\documentstyle{SibMatJ} %\TestXML \topmatter \Author Karuppusamy \Initial Y. \Sign Yadhavan Karuppusamy \Email yadhavan\@nitgoa.ac.in \AffilRef 1 \endAuthor \Author Lingeshwaran \Initial S. \Sign Shangerganesh Lingeshwaran \Email shangerganesh\@nitgoa.ac.in \AffilRef 1 \endAuthor \Author Jeyaraj \Initial M. \Sign Manimaran Jeyaraj \Email manimaran.j\@vit.ac.in \AffilRef 2 \endAuthor \Author Hendy \Initial A. \Initial S. \Email ahmed.hendy\@fsc.bu.edu.eg \AffilRef 3 \Corresponding \endAuthor \Author Abdelaliem \Initial S. \Initial M. \Initial F. \Sign Sally Mohammed Farghaly Abdelaliem \Email smfarghaly\@pnu.edu.sa \AffilRef 4 \endAuthor \Affil 1 \Division Department of Applied Sciences \Organization National Institute of Technology Goa \City Cuncolim \Country India \endAffil \Affil 2 \Division Division of Mathematics, School of Advanced Sciences, \Organization Vellore Institute of Technology \City Chennai \Country India \endAffil \Affil 3 \Division Department of Computational Mathematics and Computer Science \Organization Institute of Natural Sciences and Mathematics, Ural Federal University \City Yekaterinburg \Country Russia \endAffil \Affil 4 \Division Department of Nursing Management and Education, College of Nursing \Organization Princess Nourah bint Abdulrahman University \City Riyadh \Country Saudi Arabia \endAffil \datesubmitted March 28, 2025\enddatesubmitted \daterevised November 15, 2025\enddaterevised \dateaccepted December 8, 2025\enddateaccepted \UDclass ??? %35Q30; 35K57; 35D30; 92C17 \endUDclass \thanks Princess Nourah bint Abdulrahman University, %\? Researchers Supporting Project number (PNURSP2025R720), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. \endthanks \title The Influence of Movement of Bacteria in a Porous Medium on the Representation of Chemotaxis Navier--Stokes Model: Global Existence \endtitle \abstract This paper explores the well-posedness of the chemotaxis system involving two species featuring nonlinear diffusion in response to two stimuli, along with the Navier--Stokes equation. This system describes a biological process representing two species and the influence of chemical stimuli secreted by the species, which attract the other in a porous medium. It is proven that the system possesses global-in-time weak and bounded weak solutions in the three-dimensional spatial domain. Also, it is extended for a bounded domain with a smooth boundary. \endabstract \keywords chemotaxis, Navier--Stokes, weak solutions, two-species %\?в тексте без - \endkeywords \endtopmatter \let\epsilon\varepsilon %\? \varepsilon не было, только \epsilon всюду \head 1. Introduction \endhead Chemotaxis is an intriguing biological phenomenon observed in various organisms, from single-celled bacteria %\?в основном без артикля, по-моему to complex multicellular organisms. It refers to the directed movement of cells or organisms in response to chemical gradients in their environment. Essentially, chemotaxis allows organisms to sense and navigate towards attractant chemicals or away from repellents, facilitating vital biological processes such as finding nutrients and avoiding toxins. First, this phenomenon was modeled in [1,\,2] as $$ \cases %\? поменяли \cases справа на обычные. Спросить у авторов. u_t=\Delta u- \nabla \cdot (\chi(v) u \nabla v), \\ v_t=\Delta v-v+u, \endcases \tag1.1 $$ where $u$ refers to the density of the organism, $v$ refers to the concentration of chemical, and $\chi$ refers to the chemotactic sensitivity function. The study of similar model equations has led to many advances in understanding how organisms navigate and interact with their environment. The existence of radially symmetric equilibrium solutions and its stability of \Tag(1.1) with the growth rate incorporate cooperation and competition effects in [3]. The global existence of classical solutions and critical blow-up of \Tag(1.1) are studied in [4]. Global-in-time solutions and their asymptotic behavior of \Tag(1.1) are studied in [5] for small initial data. Considering \Tag(1.1) with nonlinear diffusion and with a power factor in the drift term, the existence of a global-in-time weak solution and its decay properties are studied in [6,\,7]. The global existence of a classical radially symmetric solution for \Tag(1.1) is established in [8]. The behavior of bacteria living in thin fluid layers near solid-air-water contact lines modeled in [9] as $$ \cases u_t + u\cdot \nabla u=\Delta u-\nabla\cdot(\chi(c) u\nabla c), \\ v_t+ u\cdot \nabla v=\Delta v-k(v)u,\\ z_t+\tau (z\cdot\nabla)z+\nabla p=\Delta z-u \nabla \phi,\quad \operatorname{div} z=0, \endcases \tag1.2 $$ where $u$ refers to the density of the bacteria, $v$ refers to the oxygen concentration, and $z,p$ refers to the velocity field and pressure of the fluid, respectively. The system \Tag(1.2) called %\?ничего не выделено \it the Keller--Segel Navier--Stokes system (KS-NS) %\? 2 раза сокращение используется and called the Keller--Segel Stokes system (KSS) %\?сокращение не используется, 1 раз используется Keller--Segel Stokes if $\tau=1$ and $\tau=0$ respectively. The global existence of a weak solution in two dimensions of \Tag(1.2) with some assumptions on bacteria density and movement established in [10]. A numerical method was found to investigate the dynamics of KS-NS in [11]. The global existence of solutions in 2D and 3D was established for \Tag(1.2) under some initial data assumptions in [12]. A local-in-time regular solution is established in [13] for KS-NS in a two-dimensional spatial domain. Also, with stronger assumptions in the chemosensitivity function, a global-in-time weak solution is established. Considering oxygen concentration in \Tag(1.2) as parabolic/hyperbolic, the local existence of a regular solution and its blow-up criterion is established in [14]. The movement of bacteria is considered as migration in a porous medium, then \Tag(1.2) becomes $$ \cases u_t + u\cdot \nabla u=\Delta u^{1+\alpha}-\nabla\cdot(\chi(c) u\nabla c), \\ v_t+ u\cdot \nabla v=\Delta v-k(v)u,\\ z_t+\tau (z\cdot\nabla)z+\nabla p=\Delta z-u \nabla \phi,\quad \operatorname{div} z=0. \endcases \tag1.3 $$ For $\tau=0$ and $\alpha \in (\frac{1}{2},1]$ in $\Bbb{R}^2$ and $\tau=1$ and $\alpha \in [0.8,1]$ in $\Bbb{R}^3$ global existence %\?the of solutions established for~\Tag(1.3) in~[15]. Further, global-in-time solution %\?a for~\Tag(1.3) in $\Bbb{R}^2$ when $\tau=0$ and $\alpha=\frac{1}{3}$ proved in~[16]. Also, the global existence of weak solution %\?a for \Tag(1.3) in three dimension for $\alpha=\frac{2}{3}$ in [17]. %\?вроде сказуемого нет Considering initial data are sufficiently regular and positive, the global existence of weak solutions of \Tag(1.3) for $\alpha>\frac{1}{7}$ with $\tau=0$ proved in [18]. Under some conditions on the initial data, the global existence of bounded weak solutions of \Tag(1.3) for any $\alpha>0$ with $\tau=0$ was proved in [19]. The global-in-time weak and bounded weak solutions under some assumptions on sensitivity functions and diffusive exponent were established in [20]. Similar studies are also performed for nonlinear diffusion operators in [21]. Besides the papers mentioned above, chemotaxis models also encompass the modeling of species evolution dynamics. It means that a species produces a chemical signal that either attracts or repels based on the concentration of chemicals. It is modeled in [22] as follows: $$ \cases u_t=\Delta u-\nabla\cdot(\chi(v) u\nabla v)+\nabla\cdot(\xi(w) u\nabla w), \\ v_t=\Delta v+\beta u - \gamma v, \\ w_t=\Delta w+\delta u - \eta w, \endcases \tag1.4 $$ where $u$ denotes to density of the species, $v$ denotes the chemoattractant concentration, and $w$ denotes the chemorepellant concentration. By assuming that $\gamma=\eta$, global solvability, blow up, and asymptotic behavior of \Tag(1.4) were studied in [23]. However, the existence of solutions of \Tag(1.4) was proved in [24] without any additional assumptions on the given data. In one-dimensional space, the global existence of classical solutions and steady states of \Tag(1.4) was proved in [25]. An attraction-repulsion chemotaxis system with logistic source was considered in [26] and established the existence of a global bounded classical solution. The global existence of a classical solution in 2D and a weak solution in 3D is established for~\Tag(1.4) with the assumption that $\xi\delta-\chi\beta>0$ in [27]. The model describing two competitive species attracted by one chemical signal is proposed in [28], $$ \cases u_t=\Delta u-\nabla\cdot(\chi_1(v) u\nabla w)+\mu_1 u (1-u-a_1v), \\ v_t=\Delta v-\nabla\cdot(\chi_2(w) v\nabla w)+\mu_2 v (1-a_2u-v), \\ \tau w_t=\Delta w+\delta u + \eta v-\gamma w, \endcases \tag1.5 $$ and discussed global existence and asymptotic behavior of the system \Tag(1.5) for $\tau=0$ with some assumptions on sensitivity functions, $a_1$ and $ a_2$. The dynamics of \Tag(1.5) and decay of species were studied in [29]. The global bounded solutions of \Tag(1.5) for $\tau=1$ and its uniqueness, asymptotic behavior for large $\mu_1$ and $ \mu_2$ were studied in [30]. The global asymptotic stability under the small assumption on chemotactic sensitivity was established in [31] for \Tag(1.5) with weak competition. Various results about global existence and steady states of \Tag(1.5) were established in [32--34]. A Chemotaxis %\?chemotaxis system involving two species and two signals with Lotka--Volterra-type kinetics can be modeled as $$ \cases u_t=\Delta u-\nabla\cdot(\chi_1(v) u\nabla v)+\mu_1 u (1-u-a_1w), \\ \tau v_t=\Delta v+\beta w - \gamma v,\\ w_t=\Delta w-\nabla\cdot(\chi_2(z) w\nabla z)+\mu_2 w (1-a_2u-w), \\ \tau z_t=\Delta z+\delta u - \eta z. \endcases \tag1.6 $$ The global classical solution, boundedness, and blow-up of \Tag(1.6) with $\tau=\mu_1=\mu_2=0$ were established in~[35]. The result was extended to a quasilinear chemotaxis model of \Tag(1.6) in [36]. Under the assumption of initial conditions, the unique global classical solution for \Tag(1.6) was obtained in [37]. A global bounded classical solution for \Tag(1.6) with $\tau=0$ in two dimensions was established in [38]. Further, a global bounded classical solution was established for \Tag(1.6) with $\tau=1$ in $n\geq 1$ under some assumptions on the chemotactic sensitivity function. The existence of a global bounded classical solution and its rates of convergence under some assumptions, sensitivity functions were obtained in [39]. The conditions assumed in [39] are improved in [40], and similar results were established. The novel contributions of this paper can be summarized as follows: \Item $\smallbullet$ Investigates a two-species system living in a fluid environment. \Item $\smallbullet$ Considers chemicals secreted by each species that attract the other species while repelling their own. \Item $\smallbullet$ Incorporated the effect of the fluid environment by modeling porous medium diffusion for both species. The proposed model offers an advantage over earlier ones by representing bacterial movement and diffusion through the nonlinear diffusion of porous medium type for bacterial movement. This provides a realistic representation of the underlying biological processes. The rest of the paper is organized as follows. In \Sec*{Section~2}, we define and formulate our proposed model. In \Sec*{Section 3}, we define the weak solution of the proposed problem \Tag(2.1) with $\tau=1$, and using a~suitable approximation problem, we deduce some uniform estimates. Further, a weak solution for \Tag(2.1) with $\tau=1$ is established. In \Sec*{Section 4}, we prove the existence of a bounded weak solution for \Tag(2.1) with $\tau=0$ in $\Bbb{R}^3$ using uniform estimates derived in the previous section. \head 2. Mathematical Model \endhead In this paper, we propose a model to describe the chemotaxis system involving two species featuring nonlinear diffusion in response to two secreted stimuli along with the Navier--Stokes equation in $\Bbb{R}^3\times[0,\Lambda)$, as $$ \cases u_t+n\cdot \nabla u=\Delta u^{1+\alpha}-\nabla\cdot(\chi_1(v) u\nabla v)+\nabla\cdot(\xi_1(z) u\nabla z), \\ v_t+n\cdot \nabla v=\Delta v+\beta w - \gamma v, \\ w_t+n\cdot \nabla w=\Delta w^{1+\alpha}-\nabla\cdot(\chi_2(z) w\nabla z)+\nabla\cdot(\xi_2(v) w\nabla v),\\ z_t+n\cdot \nabla z=\Delta z+\delta u - \eta z,\\ n_t+\tau(n\cdot\nabla)n+\nabla p=\Delta n-(u+w) \nabla \phi. \endcases \tag2.1 $$ Here $u$ and $w$ refer to the densities of two different species, $v$ refers concentration of chemical produced by~$w$, $z$ refers concentration of chemical produced by $u$, $\chi_1$, $\chi_2$ refers chemoattraction sensitivity coefficient, %\? $\xi_1$, $\xi_2$ chemorepulsion sensitivity coefficient, %\? $\alpha$, $\beta$, $\gamma$, $\delta$, and $\eta$ are positive constants. To the best of our knowledge, there is no paper in the literature which studies the solvability of the above-proposed model. This paper aims to establish the global existence of solutions of \Tag(2.1). The novelty of the proposed model \Tag(2.1) is briefed as follows: from the biological point of view, the species moving in a fluid environment and two chemical stimuli secreted by the species, which possess attraction-repulsion chemotaxis phenomena. \head 3. Weak Solutions \endhead In this section, we define the weak solution and introduce a suitable approximation problem for the proposed model \Tag(2.1) with $\tau=1$. Then we deduce a uniform estimate using the approximation problem. Using %\?по-моему, очень много using в статье the uniform estimate, we deduce weak solutions for \Tag(2.1) with $\tau=1$ in $\Bbb{R}^3$ and extend the result to a bounded domain with smooth boundary. We consider the system \Tag(2.1) with $\tau=1$ as $$ \cases u_t+n\cdot \nabla u=\Delta u^{1+\alpha}-\nabla\cdot(\chi_1(v) u\nabla v)+\nabla\cdot(\xi_1(z) u\nabla z), \\ v_t+n\cdot \nabla v=\Delta v+\beta w - \gamma v, \\ w_t+n\cdot \nabla w=\Delta w^{1+\alpha}-\nabla\cdot(\chi_2(z) w\nabla z)+\nabla\cdot(\xi_2(v) w\nabla v),\\ z_t+n\cdot \nabla z=\Delta z+\delta u - \eta z,\\ n_t+(n\cdot\nabla)n+\nabla p=\Delta n-(u+w) \nabla \phi. \endcases \tag3.1 $$ \demo{Definition 3.1} For $\alpha>0$ and $\Lambda\in (0,\infty)$, a quintuple $(u,v,w,z,n)$ is said to be a weak solution of \Tag(2.1) if \Item (i) $u,v,w,z \geq 0$, \Item (ii) $u(1+|x|+|\log u|), w(1+|x|+|\log w|)\in L^\infty(0,\Lambda;L^1(\Bbb{R}^3))$, \Item (iii) for $1\leq p \leq 1+\alpha$, $$ u,w \in L^\infty(0,\Lambda;L^p(\Bbb{R}^3))\ \text{ and }\ \nabla u^\frac{p+\alpha}{2}, \nabla w^\frac{p+\alpha}{2} \in L^2(0,\Lambda;L^2(\Bbb{R}^3)), $$ \Item (iv) $ v,z \in L^\infty(0,\Lambda;H^1(\Bbb{R}^3))\cap L^2(0,\Lambda;H^2(\Bbb{R}^3))$ and $v,z\in L^\infty(\Bbb{R}^3\times [0,\Lambda))$, \Item (v) $n$ is a vector-valued %\? on $\Bbb{R}^3\times (0,\Lambda)$ and $$ n \in L^\infty(0,\Lambda;L^2(\Bbb{R}^3)),\quad \nabla n \in L^2(0,\Lambda;L^2(\Bbb{R}^3)), $$ \Item (vi) for any $\varphi \in L^\infty(0,\Lambda;L^2(\Bbb{R}^3))$ and $\psi \in C_0^\infty(\Bbb{R}^3\times [0,\Lambda),\Bbb{R}^3)$ with $\nabla\cdot \psi=0$, $$ \align &\int\limits_0^\Lambda\int\limits_{\Bbb{R}^3}-u\varphi_t-un\cdot \nabla \varphi+\nabla u^{1+\alpha}\cdot \nabla \varphi-u \chi_1 \nabla v \cdot \nabla \varphi+u \xi_1 \nabla z \cdot \nabla \varphi \,dx\,dt=\int\limits_{\Bbb{R}^3}u_0\varphi(\cdot,0)dx, \\ &\int\limits_0^\Lambda\int\limits_{\Bbb{R}^3}-v\varphi_t-vn\cdot \nabla\varphi+\nabla v\cdot \nabla \varphi-\beta w \varphi+\gamma v \varphi\,dx\,dt=\int\limits_{\Bbb{R}^3}v_0\varphi(\cdot,0)dx, \\ &\int\limits_0^\Lambda\int\limits_{\Bbb{R}^3}-w\varphi_t-wn\cdot \nabla \varphi+\nabla w^{1+\alpha}\cdot \nabla \varphi-w \chi_2 \nabla z \cdot \nabla \varphi+w \xi_2 \nabla v \cdot \nabla \varphi \,dx\,dt=\int\limits_{\Bbb{R}^3}w_0\varphi(\cdot,0)dx, \\ &\int\limits_0^\Lambda\int\limits_{\Bbb{R}^3}-z\varphi_t-zn\cdot \nabla \varphi+\nabla z\cdot \nabla \varphi-\delta u \varphi+\eta z \varphi\,dx\,dt=\int\limits_{\Bbb{R}^3}z_0\varphi(\cdot,0)dx, \\ &\int\limits_0^\Lambda\int\limits_{\Bbb{R}^3}-n\cdot \psi_t+\nabla n\cdot \nabla \psi +((n\cdot\nabla)n)\cdot \psi+(u+w)\nabla \phi\cdot\psi\,dx\,dt=\int\limits_{\Bbb{R}^3}n_0\cdot\psi(\cdot,0)dx. \endalign $$ \enddemo The model we are addressing here is complicated as it has strong degeneracy of the diffusion terms. To reduce the complication, we consider the approximation problem of the proposed model is given as $$ \cases u_{\epsilon_t} +n_\epsilon\cdot \nabla u_\epsilon=\Delta (u_\epsilon+\epsilon)^{1+\alpha}-\nabla\cdot(\chi_1(v_\epsilon) u_\epsilon\nabla v_\epsilon)+\nabla\cdot(\xi_1(z_\epsilon) u_\epsilon\nabla z_\epsilon), \\ v_{\epsilon_t}+n_\epsilon\cdot \nabla {v_\epsilon}=\Delta v_\epsilon+\beta w_\epsilon - \gamma v_\epsilon, \\ w_{\epsilon_t}+n_\epsilon\cdot \nabla {w_\epsilon}=\Delta (w_\epsilon+\epsilon)^{1+\alpha}-\nabla\cdot(\chi_2(z_\epsilon) w_\epsilon\nabla z_\epsilon)+\nabla\cdot(\xi_2(v_\epsilon) w_\epsilon\nabla v_\epsilon),\\ z_{\epsilon_t}+n_\epsilon\cdot \nabla {z_\epsilon}=\Delta z_\epsilon+\delta u_\epsilon - \eta z_\epsilon,\\ n_{\epsilon_t}+(n_\epsilon \cdot\nabla)n_\epsilon+\nabla p_\epsilon=\Delta n_\epsilon-(u_\epsilon+w_\epsilon) \nabla \phi, \endcases \tag3.2 $$ with smooth initial conditions $$ u_{0_\epsilon}=\phi_\epsilon*u_0,\quad v_{0_\epsilon}=\phi_\epsilon*v_0, \quad w_{0_\epsilon}=\phi_\epsilon*w_0,\quad z_{0_\epsilon}=\phi_\epsilon*z_0,\ \text{ and }\ n_{0_\epsilon}=\phi_\epsilon*n_0, \tag3.3 $$ where $\phi_\epsilon$ refers usual mollifier with $\epsilon\in(0,1)$. Using standard theory of existence and regularity, \Tag(3.2) possesses local-in-time classical solution for each $\epsilon\in(0,1)$. This allows us to overcome the complications raised by degeneracy and progress in analysis. We derive some uniform estimates for the approximate problem, independent of $\epsilon$. Using that the obtained local-in-time solution extended globally and derived a weak solution for \Tag(2.1). For the simplicity of notation, hereafter, we use variables $(u_\epsilon,v_\epsilon,w_\epsilon,z_\epsilon,n_\epsilon)$ as $(u,v,w,z,n)$. We define the functionals as: $$ \aligned E(t) := \int\limits_{\Bbb{R}^3}u(t) (\log u(t) &+2\langle x \rangle)+w(t) (\log w(t) +2\langle x \rangle)\,dx+\Vert u(t)\Vert^{1+\alpha}_{1+\alpha}+\Vert w(t)\Vert^{1+\alpha}_{1+\alpha} \\ &+\Vert\nabla v(t)\Vert^2_2+\Vert\nabla z(t)\Vert^2_2+\Vert n(t)\Vert^2_2 \endaligned \tag3.4 $$ and $$ \aligned D(t):=\Vert \nabla u(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2} &+\Vert\nabla u(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v(t)\Vert^2_2 \\ &+\Vert \Delta z(t)\Vert^2_2+\Vert \nabla n(t)\Vert^2_2, \endaligned \tag3.5 $$ where $\langle x \rangle=(1+|x|^2)^\frac{1}{2}$. Now, we prove a lemma that gives an estimate for the solution of \Tag(3.2). \proclaim{Lemma 3.1} Suppose that $(u,v,w,z,n)$ is a classical solution for system \Tag(3.2) independent of $\epsilon$ and~\Tag(3.3) satisfies $$ \cases u_0(1+|x|+|\log u_0|)\text{ and } w_0(1+|x|+|\log w_0|)\in L^1(\Bbb{R}^3), \\ u_0,w_0\in L^{1+\alpha}(\Bbb{R}^3), v_0 \text{ and } z_0\in L^\infty(\Bbb{R}^3)\cap H^1(\Bbb{R}^3),\quad n_0\in L^2(\Bbb{R}^3). \endcases \tag3.6 $$ Further, assume that $$ \alpha>\frac{1}{6},\ \phi\in W^{2,\infty}(\Bbb{R}^3)\text{ and } \chi'_i,\xi_i'\in L_{\loc}^\infty %\?loc прямо сделала \quad\text{for } i\in\{1,2\}. \tag3.7 $$ Then, there exists $C>0$, independent of $\epsilon$, such that for any $01$. {\sc Case} \Par{L3.1}{(i)}: $\frac{1}{6} < \alpha\leq \frac{1}{3}$. Using $\log u$ as test function %\?the есть еще, правда, после as все может быть in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log u \,dx+\int\limits_{\Bbb{R}^3}\nabla \log u\cdot \nabla (u+\epsilon)^{1+\alpha}\,dx=\int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3} \nabla u \cdot (\xi_1 \nabla z)\,dx. \tag3.9 $$ Using $\nabla \log u\cdot (1+\alpha) u^\alpha \nabla u=\frac{4}{1+\alpha} |\nabla u^{\frac{1+\alpha}{2}}|^2$, the second term of LHS %\?принятое сокращение, так понимаю the left-hand side in above %\?the всюду estimated as $$ \int\limits_{\Bbb{R}^3}\nabla \log u\cdot \nabla (u+\epsilon)^{1+\alpha}\,dx\geq \frac{4}{1+\alpha} \Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}. \tag3.10 $$ Using $|\nabla u|=\frac{2}{1+\alpha}u^{\frac{1-\alpha}{2}}|\nabla u^{\frac{1+\alpha}{2}}|$ and Young's inequality, the first term of RHS %\?принятое сокращение, так понимаю the right-hand side, есть еще? может, артикль ставить in \Tag(3.9) estimated as $$ \int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx \leq \frac{2\overline{\chi_1}}{1+\alpha} \biggl(\epsilon \Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+C(\epsilon)\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx\biggr), \tag3.11 $$ where $\overline{\chi_1}:=\sup_{L^\infty(\Bbb{R}^3\times[0,\Lambda))}|\chi_1(v(\cdot))|$. Using $|\nabla u^{1-\alpha}|=C u^{\frac{1-3\alpha}{2}}|\nabla u^{\frac{1+\alpha}{2}}|$ and Young's inequality, the last term of RHS in above %\?the estimated as $$ \align \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx &=\int\limits_{\Bbb{R}^3}u^{1-\alpha}\nabla v\cdot\nabla v\,dx \\ &\leq C_0\biggl(\ \int\limits_{\Bbb{R}^3}|\nabla u^{1-\alpha}||\nabla v|\,dx+\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx\biggr) \\ &= C_0\biggl(\ \int\limits_{\Bbb{R}^3}C u^{\frac{1-3\alpha}{2}}|\nabla u^{\frac{1+\alpha}{2}}|\,|\nabla v|\,dx+\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx\biggr) \\ &\leq C C_0\int\limits_{\Bbb{R}^3}\epsilon|\nabla u^{\frac{1+\alpha}{2}}|^2+C(\epsilon)u^{1-3\alpha}|\nabla v|^2\,dx \\ &\qquad+C_0\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx. \endalign $$ Using above in \Tag(3.11), we get $$ \int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx \leq C_1\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+C_2\int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla v|^2\,dx+C_3 \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx. \tag3.12 $$ Similarly, we get $$ -\int\limits_{\Bbb{R}^3} \nabla u\cdot(\xi_1 \nabla z)\,dx \leq C_4\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+C_5\int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla z|^2\,dx+C_6 \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta z|\,dx. \tag3.13 $$ Using \Tag(3.10), \Tag(3.12), and \Tag(3.13) in \Tag(3.9) and choosing sufficiently small $C_1$ and $C_4$ such that $C>0$, we get $$ \aligned \frac{d}{dt}\int\limits_{\Bbb{R}^3}u \log u \,dx+C\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2} &\leq C_2\int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla v|^2\,dx +C_3 \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx \\ &\qquad+C_5\int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla z|^2\,dx+C_6 \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta z|\,dx. \endaligned \tag3.14 $$ Similarly, using $\log w$ as a test function in \EquTag{3.2}{$(3.2)_3$}, %$(3.2)_3$, we get $$ \aligned \frac{d}{dt}\int\limits_{\Bbb{R}^3}w \log w \,dx+C_0\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2} &\leq C_7 \int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla z|^2\,dx +C_8 \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta z|\,dx \\ &\qquad+C_9\int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla v|^2\,dx+C_{10} \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta v|\,dx. \endaligned \tag3.15 $$ Using $\langle x \rangle=(1+|x|^2)^\frac{1}{2}$ as test function in \EquTag{3.2}{$(3.2)_1$} %$(3.2)_1$ and simple algebraic calculations in order to bound $\log u$ in \Tag(3.14), we get $$ \aligned &\frac{d}{dt}\int\limits_{\Bbb{R}^3}\langle x \rangle u \,dx \\ &=\int\limits_{\Bbb{R}^3}un\cdot \nabla \langle x \rangle dx + \int\limits_{\Bbb{R}^3}(u+\epsilon)^{1+\alpha}\Delta\langle x \rangle\,dx+\int\limits_{\Bbb{R}^3}\nabla \langle x\rangle \cdot u \chi_1 \nabla v \,dx-\int\limits_{\Bbb{R}^3}\nabla \langle x\rangle \cdot u \xi_1 \nabla z \,dx \\ &\leq C (1+\Vert n \Vert_2^2+\Vert\nabla v \Vert_2^2+\Vert \nabla z\Vert^2_2)+(C(\epsilon)+\epsilon\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^2). \endaligned \tag3.16 $$ Similarly, to bound $\log w$ in \Tag(3.15), we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3}\langle x \rangle w \,dx \leq C_1 (1+\Vert n \Vert_2^2+\Vert\nabla z \Vert_2^2+\Vert \nabla v\Vert^2_2)+(C(\epsilon_1)+\epsilon_1\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert_2^2). \tag3.17 $$ Using $u^\alpha$ as test function in \EquTag{3.2}{$(3.2)_1$} %$(3.2)_1$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot \nabla (u+\epsilon)^{1+\alpha}\,dx=\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\xi_1 \nabla z)\,dx. \tag3.18 $$ Using $\vert \nabla u^\frac{1+2\alpha}{2}\vert^2=\frac{(1+2\alpha)^2}{4}u^\frac{2\alpha-1}{2}\vert \nabla u\vert^2$, the second term of LHS in above %\? estimated as $$ \int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot \nabla (u+\epsilon)^{1+\alpha}\,dx\geq \frac{4\alpha(1+\alpha)}{(1+2\alpha)^2}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2. \tag3.19 $$ Using Young's inequality, the first term of RHS in \Tag(3.18) is estimated as $$ \aligned \int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\chi \nabla v)\,dx &\leq C\int\limits_{\Bbb{R}^3}|\nabla u^{\frac{1+2\alpha}{2}}|(u^{\frac{1}{2}}|\nabla v|)\,dx \\ &\leq C\biggl(\epsilon \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C(\epsilon) \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx\biggr) \\ &\leq C\biggl(\epsilon \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C' \biggl(\ \int\limits_{\Bbb{R}^3}|\nabla u||\nabla v|\,dx+\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx\biggr)\biggr) \\ &= C\biggl(\epsilon \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_0\int\limits_{\Bbb{R}^3}u^{\frac{1-2\alpha}{2}}|\nabla u^{\frac{1+2\alpha}{2}}||\nabla v|dx+C'\int\limits_{\Bbb{R}^3}u|\Delta v|dx \biggr) \\ &\leq C\biggl(\epsilon \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_0\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 \\ &\qquad+C_0\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2\,dx+C'\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx \biggr) \\ &\leq C_1 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_2\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2dx+C_3 \int\limits_{\Bbb{R}^3}u|\Delta v|dx \endaligned \tag3.20 $$ Similarly, we get $$ - \int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\xi_1 \nabla z)\,dx\leq C_4 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_5\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla z|^2\,dx+C_6 \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \tag3.21 $$ Using \Tag(3.19), \Tag(3.20), and \Tag(3.21) %\?\Tag(3.19)--\Tag(3.21) in \Tag(3.18) and choosing sufficiently small $C_1$ and $C_4$ such that $C>0$, we have $$ \aligned \frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+C\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 &\leq C_2\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2\,dx+C_3 \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx \\ &\qquad+C_5\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla z|^2\,dx+C_6 \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \endaligned \tag3.22 $$ Similarly, using $w^\alpha$ as a test function in \EquTag{3.2}{$(3.2)_3$}. %$(3.2)_3$, we get $$ \aligned \frac{d}{dt}\Vert w\Vert^{1+\alpha}_{1+\alpha}+C_0\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2 &\leq C_7\int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla z|^2\,dx+C_8 \int\limits_{\Bbb{R}^3}w|\Delta z|\,dx \\ &\qquad+C_9\int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla v|^2\,dx+C_{10} \int\limits_{\Bbb{R}^3}w|\Delta v|\,dx. \endaligned \tag3.23 $$ Using $-\Delta v$ as test function in \EquTag{3.2}{$(3.2)_2$}, %$(3.2)_2$, we get $$ \frac{d}{dt}\Vert\nabla v\Vert^2_2+2\Vert \Delta v\Vert^2_2 \leq \int\limits_{\Bbb{R}^3} \Delta v \cdot (n\cdot \nabla v)\,dx -\beta\int\limits_{\Bbb{R}^3}w|\Delta v|dx+\gamma\int\limits_{\Bbb{R}^3}v|\Delta v|dx. \tag3.24 $$ The first term in %\?on есть еще, много вроде RHS of the above evaluated as $$ \int\limits_{\Bbb{R}^3} \Delta v\cdot (n\cdot\nabla v)\,dx = -\int\limits_{\Bbb{R}^3} \sum_{i,j} \partial_i v\,\partial_jv\,\partial_i n_j \,dx = \int\limits_{\Bbb{R}^3}\sum_{i,j} v\,\partial_i\partial_jv\,\partial_i n_j \,dx \leq C \Vert \nabla n \Vert_2 \Vert \Delta v \Vert_2. $$ Using above in \Tag(3.24), we get $$ \frac{d}{dt}\Vert\nabla v\Vert^2_2+2\Vert \Delta v\Vert^2_2 \leq C \Vert \nabla n \Vert_2 \Vert \Delta v \Vert_2 -\beta\int\limits_{\Bbb{R}^3}w|\Delta v|dx+\gamma\int\limits_{\Bbb{R}^3}v|\Delta v|dx. \tag3.25 $$ Similarly, using $-\Delta z$ as test function in \EquTag{3.2}{$(3.2)_4$}, %$(3.2)_4$, we get $$ \frac{d}{dt}\Vert\nabla z\Vert^2_2+2\Vert \Delta z\Vert^2_2 \leq C \Vert \nabla n \Vert_2 \Vert \Delta z \Vert_2 -\delta\int\limits_{\Bbb{R}^3}u|\Delta z|dx+\eta\int\limits_{\Bbb{R}^3}z|\Delta z|dx. \tag3.26 $$ Using $n$ as test function in \EquTag{3.2}{$(3.2)_5$}, %$(3.2)_5$, we get $$ \frac{d}{dt}\Vert n\Vert^2_2+2 \Vert\nabla n\Vert^2_2 \leq C'\int\limits_{\Bbb{R}^3}(u+w)|n|dx. \tag3.27 $$ Adding \Tag(3.14)--\Tag(3.17), \Tag(3.22), \Tag(3.23), \Tag(3.25)--\Tag(3.27) and choosing sufficiently small constants such that $C_0>0$, we get $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq C'_{0}\biggl(\ \int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla v|^2\,dx+\int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla z|^2\,dx +\int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla v|^2\,dx \\ &\qquad +\int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2\,dx+ \int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla z|^2\,dx+ \int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla v|^2\,dx \\ &\qquad +\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx+ \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta z|\,dx+ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta v|\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}u|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta v|\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}v|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}z|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}(u+w)|n|\,dx\biggr). \endaligned \tag3.28 $$ As $0\leq 1-3\alpha <\frac{2}{3}$, using the H\"older inequality and Sobolev embedding, we get $$ \aligned \int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla v|^2\,dx &\leq \cases \int\limits_{\Bbb{R}^3}(C(\epsilon_1)+\epsilon_1 u^{\frac{2}{3}})|\nabla v|^2\,dx & \text{if } \frac{1}{6}<\alpha <\frac{1}{3}, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=\frac{1}{3}. \endcases \\ &\leq \cases C(\epsilon_1)\Vert\nabla v\Vert^2_2+\epsilon_1 \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2& \text{if } \frac{1}{6}<\alpha <\frac{1}{3}, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=\frac{1}{3}. \endcases \endaligned \tag3.29 $$ Similarly, we get $$ \int\limits_{\Bbb{R}^3}u^{1-3\alpha}|\nabla z|^2\,dx \leq \cases C(\epsilon_{2})\Vert\nabla z\Vert^2_2+\epsilon_{2} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2 & \text{if } \frac{1}{6}<\alpha <\frac{1}{3}, \\ \Vert\nabla z\Vert^2_2 & \text{if } \alpha=\frac{1}{3}, \endcases \tag3.30 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla z|^2\,dx \leq \cases C(\epsilon_{3})\Vert\nabla z\Vert^2_2+\epsilon_{3} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2 & \text{if } \frac{1}{6}<\alpha <\frac{1}{3}, \\ \Vert\nabla z\Vert^2_2 & \text{if } \alpha=\frac{1}{3}, \endcases \tag3.31 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-3\alpha}|\nabla v|^2\,dx \leq \cases C(\epsilon_{4})\Vert\nabla v\Vert^2_2+\epsilon_{4} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2 & \text{if } \frac{1}{6}<\alpha <\frac{1}{3}, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=\frac{1}{3}, \endcases \tag3.32 $$ \iftex $$ \align & \int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2\,dx \leq C(\epsilon_{5})\Vert\nabla v\Vert^2_2+\epsilon_{5} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2, \tag3.33 \\ & \int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla z|^2\,dx \leq C(\epsilon_{6})\Vert\nabla z\Vert^2_2+\epsilon_{6} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2, \tag3.34 \\ & \int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla z|^2\,dx \leq C(\epsilon_{7})\Vert\nabla z\Vert^2_2+\epsilon_{7} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2, \tag3.35 \\ & \int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla v|^2\,dx \leq C(\epsilon_{8})\Vert\nabla v\Vert^2_2+\epsilon_{8} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2. \tag3.36 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla v|^2\,dx \leq C(\epsilon_{5})\Vert\nabla v\Vert^2_2+\epsilon_{5} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2, \tag3.33 $$ $$ \int\limits_{\Bbb{R}^3}u^{1-2\alpha}|\nabla z|^2\,dx \leq C(\epsilon_{6})\Vert\nabla z\Vert^2_2+\epsilon_{6} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2, \tag3.34 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla z|^2\,dx \leq C(\epsilon_{7})\Vert\nabla z\Vert^2_2+\epsilon_{7} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2, \tag3.35 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-2\alpha}|\nabla v|^2\,dx \leq C(\epsilon_{8})\Vert\nabla v\Vert^2_2+\epsilon_{8} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2. \tag3.36 $$ \fi As $\frac{4}{3}\leq \frac{6-6\alpha}{2+3\alpha}<2$, using the H\"older, Young, and Gagliardo--Nierenberg inequality, %\?inequalities we get $$ \aligned \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta v|\,dx &\leq C(\epsilon_{9})\Vert u\Vert^{2-\alpha}_{2-\alpha}+\epsilon_{9}\Vert\Delta v \Vert_2^2 \\ &\leq C(\epsilon_{9})C \Vert u_0\Vert_1^{\frac{1+4\alpha}{2+3\alpha}}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^{\frac{6-6\alpha}{2+3\alpha}} +\epsilon_{9}\Vert\Delta v \Vert_2^2 \\ &\leq C(\epsilon_{9})C\bigl(C(\epsilon_{10})\Vert u_0\Vert_1^{\frac{1+4\alpha}{2+3\alpha}}+\epsilon_{10}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^2\bigr) + \epsilon_{9}\Vert\Delta v \Vert_2^2 \\ &\qquad= C_1+C_2\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{9}\Vert\Delta v \Vert_2^2. \endaligned \tag3.37 $$ Similarly, we get \iftex $$ \align & \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta z|\,dx \leq C_3+C_4\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{11}\Vert\Delta z \Vert_2^2, \tag3.38 \\ & \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta z|\,dx \leq C_5+C_6\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{12}\Vert\Delta z \Vert_2^2, \tag3.39 \\ & \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta v|\,dx \leq C_7+C_8\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{13}\Vert\Delta v \Vert_2^2. \tag3.40 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\Delta z|\,dx \leq C_3+C_4\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{11}\Vert\Delta z \Vert_2^2, \tag3.38 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta z|\,dx \leq C_5+C_6\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{12}\Vert\Delta z \Vert_2^2, \tag3.39 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\Delta v|\,dx \leq C_7+C_8\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert_2^2+\epsilon_{13}\Vert\Delta v \Vert_2^2. \tag3.40 $$ \fi As $ \frac{3}{2}\leq\frac{6}{2+6\alpha}<2$, using the Young and Gagliardo--Nierenberg inequality, %\?inequalities we get $$ \aligned \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx &\leq C(\epsilon_{14})\Vert u\Vert^2_2 +\epsilon_{14}\Vert\Delta v\Vert_2^2 \\ &\leq C(\epsilon_{14}) C_0\Vert u_0\Vert_1^{\frac{1+6\alpha}{2+6\alpha}}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert_2^{\frac{6}{2+6\alpha}}+\epsilon_{14}\Vert\Delta v\Vert_2^2 \\ &\leq C(\epsilon_{14}) C_0\bigl(C(\epsilon_{15}) \Vert u_0\Vert_1^{\frac{1+6\alpha}{2+3\alpha}}+\epsilon_{15}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert_2^2\bigr)+\epsilon_{14}\Vert\Delta v\Vert_2^2 \\ &\qquad= C_9+C_{10}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{14}\Vert\Delta v\Vert_2^2. \endaligned \tag3.41 $$ Similarly, we get \iftex $$ \align & \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx \leq C_{11}+C_{12}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{15}\Vert\Delta z\Vert_2^2. \tag3.42 \\ & \int\limits_{\Bbb{R}^3}w|\Delta z|\,dx \leq C_{12}+C_{12}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{16}\Vert\Delta z\Vert_2^2. \tag3.43 \\ & \int\limits_{\Bbb{R}^3}w|\Delta v|\,dx \leq C_{13}+C_{12}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{17}\Vert\Delta v\Vert_2^2. \tag3.44 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx \leq C_{11}+C_{12}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{15}\Vert\Delta z\Vert_2^2. \tag3.42 $$ $$ \int\limits_{\Bbb{R}^3}w|\Delta z|\,dx \leq C_{12}+C_{12}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{16}\Vert\Delta z\Vert_2^2. \tag3.43 $$ $$ \int\limits_{\Bbb{R}^3}w|\Delta v|\,dx \leq C_{13}+C_{12}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert_2^2+\epsilon_{17}\Vert\Delta v\Vert_2^2. \tag3.44 $$ \fi Also, using Young's inequality, we get \iftex $$ \align & \int\limits_{\Bbb{R}^3}v|\Delta v|\,dx \leq C_{14}\Vert v\Vert_2^2+\epsilon_{18}\Vert\Delta v\Vert_2^2. \tag3.45 \\ &\int\limits_{\Bbb{R}^3}z|\Delta z|\,dx \leq C_{15}\Vert z\Vert_2^2+\epsilon_{19}\Vert\Delta z\Vert_2^2. \tag3.46 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}v|\Delta v|\,dx \leq C_{14}\Vert v\Vert_2^2+\epsilon_{18}\Vert\Delta v\Vert_2^2. \tag3.45 $$ $$ \int\limits_{\Bbb{R}^3}z|\Delta z|\,dx \leq C_{15}\Vert z\Vert_2^2+\epsilon_{19}\Vert\Delta z\Vert_2^2. \tag3.46 $$ \fi As $1<\frac{6}{5}<3+6\alpha$ and $0<\frac{2}{2+6\alpha}<2$, using the Young, Gagliardo--Nierenberg, and Sobolev inequality, %\?inequalities we get $$ \aligned \int\limits_{\Bbb{R}^3}u|n|\,dx &\leq C_0\Vert u\Vert_{\frac{6}{5}}\Vert n\Vert_6 \leq\epsilon_{20} \Vert u \Vert^2_{\frac{6}{5}}+C(\epsilon_{20})\Vert \nabla n\Vert_2^2 \\ &\leq C' \Vert u_0 \Vert^{\frac{3+10\alpha}{2+6\alpha}}_1 \Vert \nabla u^{\frac{1+2\alpha}{2}} \Vert^{\frac{2}{2+6\alpha}}_2+C(\epsilon_{20})\Vert \nabla n\Vert_2^2 \\ &\leq C' \Vert u_0 \Vert^{\frac{3+10\alpha}{2+6\alpha}}_1(\epsilon_{21} \Vert \nabla u^{\frac{1+2\alpha}{2}} \Vert^2_2+C(\epsilon_{21}))+C(\epsilon_{20})\Vert \nabla n\Vert_2^2 \\ &\qquad= C_{17}+C_{18}\Vert \nabla u^{\frac{1+2\alpha}{2}} \Vert^2_2+C(\epsilon_{20})\Vert \nabla n\Vert_2^2. \endaligned \tag3.47 $$ Similarly, we get $$ \int\limits_{\Bbb{R}^3}w|n|\,dx\leq C_0\Vert w\Vert_{\frac{6}{5}}\Vert n\Vert_6\leq C_{19}+C_{20}\Vert \nabla w^{\frac{1+2\alpha}{2}} \Vert^2_2+C(\epsilon_{21})\Vert \nabla n\Vert_2^2. \tag3.48 $$ Substituting \Tag(3.29)--\Tag(3.48) in \Tag(3.28) and and choosing sufficiently small constants such that $C,\overline{C}>0$, we have $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq\overline{C}(1+\Vert \nabla v\Vert_2^2+\Vert \nabla z\Vert_2^2). \endaligned $$ Integrating above with respect to $t$, we get \Tag(3.8). {\sc Case} \Par{L3.1}{(ii)}: $\frac{1}{3} < \alpha \leq 1$. Using $\log u$ as test function in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log u \,dx+\int\limits_{\Bbb{R}^3}\nabla \log u\cdot \nabla (u+\epsilon)^{1+\alpha}\,dx\leq\int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3} \nabla u \cdot (\xi_1 \nabla z)\,dx. \tag3.49 $$ As $\alpha$ does not affect \Tag(3.10) and \Tag(3.11), it holds in this case. Proceeding as similar as \Tag(3.11), we get $$ -\int\limits_{\Bbb{R}^3} \nabla u\cdot(\xi_1 \nabla z)\,dx \leq \frac{2\overline{\xi_1}}{1+\alpha}\biggl(\epsilon_1 \Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+C(\epsilon_1)\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx\biggr). \tag3.50 $$ Using \Tag(3.10), \Tag(3.11), and \Tag(3.50) in \Tag(3.49) and also choosing sufficiently small constants such that $C_1>0$, we have $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3}u \log u \,dx+C_1 \Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}\leq C_2\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx+C_3\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx. \tag3.51 $$ Similarly, we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3}w \log w \,dx+C_4 \Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}\leq C_5\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx+C_6\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx. \tag3.52 $$ Using $u^\alpha$ as test function in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+\frac{4\alpha(1+\alpha)}{(1+2\alpha)^2}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2\leq \int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\xi_1 \nabla z)\,dx. \tag3.53 $$ Using Young's inequality, the first term of RHS in the above is estimated as $$ \aligned \int\limits_{\Bbb{R}^3}\nabla u^{\alpha} &\cdot u(\chi_1 \nabla v)\,dx \leq C\int\limits_{\Bbb{R}^3}|\nabla u^{\frac{1+2\alpha}{2}}|(u^{\frac{1}{2}}|\nabla v|)\,dx \\ &\leq C\biggl(\epsilon_2 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C(\epsilon_2) \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx\biggr) \\ &= C\epsilon_2 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + CC(\epsilon_2) \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx \\ &\leq C\epsilon_2 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + CC(\epsilon_2)\epsilon_3 \biggl(\ \int\limits_{\Bbb{R}^3}|\nabla u||\nabla v|\,dx+\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx\biggr) \\ &= C\epsilon_2 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + CC(\epsilon_2)\epsilon_3 \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx +CC(\epsilon_2)\epsilon_3 \int\limits_{\Bbb{R}^3}u^{\frac{1-\alpha}{2}}|\nabla u^{\frac{1+\alpha}{2}}||\nabla v|\,dx \\ &\leq C\epsilon_2 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 +CC(\epsilon_2)\epsilon_3 \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx \\ &\qquad+ CC(\epsilon_2)\epsilon_3\biggl(\epsilon_4 \Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2 + C(\epsilon_4)\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx\biggr) \\ &= C_7\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 +C_8 \Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+ C_9\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx +C_{10} \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx. \endaligned \tag3.54 $$ Similarly, we get $$ -\int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\xi_1 \nabla z)\,dx \leq C_{11}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 +C_{12} \Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+ C_{13}\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx +C_{14} \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \tag3.55 $$ Using \Tag(3.54) and \Tag(3.55) in \Tag(3.53) and choosing sufficiently small constants such that $C>0$, we have $$ \aligned \frac{d}{dt} \Vert u\Vert^{1+\alpha}_{1+\alpha}+C \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 &\leq C_{15} \Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+ C_9\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx +C_{10} \int\limits_{\Bbb{R}^3}u|\Delta v|\,dx \\ &\qquad+C_{13}\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx +C_{14} \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \endaligned \tag3.56 $$ Similarly, we get $$ \aligned \frac{d}{dt}\Vert w \Vert^{1+\alpha}_{1+\alpha}+C'\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2 &\leq C_{16} \Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+ C_{17}\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx +C_{18} \int\limits_{\Bbb{R}^3}w|\Delta z|\,dx \\ &\qquad+C_{19}\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx +C_{20} \int\limits_{\Bbb{R}^3}w|\Delta v|\,dx. \endaligned \tag3.57 $$ As $\alpha$ does not affect \Tag(3.16), \Tag(3.17) and \Tag(3.25)--\Tag(3.27), adding those with \Tag(3.51), \Tag(3.52), \Tag(3.56), \Tag(3.57) and choosing sufficiently small constants such that $C_0,C_0'>0$, we get $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq C'_{0}\biggl(\ \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx+\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}u|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta v|\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}v|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}z|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}(u+w)|n|\,dx\biggr). \endaligned \tag3.58 $$ As $0\leq 1-\alpha <\frac{2}{3}$, using the H\"older inequality and Sobolev embedding, we get $$ \aligned \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx &\leq \cases \int\limits_{\Bbb{R}^3}\bigl(C(\epsilon_1)+\epsilon_1 u^{\frac{2}{3}}\bigr)|\nabla v|^2\,dx & \text{if } \frac{1}{3} < \alpha < 1, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=1. \endcases \\ &\leq \cases C(\epsilon_1)\Vert\nabla v\Vert^2_2+\epsilon_1 \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2 & \text{if } \frac{1}{3} < \alpha < 1, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=1. \endcases \endaligned \tag3.59 $$ Similarly, we get $$ \align \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx &\leq \cases C(\epsilon_{2})\Vert\nabla z\Vert^2_2+\epsilon_{2} \Vert u_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2 & \text{if }\frac{1}{3} < \alpha < 1, \\ \Vert\nabla z\Vert^2_2 & \text{if } \alpha=1, \endcases \\ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx &\leq \cases C(\epsilon_{3})\Vert\nabla z\Vert^2_2+\epsilon_{3} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta z \Vert_2^2 & \text{if } \frac{1}{3} < \alpha < 1, \\ \Vert\nabla z\Vert^2_2 & \text{if } \alpha=1, \endcases \\ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx &\leq \cases C(\epsilon_{4})\Vert\nabla v\Vert^2_2+\epsilon_{4} \Vert w_0\Vert^{\frac{2}{3}}\Vert\Delta v \Vert_2^2 & \text{if }\frac{1}{3} < \alpha < 1, \\ \Vert\nabla v\Vert^2_2 & \text{if } \alpha=1. \endcases \endalign $$ As $0<\frac{6}{2+6\alpha}<2$ and $ 1<\frac{6}{5}<3+6\alpha$, \Tag(3.41)--\Tag(3.48) holds. Substituting \Tag(3.59), \Tag(3.41)--\Tag(3.48) and above inequalities in \Tag(3.58) and choosing sufficiently small constants such that $C_0,\overline{C}>0$, we get $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ \leq&\overline{C}(1+\Vert \nabla v\Vert_2^2+\Vert \nabla z\Vert_2^2). \endaligned $$ Integrating above with respect to $t$, we get \Tag(3.8). {\sc Case} \Par{L3.1}{(iii)}: $\alpha > 1$. Using $\log u$ as test function in \EquTag{3.2}{$(3.2)_1$} %$(3.2)_1$ and \Tag(3.10), we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log u \,dx+\frac{4}{1+\alpha}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}\leq\int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3} \nabla u \cdot (\xi_1 \nabla z)\,dx. \tag3.60 $$ Using our assumption $\chi'\in L_{\loc}^\infty$ and $\xi'\in L_{\loc}^\infty$ in RHS of above, we get $$ \align \int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx &\leq C_1 \int\limits_{\Bbb{R}^3} u|\nabla v|^2\,dx+C_2\int\limits_{\Bbb{R}^3} u|\Delta v|\,dx, \\ -\int\limits_{\Bbb{R}^3} \nabla u\cdot(\xi_1 \nabla z)\,dx &\leq C_3 \int\limits_{\Bbb{R}^3} u|\nabla z|^2\,dx+C_4 \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \endalign $$ Using above in \Tag(3.60), we get $$ \aligned \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log u \,dx+\frac{4}{1+\alpha}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2} &\leq C_1 \int\limits_{\Bbb{R}^3} u|\nabla v|^2\,dx+C_2\int\limits_{\Bbb{R}^3} u|\Delta v|\,dx \\ &\qquad+C_3 \int\limits_{\Bbb{R}^3} u|\nabla z|^2\,dx+C_4 \int\limits_{\Bbb{R}^3}u|\Delta z|\,dx. \endaligned \tag3.61 $$ Similarly, we get $$ \aligned \frac{d}{dt}\int\limits_{\Bbb{R}^3}w\log w\,dx+\frac{4}{1+\alpha}\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2} &\leq C_5 \int\limits_{\Bbb{R}^3} w|\nabla z|^2\,dx+C_6\int\limits_{\Bbb{R}^3} w|\Delta z|\,dx \\ &\qquad+C_7 \int\limits_{\Bbb{R}^3} w|\nabla v|^2\,dx+C_8 \int\limits_{\Bbb{R}^3}w|\Delta v|\, dx. \endaligned \tag3.62 $$ Using $u^\alpha$ as test function in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+\frac{4\alpha(1+\alpha)}{(1+2\alpha)^2}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2\leq\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\xi_1 \nabla z)\,dx. \tag3.63 $$ The first term of RHS in the above estimated as $$ \int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\chi_1 \nabla v)\,dx \leq C\int\limits_{\Bbb{R}^3}|\nabla u^{\frac{1+2\alpha}{2}}|(u^{\frac{1}{2}}|\nabla v|)\,dx \leq C_5 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_6 \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx. \tag3.64 $$ Similarly, we get $$ - \int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\xi_1 \nabla z)\,dx\leq C_7 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_8 \int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx. \tag3.65 $$ Using \Tag(3.64) and \Tag(3.65) in \Tag(3.63) and choosing sufficiently small $C_5$ and $C_7$ such that $C_9>0$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+C_9\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2\leq C_6 \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx+C_8 \int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx. \tag3.66 $$ Similarly, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert w\Vert^{1+\alpha}_{1+\alpha}+C_{10}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2\leq C_{11} \int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx+ C_{12} \int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx. \tag3.67 $$ As $\alpha$ does not affect \Tag(3.16), \Tag(3.17) and \Tag(3.25)--\Tag(3.27), adding those with \Tag(3.61), \Tag(3.62), \Tag(3.66), \Tag(3.67) and choosing sufficiently small constants such that $C_0,C_0'>0$, we get $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq C'_{0}\biggl(\ \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx+\int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}u|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}u|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}w|\Delta v|\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}v|\Delta v|\,dx+\int\limits_{\Bbb{R}^3}z|\Delta z|\,dx+\int\limits_{\Bbb{R}^3}(u+w)|n|\,dx\biggr). \endaligned \tag3.68 $$ As $\frac{1+\alpha}{2}>1$, using Young's inequality, we get $$ \aligned \int\limits_{\Bbb{R}^3}u|\nabla v|^2dx &\leq \epsilon_1\Vert\nabla v\Vert^2_2+C(\epsilon_1)\int\limits_{\Bbb{R}^3}u^{\frac{1+\alpha}{2}}|\nabla v|^2\,dx \\ &=\epsilon_1\Vert\nabla v\Vert^2_2+C(\epsilon_1)\int\limits_{\Bbb{R}^3}u^{\frac{1+\alpha}{2}}\nabla v\cdot\nabla v\,dx \\ &\leq\epsilon_1\Vert\nabla v\Vert^2_2+C(\epsilon_1) C\biggl(\ \int\limits_{\Bbb{R}^3}\nabla u^{\frac{1+\alpha}{2}}\cdot \nabla v \,dx+\int\limits_{\Bbb{R}^3}u^{\frac{1+\alpha}{2}}\Delta v\,dx \biggr) \\ &\leq \epsilon_1\Vert\nabla v\Vert^2_2+C_1\bigl(\Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+\Vert\nabla v\Vert^2_2+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert\Delta v\Vert^2_2\bigr). \endaligned \tag3.69 $$ As $\frac{6\alpha}{2+3\alpha}<2$, using the Gagliardo--Nierenberg and Young inequality, %\? the third term in RHS of the above is evaluated as $$ \Vert u\Vert^{1+\alpha}_{1+\alpha}\leq C_1\Vert u_0\Vert^{\frac{2+2\alpha}{2+3\alpha}}_{1}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{\frac{6\alpha}{2+3\alpha}}_2\leq C_1C(\epsilon_2)+\epsilon_2\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^2_2. $$ Using above %\?the in \Tag(3.69), we get $$ \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx\leq C_2+C_3\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+C_4\Vert\nabla v\Vert^2_2+C_5\Vert\Delta v\Vert^2_2. \tag3.70 $$ Similarly, we get \iftex $$ \align \int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx &\leq C_6+C_7\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+C_8\Vert\nabla z\Vert^2_2+C_9\Vert\Delta z\Vert^2_2, \tag3.71 \\ \int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx &\leq C_{10}+C_{11}\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^2_2+C_{12}\Vert\nabla z\Vert^2_2+C_{13}\Vert\Delta z\Vert^2_2, \tag3.72 \\ \int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx &\leq C_{14}+C_{15}\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^2_2+C_{16}\Vert\nabla v\Vert^2_2+C_{17}\Vert\Delta v\Vert^2_2. \tag3.73 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx \leq C_6+C_7\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+C_8\Vert\nabla z\Vert^2_2+C_9\Vert\Delta z\Vert^2_2, \tag3.71 $$ $$ \int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx\leq C_{10}+C_{11}\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^2_2+C_{12}\Vert\nabla z\Vert^2_2+C_{13}\Vert\Delta z\Vert^2_2, \tag3.72 $$ $$ \int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx\leq C_{14}+C_{15}\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^2_2+C_{16}\Vert\nabla v\Vert^2_2+C_{17}\Vert\Delta v\Vert^2_2. \tag3.73 $$ \fi As $0<\frac{6}{2+6\alpha}<2$ and $ 1<\frac{6}{5}<3+6\alpha$, \Tag(3.41)--\Tag(3.48) holds. Substituting \Tag(3.70), \Tag(3.71), and \Tag(3.41)--\Tag(3.48) in \Tag(3.68) and choosing sufficiently small constants such that $C_0,\overline{C}>0$, we get $$ \align \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq\overline{C}(1+\Vert \nabla v\Vert_2^2+\Vert \nabla z\Vert_2^2). \endalign $$ Integrating above with respect to $t$, we get \Tag(3.8). \qed\enddemo %%%%%%%%%%%%%%%%%% Lemma 2 %%%%%%%%%%%%%%%%%%%%% \proclaim{Lemma 3.2} Suppose that $(u,v,w,z,n)$ is a classical solution for system \Tag(3.2) independent of $\epsilon$ and~\Tag(3.3) satisfies $$ \align &u_0(1+|x|+|\log u_0|)\ \text{ and }\ w_0(1+|x|+|\log w_0|)\in L^1(\Bbb{R}^3), \\ &u_0,w_0\in L^{1+\alpha}(\Bbb{R}^3), v_0 \ \text{ and }\ z_0\in L^\infty(\Bbb{R}^3)\cap H^1(\Bbb{R}^3),\quad n_0\in L^2(\Bbb{R}^3). \endalign $$ Further, assume that $$ \cases \alpha>\frac{1}{6},\ \phi\in W^{2,\infty}(\Bbb{R}^3)\text{ and } %\? \\ \chi'_i,\xi'_i\in L_{\loc}^\infty\text{ for } i\in\{1,2\}\text{ with }\chi'_i(\cdot)\geq \chi_{i_0}^{} %\?много \text{ for some constant } \chi_{i_0}^{}. %\? \endcases \tag3.74 $$ Then, there exist $C>0$, independent of $\epsilon$, such that for any $0 \frac{1}{6}$ already established in previous lemma. Using $\log u$ as test function in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log u \,dx+\frac{4}{1+\alpha}\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}\leq\int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3} \nabla u \cdot (\xi_1 \nabla z)\,dx. \tag3.76 $$ Using our assumption $\chi'\in L_{\loc}^\infty$ and $\chi'_1(\cdot)\geq\chi_{1_0}^{}$, %\? we get $$ \aligned \int\limits_{\Bbb{R}^3} \nabla u\cdot(\chi_1 \nabla v)dx &= -\int\limits_{\Bbb{R}^3}(\chi'_1|\nabla v|^2+\chi_1\Delta v)u\,dx \\ &\leq -\chi_{1_0}^{} %\? \int\limits_{\Bbb{R}^3}u |\nabla v|^2\,dx+C_1 \int\limits_{\Bbb{R}^3}u^{\frac{1-\alpha}{2}}|\nabla u^{\frac{1+\alpha}{2}}|\,|\nabla v|\,dx \\ &\leq -\chi_{1_0}^{}\int\limits_{\Bbb{R}^3}u |\nabla v|^2\,dx+C_2\Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+C_3\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx. \endaligned \tag3.77 $$ Similarly, we get $$ -\int\limits_{\Bbb{R}^3} \nabla u\cdot(\xi_1 \nabla z)\,dx \leq -\xi_{1_0}^{}\int\limits_{\Bbb{R}^3}u |\nabla z|^2\,dx + C_4\Vert\nabla u^{\frac{1+\alpha}{2}}\Vert^2_2+C_5\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx. \tag3.78 $$ Using \Tag(3.77) and \Tag(3.78) in \Tag(3.76) and choosing sufficiently small constants such that $C_1>0$, we get $$ \aligned \frac{d}{dt}\int\limits_{\Bbb{R}^3} u \log \ &u \,dx+C_1\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\chi_{1_0}^{}\int\limits_{\Bbb{R}^3}u |\nabla v|^2\,dx+\xi_{1_0}^{}\int\limits_{\Bbb{R}^3}u |\nabla z|^2\,dx \\ &\leq C_3\int\limits_{\Bbb{R}^3}u^{1-\alpha} |\nabla v|^2\,dx+C_5\int\limits_{\Bbb{R}^3}u^{1-\alpha} |\nabla z|^2\,dx. \endaligned \tag3.79 $$ Similarly, we get $$ \aligned \frac{d}{dt} \int\limits_{\Bbb{R}^3} w \log \ &w \,dx +C_1\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\chi_{2_0}^{}\int\limits_{\Bbb{R}^3}w |\nabla z|^2\,dx+\xi_{2_0}^{}\int\limits_{\Bbb{R}^3}w |\nabla v|^2\,dx \\ &\leq C_6\int\limits_{\Bbb{R}^3}w^{1-\alpha} |\nabla z|^2\,dx+C_7\int\limits_{\Bbb{R}^3}w^{1-\alpha} |\nabla v|^2\,dx. \endaligned \tag3.80 $$ Using $u^\alpha$ as test function in \EquTag{3.2}{$(3.2)_1$}, %$(3.2)_1$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+\frac{4\alpha(1+\alpha)}{(1+2\alpha)^2}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2\leq\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\chi_1 \nabla v)\,dx-\int\limits_{\Bbb{R}^3}\nabla u^\alpha\cdot u(\xi_1 \nabla z)\,dx. \tag3.81 $$ As $\frac{1+2\alpha}{2}>0$, using Young's inequality, the first term of RHS in the above is estimated as $$ \int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\chi_1 \nabla v)\,dx \leq \frac{2\alpha\overline{\chi_1^{}}}{1+\alpha}\int\limits_{\Bbb{R}^3}|\nabla u^{\frac{1+2\alpha}{2}}|(u^{\frac{1}{2}}|\nabla v|)\,dx \leq C_8 \Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 + C_9 \int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx. $$ Similarly, we get $$ -\int\limits_{\Bbb{R}^3}\nabla u^{\alpha}\cdot u(\xi_1 \nabla z)\,dx\leq C_{10}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 +C_{11}\int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx. $$ Using above two estimates in \Tag(3.81) and choosing sufficiently small $C_8$ and $C_{10}$ such that $C_9>0$, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert u\Vert^{1+\alpha}_{1+\alpha}+C_{14}\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2 \leq C_9\int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx +C_{11}\int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx. \tag3.82 $$ Similarly, we get $$ \frac{1}{1+\alpha}\frac{d}{dt}\Vert w\Vert^{1+\alpha}_{1+\alpha}+C_{15}\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2\leq C_{12}\int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx+C_{13}\int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx. \tag3.83 $$ As $\alpha$ does not affect \Tag(3.16), \Tag(3.17) and \Tag(3.25)--\Tag(3.27), adding those with \Tag(3.79), \Tag(3.80), \Tag(3.82), and \Tag(3.83) and choosing sufficiently small constants such that $C_0,C_0'>0$ and $C_9$, $C_{11}$, $C_{12}$, $C_{12}$, we get $$ \aligned &\frac{d}{dt}\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2 \\ &\qquad +\int\limits_{\Bbb{R}^3}u|\nabla v|^2\,dx+\int\limits_{\Bbb{R}^3}u|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w|\nabla v|^2\,dx\bigr) \\ &\leq C'_{0}\bigg(\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx+\int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx+\int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx \\ &\qquad+\int\limits_{\Bbb{R}^3}v|\Delta v|dx+\int\limits_{\Bbb{R}^3}z|\Delta z|dx-\int\limits_{\Bbb{R}^3}u|\Delta z|dx-\int\limits_{\Bbb{R}^3}w|\Delta v|dx+\int\limits_{\Bbb{R}^3}(u+w)|n|dx\bigg). \endaligned \tag3.84 $$ Using Young's inequality and choosing sufficiently small $\epsilon$, the first term of RHS in the above is estimated as $$ \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla v|^2\,dx \leq\int\limits_{\Bbb{R}^3} \bigl(C(\epsilon)+\epsilon u\bigr)|\nabla v|^2\,dx\leq C_1 \Vert \nabla v\Vert_2^2. \tag3.85 $$ Similarly, we get \iftex $$ \align \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx &\leq C_2 \Vert \nabla z\Vert_2^2, \tag3.86 \\ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx &\leq C_3 \Vert \nabla z\Vert_2^2, \tag3.87 \\ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx &\leq C_4 \Vert \nabla v\Vert_2^2. \tag3.88 \endalign $$ \else $$ \int\limits_{\Bbb{R}^3}u^{1-\alpha}|\nabla z|^2\,dx \leq C_2 \Vert \nabla z\Vert_2^2, \tag3.86 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla z|^2\,dx \leq C_3 \Vert \nabla z\Vert_2^2, \tag3.87 $$ $$ \int\limits_{\Bbb{R}^3}w^{1-\alpha}|\nabla v|^2\,dx \leq C_4 \Vert \nabla v\Vert_2^2. \tag3.88 $$ \fi As $\alpha$ does not affect \Tag(3.45)--\Tag(3.47), it holds. Substituting \Tag(3.85)--\Tag(3.88) and \Tag(3.45)--\Tag(3.47) in \Tag(3.84) and choosing sufficiently small constants such that $C_0,\overline{C}>0$, we get $$ \aligned \frac{d}{dt} &\biggl(\ \int\limits_{\Bbb{R}^3}u (\log u +2\langle x \rangle)+w (\log w +2\langle x \rangle)\,dx+\Vert u\Vert^{1+\alpha}_{1+\alpha}+\Vert w\Vert^{1+\alpha}_{1+\alpha}+\Vert\nabla v\Vert^2_2+\Vert\nabla z\Vert^2_2+\Vert n\Vert^2_2\biggr) \\ &\qquad+C_{0}\bigl(\Vert \nabla u^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v\Vert^2_2+\Vert \Delta z\Vert^2_2+\Vert \nabla n\Vert^2_2\bigr) \\ &\leq\overline{C}(1+\Vert \nabla v\Vert_2^2+\Vert \nabla z\Vert_2^2). \endaligned $$ Integrating above with respect to $t$, we get \Tag(3.75). Hence, the lemma holds. \qed\enddemo Now, we are ready to present the main findings of this paper, (i.e.) %\? the existence of weak solution for \Tag(2.1) in $\Bbb{R}^3$ and extend the same for a bounded domain with smooth boundary. \proclaim{Theorem 3.1} Assume that either \Tag(3.7) or \Tag(3.74) holds true and $(u_0,v_0,w_0,z_0,n_0)$ satisfies initial data \Tag(3.6). Then for each $\Lambda>0$, the system \Tag(3.1) possesses a weak solution $(u,v,w,z,n)$, such that $$ \aligned &\sup_{0\leq t\leq \Lambda}\biggl(\ \int\limits_{\Bbb{R}^3}u(t) (\log u(t) +2\langle x \rangle)+w(t) (\log w(t) +2\langle x \rangle)\,dx+\Vert u(t)\Vert^{1+\alpha}_{1+\alpha}+\Vert w(t)\Vert^{1+\alpha}_{1+\alpha} \\ &\qquad+\Vert\nabla v(t)\Vert^2_2+\Vert\nabla z(t)\Vert^2_2+\Vert n(t)\Vert^2_2\biggr) \\ &\qquad+\int\limits_0^\Lambda\bigl(\Vert \nabla u(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v(t)\Vert^2_2 \\ &\qquad+\Vert \Delta z(t)\Vert^2_2+\Vert \nabla n(t)\Vert^2_2\bigr)\,dt0$, the system \Tag(3.1) with \Tag(3.89) possesses a weak solution $(u,v,w,z,n)$, such that $$ \aligned &\sup_{0\leq t\leq \Lambda}\biggl(\ \int\limits_{\Bbb{R}^3}u(t) (\log u(t) +2\langle x \rangle)+w(t) (\log w(t) +2\langle x \rangle)\,dx+\Vert u(t)\Vert^{1+\alpha}_{1+\alpha}+\Vert w(t)\Vert^{1+\alpha}_{1+\alpha} \\ &\qquad+\Vert\nabla v(t)\Vert^2_2+\Vert\nabla z(t)\Vert^2_2+\Vert n(t)\Vert^2_2\biggr) \\ &\qquad+\int\limits_0^\Lambda\bigl(\Vert \nabla u(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla u(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \nabla w(t)^{\frac{1+\alpha}{2}}\Vert^{2}_{2}+\Vert\nabla w(t)^{\frac{1+2\alpha}{2}}\Vert^2_2+\Vert \Delta v(t)\Vert^2_2 \\ &\qquad+\Vert \Delta z(t)\Vert^2_2+\Vert \nabla n(t)\Vert^2_2\bigr)\,dt0$ and $\Lambda\in (0,\infty)$, a weak solution $(u,v,w,z,n)$ of \Tag(4.1) with $\tau=0$, as defined in \Par*{Definition 3.1}, is said to be a bounded weak solution if \Item (i) For any $p\in[1,\infty)$, %\?for с маленькой буквы $$ u, w \in L^\infty((0,\Lambda);\Bbb{R}^3)\text{ and } \nabla u^\frac{p+\alpha}{2},\nabla w^\frac{p+\alpha}{2}\in L^2(0,\Lambda;L^2(\Bbb{R}^3)), $$ \Item (ii) For any $q\in[2,\infty)$, %\?for с маленькой буквы $$ v,z, n \in L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3))\text{ and } v_t,z_t ,n_t \in L^q(0,\Lambda;L^q(\Bbb{R}^3)). $$ \enddemo The approximation problem of the proposed model \Tag(2.1) given by $$ \cases u_{\epsilon_t}+n_\epsilon\cdot \nabla u_\epsilon=\Delta (u_\epsilon+\epsilon)^{1+\alpha}-\nabla\cdot(\chi_1(v_\epsilon) u_\epsilon\nabla v_\epsilon)+\nabla\cdot(\xi_1(z_\epsilon) u_\epsilon\nabla z_\epsilon), \\ v_{\epsilon_t}+n_\epsilon\cdot \nabla v_\epsilon=\Delta v_\epsilon+\beta w_\epsilon - \gamma v_\epsilon, \\ w_{\epsilon_t}+n_\epsilon\cdot \nabla w_\epsilon=\Delta (w_\epsilon+\epsilon)^{1+\alpha}-\nabla\cdot(\chi_2(z_\epsilon) w_\epsilon\nabla z_\epsilon)+\nabla\cdot(\xi_2(v_\epsilon) w_\epsilon\nabla v_\epsilon),\\ z_{\epsilon_t}+n_\epsilon\cdot \nabla z_\epsilon=\Delta z_\epsilon+\delta u_\epsilon - \eta z_\epsilon,\\ n_{\epsilon_t}+\nabla p_\epsilon=\Delta n_\epsilon-(u_\epsilon+w_\epsilon) \nabla \phi, \endcases \tag4.2 $$ with smooth initial conditions given by $$ u_{0_\epsilon}=\phi_\epsilon*u_0,\quad v_{0_\epsilon}=\phi_\epsilon*v_0,\quad w_{0_\epsilon}=\phi_\epsilon*w_0,\quad z_{0_\epsilon}=\phi_\epsilon*z_0, \ \text{ and }\ n_{0_\epsilon}=\phi_\epsilon*n_0, \tag4.3 $$ where $\phi_\epsilon$ refers usual mollifier with $\epsilon\in(0,1)$. We derive some uniform estimates for the approximation problem and using that we derive a bounded weak solution for \Tag(2.1). Now, we prove a lemma that gives an estimate for the bounded weak solution of \Tag(4.2). For the simplicity of notation, hereafter, we use variables $(u_\epsilon,v_\epsilon,w_\epsilon,z_\epsilon,n_\epsilon)$ as $(u,v,w,z,n)$. \proclaim{Lemma 4.1} Suppose that $(u,v,w,z,n)$ is a classical solution for system \Tag(4.2) independent of $\epsilon$ and~\Tag(4.3) satisfies \iftex $$ \alignat2 &u_0(1+|x|+|\log u_0|),&\quad& w_0(1+|x|+|\log w_0|)\in L^1(\Bbb{R}^3), \\ &u_0 , w_0 \in L^{1+\alpha}(\Bbb{R}^3)\cup L^\infty(\Bbb{R}^3), &\quad& v_0 , z_0\in L^\infty(\Bbb{R}^3)\cap H^1(\Bbb{R}^3), \text{ and }n_0\in L^2(\Bbb{R}^3), \endalignat $$ \else $$ \gathered u_0(1+|x|+|\log u_0|),\quad w_0(1+|x|+|\log w_0|)\in L^1(\Bbb{R}^3), \\ u_0 , w_0 \in L^{1+\alpha}(\Bbb{R}^3)\cup L^\infty(\Bbb{R}^3), \quad v_0 , z_0\in L^\infty(\Bbb{R}^3)\cap H^1(\Bbb{R}^3), \text{ and }n_0\in L^2(\Bbb{R}^3), \endgathered $$ \fi and for any $q\in [2,\infty)$ $$ v_0, z_0\in W^{1,q}(\Bbb{R}^3)\text{ and }n_0\in W^{1,q}(\Bbb{R}^3). \tag4.4 $$ Further, assume that $$ \cases \alpha>\frac{1}{8},\ \phi\in W^{2,\infty}(\Bbb{R}^3),\ \gamma=\eta=0 \text{ and } %\?может, вместо and просто запятую \\ \chi'_i,\xi'_i\in L_{\loc}^\infty\text{ for } i\in\{1,2\}\text{ with }\chi'_i(\cdot)\geq \chi_{i_0}^{} %\? \text{ for some constant } \chi_{i_0}^{}. %\? \endcases \tag4.5 $$ Then, for any $t \in (0,\Lambda]$ \iftex $$ \align & u \text{ and } %\?and надо w\in L^\infty(0,\Lambda;L^p(\Bbb{R}^3)), \tag4.6 \\ & \nabla u^\frac{p+\alpha}{2}\text{ and } %\?and надо \nabla w^\frac{p+\alpha}{2}\in L^2(0,\Lambda;L^2(\Bbb{R}^3)) ,\quad 1 \leq p \leq \infty, \tag4.7 \\ & v, z, n\in L^\infty(0,\Lambda;W^{1,q}(\Bbb{R}^3))\cap L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3)),\quad 2 \leq q < \infty, \tag4.8 \\ & v_t,z_t,n_t \in L^q(0,\Lambda;L^q(\Bbb{R}^3)),\quad 2 \leq q < \infty. \tag4.9 \endalign $$ \else $$ u \text{ and } %\?and надо w\in L^\infty(0,\Lambda;L^p(\Bbb{R}^3)), \tag4.6 $$ $$ \nabla u^\frac{p+\alpha}{2}\text{ and } %\?and надо \nabla w^\frac{p+\alpha}{2}\in L^2(0,\Lambda;L^2(\Bbb{R}^3)) ,\quad 1 \leq p \leq \infty, \tag4.7 $$ $$ v, z, n\in L^\infty(0,\Lambda;W^{1,q}(\Bbb{R}^3))\cap L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3)),\quad 2 \leq q < \infty, \tag4.8 $$ $$ v_t,z_t,n_t \in L^q(0,\Lambda;L^q(\Bbb{R}^3)),\quad 2 \leq q < \infty. \tag4.9 $$ \fi \endproclaim %{$\blacksquare$} \demo{Proof} For $1 \leq p \leq 1+\alpha$, from \Par*{Lemma 4.1}, %\?нет такой Lemma 1.1, поставила пока 4.1 we have $$ \cases u \in L^\infty(0,\Lambda;L^p(\Bbb{R}^3)),\ \nabla u^\frac{p+\alpha}{2} \in L^2(0,\Lambda;L^2(\Bbb{R}^3)),\\ w \in L^\infty(0,\Lambda;L^p(\Bbb{R}^3))\text{ and } %\?может, вместо and просто запятую \nabla w^\frac{p+\alpha}{2} \in L^2(0,\Lambda;L^2(\Bbb{R}^3)). \endcases \tag4.10 $$ To prove \Tag(4.6)--\Tag(4.8), it is enough to show above holds for $\alpha>\frac{1}{8}$. Above is not available in the case that $\frac{1}{8}<\alpha <\frac{1}{6}$. Using $u^{p-1}$ as test function in \EquTag{4.2}{$(4.2)_1$}, %$(4.2)_1$, we get $$ \frac{1}{p}\frac{d}{dt}\Vert u\Vert^p_p+\int\limits_{\Bbb{R}^3}\nabla u^{p-1}\cdot \nabla(u+\epsilon)^{1+\alpha}\,dx=-\int\limits_{\Bbb{R}^3}u^{p-1}\nabla\cdot(\chi_1 u\nabla v)\,dx+\int\limits_{\Bbb{R}^3}u^{p-1}\nabla\cdot(\xi_1 u\nabla z)\,dx. \tag4.11 $$ From $\vert \nabla u^{\frac{p+\alpha}{2}} \vert_2^2=\frac{(p+\alpha)^2}{4}u^{p+\alpha-2}\vert\nabla u\vert^2$, we get $$ \int\limits_{\Bbb{R}^3}\nabla u^{p-1}\cdot \nabla(u+\epsilon)^{1+\alpha}\,dx\geq \frac{4(p-1)(1+\alpha)}{(p+\alpha)^2}\Vert \nabla u^{\frac{p+\alpha}{2}} \Vert_2^2. \tag4.12 $$ The first term in RHS of above estimated using $ u^{p-1}\nabla\cdot(\chi_1 u\nabla v)\leq \frac{2\overline{\chi_1}}{(p+\alpha)}\vert \nabla u^{\frac{p+\alpha}{2}}\vert u^{\frac{p-\alpha}{2}}\vert\nabla v \vert$ and Young's inequality as $$ \aligned -\int\limits_{\Bbb{R}^3}u^{p-1}\nabla\cdot(\chi_1 u\nabla v)\,dx & \leq \frac{2\overline{\chi_1}}{(p+\alpha)}\biggl(\epsilon\Vert \nabla u^{\frac{p+\alpha}{2}}\Vert^2_2+C(\epsilon)\int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla v \vert^2\,dx\biggr) \\ &\leq C_{1}\Vert\nabla u^{\frac{p+\alpha}{2}}\Vert^2_2+C_{2}\int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla v \vert^2\,dx. \endaligned \tag4.13 $$ Similarly, we get $$ \int\limits_{\Bbb{R}^3}u^{p-1}\nabla\cdot(\xi_1 u\nabla z)\,dx \leq C_3\Vert\nabla u^{\frac{p+\alpha}{2}}\Vert^2_2+C_4\int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla z \vert^2\,dx. \tag4.14 $$ Substituting \Tag(4.12)--\Tag(4.14) in \Tag(4.11) and choosing sufficiently small $C_1$ and $C_3$ such that $C_5>0$, we get $$ \frac{1}{p}\frac{d}{dt}\Vert u\Vert^p_p+C_5\Vert \nabla u^{\frac{p+\alpha}{2}} \Vert_2^2\leq C_{2}\int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla v \vert^2\,dx+C_4\int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla z \vert^2\,dx, \tag4.15 $$ The first term in RHS of the above estimated using the H\"older, Sobolev, and Young inequality %\? as $$ \int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla v \vert^2\,dx \leq C_6\Vert u\Vert^{p-\alpha}_p\Vert\Delta v \Vert^2_{\frac{6p}{2p+3\alpha}} \leq C_6\Bigl(\frac{\alpha}{p}+\frac{p-\alpha}{p}\Vert u\Vert_p^p\Bigr)\Vert\Delta v\Vert^2_{\frac{6p}{2p+3\alpha}}. \tag4.16 $$ Similarly, we have $$ \int\limits_{\Bbb{R}^3}u^{p-\alpha}\vert\nabla z \vert^2\,dx\leq C_7\Bigl(\frac{\alpha}{p}+\frac{p-\alpha}{p}\Vert u\Vert_p^p\Bigr)\Vert\Delta z\Vert^2_{\frac{6p}{2p+3\alpha}}. \tag4.17 $$ Substituting \Tag(4.16) and \Tag(4.17) in \Tag(4.15), we get $$ \frac{1}{p}\frac{d}{dt}\Vert u\Vert^p_p+C_5\Vert \nabla u^{\frac{p+\alpha}{2}} \Vert_2^2 \leq C_8\Bigl(\Vert\Delta v\Vert^2_{\frac{6p}{2p+3\alpha}}+\Vert\Delta z\Vert^2_{\frac{6p}{2p+3\alpha}}\Bigr)\Vert u\Vert_p^p +C_9\Bigl(\Vert\Delta v\Vert^2_{\frac{6p}{2p+3\alpha}}+\Vert\Delta z\Vert^2_{\frac{6p}{2p+3\alpha}}\Bigr). $$ Using the Gronwall inequality above, we get $$ \Vert u\Vert^p_p \leq \exp\Biggl(\ \int\limits_{0}^{t}\Vert\Delta v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}+\Vert\Delta z(s)\Vert^2_{\frac{6p}{2p+3\alpha}}\,ds\Biggr) %\\ \times \int\limits_0^t\Vert\Delta v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}+\Vert\Delta z(s)\Vert^2_{\frac{6p}{2p+3\alpha}}\,ds+\Vert u_0\Vert_p^p. $$ Hence, $u \text{ and } %\? w\in L^\infty(0,\Lambda;L^p(\Bbb{R}^3))$ for $p\in[1,\infty)$ holds, whenever the following holds $$ \int\limits_0^\Lambda \Vert\Delta v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}+\Vert\Delta z(s)\Vert^2_{\frac{6p}{2p+3\alpha}}\,ds < \infty, \quad 1+\alpha \frac{1}{3}$. Using maximal regularity estimate of heat equation %\? in \EquTag{4.2}{$(4.2)_2$}, %$(4.2)_2$, we have $$ \int\limits_0^\Lambda \Vert\Delta v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}\,ds\leq C_1 \Biggl(\Vert\nabla v_0\Vert^2_{\frac{6p}{2p+3\alpha}}+\int\limits_0^\Lambda\Vert w(s)\Vert^2_{\frac{6p}{2p+3\alpha}}ds+\int\limits_0^\Lambda\Vert n\cdot \nabla v\Vert^2_{\frac{6p}{2p+3\alpha}}ds\Biggr). \tag4.19 $$ As $\alpha>\frac{1}{3}$, for $p>1+\alpha$, the third term in RHS of the above estimated using interpolation inequality %\?the есть еще as $$ \aligned \int\limits_0^\Lambda\Vert w(s)\Vert^2_{\frac{6p}{2p+3\alpha}}\,ds &\leq C_2\int\limits_0^t\Vert w(s)\Vert_1^{2-\frac{(1+2\alpha)(4p-3\alpha)}{2p(1+3\alpha)}}\Vert w(s)\Vert_{3+6\alpha}^{\frac{(1+2\alpha)(4p-3\alpha)}{2p(1+3\alpha)}}\,ds \\ &\leq C_2\int\limits_0^\Lambda\Vert \nabla w^{\frac{1+2\alpha}{2}}(s)\Vert_2^{\frac{4}{1+3\alpha}-\frac{3\alpha}{p(1+3\alpha)}}\,ds. \endaligned \tag4.20 $$ The last term in RHS of \Tag(4.19) evaluated using $n\in L^\infty(0,\Lambda;L^6(\Bbb{R}^3))$ as $$ \int\limits_0^\Lambda\Vert n\cdot \nabla v\Vert^2_{\frac{6p}{2p+3\alpha}}ds \leq \int\limits_0^\Lambda\Vert n(s)\Vert^2_6\Vert\nabla v(s)\Vert^2_{\frac{6p}{p+3\alpha}}ds \leq C \int\limits_0^\Lambda\Vert\Delta v(s)\Vert^2_{\frac{2p}{p+\alpha}}ds. \tag4.21 $$ Using maximal regularity estimate for the heat equation in \EquTag{4.2}{$(4.2)_2$}, %$(4.2)_2$, we get $$ \aligned \int\limits_0^\Lambda \Vert\Delta v(s)\Vert^2_{\frac{2p}{p+\alpha}}\,ds &\leq C_1\Biggl(\Vert\nabla v_0\Vert^2_{\frac{2p}{p+\alpha}}+\int\limits_0^\Lambda\Vert w(s)\Vert^2_{\frac{2p}{p+\alpha}}ds+\int\limits_0^\Lambda\Vert n\cdot \nabla v\Vert^2_{\frac{2p}{p+\alpha}}ds\Biggr) \\ &\leq C_2 + C_3 \int\limits_0^\Lambda\Vert \nabla v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}ds+\int\limits_0^\Lambda\Vert w(s)\Vert^2_{\frac{2p}{p+\alpha}}ds. \endaligned \tag4.22 $$ The last term in RHS of the above is estimated as $$ \int\limits_0^\Lambda\Vert w(s)\Vert^2_{\frac{2p}{p+\alpha}}ds \leq C_4\int\limits_0^\Lambda\Vert w(s)\Vert^{2-\frac{3 (1+2 \alpha) (p-\alpha )}{2 (1+3 \alpha) p}}_1 \Vert w(s)\Vert^{2-\frac{3 (1+2 \alpha) (p-\alpha )}{2 (1+3 \alpha) p}}_{3+6\alpha} ds \leq C_4\int\limits_0^\Lambda \Vert \nabla w^{\frac{1+2\alpha}{2}}(s)\Vert_2^{\frac{3(p-\alpha)}{p(1+3\alpha)}}ds. $$ Using above in \Tag(4.22) and substituting the resulting equation in \Tag(4.21), we have $$ \int\limits_0^\Lambda\Vert n\cdot \nabla v\Vert^2_{\frac{6p}{2p+3\alpha}}ds \leq C_5+C_6 \int\limits_0^\Lambda\Vert \nabla v(s)\Vert^2_{\frac{6p}{2p+3\alpha}}ds+C_7 \int\limits_0^\Lambda \Vert \nabla w^{\frac{1+2\alpha}{2}}(s)\Vert_2^{\frac{3(p-\alpha)}{p(1+3\alpha)}}ds. \tag4.23 $$ As $\frac{6p}{2p+3\alpha}<2$, $\int\nolimits_0^\Lambda\Vert \nabla v(s)\Vert^2_q\,ds<\infty$ for $q\in[2,6]$, $\frac{3(p-\alpha)}{p(1+3\alpha)}>0$ and using \Tag(4.20) and \Tag(4.23), we have for $\max\{1+\alpha,3\alpha\}0$, we have $$ \aligned \frac{d}{dt}\Vert u\Vert^p_p+C_4\Vert \nabla u^{\frac{p+\alpha}{2}} \Vert_2^2 &\leq C_5 \int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla v|^2\,dx + C_6 \int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla z|^2\,dx \\ & \qquad+C_1\int\limits_{\Bbb{R}^3}u^{p-\alpha}|\Delta v|\,dx+C_3\int\limits_{\Bbb{R}^3}u^{p-\alpha}|\Delta z|\,dx. \endaligned \tag4.25 $$ Integrating the above with respect to $t$, we get $$ \aligned \Vert u\Vert^p_p+C_4 &\int\limits_0^t\Vert \nabla u^{\frac{p+\alpha}{2}}\Vert_2^2 ds \\ &\leq C_5 \int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla v|^2\,dx \,ds + C_6 \int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla z|^2\,dx \,ds \\ &\qquad +C_1\int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-\alpha}|\Delta v|dx \,ds+C_3\int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-\alpha}|\Delta z|\,dx \,ds +\Vert u_0\Vert^p_p \\ &\leq C_5 \int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla v|^2\,dx \,ds + C_6 \int\limits_0^t\int\limits_{\Bbb{R}^3}u^{p-3\alpha}|\nabla z|^2\,dx \,ds \\ &\qquad+C_7\int\limits_0^t\Vert u\Vert^{p-\alpha+1}_{p-\alpha+1} \,ds+C_8\int\limits_0^t\Vert\Delta v\Vert^{p-\alpha+1}_{p-\alpha+1}+\Vert\Delta z\Vert^{p-\alpha+1}_{p-\alpha+1}\,ds +\Vert u_0\Vert^p_p. \endaligned \tag4.26 $$ As $\frac{1}{8}<\alpha \leq \frac{1}{3}$ and $1\frac{1}{8}$. As proof is similar to above, proof for boundedness of $u$ and $w$ in $L^\infty-$norm is omitted. \qed\enddemo Now, we state and prove another result established in this work. \proclaim{Theorem 4.1} Assume that \Tag(4.5) holds and $(u_0,v_0,w_0,z_0,n_0)$ satisfies initial data \Tag(3.6) and \Tag(4.4). Then for each $\Lambda>0$, the system \Tag(4.1) possesses a bounded weak solution $(u,v,w,z,n)$, such that $$ \aligned \Vert u\Vert_{L^{\infty}((0,\Lambda)\times\Bbb{R}^3)} &+\Vert w\Vert_{L^{\infty}((0,\Lambda)\times\Bbb{R}^3)}+\Vert \nabla u^\frac{p+\alpha}{2}\Vert_{L^2((0,\Lambda)\times\Bbb{R}^3)}+\Vert \nabla w^\frac{p+\alpha}{2}\Vert_{L^2((0,\Lambda)\times\Bbb{R}^3)} \\ & +\Vert v\Vert_{L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3))}+\Vert z\Vert_{L^q(0,\Lambda;L^q(\Bbb{R}^3))}+\Vert n\Vert_{L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3))} \\ &+\Vert v_t\Vert_{L^q(0,\Lambda;W^{2,q}(\Bbb{R}^3))}+\Vert z_t\Vert_{L^q(0,\Lambda;L^q(\Bbb{R}^3))}+\Vert n_t\Vert_{L^q(0,\Lambda;L^q(\Bbb{R}^3))}0$, the system \Tag(4.1) with \Tag(3.89) possesses a bounded weak solution $(u,v,w,z,n)$, such that $$ \aligned \Vert u\Vert_{L^{\infty}((0,\Lambda)\times\Omega)} &+\Vert w\Vert_{L^{\infty}((0,\Lambda)\times\Omega)}+\Vert \nabla u^\frac{p+\alpha}{2}\Vert_{L^2((0,\Lambda)\times\Omega)}+\Vert \nabla w^\frac{p+\alpha}{2}\Vert_{L^2((0,\Lambda)\times\Omega)} \\ & +\Vert v\Vert_{L^q(0,\Lambda;W^{2,q}(\Omega))}+\Vert z\Vert_{L^q(0,\Lambda;L^q(\Omega))}+\Vert n\Vert_{L^q(0,\Lambda;W^{2,q}(\Omega))} \\ &+\Vert v_t\Vert_{L^q(0,\Lambda;W^{2,q}(\Omega))}+\Vert z_t\Vert_{L^q(0,\Lambda;L^q(\Omega))}+\Vert n_t\Vert_{L^q(0,\Lambda;L^q(\Omega))}