R}$, and $1\leq \kappa \leq Q/(Q-p)$, if $1\leq p< Q$; $1\leq \kappa <\infty$, if $p\geq Q$. Moreover,

whenever $f \in C_{0}^{\infty }(\overline{B_{R}})$.

Lemma 2.2

(Iterative lemma, see [25])

Let $\varphi (t)$ be a bounded nonnegative function on $[{T_{0}},T _{1}]$, where $T_{1} > {T_{0}} \ge 0$. Suppose that for any t and s, $T_{0} \leq t < s \le {T_{1}}$, $\varphi (t)$ satisfies

$$ \varphi (t) \leq \theta \varphi (s) + \frac{A}{ (s - t)^{\alpha }}+ B, $$

where $\theta,A,B$, and α are nonnegative constants with $\theta < 1$. Then for any ${T_{0}} \leq \rho < R \leq {T_{1}}$, one has

$$ \varphi (\rho ) \leq c \biggl[\frac{A}{(R - \rho )^{\alpha }}+ B \biggr], $$

where c depends only on α and θ.

The following Gehring lemma on the metric measure space $(Y,d,\mu )$ (d is a metric and μ is a doubling measure) can be found in [13, 26].

Lemma 2.3

Let $q\in [q_{0},2Q]$, where $q_{0}>1$ is fixed. Assume that functions $f,g$ are nonnegative and $g\in L_{\mathrm{loc}}^{q}(Y,\mu )$, $f\in L_{\mathrm{loc}}^{r_{0}}(Y,\mu )$, for some $r_{0}>q$. If there exist constants $b>1$ and θ such that for every ball $B\subset \sigma B\subset Y$ the following inequality holds:

then there exist nonnegative constants $\theta _{0}=\theta _{0}(q_{0},Q,C _{d},\sigma )$ and $\varepsilon _{0}=\varepsilon _{0}(b,q_{0},Q,C_{d}, \sigma )$ such that if $0<\theta <\theta _{0}$ then $g\in L_{ \mathrm{loc}}^{p}(Y,\mu )$ for $p\in [q,q+\varepsilon _{0})$ and moreover

for some positive constant $C=C(q_{0},Q,C_{d},\sigma )$.

3 Higher integrability

We first introduce two cutoff functions $\xi (x)$ and $\eta (t)$ (see to [4]) such that for any $0 < \rho < R, {B_{ \rho }} \subset {B_{R}} \subset \varOmega$,

$$\begin{aligned} &\xi (x) \in C_{0}^{\infty }(B_{R} ),\quad 0 \le \xi \le 1,\qquad \vert {X\xi } \vert \le \frac{C}{R - \rho }\quad \text{and} \quad\xi = 1 \text{ in } B_{\rho }; \\ &\eta (t) = \textstyle\begin{cases} \frac{{2t - 2 ( {{t_{0}} - \frac{{{R^{2}}}}{2}} )}}{ {{R^{2}} - {\rho ^{2}}}},& t \in ( {{t_{0}} - \frac{{{R^{2}}}}{2},{t_{0}} - \frac{{{\rho ^{2}}}}{2}} ), \\ 1, & t \in [ {{t_{0}} - \frac{{{\rho ^{2}}}}{2},{t _{0}} + \frac{{{R^{2}}}}{2}} ]. \end{cases}\displaystyle \end{aligned}$$

Setting , we denote the average of $u(x,t)$ on ${B_{R}}$ by

$$ \bar{u}(t) = { \biggl( { \int _{{B_{R}}} {{\xi ^{2}}} \,dx} \biggr)^{ - 1}} \int _{{B_{R}}} {u{\xi ^{2}}} \,dx = \frac{1}{{{N_{1}} \vert {{B_{R}}} \vert }} \int _{{B_{R}}} {u{\xi ^{2}}} \,dx. $$

Lemma 3.1

Let $u \in W_{2}^{1,1}({\varOmega _{T}},{\mathbb{R}^{N}})$ be a weak solution to (1.1). Then for any ${Q_{R}} \subset \subset {\varOmega _{T}}$, we have

$$ \int _{{B_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}} \,dx + \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \le c \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz. $$

(3.1)

Proof

Multiplying both sides of (1.1) by the test function $u-\bar{u}(t)$ and integrating on ${Q_{R}}$, we get

$$ \iint _{{Q_{R}}} { \bigl[ {u_{t}^{i} + {X_{\alpha }}^{*} \bigl( {a_{ij} ^{\alpha \beta }{X_{\beta }} {u^{j}}} \bigr)} \bigr] \bigl( {{u ^{i}} - \bar{u}(t)} \bigr)}\,dz = \iint _{{Q_{R}}} { \bigl[ {{g_{i}} + {X_{\alpha }}^{*}f_{i}^{\alpha }} \bigr] \bigl( {{u^{i}} - \bar{u}(t)} \bigr)}\,dz. $$

(3.2)

So we have

$$ \iint _{{Q_{R}}} { \bigl[ {u_{t}^{i} \bigl( {{u^{i}} - \bar{u}(t)} \bigr) + a_{ij}^{\alpha \beta }{X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}} \bigr]}\,dz = \iint _{{Q_{R}}} { \bigl[ {{g_{i}} \bigl( {{u ^{i}} - \bar{u}(t)} \bigr) + f_{i}^{\alpha }{X_{\alpha }} {u^{i}}} \bigr]}\,dz. $$

By (H1), the above can be written as

$$\begin{aligned} & \iint _{{Q_{R}}} {{{ \biggl( {\frac{1}{2}{{ \bigl\vert {{u^{i}} - \bar{u}(t)} \bigr\vert }^{2}}} \biggr)}_{t}}}\,dz + \iint _{{Q_{R}}} {A _{ij}^{\alpha \beta }{X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}}\,dz \\ &\quad = - \iint _{{Q_{R}}} {B_{ij}^{\alpha \beta }{X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}}\,dz + \iint _{{Q_{R}}} { \bigl[ {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr) + f_{i}^{\alpha }{X_{\alpha }} {u^{i}}} \bigr]}\,dz. \end{aligned}$$

(3.3)

Due to the skew-symmetry of $B_{ij}^{\alpha \beta }$,

$$ \iint _{{Q_{R}}} {B_{ij}^{\alpha \beta }{X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}}\,dz = 0. $$

(3.4)

By (H2), Hölder’s, Sobolev’s, and Young’s inequalities, we have

$$\begin{aligned} & \iint _{{Q_{R}}} {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr)}\,dz \\ &\quad \le \iint _{{Q_{R}}} { \bigl( {{g^{i}}(z) + L{{ \vert {Xu} \vert } ^{{\gamma _{0}}}}} \bigr) \bigl( {{u^{i}} - \bar{u}(t)} \bigr)}\,dz \\ &\quad \le \int _{{I_{R}}} { \biggl[ {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}} \,dx} \biggr)}^{ \frac{1}{{\tilde{q}}}}} {{ \biggl( { \int _{{B_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{\frac{{2(Q + 2)}}{Q}}}} \,dx} \biggr)}^{\frac{Q}{ {2(Q + 2)}}}}} \biggr]} \,dt \\ &\qquad{} + L \int _{{I_{R}}} { \biggl[ {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)}^{ \frac{{{\gamma _{0}}}}{2}}} {{ \biggl( { \int _{{B_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{\frac{2}{{2 - {\gamma _{0}}}}}}} \,dx} \biggr)} ^{\frac{{2 - {\gamma _{0}}}}{2}}}} \biggr]} \,dt \\ &\quad \le \int _{{I_{R}}} { \biggl[ {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}} \,dx} \biggr)}^{ \frac{1}{{\tilde{q}}}}}c{R^{\frac{2}{{Q + 2}}}} {{ \biggl( { \int _{{B _{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)}^{\frac{1}{2}}}} \biggr]} \,dt \\ &\qquad{} + \int _{{I_{R}}} { \biggl[ {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)}^{\frac{{{\gamma _{0}}}}{2}}}c{R^{\frac{ {Q + 2 - Q{\gamma _{0}}}}{2}}} {{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)}^{\frac{1}{2}}}} \biggr]} \,dt \\ &\quad \le {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}}}\,dz + \varepsilon {R^{\frac{4}{Q}}} \int _{{I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)} ^{\frac{{Q + 2}}{Q}}}} \,dt \\ &\qquad{}+ c{R^{\frac{{Q + 2 - Q{\gamma _{0}}}}{2}}}\sup_{I_{R}} { \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)^{\frac{ {{\gamma _{0}} - 1}}{2}}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad \le {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}}}\,dz + \varepsilon {R^{\frac{4}{Q}}}\sup_{I_{R}} { \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)^{ \frac{2}{Q}}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\qquad{} + c{R^{\frac{{Q + 2 - Q{\gamma _{0}}}}{2}}}\sup_{I_{R}} { \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)^{\frac{ {{\gamma _{0}} - 1}}{2}}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz, \end{aligned}$$

(3.5)

and

$$\begin{aligned} \iint _{{Q_{R}}} {f_{i}^{\alpha }{X_{\alpha }} {u^{i}}}\,dz &\le \iint _{{Q_{R}}} { \bigl\vert {g_{i}^{\alpha }(z)} \bigr\vert \vert {Xu} \vert }\,dz + L \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ & \leq {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz + (\varepsilon + L) \iint _{{Q_{R}}} {{{ \vert {Xu} \vert } ^{2}}}\,dz. \end{aligned}$$

(3.6)

Inserting (3.4), (3.5), and (3.6) into (3.3), and by (H1), we get

$$\begin{aligned} & \int _{{B_{R}}} {\frac{1}{2}{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert } ^{2}}} \,dx + \varLambda \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad \le {c_{\varepsilon }} \iint _{{Q_{R}}} \vert \tilde{g} \vert ^{\tilde{q}}\,dz + {c_{\varepsilon }} \iint _{{{Q}_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert } ^{2}}}\,dz + \theta \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz, \end{aligned}$$

where $\theta = \varepsilon {R^{\frac{4}{Q}}}\sup_{I_{R}} { ( {\int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} )^{ \frac{2}{Q}}} + c{R^{\frac{{Q + 2 - Q{\gamma _{0}}}}{2}}}\sup_{I_{R}} { ( {\int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} ) ^{\frac{{{\gamma _{0}} - 1}}{2}}} + \varepsilon + L$. Because $L < \varLambda$, by choosing $\varepsilon,R$ small enough we can get that $\theta < \varLambda$. So using Lemma 2.2, we complete the proof. □

Lemma 3.2

Let $u \in W_{2}^{1,1}({\varOmega _{T}},{\mathbb{R}^{N}})$ be a weak solution of (1.1). Then for any $0 < \rho < R$, $Q_{R} \subset \subset \varOmega _{T}$, we have

$$\begin{aligned} &\sup_{I_{\rho }} \int _{{B_{\rho }}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}} \,dx + \iint _{{Q_{\rho }}} {{{ \vert {Xu} \vert } ^{2}}}\,dz \\ &\quad \leq \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + c \biggl( {\frac{{{R^{3}}}}{{{{(R - \rho )}^{2}}}} + 1} \biggr) \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert } ^{2}}} \bigr)}\,dz. \end{aligned}$$

(3.7)

Proof

Let ${B_{\rho }} \subset {B_{R}} \subset \varOmega$. Multiplying both sides of (1.1) by the test function $( {u - \bar{u}(t)} )\times {\xi ^{2}}(x)\eta (t)$ and integrating on ${Q'_{R}} = {B_{R}}( {x_{0}}) \times ({t_{0}} - \frac{{{R^{2}}}}{2},s]$ ($s \le {t_{0}} + \frac{ {{R^{2}}}}{2}$), we get

$$\begin{aligned} & \iint _{{{Q'}_{R}}} { \bigl[ {u_{t}^{i} + {X_{\alpha }}^{*} \bigl( {a_{ij}^{\alpha \beta }{X_{\beta }} {u^{j}}} \bigr)} \bigr] \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta }\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \bigl[ {{g_{i}} + {X_{\alpha }}^{*}f_{i}^{ \alpha }} \bigr] \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}} \eta }\,dz. \end{aligned}$$

(3.8)

By (H1), one has

$$\begin{aligned} &\iint _{{{Q'}_{R}}} { \bigl[ {u_{t}^{i} + X_{\alpha }^{*} \bigl( {a_{ij}^{\alpha \beta }{X_{\beta }} {u^{j}}} \bigr)} \bigr] \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta }\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \bigl[ {u_{t}^{i} \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta + a_{ij}^{\alpha \beta }{X_{\beta }} {u^{j}} {X_{\alpha }} \bigl( { \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}} \eta } \bigr)} \bigr]}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \bigl[ {u_{t}^{i} \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta + a_{ij}^{\alpha \beta }{\xi ^{2}}\eta {X_{ \alpha }} {u^{i}} {X_{\beta }} {u^{j}} + 2a_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}} \bigr]}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \biggl[ {{{ \biggl( {\frac{1}{2}{{ \bigl\vert {{u ^{i}} - \bar{u}(t)} \bigr\vert }^{2}} \eta } \biggr)}_{t}} {\xi ^{2}} - \frac{1}{2}{{ \bigl\vert {{u^{i}} - \bar{u}(t)} \bigr\vert }^{2}} { \xi ^{2}} {\eta _{t}} + A_{ij}^{\alpha \beta }{ \xi ^{2}}\eta {X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}} \biggr]\,dz} \\ &\qquad{}+ \iint _{{{Q'}_{R}}} {B_{ij}^{\alpha \beta }{\xi ^{2}}\eta {X _{\alpha }} {u^{i}} {X_{\beta }} {u^{j}} + 2a_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}\,dz}, \end{aligned}$$

and

$$\begin{aligned} &\iint _{{{Q'}_{R}}} { \bigl[ {{g_{i}} + {X_{\alpha }}^{*}f_{i} ^{\alpha }} \bigr] \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}} \eta }\,dz \\ &\quad= \iint _{{{Q'}_{R}}} { \bigl[ {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr) {\xi ^{2}}\eta + f_{i}^{\alpha }{X_{\alpha }} \bigl( { \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta } \bigr)} \bigr]}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \bigl[ {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr) {\xi ^{2}}\eta + f_{i}^{\alpha }{\xi ^{2}}\eta {X_{\alpha }} {u^{i}} + 2 \xi \eta \bigl( {{u^{i}} - \bar{u}(t)} \bigr)f_{i}^{\alpha }{X_{ \alpha }}\xi } \bigr]}\,dz. \end{aligned}$$

By the above, (3.8) can be written as

$$\begin{aligned} & \iint _{{{Q'}_{R}}} {{{ \biggl( {\frac{1}{2}{{ \bigl\vert {{u^{i}} - \bar{u}(t)} \bigr\vert }^{2}}\eta } \biggr)}_{t}} {\xi ^{2}}}\,dz + \iint _{{{Q'}_{R}}} {A_{ij}^{\alpha \beta }{\xi ^{2}}\eta {X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} { \biggl[ {\frac{1}{2}{{ \bigl\vert {{u^{i}} - \bar{u}(t)} \bigr\vert }^{2}} {\xi ^{2}} {\eta _{t}} - B_{ij}^{\alpha \beta }{ \xi ^{2}}\eta {X_{\alpha }} {u^{i}} {X_{\beta }} {u^{j}}} \biggr]}\,dz \\ &\qquad{} - 2 \iint _{{{Q'}_{R}}} {a_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\qquad{} + \iint _{{{Q'}_{R}}} { \bigl[ {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr){\xi ^{2}}\eta + 2\xi \eta \bigl( {{u^{i}} - \bar{u}(t)} \bigr)f_{i}^{\alpha }{X_{\alpha }} \xi }+ f_{i}^{\alpha }{\xi ^{2}}\eta {X_{\alpha }} {u^{i}} \bigr]}\,dz. \end{aligned}$$

(3.9)

Due to the skew-symmetry of $B_{ij}^{\alpha \beta }$,

$$ \iint _{{{Q'}_{R}}} {{{\bigl(B_{ij}^{\alpha \beta } \bigr)}_{R}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz = 0. $$

(3.10)

By (H1), (3.10) and Young’s inequality, we have

$$\begin{aligned} & \iint _{{{Q'}_{R}}} {a_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} {A_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz + \iint _{{{Q'}_{R}}} {B_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\quad = \iint _{{{Q'}_{R}}} {A_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\qquad{} + \iint _{{{Q'}_{R}}} { \bigl( {B_{ij}^{\alpha \beta } - {{\bigl(B_{ij}^{\alpha \beta }\bigr)}_{R}}} \bigr) \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\quad \le {\varLambda ^{ - 1}} \iint _{{{Q'}_{R}}} { \bigl\vert {u - \bar{u}(t)} \bigr\vert \vert {X\xi } \vert \vert {Xu} \vert \xi \eta }\,dz \\ &\qquad{} + \iint _{{{Q'}_{R}}} { \bigl\vert {B_{ij}^{\alpha \beta } - {{\bigl(B_{ij}^{\alpha \beta }\bigr)}_{R}}} \bigr\vert \bigl\vert {u - \bar{u}(t)} \bigr\vert \vert {X\xi } \vert \vert {Xu} \vert \xi \eta }\,dz \\ &\quad \le {c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}} {{ \vert {X\xi } \vert }^{2}}\eta }\,dz + 2 \varepsilon \iint _{{{Q'}_{R}}} {{{ \vert {Xu} \vert }^{2}} {\xi ^{2}}\eta }\,dz \\ &\qquad{} + {c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \bigl\vert {B_{ij} ^{\alpha \beta } - {{\bigl(B_{ij}^{\alpha \beta } \bigr)}_{R}}} \bigr\vert }^{2}} {{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}} {{ \vert {X\xi } \vert } ^{2}}\eta }\,dz. \end{aligned}$$

(3.11)

By Hölder’s and Sobolev’s inequalities, we have

$$\begin{aligned} & \iint _{{{Q'}_{R}}} {{{ \bigl\vert {B_{ij}^{\alpha \beta } - {{\bigl(B _{ij}^{\alpha \beta }\bigr)}_{R}}} \bigr\vert }^{2}} {{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz \\ &\quad \le { \biggl( { \iint _{{{Q'}_{R}}} {{{ \bigl\vert {B_{ij}^{\alpha \beta } - {{\bigl(B_{ij}^{\alpha \beta }\bigr)}_{R}}} \bigr\vert }^{Q}}}\,dz} \biggr)^{ \frac{2}{Q}}} { \biggl( { \iint _{{{Q'}_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{\frac{{2Q}}{{Q - 2}}}}}\,dz} \biggr)^{\frac{{Q - 2}}{Q}}} \\ &\quad \le c{ \vert {{Q_{R}}} \vert ^{\frac{2}{Q}}} \cdot \Vert B \Vert _{\mathrm{BMO}}^{2}{ \biggl( { \int _{{I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}\,dx} } \biggr)}^{\frac{Q}{ {Q - 2}}}}\,dt} } \biggr)^{\frac{{Q - 2}}{Q}}} \\ &\quad \le c \Vert B \Vert _{\mathrm{BMO}}^{2}{R^{3}} { \biggl( { \int _{{I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} } \biggr)}^{2}}\,dt} } \biggr)^{\frac{1}{2}}}. \end{aligned}$$

(3.12)

Putting (3.12) into (3.11), we get

$$\begin{aligned} & \iint _{{{Q'}_{R}}} {a_{ij}^{\alpha \beta } \bigl( {{u^{i}} - \bar{u}(t)} \bigr)\xi \eta {X_{\alpha }}\xi {X_{\beta }} {u^{j}}}\,dz \\ &\quad \le {c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}} {{ \vert {X\xi } \vert }^{2}}\eta }\,dz + 2 \varepsilon \iint _{{{Q'}_{R}}} {{{ \vert {Xu} \vert }^{2}} {\xi ^{2}}\eta }\,dz \\ &\qquad{} + \frac{{{c_{\varepsilon }} \Vert B \Vert _{ \mathrm{BMO}}^{2}{R^{3}}}}{{{{(R - \rho )}^{2}}}}{ \biggl( { \int _{ {I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}\,dx} } \biggr)}^{2}}\,dt} } \biggr)^{\frac{1}{2}}}. \end{aligned}$$

(3.13)

Using properties of $\xi (x),\eta (t)$ and (3.5),

$$\begin{aligned} & \iint _{{{Q'}_{R}}} {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr) {\xi ^{2}}\eta }\,dz \\ &\quad \le \iint _{{Q_{R}}} {{g_{i}} \bigl( {{u^{i}} - \bar{u}(t)} \bigr)}\,dz \\ &\quad \le {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}}}\,dz + \varepsilon {R^{\frac{4}{Q}}}\sup_{I_{R}} { \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)^{ \frac{2}{Q}}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\qquad{}+ c{R^{\frac{{Q + 2 - Q{\gamma _{0}}}}{2}}}\sup_{I_{R}} { \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} \biggr)^{\frac{ {{\gamma _{0}} - 1}}{2}}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz. \end{aligned}$$

(3.14)

By (H2), Hölder’s and Young’s inequalities,

$$\begin{aligned} & \iint _{{{Q'}_{R}}} { \bigl[ {2\xi \eta \bigl( {{u^{i}} - \bar{u}(t)} \bigr)f_{i}^{\alpha }{X_{\alpha }}\xi + f_{i}^{\alpha } {\xi ^{2}}\eta {X_{\alpha }} {u^{i}}} \bigr]}\,dz \\ &\quad \le 2 \iint _{{{Q'}_{R}}} { \bigl\vert {u - \bar{u}(t)} \bigr\vert \bigl\vert {g_{i}^{\alpha }(z)} \bigr\vert \vert {X\xi } \vert \xi \eta }\,dz + 2L \iint _{{{Q'}_{R}}} { \bigl\vert {u - \bar{u}(t)} \bigr\vert \vert {Xu} \vert \vert {X\xi } \vert \xi \eta }\,dz \\ &\qquad{} + \iint _{{{Q'}_{R}}} { \bigl\vert {g_{i}^{\alpha }(z)} \bigr\vert \vert {Xu} \vert {\xi ^{2}}\eta }\,dz + L \iint _{{{Q'}_{R}}} {{{ \vert {Xu} \vert }^{2}} {\xi ^{2}}\eta }\,dz \\ &\quad \le 2{c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}} {{ \vert {X\xi } \vert }^{2}}\eta }\,dz + {c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}} {\xi ^{2}}\eta }\,dz \\ &\qquad{} + (2\varepsilon + L) \iint _{{{Q'}_{R}}} {{{ \vert {Xu} \vert } ^{2}} { \xi ^{2}}\eta }\,dz. \end{aligned}$$

(3.15)

Inserting (3.13), (3.14), and (3.15) into (3.9), and by (H1), (3.3), (3.4), and Young’s inequality, we get

$$\begin{aligned} &\int _{{B_{R}}} {\frac{1}{2}{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert } ^{2}} {\xi ^{2}}\eta } \,dx + \varLambda \iint _{{{Q'}_{R}}} {{{ \vert {Xu} \vert } ^{2}} { \xi ^{2}}\eta }\,dz \\ &\quad\le \iint _{{{Q'}_{R}}} {\frac{1}{2}{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert } ^{2}} {\xi ^{2}} {\eta _{t}}}\,dz + 3{c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}} {{ \vert {X\xi } \vert } ^{2}}\eta }\,dz \\ &\qquad{}+ \frac{{{c_{\varepsilon }} \Vert B \Vert _{ \mathrm{BMO}}^{2}{R^{3}}}}{{{{(R - \rho )}^{2}}}}{ \biggl( { \int _{ {I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}\,dx} } \biggr)}^{2}}\,dt} } \biggr)^{\frac{1}{2}}} + {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz \\ &\qquad{}+ {c_{\varepsilon }} \iint _{{{Q'}_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}} {\xi ^{2}}\eta }\,dz + {\theta _{1}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz, \end{aligned}$$

where ${\theta _{1}} = \varepsilon {R^{\frac{4}{Q}}}\sup_{I_{R}} { ( {\int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} )^{ \frac{2}{Q}}} + c{R^{\frac{{Q + 2 - Q{\gamma _{0}}}}{2}}}\sup_{I_{R}} { ( {\int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}} \,dx} ) ^{\frac{{{\gamma _{0}} - 1}}{2}}} + 4\varepsilon + L$. Employing properties of $\xi (x),\eta (t)$, (H1), and since $\frac{1}{R^{2}-\rho ^{2}}\leq \frac{C}{(R-\rho )^{2}}$, we have

$$\begin{aligned} & \frac{1}{2}\mathop{\sup } _{{I_{\rho }}} \int _{{B_{\rho }}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}} \,dx + \varLambda \iint _{{Q_{\rho }}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad\le \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + \frac{{{c_{\varepsilon }} \Vert B \Vert _{\mathrm{BMO}}^{2}{R^{3}}}}{{{{(R - \rho )}^{2}}}}{ \biggl( { \int _{{I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} } \biggr)}^{2}}\,dt} } \biggr)^{\frac{1}{2}}} \\ &\qquad{}+ {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}}}\,dz + {c_{\varepsilon }} \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz + {\theta _{1}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz. \end{aligned}$$

Because $L < \varLambda$, by choosing $\varepsilon,R$ small enough we can get that ${\theta _{1}} < \varLambda$, so Lemma 2.2 yields

$$\begin{aligned} & \mathop{\sup } _{{I_{\rho }}} \int _{{B_{\rho }}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}} \,dx + \iint _{{Q_{\rho }}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad \le \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz \\ &\qquad{}+ \frac{{c \Vert B \Vert _{\mathrm{BMO}}^{2}{R^{3}}}}{ {{{(R - \rho )}^{2}}}} \biggl(\sup_{I_{R}} \int _{{B_{R}}} {{{ \vert {Xu} \vert }^{2}}\,dx} \biggr)^{\frac{1}{2}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr)^{\frac{1}{2}}}. \end{aligned}$$

By (3.1),

$$ \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \le c \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz + c \iint _{{{Q}_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz. $$

Then

$$\begin{aligned} & \mathop{\sup } _{{I_{\rho }}} \int _{{B_{\rho }}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}} \,dx + \iint _{{Q_{\rho }}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad \le \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz \\ &\qquad{}+ \frac{{c \Vert B \Vert _{\mathrm{BMO}}^{2}{R^{3}} \mathop{\sup } _{{I_{R} }} \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} }}{{{{(R - \rho )}^{2}}}}{ \biggl( { \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz} \biggr)^{ \frac{1}{2}}} \\ &\quad \le \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}}\,dz + c \iint _{{Q_{R}}} {{{ \vert {\tilde{\tilde{g}}} \vert }^{2}}}\,dz \\ &\qquad{}+ \frac{{c \Vert B \Vert _{\mathrm{BMO}}^{2}{R^{3}} \mathop{\sup } _{{I_{R}}} \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} }}{{{{(R - \rho )}^{2}}{{ ( { \Vert {\tilde{g}} \Vert _{{L^{\tilde{q}}}}^{\tilde{q}} + \Vert {\tilde{\tilde{g}}} \Vert _{{L^{2}}}^{2}} )}^{\frac{1}{2}}}}} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz \\ &\quad\le \frac{c}{{{{(R - \rho )}^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + c \biggl( {\frac{{{R^{3}}}}{{{{(R - \rho )}^{2}}}} + 1} \biggr) \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert } ^{2}}} \bigr)}\,dz. \end{aligned}$$

The proof is completed. □

Lemma 3.3

Let $u \in W_{2}^{1,1}({\varOmega _{T}},{\mathbb{R}^{N}})$ be a weak solution of (1.1). Then there exists a positive constant $\varepsilon _{0}$ such that for any $p \in [ {2,2 + \tilde{q} {\varepsilon _{0}}} )$, we have $u \in L_{\mathrm{loc}}^{\frac{ {p\gamma }}{2}}({Q_{T}}),Xu \in L_{\mathrm{loc}}^{p}({Q_{T}})$, and for any ${Q_{2R}} \subset \subset {Q_{T}}$,

$$ \frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{p}}}\,dz \leq c \biggl[ \biggl(\frac{1}{ \vert Q_{2R} \vert } \iint _{Q_{2R}} \vert Xu \vert ^{2}\,dz \biggr)^{\frac{p}{2}}+ \frac{1}{ \vert Q_{2R} \vert } \iint _{Q_{2R}} \bigl( \vert \tilde{g} \vert ^{\tilde{q}} + \vert \tilde{\tilde{g}} \vert ^{2} \bigr) ^{\frac{p}{2}}\,dz \biggr]. $$

Proof

By (3.7) and Sobolev’s inequality,

$$\begin{aligned} & \mathop{\sup } _{{I_{4R/5}}} { \biggl( { \int _{{B_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}\,dx} } \biggr)^{\frac{1}{2}}} \\ & \quad\le { \biggl( {\frac{c}{{{R^{2}}}} \iint _{{Q_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz} \biggr)^{\frac{1}{2}}} + c{ \biggl( { \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz} \biggr) ^{\frac{1}{2}}} \\ &\quad \le c{ \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr) ^{\frac{1}{2}}} + c{ \biggl( { \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}}+ {{ \vert {\tilde{\tilde{g}}} \vert } ^{2}}} \bigr)}\,dz} \biggr)^{\frac{1}{2}}}. \end{aligned}$$

(3.16)

By Hölder’s and Sobolev’s inequalities, it follows

$$\begin{aligned} & { \int _{{I_{4R/5}}} { \biggl( { \int _{{B_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}\,dx} } \biggr)} ^{\frac{1}{2}}}\,dt \\ &\quad \le { \int _{{I_{R}}} { \biggl( { \int _{{B_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{\tilde{q}}}\,dx} } \biggr)} ^{\frac{1}{{2 \tilde{q}}}}} { \biggl( { \int _{{B_{R}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert } ^{\gamma }}\,dx} } \biggr)^{\frac{1}{{2\gamma }}}}\,dt \\ &\quad \le c{R^{\frac{1}{{\tilde{q}}}}} \int _{{I_{R}}} {{{ \biggl( { \int _{{B _{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}\,dx} } \biggr)}^{\frac{1}{ {2\tilde{q}}}}} {{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} } \biggr)}^{\frac{1}{4}}}} \,dt \\ &\quad \le c{R^{\frac{1}{{\tilde{q}}}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr)^{\frac{1}{{2\tilde{q}}}}} { \biggl( { \int _{{I_{R}}} {{{ \biggl( { \int _{{B_{R}}} {{{ \vert {Xu} \vert } ^{2}}\,dx} } \biggr)}^{\frac{1}{2}\frac{{\tilde{q}}}{{2\tilde{q} - 1}}}}} \,dt} \biggr)^{\frac{{2\tilde{q} - 1}}{{2\tilde{q}}}}} \\ &\quad \le c{R^{\frac{3}{2}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert } ^{\tilde{q}}}}\,dz} \biggr)^{\frac{1}{{2\tilde{q}}}}} { \biggl( { \iint _{{Q _{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr)^{\frac{1}{4}}}, \end{aligned}$$

(3.17)

where $\gamma =\frac{2(Q+2)}{Q}$. By (3.16) and (3.17),

$$\begin{aligned} & \iint _{{Q_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz \\ &\quad = \int _{{I_{4R/5}}} { \biggl( { \int _{{B_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}\,dx} } \biggr)} \,dt \\ &\quad \le \mathop{\sup } _{{I_{4R/5}}} { \biggl( { \int _{{B_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}\,dx} } \biggr)^{\frac{1}{2}}} \cdot \biggl( {{{ \int _{{I_{4R/5}}} { \biggl( { \int _{{B_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}\,dx} } \biggr)} }^{ \frac{1}{2}}}\,dt} \biggr) \\ &\quad \le c{R^{\frac{3}{2}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert } ^{\tilde{q}}}}\,dz} \biggr)^{\frac{1}{{2\tilde{q}}}}} { \biggl( { \iint _{{Q _{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr)^{\frac{3}{4}}} \\ &\qquad{} + c{R^{\frac{3}{2}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr)^{\frac{1}{{2\tilde{q}}}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr) ^{\frac{1}{4}}} { \biggl( { \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz} \biggr)^{\frac{1}{2}}} \\ &\quad \equiv {I_{1}} + {I_{2}}. \end{aligned}$$

(3.18)

By Young’s inequality,

$$\begin{aligned} &{I_{1}} \le {c_{\varepsilon }} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr)^{\frac{2}{{\tilde{q}}}}} + \varepsilon {R^{2}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz, \\ &{I_{2}}\leq \varepsilon R{ \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr)^{\frac{1}{{\tilde{q}}}}} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr) ^{\frac{1}{2}}} + {c_{\varepsilon }} {R^{2}} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz \\ & \phantom{{I_{2}}}\leq {c_{\varepsilon }} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert } ^{\tilde{q}}}}\,dz} \biggr)^{\frac{2}{{\tilde{q}}}}} + \varepsilon {R^{2}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz + {c_{\varepsilon }} {R^{2}} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz. \end{aligned}$$

Inserting the estimates of $I_{1}$ and $I_{2}$ into (3.18), we get

$$\begin{aligned} & \iint _{{Q_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz \\ &\quad \le {c_{\varepsilon }} { \biggl( { \iint _{{Q_{R}}} {{{ \vert {Xu} \vert } ^{\tilde{q}}}}\,dz} \biggr)^{\frac{2}{{\tilde{q}}}}} + \varepsilon {R^{2}} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz + {c_{\varepsilon }} {R^{2}} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert } ^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz. \end{aligned}$$

(3.19)

By (3.7) and (3.19),

$$\begin{aligned} & \frac{1}{{ \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{3R/4}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\quad \le \frac{c}{{{R^{2}}}}\frac{1}{{ \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{4R/5}}} {{{ \bigl\vert {u - \bar{u}(t)} \bigr\vert }^{2}}}\,dz + \frac{c}{ { \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{4R/5}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert } ^{2}}} \bigr)}\,dz \\ &\quad \le \frac{{{c_{\varepsilon }}{{ \vert {{Q_{R}}} \vert }^{\frac{2}{ {\tilde{q}}}}}}}{{ \vert {{Q_{3R/4}}} \vert {R^{2}}}}{ \biggl( {\frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr)^{\frac{2}{{\tilde{q}}}}} + \frac{ \varepsilon }{{ \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\qquad{} + \frac{{{c_{\varepsilon }}}}{{ \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz \\ &\quad \le {c_{\varepsilon }} { \biggl( {\frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\tilde{q}}}}\,dz} \biggr) ^{\frac{2}{{\tilde{q}}}}} + \frac{\varepsilon }{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz \\ &\qquad{} + \frac{{{c_{\varepsilon }}}}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz. \end{aligned}$$

(3.20)

Let $\hat{g} = { \vert {Xu} \vert ^{\tilde{q}}}$ ($\hat{q} = \frac{2}{ {\tilde{q}}} = \frac{{Q + 4}}{{Q + 2}} > 1$), $\hat{f} = { ( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} )^{\frac{{\tilde{q}}}{2}}}$, then the above can be written as

$$ \frac{1}{{ \vert {{Q_{3R/4}}} \vert }} \iint _{{Q_{3R/4}}} {{{\hat{g}} ^{\hat{q}}}}\,dz \le c \biggl[ {{{ \biggl( {\frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {\hat{g}}\,dz} \biggr)}^{\hat{q}}} + \frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{\hat{f}}^{\hat{q}}}}\,dz} \biggr] + \frac{\varepsilon }{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{\hat{g}}^{\hat{q}}}}\,dz. $$

By Lemma 2.3, we know that there exists a positive constant $\varepsilon _{0}$ such that for any $\hat{p} \in [\hat{q},\hat{q} + {\varepsilon _{0}})$,

$$\begin{aligned} &{ \biggl( {\frac{1}{{ \vert {{Q_{R}}} \vert }} \iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{\hat{p}\tilde{q}}}}\,dz} \biggr)^{\frac{1}{ {\hat{p}}}}} \\ &\quad \le c \biggl[ {{{ \biggl( {\frac{1}{{ \vert {{Q_{2R}}} \vert }} \iint _{{Q_{2R}}} {{{ \vert {Xu} \vert }^{2}}}\,dz} \biggr)}^{\frac{ {\tilde{q}}}{2}}} + {{ \biggl( {\frac{1}{{ \vert {{Q_{2R}}} \vert }} \iint _{{Q_{2R}}} {{{ \bigl( {{{ \vert {\tilde{g}} \vert }^{ \tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)} ^{\frac{{\hat{p}\tilde{q}}}{2}}}}\,dz} \biggr)}^{\frac{1}{{\hat{p}}}}}} \biggr]. \end{aligned}$$

Letting $p = \hat{p}\tilde{q} \in [ {2,2 + \tilde{q}{\varepsilon _{0}}} )$, we finish the proof. □

Proof of Theorem 1.1

By (3.1), Lemma 3.3, and Hölder’s inequality, we have

$$\begin{aligned} &\iint _{{Q_{R}}} {{{ \vert {Xu} \vert }^{p}}}\,dz \\ &\quad \le c \vert {{Q_{R}}} \vert \biggl[ {{{ \biggl( { \frac{1}{{ \vert {{Q_{2R}}} \vert }} \iint _{{Q_{2R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}\,dz} \biggr)}^{\frac{p}{2}}} + \frac{1}{{ \vert {{Q_{2R}}} \vert }}{{ \iint _{{Q_{2R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{\tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)}}^{\frac{p}{2}}}\,dz} \biggr] \\ &\quad \le c{ \iint _{{Q_{2R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{ \tilde{q}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{2}}} \bigr)} ^{\frac{p}{2}}}\,dz \le c \iint _{{Q_{2R}}} { \bigl( {{{ \vert {\tilde{g}} \vert }^{p{q_{0}}}} + {{ \vert {\tilde{\tilde{g}}} \vert }^{p}}} \bigr)}\,dz \\ &\quad \le c \bigl( { \Vert {\tilde{g}} \Vert _{{L^{p{q_{0}}}}}^{p {q_{0}}} + \Vert {\tilde{\tilde{g}}} \Vert _{{L^{p}}}^{p}} \bigr). \end{aligned}$$

The proof is completed. □