Local C^0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

Abstract

In this article, when $2\le p\le 4$, we establish the $C_{\mathrm{loc}}^{0,1}$-regularity of weak solutions to the degenerate parabolic p-Laplacian equation ${\partial }_{t}u=-{\displaystyle \sum _{i=1}^6}{X}_i^{*}(|{\nabla }_{\mathcal{H}}u{|}^{p-2}{X}_{i}u)$ on the group SU(3) granted with horizontal vector fields $X_1, \dots , X_6$. Compared to the Heisenberg group, ${\mathbb{H}}^n$, we obtained the optimal range of p; that is, $2\le p\le 4$.

1. Introduction

The study of the regularity for partial differential equations involving the p-Laplacian operator has always been a hot topic. In the Euclidean space, the $C^{0,1}$, $C^{1,\alpha }$, $W^{2,2}$-regularities and other second-order Sobolev regularities for the p-Laplacian equation have been proved in [1,2,3,4,5,6,7]. In recent years, there has been significant progress in the study of the regularity for the p-Laplacian equation in sub-Riemannian manifolds. Many scholars have made outstanding contributions. In the Heisenberg group, ${\mathbb{H}}^n$, Domokos-Manfredi [8,9], Manfredi-Mingione [10], Migione et al. [11], Ricciotti [12], and Zhong-Mukherjee [13,14] established the $C^{0,1}$ and $C^{1,\alpha }$-regularities for the p-Laplacian equation in the full range $1<p<\infty$; Domokos [15] and Lie et al. [16] proved the $W^{2,2}$-regularity for the p-Laplacian equation in the range of $1<p<3+\frac{1}{n - 1}$ with $n\ge 2$. In the group SU(3), the $C^{0,1}$, $C^{1,\alpha }$, and $W^{2,2}$-regularities of the p-Laplacian equation were established by [17,18]. The method in [13,14] is extended by Citti-Mukherjee [19] to include Hörmander vector fields of step two, and the $C^{0,1}$ and $C^{1,\alpha }$-regularities for the p-Laplacian equation have been successfully established. The $C^{1,\alpha }$-regularity for inhomogeneous quasi-linear equations on the Heisenberg group ${\mathbb{H}}^n$ were established by [20,21] when $2-\frac{1}{2n + 2}<p<\infty$. New ideas and perspectives behind the development of research on regularity include certain hybrid-type Caccioppoli-type inequalities, as first proposed and introduced by Zhong [13]. In comparison, for the degenerate parabolic p-Laplacian equation, such inequalities are not applicable due to the differences in homogeneity between the time and spatial derivatives. Therefore, we need to find and create new methods and techniques to establish more suitable Caccioppoli-type inequalities.

In this study paper, we propose a new method to construct a crucial Caccioppoli-type inequality. Based on the inequality, when $2\le p\le 4$, we establish the $C^{0,1}$-regularity for the parabolic p-Laplacian equation on the group SU(3). To be specific, we focus on a special type of unitary group composed of $3\times 3$ complex matrices. We denote by SU(3) this unitary group and endow it with horizontal vector fields $X_1,X_2, \dots , X_6$. More exhaustive geometries and properties of SU(3) are shown in Section 2. We select an open domain $\mathsf{\Omega }$ in the group SU(3). For $T>0$, we define a cylinder $\mathcal{Q}=\mathsf{\Omega }\times (0, T)$, as first proposed in [22]. In $\mathcal{Q}$, we consider the following equation:

$${\partial }_{t}u=-\sum _{i=1}^6{X}_i^{*}{\mathcal{A}}_i({\nabla }_{\mathcal{H}}u)\mathrm{in}\mathcal{Q}=\mathsf{\Omega }\times (0, T).$$

(1)

Here, $X_i^{*}$ is the formal adjoint of $X_i$; ${\nabla }_{\mathcal{H}}=(X_1,X_2, \dots , X_6)$ is the horizontal gradient; the vector function $\mathcal{A}≔({\mathcal{A}}_1, \dots , {\mathcal{A}}_6)\in C^2({\mathbb{R}}^6,{\mathbb{R}}^6)$ meets the following condition:

$$\left\{\begin{array}{c}{\upsilon }^{\prime }{|\zeta |}^{p-2}{|\varrho |}^2\le {\displaystyle \sum _{i,j=1}^6}{\mathcal{A}}_{i,{\zeta }_j}(\zeta ){\varrho }_i{\varrho }_j\le {\mathrm{{\rm Y}}}^{\prime }{|\zeta |}^{p-2}{|\varrho |}^2,\hfill \\ |{\mathcal{A}}_i(\zeta )|\le {\mathrm{{\rm Y}}}^{\prime }{|\zeta |}^{p-1}.\hfill \end{array}\right.$$

(2)

for every $\zeta , \varrho \in {\mathbb{R}}^6$, where ${\mathcal{A}}_{i,{\zeta }_j}(\zeta )≔{\partial }_{{\zeta }_j}{\mathcal{A}}_i(\zeta )$, $p\in [2,\infty )$ and $0<{\upsilon }^{\prime }\le {\mathrm{{\rm Y}}}^{\prime }<\infty$. If, for every function $\psi \in C_0^{\infty }(\mathcal{Q})$, the equation

$${\int }_0^T{\int }_{\mathsf{\Omega }}{\partial }_{t}u\psi \mathrm{d}x\mathrm{d}t=-{\int }_0^T{\int }_{\mathsf{\Omega }}\sum _{i=1}^6{\mathcal{A}}_i({\nabla }_{\mathcal{H}}u)X_i\psi \mathrm{d}x\mathrm{d}t$$

(3)

holds true, then we name the function $u\in L^p((0, T), W_{\mathcal{H},\mathrm{loc}}^{1,p}(\mathsf{\Omega }))$ as a weak solution to Equation (1). Here, $W_{\mathcal{H},\mathrm{loc}}^{1,p}(\mathsf{\Omega })$ is the first-order p-th integrable horizontal local Sobolev space, which is composed of total functions $f\in L_{\mathrm{loc}}^p(\mathsf{\Omega })$, whose distributional horizontal gradients are ${\nabla }_{\mathcal{H}}f\in L_{\mathrm{loc}}^p(\mathsf{\Omega })$. In the classic case, $\mathcal{A}(\xi )={|\xi |}^{p-2}\xi$, Equation (1) becomes the parabolic p-Laplacian equation:

$${\partial }_{t}u=-\sum _{i=1}^6{X}_i^{*}(|{\nabla }_{\mathcal{H}}u{|}^{p-2}{X}_{i}u).$$

The study of the parabolic p-Laplacian equation originated from DiBenedetto-Friedman [22]. They established the $C^{1,\alpha }$-regularity of the weak solution in the Euclidean space; Wiegner [23] also proved the same result. For more exhaustive results on the parabolic p-Laplacian equation and more general cases in the Euclidean space, we refer to the book by DiBenedetto [24]. For the study of the parabolic p-Laplacian equation in the sub-Riemannian manifold, Capogna et al. [25] established, when $2\le p<\infty$, the $C^{\infty }$-regularity of the weak solution to the non-degenerate parabolic p-Laplacian equation in the Heisenberg group ${\mathbb{H}}^n$, as follows:

$${\partial }_{t}u=\sum _{i=1}^{2n}{X}_i\left(\right(1+{|Xu|}^2{)}^{\frac{p-2}{2}}{X}_{i}u).$$

Recently, for the degenerate parabolic p-Laplacian equation in the Heisenberg group, ${\mathbb{H}}^n$, when $2\le p\le 4$, Capogna et al. [26] established the $C^{0,1}$-regularity of the weak solution.

In this study paper, we focus on the $C^{0,1}$-regularity of the weak solution u to (3) on SU(3). As a consequence, when $2\le p\le 4$, we establish the $C_{\mathrm{loc}}^{0,1}$-regularity of u; that is, ${\nabla }_{\mathcal{H}}u\in L_{\mathrm{loc}}^{\infty }$. See Theorem 1 below for details.

Theorem 1.

Suppose $u\in L^p((0, T), W_{\mathcal{H},\mathrm{loc}}^{1,p}(\mathsf{\Omega }))$ is a weak solution to (1), satisfying condition (2), in $\mathcal{Q}=\mathsf{\Omega }\times (0,T)$. Then, ${\nabla }_{\mathcal{H}}u\in L_{\mathrm{loc}}^{\infty }(\mathcal{Q})$ for $2\le p\le 4$. Moreover, when $2\le p\le 4$, for every ${\mathcal{Q}}_{\mu ,2r}\subset {\mathcal{Q}}_{\mu ,2r_0}\subset \mathcal{Q}$, we have the following:

$$\underset{{\mathcal{Q}}_{\mu ,r}}{sup}|{\nabla }_{\mathcal{H}}u|\le C{\mu }^{\frac{1}{2}}max\left({\left(\frac{1}{\mu r^{N+2}}\int {\int }_{{\mathcal{Q}}_{\mu ,2r}}(1 + |{\nabla }_{\mathcal{H}}u{{|}^2)}^{\frac{p}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{2}}, {\mu }^{\frac{p}{2(2-p)}}\right),$$

(4)

where $C=C(p,\upsilon ,\mathrm{{\rm Y}},r_0)>0$, ${\mathcal{Q}}_{\mu ,r}≔B(x_0,r)\times (t_0-\mu r^2,t_0)$ and $N=10$ is the homogeneous dimension of SU(3).

Consequently, when $2\le p\le 4$, the weak solution to the parabolic p-Laplacian equation on SU(3) has the $C^{0,1}$-regularity and satisfies (4).

To prove Theorem 1, it requires us to contemplate the following regularized equation:

$${\partial }_t{u}_{\sigma }=\sum _{i=1}^6{X}_i{\mathcal{A}}^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })\mathrm{in}\mathcal{Q};u_{\sigma }=u\mathrm{on}{\partial }_p\mathcal{Q},$$

(5)

where u is a weak solution to (1), and ${\partial }_p\mathcal{Q}=\mathsf{\Omega }\times \{t=0\}\cup \partial \mathsf{\Omega }\times (0,T)$ is the parabolic boundary of the cylinder $\mathcal{Q}$, with the following condition:

$$\left\{\begin{array}{c}{\upsilon (\sigma +|\zeta |}^2{)}^{\frac{p-2}{2}}{|\varrho |}^2\le {\displaystyle \sum _{i,j=1}^6}{\mathcal{A}}_{i,{\zeta }_j}^{\sigma }(\zeta ){\varrho }_i{\varrho }_j\le {\mathrm{{\rm Y}}(\sigma +|\zeta |}^2{)}^{\frac{p-2}{2}}{|\varrho |}^2,\hfill \\ |{\mathcal{A}}_i^{\sigma }{(\zeta )|\le \mathrm{{\rm Y}}(\sigma +|\zeta |}^2{)}^{\frac{p-1}{2}}.\hfill \end{array}\right.$$

(6)

for every $\zeta , \varrho \in {\mathbb{R}}^6$, where $\sigma \in (0,1]$, ${\mathcal{A}}_{i,{\zeta }_j}^{\sigma }(\zeta )≔{\partial }_{{\zeta }_j}{\mathcal{A}}_i^{\sigma }(\zeta )$, $p\in [2,\infty )$ and $0<\upsilon \le \mathrm{{\rm Y}}<\infty$. Here, from [17], since ${\left\{X_i\right\}}_{1\le i\le 6}$ are the left-invariant vector fields, we have $X_i^{*}=-X_i$. Simultaneously, we also need to consider the Riemannian approximation equation (see Section 2 for details):

$${\partial }_t{u}_{ϵ}=\sum _{i=1}^8{X}_i^{ϵ}{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})\mathrm{in}\mathcal{Q};u_{ϵ}=u_{\sigma }\mathrm{on}{\partial }_p\mathcal{Q},$$

(7)

where $u_{\sigma }$ is a weak solution to (5), with the following condition:

$$\left\{\begin{array}{c}{\upsilon (\sigma +|\zeta |}^2{)}^{\frac{p-2}{2}}{|\varrho |}^2\le {\displaystyle \sum _{i,j=1}^8}{\mathcal{A}}_{i,{\zeta }_j}^{\sigma ,ϵ}(\zeta ){\varrho }_i{\varrho }_j\le {\mathrm{{\rm Y}}(\sigma +|\zeta |}^2{)}^{\frac{p-2}{2}}{|\varrho |}^2,\hfill \\ |{\mathcal{A}}_i^{\sigma ,ϵ}{(\zeta )|\le \mathrm{{\rm Y}}(\sigma +|\zeta |}^2{)}^{\frac{p-1}{2}}.\hfill \end{array}\right.$$

(8)

for every $\zeta , \varrho \in {\mathbb{R}}^8$, where ${\mathcal{A}}_{i,{\zeta }_j}^{\sigma ,ϵ}(\zeta )≔{\partial }_{{\zeta }_j}{\mathcal{A}}_i^{\sigma ,ϵ}(\zeta )$, $p\in [2,\infty )$ and $0<\upsilon \le \mathrm{{\rm Y}}<\infty$. Above, $\upsilon , \mathrm{{\rm Y}}$ depend only on ${\upsilon }^{\prime }, {\mathrm{{\rm Y}}}^{\prime }$. Let $u_{ϵ}$ be a weak solution to (7). When $2\le p<\infty$, we write ${\mathcal{A}}^{\sigma }(\zeta )=\mathcal{A}(\zeta )+\upsilon {\sigma }^{\frac{p-2}{2}}\zeta$ and ${\mathcal{A}}_i^{\sigma ,ϵ}(\zeta )={\tilde{\mathcal{A}}}_i({\zeta }_{\mathbb{H}}){+ \upsilon (\sigma +|\zeta |}_{ϵ}^2{)}^{\frac{p-2}{2}}{\zeta }_i$; see ([26], Section 2) for details. The Riemannian approximation technique has become a mature technique widely used in studying equations; see [17,19,25,26] for the definition and more details of the technique. It is proven in [25,26] that ${\mathcal{A}}^{\sigma }\to \mathcal{A}$ and $u_{\sigma }\to u$ as $\sigma \to 0$, and that ${\mathcal{A}}^{\sigma ,ϵ}\to {\mathcal{A}}^{\sigma }$ and $u_{ϵ}\to u_{\sigma }$ as $ϵ\to 0$; also see [13,14,17,19] for an example.

Hence, to obtain Theorem 1, we only need to prove that ${\left\{u_{ϵ}\right\}}_{\sigma ,ϵ\in (0,1]}$ have the following $C_{\mathrm{loc}}^{0,1}$-regularity uniformly in $\sigma ,ϵ\in (0, 1]$. Finally, letting $ϵ\to 0, \sigma \to 0$, from the following theorem, we can apply the standard method as [25,26] to derive Theorem 1.

Theorem 2.

Assume that $u_{ϵ}\in L^p((0, T),W_{\mathcal{H},\mathrm{loc}}^{1,p}(\mathsf{\Omega }))$ is a weak solution to (7) with condition (8), in $\mathcal{Q}=\mathsf{\Omega }\times (0,T)$. If $2\le p\le 4$, then ${\nabla }^{ϵ}u\in L_{\mathrm{loc}}^{\infty }(\mathcal{Q})$. Moreover, when $2\le p\le 4$, for any ${\mathcal{Q}}_{\mu ,2r}\subset {\mathcal{Q}}_{\mu ,2r_0}\subset \mathcal{Q}$, we have the following:

$$\underset{{\mathcal{Q}}_{\mu ,r}}{sup}|{\nabla }^{ϵ}{u}_{ϵ}|\le C{\mu }^{\frac{1}{2}}max\left({\left(\frac{1}{\mu r^{N+2}}\int {\int }_{{\mathcal{Q}}_{\mu ,2r}}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{2}}, {\mu }^{\frac{p}{2(2-p)}}\right),$$

(9)

where $C=C(p,\upsilon ,\mathrm{{\rm Y}},r_0)>0$ and ${\mathcal{Q}}_{\mu ,r}≔B_{ϵ}(x_0,r)\times (t_0-\mu r^2,t_0)$.

The proof of Theorem 2 relies on Moser’s iteration; see Section 4 for details. The key point, by the approach in [25,26], is to establish a crucial Caccioppoli-type estimate for ${\nabla }^{ϵ}{u}_{ϵ}$ involving ${\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}$ (see Lemma 6). To obtain the crucial Caccioppoli-type estimate, when $2\le p\le 4$, we establish two Caccioppoli-type inequalities for ${\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}$ and ${\nabla }_{\mathcal{R}}{u}_{ϵ}$ in Lemmas 4 and 5, proven in Section 3. Applying Lemma 5 to re-estimate the integral terms on the right hand of (26) in Lemma 4, we prove the crucial Caccioppoli-type estimate in Section 3.

Consequently, we construct a crucial Caccioppoli-type inequality (38). Based on the inequality we establish, when $2\le p\le 4$, the $C^{0,1}$-regularity for the parabolic p-Laplacian equation on the group SU(3). Compared to the Heisenberg group ${\mathbb{H}}^n$, our new result achieves the same range of p as [26]. Unfortunately, the $C^{0,1}$-regularity for the range $p\in (1,2)\cup (4,\infty )$ cannot be achieved with our current technology because our argument rests in a crucial way on Lemma 5 with the condition $p\in [2, 4]$. The difficulties in the proof arise from handling and estimating integral terms involving ${\nabla }_{\mathcal{R}}{\nabla }^{ϵ}{u}_{ϵ}$. In the Heisenberg group ${\mathbb{H}}^n$, there exists the property that $[X_i,R]=0$; however, it does not hold true on SU(3). For example, $[X_1,R_7]=4X_2$ (see Table 1). This means that we need to handle more integral terms when estimating integral terms involving ${\nabla }_{\mathcal{R}}{\nabla }^{ϵ}{u}_{ϵ}$. Our approach can also be applied to more general sub-Riemannian manifolds. For instance, it can be used with a special class of semi-simple Lie groups as proposed in [17], and Hörmander vector fields of step two as discussed in [19], to establish the regularity for the parabolic p-Laplacian equation. Technically speaking, our method can also be extended to other types of partial differential equations, for example, the non-homogeneous equation ${\partial }_{t}u=-{\displaystyle \sum _{i=1}^6}{X}_i^{*}{\mathcal{A}}_i({\nabla }_{\mathcal{H}}u)+\mathcal{B}(x,t,u,{\nabla }_{\mathcal{H}}u)$. The establishment of the regularity for the range of $p\in (1,2)\cup (4,\infty )$ will be the focus and difficulty of our next work.

Table 1. Lie bracket on SU(3).

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{R}}_7$	${\mathit{R}}_8$
$X_1$	0	$-R_7$	$X_5$	$-X_6$	$-X_3$	$X_4$	$4X_2$	$2X_2$
$X_2$	$R_7$	0	$X_6$	$X_5$	$-X_4$	$-X_3$	$-4X_1$	$-2X_1$
$X_3$	$-X_5$	$-X_6$	0	$-R_8$	$X_1$	$X_2$	$2X_4$	$4X_4$
$X_4$	$X_6$	$-X_5$	$R_8$	0	$X_2$	$-X_1$	$-2X_3$	$-4X_3$
$X_5$	$X_3$	$X_4$	$-X_1$	$-X_2$	0	$R_8-R_7$	$2X_6$	$-2X_6$
$X_6$	$-X_4$	$X_3$	$-X_2$	$X_1$	$R_7-R_8$	0	$-2X_5$	$2X_5$
$R_7$	$-4X_2$	$4X_1$	$-2X_4$	$2X_3$	$-2X_6$	$2X_5$	0	0
$R_8$	$-2X_2$	$2X_1$	$-4X_4$	$4X_3$	$2X_6$	$-2X_5$	0	0

2. Preliminaries

The group SU(3) is a special type of unitary group composed of $3\times 3$ complex matrices; that is,

$$\mathrm{SU}(3)≔\{A\in \mathrm{GL}(3,\mathbb{C}):A·A^{*}=E,detA=1\},$$

where E is the identity matrix. The Lie algebra of SU(3) is defined by the following:

$$\mathrm{su}(3)≔\{B\in \mathrm{gl}(3,\mathbb{C}):B+B^{*}=0,\mathrm{tr}B=0\}$$

granted with the inner product $⟨B,C⟩≔-\frac{1}{2}\mathrm{tr}(BC).$

The two-dimensional maximal torus on the group SU(3) is provided by the following:

$$\mathbb{S}≔\left\{\left(\begin{array}{ccc}{e}^{i{s}_1}& 0& 0\\ 0& e^{i{s}_2}& 0\\ 0& 0& e^{i{s}_3}\end{array}\right): s_1, s_2, s_3\in \mathbb{R}, s_1+s_2+s_3=0\right\},$$

whose Lie algebra is as follows:

$$\mathcal{S}≔\left\{\left(\begin{array}{ccc}i{s}_1& 0& 0\\ 0& i{s}_2& 0\\ 0& 0& i{s}_3\end{array}\right): s_1, s_2, s_3\in \mathbb{R}, s_1+s_2+s_3=0\right\}$$

is selected as the Cartan subalgebra. The following Gell–Mann matrices form a set of the orthogonal basis of su(3), namely the following:

$$\begin{array}{c}{X}_1=\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right),X_2=\left(\begin{array}{ccc}0& i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right),X_3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right),\hfill \\ \\ X_4=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& -i& 0\end{array}\right),X_5=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right),X_6=\left(\begin{array}{ccc}0& 0& i\\ 0& 0& 0\\ i& 0& 0\end{array}\right),\hfill \\ \\ S_1=\left(\begin{array}{ccc}-i& 0& 0\\ 0& i& 0\\ 0& 0& 0\end{array}\right),S_2=\left(\begin{array}{ccc}-\frac{i}{\sqrt{3}}& 0& 0\\ 0& -\frac{i}{\sqrt{3}}& 0\\ 0& 0& \frac{2i}{\sqrt{3}}\end{array}\right).\hfill \end{array}$$

The following two vector fields are generated from $[X_1, X_2]$ and $[X_3, X_4]$, respectively; that is,

$$R_7=-[X_1,X_2]=\left(\begin{array}{ccc}-2i& 0& 0\\ 0& 2i& 0\\ 0& 0& 0\end{array}\right)\mathrm{and}{R}_8=-[X_3,X_4]=\left(\begin{array}{ccc}0& 0& 0\\ 0& 2i& 0\\ 0& 0& -2i\end{array}\right).$$

Since $S_1=\frac{1}{2}{R}_7$ and $S_2=\frac{1}{2\sqrt{3}}{R}_7-\frac{1}{\sqrt{3}}{R}_8$, the vertical vector fields $R_7, R_8$ form a set of orthogonal basis of $\mathcal{S}$. Hence, the vertical gradient is defined by ${\nabla }_{\mathcal{R}}≔(R_7, R_8)$.

We recall the Riemannian approximation technique. Given $ϵ\in (0, 1]$, we define the Riemannian approximation to the vector fields $X_1, X_2, \dots , X_6$, as

$$X_1^{ϵ}=X_1, X_2^{ϵ}=X_2, \dots , X_6^{ϵ}=X_6, X_7^{ϵ}=ϵR_7, X_8^{ϵ}=ϵR_8.$$

From which, we denote ${\nabla }^{ϵ}=(X_1, \dots , X_6,ϵR_7,ϵR_8)$ as the gradient,

The following table ([17], Table 2.1), shows the total Lie bracket for any two vector fields belonging to $\{X_1, \dots , X_6,R_7,R_8\}$.

Table 1 shows that

$$[X_i,X_j]={\upsilon }_{i,j}^{(k)}{X}_k+{\theta }_{i,j}^{(l)}{R}_l,[X_i,R_j]={\vartheta }_{i,j}^{(k)}{X}_k,[R_i,R_j]=0,$$

(10)

and that

$$[X_i^{ϵ},X_j^{ϵ}]={\upsilon }_{i,j}^{(k)}{X}_k+{\theta }_{i,j}^{(l)}{R}_l,$$

(11)

where ${\upsilon }_{i,j}^{(k)},{\theta }_{i,j}^{(l)},{\vartheta }_{i,j}^{(k)}\in \mathbb{R}$ are constants determined entirely by Table 1. From Table 1, it is not difficult for us to discover that the horizontal subspace $\mathcal{H}$ in SU(3) is generated by the set of orthogonal bases $\{X_1, X_2, \dots , X_6\}$ satisfying the Hörmander condition. Hence, the horizontal gradient is defined by ${\nabla }_{\mathcal{H}}=(X_1, X_2, \dots , X_6)$. Here, the basis $\{X_1, X_2, \dots , X_6\}$ is left-invariant due to the left-invariance of the Gell–Mann matrices. To summarize, the basis $\{X_1, X_2, \dots , X_6\}$ generates the horizontal distribution of a sub-Riemannian manifold.

3. Several Caccioppoli-Type Inequalities and a Crucial Caccioppoli-Type Estimate

In this section, we establish the crucial Caccioppoli-type estimate for ${\nabla }^{ϵ}{u}_{ϵ}$ involving ${\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}$ and some Caccioppoli-type inequalities, which are uniform in $\sigma ,ϵ\in (0, 1]$. The following two lemmas are prerequisites for the proofs of subsequent lemmas.

Lemma 1.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, $v_l^{ϵ}=X_l^{ϵ}{u}_{ϵ}$, with $l=1, \dots , 8$, solves

$${\partial }_t{v}_l^{ϵ}=\sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ})+\sum _{i=1}^8[X_l^{ϵ},X_i^{ϵ}]{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}).$$

(12)

Proof. 

From (7), by the Lie bracket, we have the following:

$$\begin{array}{cc}\hfill {\partial }_t{v}_l^{ϵ}& =X_l^{ϵ}{\partial }_t{u}_{ϵ}=\sum _{i=1}^8{X}_l^{ϵ}(X_i^{ϵ}{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}))\hfill \\ \hfill & =\sum _{i=1}^8{X}_i^{ϵ}(X_l^{ϵ}{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}))+\sum _{i=1}^8[X_l^{ϵ},X_i^{ϵ}]{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})\hfill \\ \hfill & =\sum _{i=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ})+\sum _{i,j=1}^8[X_l^{ϵ},X_i^{ϵ}]{\mathcal{A}}_i^{\delta ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})\hfill \end{array}$$

□

Lemma 2.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, $R_l{u}_{ϵ}$, with $l=7, 8$ solves the following:

$$\begin{array}{cc}\hfill {\partial }_t{R}_l{u}_{ϵ}=& \sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}{R}_l{u}_{ϵ})+\sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[R_l,X_j^{ϵ}]u_{ϵ})\hfill \\ \hfill & +\sum _{i=1}^8[R_l,X_i^{ϵ}]{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}).\hfill \end{array}$$

(13)

Proof. 

Letting $v_l^{ϵ}=ϵR_l{u}_{ϵ}$ in Lemma 1, we have the following:

$${\partial }_t{R}_l{u}_{ϵ}=\sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})R_l{X}_j^{ϵ}{u}_{ϵ})+\sum _{i=1}^8[R_l,X_i^{ϵ}]{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}).$$

From this, by $R_l{X}_j^{ϵ}=X_j^{ϵ}{R}_l+[R_l,X_j^{ϵ}]$, we obtain (13). □

3.1. Several Caccioppoli-Type Inequalities

The following lemma provides a Caccioppoli-type inequality for ${\nabla }_{\mathcal{R}}{u}_{ϵ}$ involving ${\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}$.

Lemma 3.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, when $p\in (1,\infty )$, for every $\gamma \ge 0$ and every $\varrho \in C^1([0,T],C_0^{\infty }(\mathsf{\Omega }))$, we have the following:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2{\varrho }^{4+\gamma }{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +2}{\varrho }^{3+\gamma }|{\partial }_t\varrho |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(\gamma +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\varrho |}^2{\varrho }^{2+\gamma }{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(\gamma +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p}{2}}{\varrho }^{4+\gamma }{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(14)

where $C=C(\upsilon ,\mathrm{{\rm Y}})>0$.

Proof. 

Applying $\psi ={\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}$ to test (13), we obtain the following:

$$\begin{array}{cc}\hfill L^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t{R}_l{u}_{ϵ}{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}{R}_l{u}_{ϵ}){\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{X}_i^{ϵ}({\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[R_l,X_j^{ϵ}]u_{ϵ}){\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8[R_l,X_i^{ϵ}]{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t=S_1^l+S_2^l+S_3^l.\hfill \end{array}$$

(15)

For $L^l$, integrating by parts, we have the following:

$$\sum _{l=7}^8{L}^l=\frac{1}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t(|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +2}){\varrho }^2\mathrm{d}x\mathrm{d}t=-\frac{2}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma +2}\varrho {\partial }_t\varrho \mathrm{d}x\mathrm{d}t,$$

which yields

$$|\sum _{l=7}^8{L}^l|\le \frac{2}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +2}\varrho |{\partial }_t\varrho |\mathrm{d}x\mathrm{d}t.$$

(16)

For $S_1^l$, integrating by parts, we have the following:

$$\begin{array}{cc}\hfill \sum _{l=7}^8{S}_1^l=& -\sum _{l=7}^8{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}{R}_l{u}_{ϵ}2\varrho X_i^{ϵ}\varrho {|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - \sum _{l=7}^8{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}{R}_l{u}_{ϵ}{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{X}_i^{ϵ}{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - \gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}(|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2){\varrho }^2|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma -2}{X}_i^{ϵ}(|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -S_{11}-S_{12}-S_{13}.\hfill \end{array}$$

(17)

For $S_2^l$, integrating by parts, we have the following:

$$\begin{array}{cc}\hfill S_2^l=& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[R_l,X_j^{ϵ}]u_{ϵ}2\varrho X_i^{ϵ}\varrho {|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[R_l,X_j^{ϵ}]u_{ϵ}{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{X}_i^{ϵ}{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - \gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[R_l,X_j^{ϵ}]u_{ϵ}{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma -2}\sum _{k=7}^8{R}_k{u}_{ϵ}{X}_i^{ϵ}{R}_k{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -S_{21}^l-S_{22}^l-S_{23}^l.\hfill \end{array}$$

(18)

For $S_2^l$, integrating by parts, we have the following:

$$\begin{array}{cc}\hfill S_3^l=& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})2\varrho [R_l,X_i^{ϵ}]\varrho {|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }{R}_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2[R_l,X_i^{ϵ}]R_l{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - \gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2{R}_l{u}_{ϵ}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma -2}\sum _{k=7}^8{R}_k{u}_{ϵ}[R_l,X_i^{ϵ}]R_k{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -S_{31}^l-S_{32}^l-S_{33}^l.\hfill \end{array}$$

(19)

Combining (15) and (17)–(19), we obtain the following:

$$S_{12}+S_{13}=-\sum _{l=7}^8{L}^l-S_{11}-\sum _{l=7}^8\sum _{k=1}^3(S_{2k}^l+S_{3k}^l).$$

(20)

Now, we use the condition inequality to estimate each term in (20) separately. Applying condition (8) to estimate $S_{12}$, we obtain the following:

$$S_{12}\ge \upsilon {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }\mathrm{d}x\mathrm{d}t.$$

(21)

Applying condition (8) to estimate $S_{13}$, we obtain the following:

$$S_{13}\ge \frac{\gamma \upsilon }{4}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}(|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2{)|}^2{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma -2}\mathrm{d}x\mathrm{d}t\ge 0.$$

(22)

Applying condition (8) to estimate $S_{13}$, we obtain the following:

$$|S_{11}|\le 2\mathrm{{\rm Y}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|\varrho |{\nabla }^{ϵ}\varrho ||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +1}\mathrm{d}x\mathrm{d}t.$$

(23)

Applying condition (8) to estimate $S_{21}^l$ and $S_{31}^l$, we obtain the following:

$$|\sum _{l=7}^8{S}_{21}^l| + |\sum _{l=7}^8{S}_{31}^l| \le 4\mathrm{{\rm Y}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1}{2}}\varrho |{\nabla }^{ϵ}\varrho ||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +1}\mathrm{d}x\mathrm{d}t.$$

(24)

Applying condition (8) to estimate $S_{22}^l$, $S_{32}^l$, $S_{23}^l$, and $S_{33}^l$, we obtain the following:

$$\begin{array}{cc}\hfill & |\sum _{l=7}^8{S}_{22}^l| + |\sum _{l=7}^8{S}_{32}^l| + |\sum _{l=7}^8{S}_{23}^l| + |\sum _{l=7}^8{S}_{33}^l|\hfill \\ \hfill & \le 4\mathrm{{\rm Y}}(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1}{2}}{\varrho }^2|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(25)

Combining (16) and (20)–(25), by Young’s inequality, we obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^2{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le \frac{C}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma +2}\varrho |{\partial }_t\varrho |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }^{ϵ}{\varrho |}^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(\gamma +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p}{2}}{\varrho }^2{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{\gamma }\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

where $C=C(\upsilon ,\mathrm{{\rm Y}})>0$. Setting $\varrho \to {\varrho }^{2+\gamma /2}$ in the above inequality, we obtain (14). □

The following lemma provides a Caccioppoli-type inequality for ${\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}$.

Lemma 4.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, when $p\in (1,\infty )$, for every $\gamma \ge 0$ and every $\varrho \in C^1([0,T],C_0^{\infty }(\mathsf{\Omega }))$, we have the following:

$$\begin{array}{cc}\hfill & \frac{1}{\gamma +2}\underset{t_1<t<t_2}{sup}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma +2}{2}}{\varrho }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}{|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(\gamma +1)}^2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p+\gamma }{2}}({\varrho }^2 + |{\nabla }^{ϵ}{\varrho |}^2+\varrho |{\nabla }_{\mathcal{R}}\varrho |)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C{(\gamma +1)}^4{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+\frac{C}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma +2}{2}}|{\partial }_t\varrho |\varrho \mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(26)

where $C=C(n,p,\upsilon ,\mathrm{{\rm Y}})>0$.

Proof. 

Applying $\psi ={\varrho }^2(\sigma +|{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_l^{ϵ}{u}_{ϵ}$ to test (12), then integrating by parts, we obtain the following:

$$\begin{array}{cc}\hfill L^l=& \frac{1}{2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{\partial }_t({(X_l^{ϵ}{u}_{ϵ})}^2){\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ}2\varrho X_i^{ϵ}\varrho (\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ}{\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_i^{ϵ}{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -\frac{\gamma }{2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ}{\varrho }^2{X}_l^{ϵ}{u}_{ϵ}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma -2}{2}}{X}_i^{ϵ}(|{\nabla }^{ϵ}{u}_{ϵ}{|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})2\varrho [X_l^{ϵ},X_i^{ϵ}]\varrho (\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}[X_l^{ϵ},X_i^{ϵ}]X_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -\gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2{X}_l^{ϵ}{u}_{ϵ}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma -2}{2}}\sum _{k=1}^8{X}_k^{ϵ}{u}_{ϵ}[X_l^{ϵ},X_i^{ϵ}]X_k^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =-S_1^l-S_2^l-S_3^l-S_4^l-S_5^l-S_6^l.\hfill \end{array}$$

(27)

For $S_2^l$, we use $X_l^{ϵ}{X}_j^{ϵ}=X_j^{ϵ}{X}_l^{ϵ}+[X_l^{ϵ},X_j^{ϵ}]$ to obtain the following:

$$\begin{array}{cc}\hfill S_2^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}{X}_l^{ϵ}{u}_{ϵ}{\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_i^{ϵ}{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[X_l^{ϵ},X_j^{ϵ}]u_{ϵ}{\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_i^{ϵ}{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& S_{21}^l+S_{22}^l.\hfill \end{array}$$

(28)

For $S_3^l$, we use $X_l^{ϵ}{X}_j^{ϵ}=X_j^{ϵ}{X}_l^{ϵ}+[X_l^{ϵ},X_j^{ϵ}]$ to obtain the following:

$$\begin{array}{cc}\hfill S_3^l=& \frac{\gamma }{4}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_j^{ϵ}({(X_l^{ϵ}{u}_{ϵ})}^2){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma -2}{2}}{X}_i^{ϵ}(|{\nabla }^{ϵ}{u}_{ϵ}{|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \frac{\gamma }{2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})[X_l^{ϵ},X_j^{ϵ}]u_{ϵ}{\varrho }^2{X}_l^{ϵ}{u}_{ϵ}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma -2}{2}}{X}_i^{ϵ}(|{\nabla }^{ϵ}{u}_{ϵ}{|}^2)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& S_{31}^l+S_{32}^l.\hfill \end{array}$$

(29)

Combining (27)–(29), we obtain

$$L^l+S_{21}^l+S_{31}^l=-S_1^l-S_{22}^l-S_{32}^l-S_4^l-S_5^l-S_6^l.$$

(30)

Now, we use the condition inequality to estimate each term in (30) separately. To bound the first term in the left hand of (26), we note the following:

$$\frac{1}{\gamma +2}\left({\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma +2}{2}}{\varrho }^2\mathrm{d}x\right){|}_{t_1}^{t_2}=2L^l+\frac{2}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma +2}{2}}\varrho {\partial }_t\varrho \mathrm{d}x\mathrm{d}t.$$

Applying condition (8) to estimate $S_{21}^l$, we obtain the following:

$$\sum _{l=1}^8{S}_{21}^l\ge \upsilon {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}{|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t.$$

Applying condition (8) to estimate $S_{31}^l$, we obtain the following:

$$\sum _{l=1}^8{S}_{31}^l\ge \frac{\upsilon \gamma }{4}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-4+\gamma }{2}}|{\nabla }^{ϵ}(|{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t.$$

Applying condition (8) to estimate $S_1^l$, we obtain the following:

$$|S_1^l|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|\varrho |{\nabla }^{ϵ}\varrho |\mathrm{d}x\mathrm{d}t,$$

where $C=C(\mathrm{{\rm Y}})>0$. Applying condition (8) to estimate $S_{22}^l$ and $S_{32}^l$, we obtain the following:

$$|S_{22}^l| + |S_{32}^l|\le C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}(|{\nabla }^{ϵ}{u}_{ϵ}| + |{\nabla }_{\mathcal{R}}{u}_{ϵ}|){\varrho }^2|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t,$$

where $C=C(\mathrm{{\rm Y}})>0$. Applying condition (8) to estimate $S_4^l$, we obtain the following:

$$|S_4^l| \le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p+\gamma }{2}}\varrho (|{\nabla }^{ϵ}\varrho | + |{\nabla }_{\mathcal{R}}\varrho |)\mathrm{d}x\mathrm{d}t,$$

where $C=C(\mathrm{{\rm Y}})>0$.

Below, we estimate $S_5^l$. We use (11) to $S_5^l$ and obtain the following:

$$\begin{array}{cc}\hfill S_5^l=& {\upsilon }_k{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_k{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + {\theta }_m{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{R}_m{X}_l^{ϵ}{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& {\upsilon }_k{S}_{51}^l+{\theta }_m{S}_{52}^l,\hfill \end{array}$$

where ${\upsilon }_k, {\theta }_m$ are constants completely determined by Table 1. Applying condition (8) to estimate $S_{51}^l$, we obtain the following:

$$|S_{51}^l| \le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2\mathrm{d}x\mathrm{d}t,$$

where $C=C(\mathrm{{\rm Y}})>0$. For $S_{52}^l$, by $R_m{X}_l^{ϵ}=X_l^{ϵ}{R}_m+[R_m,X_l^{ϵ}]$, we obtain the following:

$$\begin{array}{cc}\hfill S_{52}^l=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{X}_l^{ϵ}{R}_m{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}[R_m,X_l^{ϵ}]u_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& S_{521}^l+S_{522}^l.\hfill \end{array}$$

Using (10) to $S_{522}^l$, by condition (8), we obtain the following:

$$|S_{522}^l| \le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}{\varrho }^2\mathrm{d}x\mathrm{d}t,$$

where $C=C(\mathrm{{\rm Y}})>0$. For $S_{521}^l$, integrating by parts, we have the following:

$$\begin{array}{cc}\hfill S_{521}^l=& -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i,j=1}^8{\mathcal{A}}_{i,{\xi }_j}^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})X_l^{ϵ}{X}_j^{ϵ}{u}_{ϵ}{\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{R}_m{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - \frac{\gamma }{2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ}){\varrho }^2(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma -2}{2}}\sum _{k=1}^8{X}_k^{ϵ}{u}_{ϵ}{X}_l^{ϵ}{X}_k^{ϵ}{u}_{ϵ}{R}_m{u}_{ϵ}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & - 2{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}\sum _{i=1}^8{\mathcal{A}}_i^{\sigma ,ϵ}({\nabla }^{ϵ}{u}_{ϵ})\varrho X_l^{ϵ}\varrho (\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{\gamma }{2}}{R}_m{u}_{ϵ}\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Applying condition (8) to estimate $S_{521}^l$, we obtain the following:

$$\begin{array}{cc}\hfill |S_{521}^l| \le & C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2|{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma +|{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}\varrho |{\nabla }^{ϵ}\varrho ||{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

where $C=C(\mathrm{{\rm Y}})>0$. Combining these estimates, we obtain the estimate of $S_5^l$, as follows:

$$\begin{array}{cc}\hfill |S_5^l| \le & C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2|{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}\varrho |{\nabla }^{ϵ}\varrho ||{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

where $C=C(\mathrm{{\rm Y}})>0$.

For $S_6^l$, we use the same method as estimating $S_5^l$ and obtain the following:

$$\begin{array}{cc}\hfill |S_6^l| \le & C\gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C\gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C\gamma (\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|{\varrho }^2|{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C\gamma {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-1+\gamma }{2}}\varrho |{\nabla }^{ϵ}\varrho ||{\nabla }_{\mathcal{R}}{u}_{ϵ}|\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

where $C=C(\mathrm{{\rm Y}})>0$.

Combining all estimates to (30), then by Young’s inequality, we obtain (26). □

Based on Lemma 3, we obtain the following lemma, which provides two Caccioppoli-type inequalities for ${\nabla }_{\mathcal{R}}{u}_{ϵ}$.

Lemma 5.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, when $p\in [2, 4]$, for every $\gamma \ge 0$ and every $\varrho \in C^1([0, T],C_0^{\infty }(\mathsf{\Omega }))$, we have the following:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p+\gamma }{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C(p+\gamma )\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+C(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2}{2}}|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma }{|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|}^2{\varrho }^{4+\gamma }\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(31)

where $C=C(p)>0$;

$$\begin{array}{cc}\hfill & {\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p+\gamma }{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{p+\gamma }}\hfill \\ \hfill & \le C{(p+\gamma )}^2(\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}+{\parallel \varrho \parallel }_{L^{\infty }}){\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{p+\gamma }}\hfill \\ \hfill & \phantom{\le }+ C(p+\gamma )\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}^{\frac{1}{2}}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{2(p+\gamma )}}{\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{4-p}{2(p+\gamma )}},\hfill \end{array}$$

(32)

where $C=C(\upsilon ,\mathrm{{\rm Y}})>0$.

Proof. 

First, we prove (31). Denote

$$U≔{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p+\gamma }{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t,V≔\int {\int }_{\mathrm{spt}(\varrho )}(\sigma +|{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t.$$

According to Table 1, we write the following:

$$\begin{array}{cc}\hfill & R_7{u}_{ϵ}=-[X_1,X_2]u_{ϵ}=X_2{X}_1{u}_{ϵ}-X_1{X}_2{u}_{ϵ}\hfill \\ \hfill & R_8{u}_{ϵ}=-[X_3,X_4]u_{ϵ}=X_4{X}_3{u}_{ϵ}-X_3{X}_4{u}_{ϵ}.\hfill \end{array}$$

From this and ${\nabla }_{\mathcal{R}}{u}_{ϵ}=(R_7{u}_{ϵ}, R_8{u}_{ϵ})$, we rewrite $|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p+\gamma }$ as

$$\begin{array}{cc}\hfill |{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p+\gamma }=& |{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-2+\gamma }({(R_7{u}_{ϵ})}^2+{(R_8{u}_{ϵ})}^2)\hfill \\ \hfill =& |{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-2+\gamma }(R_7{u}_{ϵ}(X_2{X}_1{u}_{ϵ}-X_1{X}_2{u}_{ϵ})+R_8{u}_{ϵ}(X_4{X}_3{u}_{ϵ}-X_3{X}_4{u}_{ϵ})).\hfill \end{array}$$

Then U can be written as follows:

$$\begin{array}{cc}\hfill U=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p-2+\gamma }{R}_7{u}_{ϵ}(X_2{X}_1{u}_{ϵ}-X_1{X}_2{u}_{ϵ}){\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p-2+\gamma }{R}_8{u}_{ϵ}(X_4{X}_3{u}_{ϵ}-X_3{X}_4{u}_{ϵ}){\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t=U_1+U_2.\hfill \end{array}$$

For $U_1$, we integrate by parts to obtain the following:

$$\begin{array}{cc}\hfill U_1=& -(p-2-\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-3+\gamma }{R}_7{u}_{ϵ}{\varrho }^{p+\gamma }(X_2|{\nabla }_{\mathcal{R}}{u}_{ϵ}|X_1{u}_{ϵ}-X_1|{\nabla }_{\mathcal{R}}{u}_{ϵ}|X_2{u}_{ϵ})\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p-2+\gamma }{\varrho }^{p+\gamma }(X_2{R}_7{u}_{ϵ}{X}_1{u}_{ϵ}-X_1{R}_7{u}_{ϵ}{X}_2{u}_{ϵ})\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p-2+\gamma }{R}_7{u}_{ϵ}{\varrho }^{p-1+\gamma }(X_2\varrho X_1{u}_{ϵ}-X_1\varrho X_2{u}_{ϵ})\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill U_1\le & 2(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-2+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + 2(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-1+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}||{\nabla }^{ϵ}\varrho |{\varrho }^{p-1+\gamma }\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

In the same way, we obtain the estimate of $U_2$, as follows:

$$\begin{array}{cc}\hfill U_2\le & 2(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-2+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + 2(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-1+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}||{\nabla }^{ϵ}\varrho |{\varrho }^{p-1+\gamma }\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill U\le & 4(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-2+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + 4(p+\gamma ){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}||{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{p-1+\gamma }|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}||{\nabla }^{ϵ}\varrho |{\varrho }^{p-1+\gamma }\mathrm{d}x\mathrm{d}t=L_1+L_2.\hfill \end{array}$$

(33)

Below, we estimate $L_1$ and $L_2$. For $L_1$, applying Hölder’s inequality, we have the following:

$$L_1\le 4(p+\gamma ){\chi }^{\frac{1}{2}}{V}^{\frac{4-p}{2(p+\gamma )}}{U}^{\frac{2p-4+\gamma }{2(p+\gamma )}},$$

(34)

where

$$\chi ≔{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\nabla }^{ϵ}{u}_{ϵ}{|}^{p-2}|{\nabla }_{\mathcal{R}}{u}_{ϵ}{|}^{\gamma }{|{\nabla }^{ϵ}{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^2{\varrho }^{4+\gamma }\mathrm{d}x\mathrm{d}t.$$

(35)

For $L_2$, applying Hölder’s inequality, we have the following:

$$L_2\le 4(p+\gamma ){\parallel {\nabla }^{ϵ}\varrho \parallel }_{L^{\infty }}{V}^{\frac{1}{p+\gamma }}{U}^{\frac{p-1+\gamma }{p+\gamma }}.$$

(36)

Thus,

$$U\le 4(p+\gamma ){\chi }^{\frac{1}{2}}{V}^{\frac{4-p}{2(p+\gamma )}}{U}^{\frac{2p-4+\gamma }{2(p+\gamma )}}+4(p+\gamma ){\parallel {\nabla }^{ϵ}\varrho \parallel }_{L^{\infty }}{V}^{\frac{1}{p+\gamma }}{U}^{\frac{p-1+\gamma }{p+\gamma }}.$$

(37)

From this, by Young’s inequality, we obtain (31).

Second, we prove (32). Applying Lemma 3 to re-estimate $\mathcal{M}$ defined in (35), then we apply Hölder’s inequality to obtain the following:

$$\begin{array}{cc}\hfill \chi \le & C{(\gamma +1)}^2\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2{V}^{\frac{p-2}{p+\gamma }}{U}^{\frac{\gamma +2}{p+\gamma }}+C\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{U}^{\frac{\beta +2}{p+\gamma }}\hfill \\ \hfill & + C{(\gamma +1)}^2{\parallel \varrho \parallel }_{L^{\infty }}^4{V}^{\frac{p}{p+\gamma }}{U}^{\frac{\gamma }{p+\gamma }},\hfill \end{array}$$

where $C=C(\upsilon ,\mathrm{{\rm Y}})>0$. This, with (34), yields the following:

$$\begin{array}{cc}\hfill L_1\le & C{(p+\gamma )}^2\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}{V}^{\frac{1}{p+\gamma }}{U}^{\frac{p-1+\gamma }{p+\gamma }}+C(p+\gamma )\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}^{\frac{1}{2}}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{2(p+\gamma )}}{V}^{\frac{4-p}{2(p+\gamma )}}{U}^{\frac{p-1+\gamma }{p+\gamma }}\hfill \\ \hfill & + C{(p+\gamma )}^2{\parallel \varrho \parallel }_{L^{\infty }}^2{V}^{\frac{2}{p+\gamma }}{U}^{\frac{p-2+\gamma }{p+\gamma }}.\hfill \end{array}$$

Combining the above inequality, (33) and (36), we obtain the following:

$$\begin{array}{cc}\hfill U\le & C{(p+\gamma )}^2\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}{V}^{\frac{1}{p+\gamma }}{U}^{\frac{p-1+\gamma }{p+\gamma }}+C(p+\gamma )\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}^{\frac{1}{2}}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{2(p+\gamma )}}{V}^{\frac{4-p}{2(p+\gamma )}}{U}^{\frac{p-1+\gamma }{p+\gamma }}\hfill \\ \hfill & + C{(p+\gamma )}^2{\parallel \varrho \parallel }_{L^{\infty }}^2{V}^{\frac{2}{p+\gamma }}{U}^{\frac{p-2+\gamma }{p+\gamma }}.\hfill \end{array}$$

Dividing both sides of the above inequality by $U^{\frac{p-2+\gamma }{p+\gamma }}$ simultaneously, we apply Young’s inequality to obtain the following:

$$\begin{array}{cc}\hfill U^{\frac{2}{p+\gamma }}\le & C{(p+\gamma )}^4\parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2{V}^{\frac{2}{p+\gamma }}+C{(p+\gamma )}^2\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{V}^{\frac{4-p}{p+\gamma }}\hfill \\ \hfill & + C{(p+\gamma )}^2{\parallel \varrho \parallel }_{L^{\infty }}^2{V}^{\frac{2}{p+\gamma }},\hfill \end{array}$$

which implies (32). □

3.2. A crucial Caccioppoli-Type Estimate

Based on Lemmas 4 and 5, we obtain the crucial Caccioppoli-type estimate for ${\nabla }^{ϵ}{u}_{ϵ}$ involving ${\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}$.

Lemma 6.

Suppose $u_{ϵ}$ is a weak solution to (7). Then, when $p\in [2, 4]$, for every $\gamma \ge 0$ and $\varrho \in C^1([0, T],C_0^{\infty }(\mathsf{\Omega }))$, we have the following:

$$\begin{array}{cc}\hfill & \underset{t_1<t<t_2}{sup}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma +2}{2}}{\varrho }^2\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}{|{\nabla }^{ϵ}{\nabla }^{ϵ}{u}_{ϵ}|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\gamma )}^9{(\parallel \varrho \parallel }_{L^{\infty }}^2 + \parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2 + \parallel \varrho {\nabla }_{\mathcal{R}}{\varrho \parallel }_{L^{\infty }})\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(p+\gamma )}^7\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +2}{p+\gamma }},\hfill \end{array}$$

(38)

where $C=C(p, \upsilon , \mathrm{{\rm Y}})>0$.

Proof. 

To obtain (38), we need to re-estimate each integral term on the right-hand side of (26), separately.

First, we bind the second integral term on the hand side of (26). Applying Hölder’s inequality, then by (32) in Lemma 5, we obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{p-2+\gamma }{2}}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^2{\varrho }^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{p-2+\gamma }{p+\gamma }}{\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }_{\mathcal{R}}{u}_{ϵ}|}^{p+\gamma }{\varrho }^{p+\gamma }\mathrm{d}x\mathrm{d}t\right)}^{\frac{2}{p+\gamma }}\hfill \\ \hfill & \le C{(p+\gamma )}^4{(\parallel \varrho \parallel }_{L^{\infty }}^2 + \parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2)\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(p+\gamma )}^2\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +2}{p+\gamma }},\hfill \end{array}$$

where $C=C(p,\upsilon ,\mathrm{{\rm Y}})>0$.

Second, we bind the final integral term on the hand side of (26). We apply Hölder’s inequality to obtain the following:

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{|}^2{)}^{\frac{\gamma +2}{2}}|{\partial }_t\varrho |\varrho \mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le \parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +2}{p+\gamma }}.\hfill \end{array}$$

Combining the above estimates and (26), we obtain the following: (38). □

4. Proof of Theorem 2

In this section, we apply the crucial Caccioppoli-type estimate to prove Theorem 2.

The Proof of Theorem 2.

For every non-negative cut-off function $\varrho \in C^1([0,T],C_0^{\infty }(\mathsf{\Omega }))$ vanishing on the parabolic boundary of $\mathcal{Q}$, satisfying $|\varrho |\le 1$ in $\mathcal{Q}$, and for any $\gamma \ge 0$, we denote the following:

$$w≔(\sigma +|{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{p+\gamma }{4}}{\varrho }^2.$$

Then, (38) is rewritten as follows:

$$\begin{array}{cc}\hfill & \underset{t_1<t<t_2}{sup}{\int }_{\mathsf{\Omega }}{w}^{\frac{2(\gamma +2)}{p+\gamma }}\mathrm{d}x+{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }^{ϵ}w|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{(p+\gamma )}^9{(\parallel \varrho \parallel }_{L^{\infty }}^2 + \parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2 + \parallel \varrho {\nabla }_{\mathcal{R}}\varrho {\parallel }_{L^{\infty }})\int {\int }_{\mathrm{spt}(\varrho )}{w}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \phantom{\le }+ C{(p+\gamma )}^7\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{\left(\int {\int }_{\mathrm{spt}(\varrho )}{w}^2\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +2}{p+\gamma }}.\hfill \end{array}$$

(39)

We denote $q≔2+\frac{4(\gamma + 2)}{N(p + \gamma )}$, where $N=10$ is the homogeneous dimension of SU(3). Applying Hölder’s inequality, we apply the Sobolev inequality to obtain the following:

$$\begin{array}{cc}\hfill {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{w}^q\mathrm{d}x\mathrm{d}t\le & {\int }_{t_1}^{t_2}{\left({\int }_{\mathsf{\Omega }}{w}^{\frac{2(\gamma +2)}{p+\gamma }}\mathrm{d}x\right)}^{\frac{2}{N}}{\left({\int }_{\mathsf{\Omega }}{w}^{\frac{2N}{N-2}}\mathrm{d}x\right)}^{\frac{N-2}{N}}\mathrm{d}t\hfill \\ \hfill \le & C{\left(\underset{t_1<t<t_2}{sup}{\int }_{\mathsf{\Omega }}{w}^{\frac{2(\gamma +2)}{p+\gamma }}\mathrm{d}x\right)}^{\frac{2}{N}}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\nabla }^{ϵ}w|}^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

which, together with (39), yields the following:

$$\begin{array}{cc}\hfill {\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{w}^q\mathrm{d}x\mathrm{d}t\right)}^{\frac{N}{N+2}}\le & C{(p+\gamma )}^9{(\parallel \varrho \parallel }_{L^{\infty }}^2 + \parallel {\nabla }^{ϵ}{\varrho \parallel }_{L^{\infty }}^2 + \parallel \varrho {\nabla }_{\mathcal{R}}\varrho {\parallel }_{L^{\infty }})\int {\int }_{\mathrm{spt}(\varrho )}{w}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + C{(p+\gamma )}^7\parallel \varrho {\partial }_t{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{p-2}{p+\gamma }}{\left(\int {\int }_{\mathrm{spt}(\varrho )}{w}^2\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +2}{p+\gamma }},\hfill \end{array}$$

(40)

where $C=C(p, \upsilon , \mathrm{{\rm Y}})>0$.

For any $\mu , r>0$, we define the parabolic cylinder ${\mathcal{Q}}_{\mu ,r}≔B_{ϵ}(x_0,r)\times (t_0-\mu r^2,t_0)$. Given any ${\mathcal{Q}}_{\mu ,2r}\subset {\mathcal{Q}}_{\mu ,2r_0}\subset \mathcal{Q}$, we denote $r_i=(1+2^{-i})r$ and ${\gamma }_i=2({\kappa }^i-1)$ with $\kappa =\frac{N + 2}{N}$ such that

$$p+{\gamma }_{i+1}=(p+{\gamma }_i)\left(1+\frac{2({\gamma }_i+2)}{N(p+{\gamma }_i)}\right),i=0, 1, 2, \dots ;$$

we write ${\mathcal{Q}}_i={\mathcal{Q}}_{\mu ,r_i}$ with ${\mathcal{Q}}_0={\mathcal{Q}}_{\mu ,2r}$ and ${\mathcal{Q}}_{\infty }={\mathcal{Q}}_{\mu ,r}$, then choose a standard parabolic cut-off function ${\varrho }_i\in C^{\infty }({\mathcal{Q}}_i)$ satisfying the following:

$$\left\{\begin{array}{c}{\varrho }_i=1\mathrm{in}{\mathcal{Q}}_{i+1},\hfill \\ |{\nabla }^{ϵ}{\varrho }_i|\le \frac{2^{i+8}}{r},|{\nabla }_{\mathcal{R}}{\varrho }_i|\le \frac{2^{2i+8}}{r^2},|{\partial }_t{\varrho }_i|\le \frac{2^{2i+8}}{\mu r^2}\mathrm{in}{\mathcal{Q}}_i.\hfill \end{array}\right.$$

By (40) with $\varrho ={\varrho }_i$ and $\gamma ={\gamma }_i$, writing ${\vartheta }_i=p+{\gamma }_i=p-2+2{\kappa }^i$, we obtain the following:

$$\begin{array}{cc}\hfill & {\left(\int {\int }_{{\mathcal{Q}}_{i+1}}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{{\vartheta }_{i+1}}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{N}{N+2}}\hfill \\ \hfill & \le C{2}^{2i}{\vartheta }_i^9(r^{-2}+1)\left[{\left(\int {\int }_{{\mathcal{Q}}_i}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{{\vartheta }_i}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{p-2}{{\vartheta }_i}}+{\mu }^{-1}{(\mu r^{N+2})}^{\frac{p-2}{{\vartheta }_i}}\right]\hfill \\ \hfill & \phantom{\le }\times {\left(\int {\int }_{{\mathcal{Q}}_i}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{{\vartheta }_i}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{{\vartheta }_i-p+2}{{\vartheta }_i}},\hfill \end{array}$$

(41)

where $C=C(p, \upsilon , \mathrm{{\rm Y}})>0$. To simplify writing, we denote

$${\chi }_i={\left(\int {\int }_{{\mathcal{Q}}_i}(\sigma + |{\nabla }^{ϵ}{u}_{ϵ}{{|}^2)}^{\frac{{\vartheta }_i}{2}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{{\vartheta }_i}}.$$

Then (41) becomes

$${\chi }_{i+1}^{\frac{{\vartheta }_{i+1}}{\kappa }}\le C{\mu }^{\frac{2}{N+2}}{2}^{2i}{\vartheta }_i^9({\chi }_i^{p-2}+{\mu }^{-1}){\chi }_i^{{\vartheta }_i-p+2},$$

where $C=C(p, \upsilon , \mathrm{{\rm Y}}, r_0)=C(p, \upsilon , \mathrm{{\rm Y}})(1+r_0^2)>0$. From this, letting ${\overline{\chi }}_i=max({\chi }_i, {\mu }^{\frac{1}{2-p}})$, we obtain the following:

$${\overline{\chi }}_{i+1}^{\frac{{\vartheta }_{i+1}}{\kappa }}\le C{\mu }^{\frac{2}{N+2}}{2}^{2i}{\vartheta }_i^9{\overline{\chi }}_i^{{\vartheta }_i}.$$

(42)

Without loss of generality, we may assume $C=C(p, \upsilon , \mathrm{{\rm Y}}, r_0)\ge 1$. Iterating (42), we have the following:

$${\overline{\chi }}_{i+1}\le \left(\prod _{j=0}^i{K}_j^{\frac{{\kappa }^{i+1-j}}{{\vartheta }_{i+1}}}\right){\overline{\chi }}_0^{\frac{{\vartheta }_0{\kappa }^{i+1}}{{\vartheta }_{i+1}}},$$

where $K_j=C{\mu }^{\frac{2}{N+2}}{2}^{2j}{\vartheta }_j^9$, ${\vartheta }_i=p-2+2{\kappa }^i$ and $\kappa =\frac{N + 2}{N}$. From this, letting $i\to \infty$, we obtain the following:

$${\overline{\chi }}_{\infty }≔\underset{i\to \infty }{lim\; sup}{\overline{\chi }}_i\le C{\mu }^{\frac{1}{2}}{\overline{\chi }}_0^{\frac{p}{2}},$$

(43)

where $C=C(p, \upsilon , \mathrm{{\rm Y}}, r_0)>0$. Since ${sup}_{{\mathcal{Q}}_{\mu ,r}}|{\nabla }^{ϵ}{u}_{ϵ}|\le {\overline{\chi }}_{\infty }$, combining (43), we obtain (9). □

5. Higher Integrability of ${\partial }_{\mathit{t}}\mathit{u}$

In this section, based on Theorem 1, when $2\le p\le 4$, we prove the higher integrability of ${\partial }_{t}u$. Setting $\sigma \to 0$ in the following theorem, we gain ${\partial }_{t}u\in L_{\mathrm{loc}}^q$ for any $1\le q<\infty$.

Theorem 3.

Suppose $u_{\sigma }$ is a weak solution to (5) in $\mathsf{\Omega }\times (0,T)$. Then, when $2\le p\le 4$, we have ${\partial }_t{u}_{\sigma }\in L_{\mathrm{loc}}^q(\mathsf{\Omega }\times (0, T))$ for any $q\in [1,\infty )$. Moreover, when $p\in [2,4]$, for every $\gamma \ge 0$ and every $\varrho \in C^1([0, T],C_0^{\infty }(\mathsf{\Omega }))$, we have

$$\begin{array}{cc}\hfill & {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma +2}{\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C^{\gamma +2}{(\gamma +2)}^{\gamma +2}|\mathrm{spt}(\varrho )|({\chi }^{2p-2}\parallel {\nabla }_{\mathcal{H}}{\varrho \parallel }_{L^{\infty }}^2+{\chi }^p\parallel \varrho {\partial }_t\varrho {{\parallel }_{L^{\infty }})}^{\frac{\gamma +2}{2}},\hfill \end{array}$$

(44)

where $C=C(p, \upsilon , \mathrm{{\rm Y}})>0$ and $\chi ={sup}_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{{|}^2)}^{\frac{1}{2}}$.

Proof. 

For any $\gamma \ge 0$, from (5), we have

$$|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}={|{\partial }_t{u}_{\sigma }|}^{\gamma }{\partial }_t{u}_{\sigma }\sum _{i=1}^6{X}_i({\mathcal{A}}^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })).$$

From this, integrating by parts, we have

$$\begin{array}{cc}\hfill L=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}{\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t={\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma }{\partial }_t{u}_{\sigma }\sum _{i=1}^6{X}_i({\mathcal{A}}_i^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })){\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -(\gamma +2){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma }{\partial }_t{u}_{\sigma }\sum _{i=1}^6{\mathcal{A}}_i^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma }){\varrho }^{\gamma +1}{X}_i\varrho \mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma }{X}_i{\partial }_t{u}_{\sigma }\sum _{i=1}^6{\mathcal{A}}_i^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma }){\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t=I_1+I_2.\hfill \end{array}$$

(45)

We apply condition (6) and Hölder’s inequality to obtain the following:

$$\begin{array}{cc}\hfill |I_1| \le & C(\gamma +2){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-1}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma +1}{\varrho }^{\gamma +1}|{\nabla }_{\mathcal{H}}\varrho |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & C(\gamma +2){\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma +2}{\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma +1}{\gamma +2}}\hfill \\ \hfill & \times {\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{(p-1)(\gamma +2)}{2}}{|{\nabla }_{\mathcal{H}}\varrho |}^{\gamma +2}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{\gamma +2}}\hfill \\ \hfill \le & C(\gamma +2)\parallel {\nabla }_{\mathcal{H}}{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{1}{\gamma +2}}{\chi }^{p-1}{L}^{\frac{\gamma +1}{\gamma +2}};\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill |I_2| \le & C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-1}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma }|{\nabla }_{\mathcal{H}}{\partial }_t{u}_{\sigma }|{\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & C(\gamma +1){\left({\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma +2}{\varrho }^{\gamma +2}\mathrm{d}x\mathrm{d}t\right)}^{\frac{\gamma }{2(\gamma +2)}}\hfill \\ \hfill & \times {\left(\int {\int }_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{{|}^2)}^{\frac{p(\gamma +2)}{4}}\mathrm{d}x\mathrm{d}t\right)}^{\frac{1}{\gamma +2}}{J}^{\frac{1}{2}}\hfill \\ \hfill \le & {C(\gamma +1)|\mathrm{spt}(\varrho )|}^{\frac{1}{\gamma +2}}{\chi }^{\frac{p}{2}}{L}^{\frac{\gamma }{2(\gamma +2)}}{J}^{\frac{1}{2}},\hfill \end{array}$$

(47)

where $\chi ={sup}_{\mathrm{spt}(\varrho )}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{{|}^2)}^{\frac{1}{2}}$ and

$$J={\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-2}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma }{|{\nabla }_{\mathcal{H}}{\partial }_t{u}_{\sigma }|}^2{\varrho }^{\gamma +4}\mathrm{d}x\mathrm{d}t.$$

Below, we estimate J. Differentiating (5) with respect to t, we obtain the following:

$${\partial }_t({\partial }_t{u}_{\sigma })=\sum _{i=1}^6{X}_i({\partial }_t{\mathcal{A}}_i^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma }))=\sum _{i,j=1}^6{X}_i({\mathcal{A}}_{i,{\xi }_j}^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })X_j{\partial }_t{u}_{\sigma }).$$

(48)

Applying $\psi =|{\partial }_t{u}_{\sigma }{|}^{\gamma }{\partial }_t{u}_{\sigma }{\varrho }^{\gamma +4}$ to test (48), we integrate by parts to obtain the following:

$$\begin{array}{cc}\hfill \mathcal{L}=& {\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t({\partial }_t{u}_{\sigma }){|{\partial }_t{u}_{\sigma }|}^{\gamma }{\partial }_t{u}_{\sigma }{\varrho }^{\gamma +4}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill =& -(\gamma +1)\sum _{i,j=1}^6{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\mathcal{A}}_{i,{\xi }_j}^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })X_j{\partial }_t{u}_{\sigma }{|{\partial }_t{u}_{\sigma }|}^{\gamma }{X}_i{\partial }_t{u}_{\sigma }{\varrho }^{\gamma +4}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -(\gamma +4)\sum _{i,j=1}^6{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\mathcal{A}}_{i,{\xi }_j}^{\sigma }({\nabla }_{\mathcal{H}}{u}_{\sigma })X_j{\partial }_t{u}_{\sigma }{|{\partial }_t{u}_{\sigma }|}^{\gamma }{\partial }_t{u}_{\sigma }{\varrho }^{\gamma +3}{X}_i\varrho \mathrm{d}x\mathrm{d}t=-S_1-S_2.\hfill \end{array}$$

Thus,

$$S_1=-\mathcal{L}-S_2.$$

For $\mathcal{L}$, we integrate by parts to obtain

$$\mathcal{L}=\frac{1}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{\partial }_t(|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}){\varrho }^{\gamma +4}\mathrm{d}x\mathrm{d}t=-\frac{\gamma +4}{\gamma +2}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}{|{\partial }_t{u}_{\sigma }|}^{\gamma +2}{\varrho }^{\gamma +3}{\partial }_t\varrho \mathrm{d}x\mathrm{d}t,$$

which, together with condition (6), yields

$$|\mathcal{L}|\le C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}{\varrho }^{\gamma +3}|{\partial }_t\varrho |\mathrm{d}x\mathrm{d}t.$$

For $S_1$, condition (6) implies

$$S_1\ge \upsilon (\gamma +1)J.$$

For $S_2$, by condition (6), by Young’s inequality, we have the following:

$$\begin{array}{cc}\hfill |S_2| \le & C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-2}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma +1}|{\nabla }_{\mathcal{H}}{\partial }_t{u}_{\sigma }|{\varrho }^{\gamma +3}|{\nabla }_{\mathcal{H}}\varrho |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & \frac{\upsilon (\gamma +1)}{2}J+C(\gamma +1){\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-2}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}{\varrho }^{\gamma +2}{|{\nabla }_{\mathcal{H}}\varrho |}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Combining these estimates, we obtain the estimate of J, as follows:

$$\begin{array}{cc}\hfill J\le & C{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}(\sigma + |{\nabla }_{\mathcal{H}}{u}_{\sigma }{|}^2{)}^{\frac{p-2}{2}}|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}{\varrho }^{\gamma +2}{|{\nabla }_{\mathcal{H}}\varrho |}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & + \frac{C}{\gamma +1}{\int }_{t_1}^{t_2}{\int }_{\mathsf{\Omega }}|{\partial }_t{u}_{\sigma }{|}^{\gamma +2}{\varrho }^{\gamma +3}|{\partial }_t\varrho |\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & C({\chi }^{p-2}\parallel {\nabla }_{\mathcal{H}}\varrho {\parallel }_{L^{\infty }}^2+\frac{1}{\gamma +1}\parallel \varrho {\partial }_t\varrho {\parallel }_{L^{\infty }})L.\hfill \end{array}$$

(49)

Combining (47) and (49), we obtain the following:

$$|I_2{| \le C(\gamma +1)|\mathrm{spt}(\varrho )|}^{\frac{1}{\gamma +2}}{\chi }^{\frac{p}{2}}{L}^{\frac{\gamma +1}{\gamma +2}}({\chi }^{p-2}\parallel {\nabla }_{\mathcal{H}}{\varrho \parallel }_{L^{\infty }}^2+\parallel \varrho {\partial }_t\varrho {{\parallel }_{L^{\infty }})}^{\frac{1}{2}}.$$

(50)

Combining (45), (46) and (50), we obtain the following:

$$\begin{array}{cc}\hfill L\le & C(\gamma +2)\parallel {\nabla }_{\mathcal{H}}{\varrho \parallel }_{L^{\infty }}{|\mathrm{spt}(\varrho )|}^{\frac{1}{\gamma +2}}{\chi }^{p-1}{L}^{\frac{\gamma +1}{\gamma +2}}\hfill \\ \hfill & +{C(\gamma +1)|\mathrm{spt}(\varrho )|}^{\frac{1}{\gamma +2}}{\chi }^{\frac{p}{2}}{L}^{\frac{\gamma +1}{\gamma +2}}({\chi }^{p-2}\parallel {\nabla }_{\mathcal{H}}{\varrho \parallel }_{L^{\infty }}^2+\parallel \varrho {\partial }_t\varrho {{\parallel }_{L^{\infty }})}^{\frac{1}{2}}.\hfill \end{array}$$

From this, we obtain (44). □

6. Conclusions

In this article, we construct a crucial Caccioppoli-type inequality (38). Based on the inequality, when $p\in [2, 4]$, we built up the $C_{\mathrm{loc}}^{0,1}$-regularity of weak solutions to the degenerate parabolic p-Laplacian equation on the group SU(3) granted with the horizontal vector fields $X_1, \dots , X_6$. Compared to the Heisenberg group ${\mathbb{H}}^n$, our new result achieves the same range of p as [26]. Unfortunately, the $C^{0,1}$-regularity for the range $p\in (1, 2)\cup (4, \infty )$ cannot be achieved with our current technology because our argument rests in a crucial way on Lemma 5 with the condition $p\in [2, 4]$. Our approach can also be used for more general sub-Riemannian manifolds, for instance, a special class of the semi-simple Lie group proposed in [17] and Hörmander vector fields of step two in [19], to establish regularity for the parabolic p-Laplacian equation. Technically speaking, our method can also be extended to other types of partial differential equations, for example, the non-homogeneous equation ${\partial }_{t}u=-{\displaystyle \sum _{i=1}^6}{X}_i^{*}{\mathcal{A}}_i({\nabla }_{\mathcal{H}}u)+\mathcal{B}(x,t,u,{\nabla }_{\mathcal{H}}u)$. The establishment of the regularity for the range $p\in (1,2)\cup (4,\infty )$ will be the focus and difficulty of our next work.

In conclusion, the results shown in this article are original. We believe that our results will be widely applied in the study of regularities for equations involving the p-Laplacian operator and other areas of applied science.