© Wuhan University 2026

0 Introduction

We consider the following model of a Keller-Segel-Navier-Stokes with subcritical sensitivity^[1-2]:

${\partial }_{t}u+u\cdot \nabla u+\nabla \pi -\mathrm{\Delta }u=n\nabla \varphi ,$(1)

$\mathrm{d}\mathrm{i}\mathrm{v}u=\mathrm{0},$(2)

$\begin{array}{c}{\partial }_{t}n+u\cdot \nabla n-\mathrm{\Delta }n=-\nabla \cdot (nS(x,n,p,q)\cdot \nabla p)\\ -\nabla \cdot (nS(x,n,p,q)\cdot \nabla q),\end{array}$(3)

${\partial }_{t}p+u\cdot \nabla p-\mathrm{\Delta }p=-np,$(4)

${\partial }_{t}q+u\cdot \nabla q-\mathrm{\Delta }q+q=n\mathrm{i}\mathrm{n}\mathrm{\Omega }\times (\mathrm{0},\mathrm{\infty }),$(5)

$u=\mathrm{0},\frac{\partial n}{\partial v}=\frac{\partial p}{\partial v}=\frac{\partial q}{\partial v}=\mathrm{0}\mathrm{o}\mathrm{n}\partial \mathrm{\Omega }\times (\mathrm{0},\mathrm{\infty }),$(6)

$(u,n,p,q)(\cdot ,\mathrm{0})=(u_{\mathrm{0}},n_{\mathrm{0}},p_{\mathrm{0}},q_{\mathrm{0}})(\cdot )\mathrm{i}\mathrm{n}\mathrm{\Omega }\subset {\mathbb{R}}^{\mathrm{2}},$(7)

where $u$ is the velocity of the fluid, $\pi$ is the pressure, $n,p$ and $q$ denote the density of amoebae, oxygen and chemical attractant, respectively. The smooth function $\varphi :=\varphi (x)$ is a potential. $S$ is the chemotactic sensitivity. $\mathrm{\Omega }$ is a bounded convex domain with smooth boundary $\partial \mathrm{\Omega }$, $v$ is the unit outward normal vector to $\partial \mathrm{\Omega }$.

The two-dimensional Keller-Segel-Navier-Stokes system serves as a fundamental mathematical model for characterizing the dynamic evolution of chemotactic biological populations in fluid environments^[3-4]. Its physical foundation stems from bidirectional coupling between chemotactic behavior and fluid motion: biological entities form aggregation patterns through sensing chemical concentration gradients (Keller-Segel mechanism), while viscous fluid flow (Navier-Stokes equations) alters population distribution via turbulent or laminar transport. The subcritical sensitivity condition further regulates biological response intensity to prevent solution blowup. This study establishes the theoretical conditions ensuring global existence of strong solutions for the two-dimensional Keller-Segel-Navier-Stokes system under the subcritical sensitivity condition in bounded domains. Beyond providing a supplementary analytical approach for chemotaxis-fluid systems, the research advances understanding of nonlinear stability mechanisms in biological-fluid interactions through energy estimation methods for coupled systems. Its practical significance lies in offering theoretical support for biomedical and environmental engineering applications, including tumor cell migration modeling, algal bloom prediction, and microfluidic chip design. Methodologically, the developed techniques for a priori estimates and compactness arguments provide transferable mathematical tools for interdisciplinary research in magnetohydrodynamics and other complex systems.

Recent advances in mathematical modeling of chemotaxis-fluid systems have yielded significant insights. Bellomo et al^[5] conducted a comprehensive analysis of Keller-Segel models and their modifications, deriving macroscopic formulations through kinetic theory that prevent non-physical solution blow-up. Kiselev and Ryzhik^[6] demonstrated that chemotactic interactions can achieve near-complete biological reaction efficiency (>99%), with this enhancement being independent of reaction amplitude, thereby revealing fundamental nonlinear coupling mechanisms. In two-dimensional domains, Wang et al^[7] established global weak solution existence for the chemotaxis-Navier-Stokes system under subcritical sensitivity conditions ($\alpha >\frac{\mathrm{1}}{\mathrm{3}}$), providing methodological foundations for our strong solution analysis. For three-dimensional cases, Fan and Li^[8] proved vanishing viscosity limits using $L^p$-energy methods, though their approach did not address stability regulation by subcritical parameters. This study presents a proof of global strong solution existence for two-dimensional systems with subcritical sensitivity (where $\chi$ denotes the chemotactic sensitivity coefficient and ${\chi }^{\mathrm{*}}$ is the critical threshold, satisfying $\chi <{\chi }^{\mathrm{*}}$) in bounded domains. Through hierarchical energy estimates and Poincaré inequality control, we reveal the deterministic role of subcritical conditions in maintaining system stability.

When $u=\mathrm{0}$, system (3), (4) and (5) reduces to the Keller-Segel system^[9-11], which received many studies^[12-18].

Fan and Zhao^[19] established some regularity criteria when $q=\mathrm{0}$.

We assume that $S\in C^{\mathrm{2}}(\overline{\mathrm{\Omega }}\times {[\mathrm{0},\mathrm{\infty })}^{\mathrm{3}})$ has the property that there exist $S_{\mathrm{0}}\ge \mathrm{0}$ and $\alpha >\mathrm{0}$ fulfilling

$\begin{array}{l}\left|S(x,n,p,q)\right|\le S_{\mathrm{0}}{(\mathrm{1}+n)}^{-\alpha }\\ \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x\in \overline{\mathrm{\Omega }},n\ge \mathrm{0},p\ge \mathrm{0}\mathrm{a}\mathrm{n}\mathrm{d}q\ge \mathrm{0},\end{array}$(8)

where we evidently may assume without loss of generality that $\alpha <\frac{\mathrm{1}}{\mathrm{2}}$ .

When $p=\mathrm{0}$, Wang et al^[7] showed the global existence of strong solutions.

The aim of this paper is to generalize the results in Ref. [7] to the case $p\ne \mathrm{0}$, we will prove

Theorem 1   Let $u_{\mathrm{0}}\in H_{\mathrm{0}}^{\mathrm{1}}\bigcap H^{\mathrm{2}},n_{\mathrm{0}},p_{\mathrm{0}},q_{\mathrm{0}}\in H^{\mathrm{2}},$ $\mathrm{d}\mathrm{i}\mathrm{v}{u}_{\mathrm{0}}=\mathrm{0},n_{\mathrm{0}},p_{\mathrm{0}},q_{\mathrm{0}}\ge \mathrm{0},$ in $\mathrm{\Omega }$ and $\frac{\partial n_{\mathrm{0}}}{\partial v}=\frac{\partial p_{\mathrm{0}}}{\partial v}=\frac{\partial q_{\mathrm{0}}}{\partial v}=\mathrm{0}$ on $\partial \mathrm{\Omega }$. Suppose that $\varphi :=\varphi (x)$ is a smooth function and (8) holds true. Then the problem (1)-(7) has a unique strong solution $(u,n,p,q)$ satisfying

$\begin{array}{l}u,n,p,q\in L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{2}})\bigcap L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{3}}),\\ {\partial }_{t}u,{\partial }_{t}n,{\partial }_{t}p,{\partial }_{t}q\in L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})\bigcap L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})\end{array}$(9)

for any given $T>\mathrm{0}$.

1 Proof of Theorem 1

This section is devoted to the proof of Theorem 1. It is easy to show the local well-posedness of smooth solutions, we only need to prove some a priori estimates (9).

First, by the maximum principle, and integrating (3) over $\mathrm{\Omega }\times (\mathrm{0},t),$we get

$n,p,q\ge \mathrm{0},p\le C,\mathrm{a}\mathrm{n}\mathrm{d}\int n\mathrm{d}x=\int n_{\mathrm{0}}\mathrm{d}x.$(10)

Integrating (5) over Ω, we see that

$\frac{\mathrm{d}}{\mathrm{d}t}\int q\mathrm{d}x+\int q\mathrm{d}x=\int n\mathrm{d}x=\int n_{\mathrm{0}}\mathrm{d}x,$

which gives

$\int q\mathrm{d}x\le {\mathrm{e}}^{-t}\int q_{\mathrm{0}}\mathrm{d}x+(\mathrm{1}-{\mathrm{e}}^{-t})\int n_{\mathrm{0}}\mathrm{d}x.$(11)

Testing (3) by $n^{\mathrm{2}\alpha -\mathrm{1}}$, using (2) and (8), we find that

$\begin{array}{l}-\frac{\mathrm{1}}{\mathrm{2}\alpha }\frac{\mathrm{d}}{\mathrm{d}t}\int n^{\mathrm{2}\alpha }\mathrm{d}x+\frac{\mathrm{1}-\mathrm{2}\alpha }{{\alpha }^{\mathrm{2}}}{\int \left|\nabla n^{\alpha }\right|}^{\mathrm{2}}\mathrm{d}x\\ =-\int nS\nabla p\cdot \nabla n^{\mathrm{2}\alpha -\mathrm{1}}\mathrm{d}x-\int nS\nabla q\cdot \nabla n^{\mathrm{2}\alpha -\mathrm{1}}\mathrm{d}x\\ \le C\int \left|\nabla n^{\alpha }\cdot \nabla p\right|\mathrm{d}x+\int C\left|\nabla n^{\alpha }\cdot \nabla q\right|\mathrm{d}x\\ \le \frac{\mathrm{1}}{\mathrm{8}}\frac{\mathrm{1}-\mathrm{2}\alpha }{{\alpha }^{\mathrm{2}}}\int {\left|\nabla n^{\alpha }\right|}^{\mathrm{2}}\mathrm{d}x+C_{\mathrm{1}}\int {\left|\nabla p\right|}^{\mathrm{2}}\mathrm{d}x+C_{\mathrm{2}}{\int \left|\nabla q\right|}^{\mathrm{2}}\mathrm{d}x.\end{array}$(12)

Testing (4) by $p$ and using (2) and (10), we deduce that

$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int p^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla p\right|}^{\mathrm{2}}\mathrm{d}x+\int n{p}^{\mathrm{2}}\mathrm{d}x=\mathrm{0},$

which yields

${\int }_{\mathrm{0}}^T{\int \left|\nabla p\right|}^{\mathrm{2}}\mathrm{d}x\mathrm{d}t\le C.$(13)

Testing (5) by $q$ and using (2), we infer that

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int q^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla q\right|}^{\mathrm{2}}\mathrm{d}x+\int q^{\mathrm{2}}\mathrm{d}x=\int nq\mathrm{d}x\\ \le {\Vert n\Vert }_{L^{\theta }}{\Vert q\Vert }_{L^{\frac{\theta }{\theta -\mathrm{1}}}}\le C{\Vert n\Vert }_{L^{\theta }}{\Vert q\Vert }_{H^{\mathrm{1}}}\\ \le \frac{\mathrm{1}}{\mathrm{2}}{\Vert q\Vert }_{H^{\mathrm{1}}}^{\mathrm{2}}+C{\Vert n\Vert }_{L^{\theta }}^{\mathrm{2}}.\end{array}$(14)

On the other hand, we first impose the condition that

$\theta <\frac{\mathrm{1}}{\mathrm{1}-\alpha }.$(15)

Then, by applying Young's inequality under this constraint, we observe that

${\Vert n\Vert }_{L^{\theta }}^{\mathrm{2}}={\Vert n^{\alpha }\Vert }_{L^{\frac{\theta }{\alpha }}}^{\frac{\mathrm{2}}{\alpha }}\le C({\Vert n^{\alpha }\Vert }_{L^{\frac{\mathrm{1}}{\alpha }}}^{\frac{\mathrm{2}}{\alpha \theta }}{\Vert \nabla n^{\alpha }\Vert }_{L^{\mathrm{2}}}^{\frac{\mathrm{2}(\theta -\mathrm{1})}{\alpha \theta }}+{\Vert n^{\alpha }\Vert }_{L^{\frac{\mathrm{1}}{\alpha }}}^{\frac{\mathrm{2}}{\alpha }})$

$\le C{\Vert \nabla n^{\alpha }\Vert }_{L^{\mathrm{2}}}^{\frac{\mathrm{2}}{\alpha }(\mathrm{1}-\frac{\mathrm{1}}{\theta })}+C\le \frac{\mathrm{1}}{\mathrm{8}}\frac{\mathrm{1}-\mathrm{2}\alpha }{{\alpha }^{\mathrm{2}}}{\Vert \nabla n^{\alpha }\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C.$(16)

Summing up (12) and (14), using (13), (16) and the Gronwall inequality, we have

$\int q^{\mathrm{2}}\mathrm{d}x+{\int }_{\mathrm{0}}^T{\int (\left|\nabla n^{\alpha }\right|}^{\mathrm{2}}+{\left|\nabla q\right|}^{\mathrm{2}})\mathrm{d}x\mathrm{d}t\le C.$(17)

Using the Gagliardo-Nirenberg inequality and (17), we have

$\begin{array}{l}{\int }_{\mathrm{0}}^T{\Vert n\Vert }_{L^r}^{\frac{\mathrm{2}r\alpha }{r-\mathrm{1}}}\mathrm{d}t={\int }_{\mathrm{0}}^T{\Vert n^{\alpha }\Vert }_{L^{\frac{r}{\alpha }}}^{\frac{\mathrm{2}r}{r-\mathrm{1}}}\mathrm{d}t\\ \le C{\int }_{\mathrm{0}}^T{\Vert n^{\alpha }\Vert }_{L^{\frac{\mathrm{1}}{\alpha }}}^{\frac{\mathrm{2}}{r-\mathrm{1}}}{\Vert \nabla n^{\alpha }\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}\mathrm{d}t+C{\int }_{\mathrm{0}}^T{\Vert n^{\alpha }\Vert }_{L^{\frac{\mathrm{1}}{\alpha }}}^{\frac{\mathrm{2}r}{r-\mathrm{1}}}\mathrm{d}t\\ \le C+C{\int }_{\mathrm{0}}^T{\Vert \nabla n^{\alpha }\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}\mathrm{d}t\le C[for][all]\mathrm{1}<r<\mathrm{\infty },\end{array}$(18)

and

${\int }_{\mathrm{0}}^T{\Vert n\Vert }_{L^{\frac{\mathrm{1}}{\mathrm{1}-\alpha }}}^{\mathrm{2}}\mathrm{d}t\le C.$(19)

Testing (1) by $u$ and using (2) and (19), we have

$\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int {\left|u\right|}^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla u\right|}^{\mathrm{2}}\mathrm{d}x=\int n\nabla \phi \cdot u\mathrm{d}x\\ \le {\Vert \nabla \phi \Vert }_{L^{\mathrm{\infty }}}{\Vert n\Vert }_{L^{\frac{\mathrm{1}}{\mathrm{1}-\alpha }}}{\Vert u\Vert }_{L^{\frac{\mathrm{1}}{\alpha }}}\end{array}$

$\le C{\Vert n\Vert }_{L^{\frac{\mathrm{1}}{\mathrm{1}-\alpha }}}{\Vert \nabla u\Vert }_{L^{\mathrm{2}}}\le \frac{\mathrm{1}}{\mathrm{2}}{\Vert \nabla u\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C{\Vert n\Vert }_{L^{\frac{\mathrm{1}}{\mathrm{1}-\alpha }}}^{\mathrm{2}},$

which implies

${\Vert u\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert u\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})}\le C.$(20)

Testing (5) by $q^{r-\mathrm{1}}(r\ge \mathrm{2})$ and using (2), (18) and by the same calculations as those in Ref. [5], we have

$\int q^r\mathrm{d}x\le C\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{2}\le r<\mathrm{\infty }.$(21)

We omit the details here.

Testing (3) by $\mathrm{l}\mathrm{n}n$, using (2), and by the same calculations as that in Ref. [5], we have

${\Vert (p,q)\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{1}})}+{\Vert (p,q)\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{2}})}\le C,$(22)

${\Vert n\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert \nabla \sqrt[]{n}\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;L^{\mathrm{2}})}\le C,$(23)

and hence we omit the details here.

Using (23), testing (1) by $\nabla \mathrm{\pi }-\mathrm{\Delta }u$ we easily get

${\Vert u\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{1}})}+{\Vert u\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;L^{\mathrm{2}})}\le C.$(24)

Testing (3) by $n^{\mathit{\ell }-\mathrm{1}}(\mathit{\ell }\ge \mathrm{2})$, using (2) and (22), we have

$\begin{array}{l}\frac{\mathrm{1}}{\mathit{\ell }}\frac{\mathrm{d}}{\mathrm{d}t}\int n^{\mathrm{\ell }}\mathrm{d}x+\frac{\mathrm{4}(\mathit{\ell }-\mathrm{1})}{{\mathit{\ell }}^{\mathrm{2}}}\int {\left|\nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\right|}^{\mathrm{2}}\mathrm{d}x=\int nS\nabla (p+q)\nabla n^{\mathit{\ell }-\mathrm{1}}\mathrm{d}x\\ \le C\int \left|\nabla (p+q)\right|n^{{}^{\frac{\mathrm{\ell }}{\mathrm{2}}}}\left|\nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\right|\mathrm{d}x\le C{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}{\Vert n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{4}}}{\Vert \nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}\\ \le C{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}({\Vert n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}^{\frac{\mathrm{1}}{\mathrm{2}}}{\Vert \nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}^{\frac{\mathrm{1}}{\mathrm{2}}}+{\Vert n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}){\Vert \nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}\\ \le \frac{\mathit{\ell }-\mathrm{1}}{{\mathit{\ell }}^{\mathrm{2}}}{\Vert \nabla n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C({\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}^{\mathrm{4}}+\mathrm{1}){\Vert n^{\frac{\mathrm{\ell }}{\mathrm{2}}}\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}},\end{array}$

which gives

$\int n^{\mathit{\ell }}\mathrm{d}x+{{\int }_{\mathrm{0}}^T\int \left|\nabla n\right|}^{\mathrm{2}}\mathrm{d}x\mathrm{d}t\le C\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{2}\le \mathrm{\ell }<{\infty }.$(25)

By the standard $L^{\mathrm{\infty }}$-estimate of heat equations, it follows from (5), (25) and (24) that

${\Vert q\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{\infty }})}\le C.$(26)

It follows from (4), (5), (22) and (24) that

${\Vert ({\partial }_{t}p,{\partial }_{t}q)\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;L^{\mathrm{2}})}\le C.$(27)

Applying ${\partial }_t$ to (1), testing by ${\partial }_{t}u$, using (2), (3), (24), (25) and (22), we have

$\begin{array}{l}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int {\left|{\partial }_{t}u\right|}^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla {\partial }_{t}u\right|}^{\mathrm{2}}\mathrm{d}x\\ =-\int {\partial }_{t}u\cdot \nabla u\cdot {\partial }_{t}u\mathrm{d}x+\int {\partial }_{t}n\cdot \nabla \phi {\partial }_{t}u\mathrm{d}x=-\int {\partial }_{t}u\cdot \nabla u\cdot {\partial }_{t}u\mathrm{d}x\end{array}$

$+\int (\mathrm{\Delta }n-\mathrm{d}\mathrm{i}\mathrm{v}(un+nS\cdot \nabla p+nS\cdot \nabla q))\nabla \phi {\partial }_{t}u\mathrm{d}x$

$=-\int {\partial }_{t}u\cdot \nabla u\cdot {\partial }_{t}u\mathrm{d}x+\int (-\nabla n+un+nS(\nabla p+\nabla q))\nabla (\nabla \phi {\partial }_{t}u)\mathrm{d}x$

$\begin{array}{l}\le {\Vert \nabla u\Vert }_{L^{\mathrm{2}}}{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{4}}}^{\mathrm{2}}+C({\Vert \nabla n\Vert }_{L^{\mathrm{2}}}+{\Vert u\Vert }_{L^{\mathrm{4}}}{\Vert n\Vert }_{L^{\mathrm{4}}}\\ +{\Vert n\Vert }_{L^{\mathrm{4}}}{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}){\Vert \nabla {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}\end{array}$

$\le C{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{4}}}^{\mathrm{2}}+C({\Vert \nabla n\Vert }_{L^{\mathrm{2}}}+\mathrm{1}+{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}){\Vert \nabla {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}$

$\le \frac{\mathrm{1}}{\mathrm{4}}{\Vert \nabla {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C{\Vert \nabla n\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C+C{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}^{\mathrm{2}},$

which leads to

${\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})}\le C.$(28)

And thus

${\Vert u\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert u\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{3}})}\le C.$(29)

Applying ${\partial }_t$ to (4), testing by ${\partial }_{t}p$, using (2), (10), (28), (29), (25) and (22), we have

$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int {\left({\partial }_{t}p\right)}^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla {\partial }_{t}p\right|}^{\mathrm{2}}\mathrm{d}x+\int n{\left({\partial }_{t}p\right)}^{\mathrm{2}}\mathrm{d}x$

$=-\int {\partial }_{t}u\cdot \nabla p\cdot {\partial }_{t}p\mathrm{d}x-\int {\partial }_{t}np{\partial }_{t}p\mathrm{d}x$

$=\int {\partial }_{t}u\cdot p\cdot \nabla {\partial }_{t}p\mathrm{d}x+\int \mathrm{d}\mathrm{i}\mathrm{v}(un+nS\nabla p+nS\nabla q-\nabla n)p{\partial }_{t}p\mathrm{d}x$

$=\int {\partial }_{t}u\cdot p\cdot \nabla {\partial }_{t}p\mathrm{d}x-\int (un+nS\nabla p+nS\nabla q-\nabla n)\nabla (p{\partial }_{t}p)\mathrm{d}x$

$\begin{array}{l}\le {\Vert p\Vert }_{L^{\mathrm{\infty }}}{\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}{\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}+C({\Vert u\Vert }_{L^{\mathrm{4}}}{\Vert n\Vert }_{L^{\mathrm{4}}}+C{\Vert n\Vert }_{L^{\mathrm{4}}}{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}})\\ +{\Vert \nabla n\Vert }_{L^{\mathrm{2}}}\times ({\Vert p\Vert }_{L^{\mathrm{\infty }}}{\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}+{\Vert \nabla p\Vert }_{L^{\mathrm{4}}}{\Vert {\partial }_{t}p\Vert }_{L^{\mathrm{4}}})\end{array}$

$\begin{array}{l}\le C{\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}+C(\mathrm{1}+{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}+{\Vert \nabla n\Vert }_{L^{\mathrm{2}}})\\ ({\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}+{\Vert \nabla p\Vert }_{L^{\mathrm{4}}}{\Vert {\partial }_{t}p\Vert }_{L^{\mathrm{4}}})\end{array}$

$\begin{array}{l}\le \frac{\mathrm{1}}{\mathrm{8}}{\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C+C{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}^{\mathrm{4}}+C{\Vert \nabla p\Vert }_{L^{\mathrm{4}}}^{\mathrm{4}}\\ +C{\Vert \nabla n\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C{\Vert \nabla n\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}{\Vert {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}},\end{array}$

which gives

${\Vert {\partial }_{t}p\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert {\partial }_{t}p\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})}\le C,$(30)

and hence

${\Vert p\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert p\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{3}})}\le C.$(31)

Similarly to (30)-(31), we have

${\Vert {\partial }_{t}q\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert {\partial }_{t}q\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})}\le C,$(32)

${\Vert q\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert q\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{3}})}\le C.$(33)

Similarly to (26), we have

${\Vert n\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{\infty }})}\le C.$(34)

Now testing (3) by ${\partial }_{t}n-\mathrm{\Delta }n,$it is easy to verify that

${\Vert n\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{1}})}+{\Vert n\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert {\partial }_{t}n\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;L^{\mathrm{2}})}\le C,$(35)

we omit the details here.

Applying ${\partial }_t$ to (3), testing by ${\partial }_{t}n$, using (2), (34), (35) and (28)-(33), we arrive at

$\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int {\left({\partial }_{t}n\right)}^{\mathrm{2}}\mathrm{d}x+{\int \left|\nabla {\partial }_{t}n\right|}^{\mathrm{2}}\mathrm{d}x$

$=-\int {\partial }_{t}u\cdot \nabla n\cdot {\partial }_{t}n\mathrm{d}x+\int {\partial }_t\left[nS\nabla (p+q)\right]\cdot \nabla {\partial }_{t}n\mathrm{d}x$

$=-\int {\partial }_{t}u\cdot n\nabla {\partial }_{t}n\mathrm{d}x+\int [{\partial }_{t}nS\nabla (p+q)+n(\frac{\partial S}{\partial n}{\partial }_{t}n+\frac{\partial S}{\partial p}{\partial }_{t}p+\frac{\partial S}{\partial q}{\partial }_{t}q)\nabla (p+q)+nS\nabla ({\partial }_{t}p+{\partial }_{t}q)]\nabla {\partial }_{t}n\mathrm{d}x$

$\le {\Vert {\partial }_{t}u\Vert }_{L^{\mathrm{2}}}{\Vert n\Vert }_{L^{\mathrm{\infty }}}{\Vert \nabla {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}+C({\Vert ({\partial }_{t}n,{\partial }_{t}p,{\partial }_{t}q)\Vert }_{L^{\mathrm{4}}}{\Vert \nabla (p+q)\Vert }_{L^{\mathrm{4}}}$

$+{\Vert \nabla {\partial }_{t}p\Vert }_{L^{\mathrm{2}}}+{\Vert \nabla {\partial }_{t}q\Vert }_{L^{\mathrm{2}}}){\Vert \nabla {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}$

$\le C{\Vert \nabla {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}+C({\Vert {\partial }_{t}n\Vert }_{L^{\mathrm{4}}}+{\Vert {\partial }_{t}p\Vert }_{H^{\mathrm{1}}}+{\Vert {\partial }_{t}q\Vert }_{H^{\mathrm{1}}}){\Vert \nabla {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}$

$\le \frac{\mathrm{1}}{\mathrm{8}}{\Vert \nabla {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C+C{\Vert {\partial }_{t}n\Vert }_{L^{\mathrm{2}}}^{\mathrm{2}}+C{\Vert {\partial }_{t}p\Vert }_{H^{\mathrm{1}}}^{\mathrm{2}}+C{\Vert {\partial }_{t}q\Vert }_{H^{\mathrm{1}}}^{\mathrm{2}},$

which implies

${\Vert {\partial }_{t}n\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;L^{\mathrm{2}})}+{\Vert {\partial }_{t}n\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{1}})}\le C,$(36)

whence

${\Vert n\Vert }_{L^{\mathrm{\infty }}(\mathrm{0},T;H^{\mathrm{2}})}+{\Vert n\Vert }_{L^{\mathrm{2}}(\mathrm{0},T;H^{\mathrm{3}})}\le C$(37)

This completes the proof.

References