Smooth Controllability of the Navier–Stokes Equation with Navier Conditions: Application to Lagrangian Controllability

Abstract

We deal with the 3D Navier–Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact null controllability result in Coron et al. (J Eur Math Soc 22:1625–1673, 2020) from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier–Stokes equations the recent results (Glass and Horsin in J Math Pures Appl (9) 93:61–90, 2010; Glass and Horsin in SIAM J Control Optim 50: 2726–2742, 2012; Horsin and Kavian in ESAIM Control Optim Calc Var 23:1179–1200, 2017) in the case of the Euler equations and the result (Glass and Horsin in ESAIM Control Optim Calc Var 22:1040–1053, 2016) in the case of the steady Stokes equations.

Appendix A. On the Regularization of the Uncontrolled Strong Solutions to the Navier–Stokes Equations with Navier Boundary Conditions

In this appendix we prove a regularization result of the uncontrolled strong solutions to the Navier–Stokes equations with Navier boundary conditions on the whole boundary $\partial \Omega$, that is, to the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+u\cdot \nabla u- \Delta u +\nabla p=0, \quad \text{ and } \quad \mathrm {div} \,u=0 \quad \text{ in } \Omega ,\\ u\cdot {\mathbf {n}}=0 \quad \text{ and } \quad {\mathcal {N}}(u)=0\quad \text{ on } \partial \Omega , \\ u=u_0\quad \text{ at } t=0 . \end{array}\right. } \end{aligned}$$

(A.1)

Theorem A.1

Let $T>0$, p in ${\mathbb {N}}^*$ and $R>0$. Then there exists a continuous function $C_{T,p,R}$ from $[0,+\infty )$ to $[0,+\infty )$ with $C_{T,p,R}(0)=0$, such that there exists $T_1$ in (0, T) and for any $u_0$ in $H^1(\Omega )$, with $\Vert u_0\Vert _{H^1(\Omega )} \le R$, divergence free and tangent to $\partial \Omega$, the unique strong solution u in $C([0,T_1];H^1(\Omega ))\cap L^2([0,T_1];H^2(\Omega ))$ to (A.1) satisfies

$$\begin{aligned}&\sum _{0\le j \le \frac{p}{2}}\bigl \Vert t^{\frac{p-1}{2}}\partial _t^ju\bigr \Vert _{L^\infty _{T_1}(H^{p-2j}(\Omega ))} +\sum _{0\le j \le \frac{p+1}{2}}\bigl \Vert t^{\frac{p-1}{2}}\partial _t^ju\bigr \Vert _{L^2_{T_1}(H^{p+1-2j}(\Omega ))}\nonumber \\&\quad \le C_{p,T_1,R}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned}$$

(A.2)

As recalled in Sect. 2.1 The goal of this section is to present the proof of Theorem 2.1. The local-in-time existence and uniqueness of strong solutions with $H^1$ initial data is classical. The interest of Theorem A.1 is to detail the regularization in time of this strong solution near the time zero. In particular it implies the part of Theorem 2.1 regarding the regularization.

Proof

We will proceed by induction on p. We start with recalling how to prove the case $p=1$, by proving first a $L^2(\Omega )$ energy estimate and then a $H^1(\Omega )$ energy estimate.

$\bullet$ ${\underline{L^{2}(\Omega )\,\, \hbox {energy}\,\, \hbox {estimate}}}$

Indeed, we first get, by taking $L^2(\Omega )$ inner product of the u equation in (A.1) with u,  that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u(t)\Vert _{L^2(\Omega )}^2+\left( u\cdot \nabla u | u\right) _{L^2(\Omega )}-\left( \Delta u | u\right) _{L^2(\Omega )}+\left( \nabla p | u\right) _{L^2(\Omega )}=0. \end{aligned}$$

(A.3)

Here and in all that follows, we always denote $(f | g)_{L^2(\Omega )}:=\int _{\Omega } f g\,dx.$

Due to $\mathrm {div} \,u=0$ and $u\cdot \mathbf{n}|_{\partial \Omega }=0,$ we have

$$\begin{aligned} \left( u\cdot \nabla u | u\right) _{L^2(\Omega )}=0=\left( \nabla p | u\right) _{L^2(\Omega )}. \end{aligned}$$

Moreover it follows from Stokes formula that

$$\begin{aligned} \begin{aligned} -\left( \Delta u | u\right) _{L^2(\Omega )} =&\int _{\partial \Omega }\left[ (\nabla \times u)\times u\right] \cdot \mathbf{n}\,dS+\int _{\Omega }|\nabla \times u|^2\,dx. \end{aligned} \end{aligned}$$

By inserting the above equalities into (A.3), we obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u(t)\Vert _{L^2(\Omega )}^2+\Vert \nabla \times u\Vert _{L^2(\Omega )}^2=\int _{\partial \Omega }\left[ u\times (\nabla \times u)\right] \cdot \mathbf{n}\,dS. \end{aligned}$$

(A.4)

Let us denote by $M_{\mathrm{w}}$ the shape operator associated with $\Omega$. Recall that, since $\Omega$ is smooth, the shape operator $M_{\mathrm{w}}$ is smooth and for any x in $\partial \Omega ,$ it defines a self-adjoint operator with values in the tangent space $T_x$. Then we have the following result, see [1, 12].

Lemma A.2

For any smooth divergence free vector field u satisfying $u\cdot \mathbf{n}=0$ on $\partial \Omega ,$ we have

$$\begin{aligned} \left[ D(u)\mathbf{n}+M_{\mathrm{w}}u\right] _{\text{ tan }}=\frac{1}{2}(\nabla \times u)\times \mathbf{n}. \end{aligned}$$

(A.5)

However, due to ${\mathcal {N}}(u)|_{\partial \Omega }=0,$ we deduce from Lemma A.2 that

$$\begin{aligned} \begin{aligned} \left[ u \times (\nabla \times u)\right] \cdot \mathbf{n}\bigr |_{\partial \Omega }=&u\cdot \left[ (\nabla \times u)\times \mathbf{n}\right] \bigr |_{\partial \Omega }\\ =&2\left[ (M_{\mathrm{w}}-M)u\right] _{\text{ tan }}\cdot u\bigr |_{\partial \Omega }\\ =&2\left[ (M_{\mathrm{w}}-M)u\right] \cdot u\bigr |_{\partial \Omega }, \end{aligned} \end{aligned}$$

(A.6)

where we used $u\cdot \mathbf{n}\bigr |_{\partial \Omega }=0$ in the last step. Then by applying Stokes formula and Young’s inequality, we find that for any $\lambda >0,$ there exists $C_\lambda$ so that

$$\begin{aligned} \begin{aligned} \bigl |\int _{\partial \Omega }\left[ (\nabla \times u)\times u\right] \cdot \mathbf{n}\,dS\bigr |=&2\bigl |\int _{\Omega }\mathrm {div} \,\left[ \bigl ((M_{\mathrm{w}}-M)u\cdot u\bigr )\mathbf{n}\right] \,dx\bigr |\\ \le&\lambda \Vert \nabla u\Vert _{L^2(\Omega )}^2+C_\lambda \Vert u\Vert _{L^2(\Omega )}^2, \end{aligned} \end{aligned}$$

(A.7)

On the other hand, due to $\mathrm {div} \,u=0$ in $\Omega$ and $u\cdot \mathbf{n}|_{\partial \Omega }=0,$ we deduce from Korn’s type inequality (see [10] for instance) that there exists a positive constant $C_\Omega$ so that

$$\begin{aligned} \Vert \nabla \times u\Vert _{L^2(\Omega )}^2\ge \frac{1}{C_\Omega } \Vert u\Vert _{H^1(\Omega )}^2-\Vert u\Vert _{L^2(\Omega )}^2. \end{aligned}$$

(A.8)

By inserting the estimates, (A.7) and (A.8), into (A.4) and taking $\lambda =\frac{1}{2C_\Omega }$ in the resulting inequality, we achieve

$$\begin{aligned} \frac{d}{dt}\Vert u(t)\Vert _{L^2(\Omega )}^2+\frac{1}{C_\Omega } \Vert u\Vert _{H^1(\Omega )}^2\le C \Vert u\Vert _{L^2(\Omega )}^2. \end{aligned}$$

(A.9)

Applying Gronwall’s inequality gives rise to

$$\begin{aligned} \Vert u\Vert _{L^\infty _t(L^2(\Omega ))}^2+\frac{1}{C_\Omega } \Vert u\Vert _{L^2_t(H^1(\Omega ))}^2 \le \Vert u_0\Vert _{L^2(\Omega )}^2 e^{Ct}. \end{aligned}$$

(A.10)

$\bullet$ ${\underline{H^{1}(\Omega )\,\, \hbox {energy}\,\, \hbox {estimate}}}$

By taking $L^2(\Omega )$ inner product of the u equation of (A.1) with $\partial _tu,$ we get

$$\begin{aligned} \Vert \partial _tu\Vert _{L^2(\Omega )}^2-\left( \Delta u | \partial _t u\right) _{L^2(\Omega )}+\left( \nabla p | \partial _t u\right) _{L^2(\Omega )}=-\left( u\cdot \nabla u | \partial _t u\right) _{L^2(\Omega )}. \end{aligned}$$

(A.11)

Notice that $\partial _tu\cdot \mathbf{n}|_{\partial \Omega }=0,$ by applying Stokes formula and along the same line to the proof of (A.6), we obtain

$$\begin{aligned} \begin{aligned} -\left( \Delta u | \partial _t u\right) _{L^2(\Omega )}=&\int _{\partial \Omega }[(\nabla \times u)\times \partial _tu]\cdot \mathbf{n}\,dS+\int _{\Omega }(\nabla \times u) \cdot (\nabla \times \partial _tu)\,dx\\ =&2\int _{\partial \Omega } \partial _tu(M-M_\mathrm{w})u\,dS+\frac{1}{2}\frac{d}{dt}\int _{\Omega }|\nabla \times u|^2\,dx, \end{aligned} \end{aligned}$$

which together with the facts: M is a symmetric matrix and $M_{\mathrm{w}}$ is a self-adjoint operator on $T_x,$ ensures that

$$\begin{aligned} -\left( \Delta u | \partial _t u\right) _{L^2(\Omega )}=\frac{d}{dt}\Bigl (\int _{\partial \Omega } u(M-M_{\mathrm{w}})u\,dS+\frac{1}{2}\int _{\Omega }|\nabla \times u|^2\,dx\Bigr ). \end{aligned}$$

Again due to $\partial _tu\cdot \mathbf{n}|_{\partial \Omega }=0,$ one has

$$\begin{aligned} \left( \nabla p | \partial _t u\right) _{L^2(\Omega )}=0. \end{aligned}$$

By inserting the above equalities into (A.11), we achieve

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\Bigl (\int _{\partial \Omega } u(M-M_\mathrm{w})u\,dS+\frac{1}{2}\int _{\Omega }|\nabla \times u|^2\,dx\Bigr )+\Vert \partial _tu\Vert _{L^2(\Omega )}^2 =-\left( u\cdot \nabla u | \partial _t u\right) _{L^2(\Omega )}\\&\quad \le \Vert u\Vert _{L^6(\Omega )}\Vert \nabla u\Vert _{L^3(\Omega )}\Vert \partial _tu\Vert _{L^2(\Omega )}\\&\quad \le C\Vert u\Vert _{H^1(\Omega )}\Vert \nabla u\Vert _{L^2(\Omega )}^{\frac{1}{2}}\Vert \nabla u\Vert _{H^1(\Omega )}^{\frac{1}{2}}\Vert \partial _tu\Vert _{L^2(\Omega )}. \end{aligned} \end{aligned}$$

Applying Young’s inequality yields

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\Bigl (\int _{\partial \Omega } u(M-M_{\mathrm{w}})u\,dS+\frac{1}{2}\int _{\Omega }|\nabla \times u|^2\,dx\Bigr )+\frac{3}{4}\Vert \partial _tu\Vert _{L^2(\Omega )}^2\\&\quad \le C_\lambda \bigl (1+\Vert u\Vert _{H^1(\Omega )}^4\bigr )\Vert \nabla u\Vert _{L^2(\Omega )}^2+\lambda \Vert \nabla ^2 u\Vert _{L^2(\Omega )}^2. \end{aligned} \end{aligned}$$

(A.12)

Moreover in view of (A.1), we write

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta u +\nabla p=-\partial _t u-u\cdot \nabla u \\ \mathrm {div} \,u=0 \quad \text{ in } \Omega ,\\ u\cdot {\mathbf {n}}=0 \quad \text{ and } \quad {\mathcal {N}}(u)=0\quad \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(A.13)

The following type of Cattabriga-Solonnikov estimate can be proved along the same line to that of Theorem 2.2 in [32]:

Lemma A.3

Let k be a non-negative integer and $\Omega$ be a bounded domain with sufficiently smooth boundary. Let f in $H^k(\Omega )$ and g in $H^{k+1}(\Omega )$ with $\int _\Omega g\,dx=0.$ Then the non-homogeneous Stokes problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta u +\nabla p=f \\ \mathrm {div} \,u=g \quad \text{ in } \Omega ,\\ u\cdot {\mathbf {n}}=0 \quad \text{ and } \quad {\mathcal {N}}(u)=0\quad \text{ on } \partial \Omega \end{array}\right. } \end{aligned}$$

has a unique solution (u, p) so that

$$\begin{aligned} \Vert \nabla ^2u\Vert _{H^k(\Omega )}+\Vert \nabla p\Vert _{H^k(\Omega )}\le C\bigl (\Vert f\Vert _{H^k(\Omega )}+\Vert \nabla g\Vert _{H^k(\Omega )}\bigr ). \end{aligned}$$

(A.14)

Then it follows from Lemma A.3 and (A.13) that

$$\begin{aligned} \begin{aligned} \Vert \nabla ^2 u\Vert _{L^2(\Omega )}\le&C\bigl (\Vert \partial _tu\Vert _{L^2(\Omega )}+\Vert u\cdot \nabla u\Vert _{L^2(\Omega )}\bigr )\\ \le&C\bigl (\Vert \partial _tu\Vert _{L^2(\Omega )}+\Vert u\Vert _{H^1(\Omega )}\Vert \nabla u\Vert _{L^2(\Omega )}^{\frac{1}{2}}\Vert \nabla u\Vert _{H^1(\Omega )}^{\frac{1}{2}}\bigr ), \end{aligned} \end{aligned}$$

from which, we infer

$$\begin{aligned} \Vert \nabla u\Vert _{H^1(\Omega )}\le C\bigl (\Vert \partial _tu\Vert _{L^2(\Omega )}+(1+\Vert u\Vert _{H^1(\Omega )}^2)\Vert \nabla u\Vert _{L^2(\Omega )}\bigr ). \end{aligned}$$

(A.15)

By substituting (A.15) into (A.12) and then taking $\lambda =\frac{1}{4C},$ we achieve

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\Bigl (\int _{\partial \Omega } u(M-M_{\mathrm{w}})u\,dS+\frac{1}{2}\int _{\Omega }|\nabla \times u|^2\,dx\Bigr )+&\frac{1}{2}\Vert \partial _tu\Vert _{L^2(\Omega )}^2\\ \le&C\bigl (1+\Vert u\Vert _{H^1(\Omega )}^4\bigr )\Vert \nabla u\Vert _{L^2(\Omega )}^2. \end{aligned} \end{aligned}$$

(A.16)

On the other hand, it follows from trace inequality (5.25) that

$$\begin{aligned} \begin{aligned} \bigl |\int _{\partial \Omega } u(M-M_{\mathrm{w}})u\,dS\bigr |\le C\Vert u\Vert _{L^2(\partial \Omega )}^2\le&C\bigl (\Vert u\Vert _{L^2(\Omega )}^2+\Vert u\Vert _{L^2(\Omega )}\Vert \nabla u\Vert _{L^2(\Omega )}\bigr )\\ \le&\frac{1}{4C_\Omega }\Vert u\Vert _{H^1(\Omega )}^2+C\Vert u\Vert _{L^2(\Omega )}^{2}, \end{aligned} \end{aligned}$$

so that in view of (A.8), there exists a large enough constant K which satisfies

$$\begin{aligned} E_1(u):= K\Vert u\Vert _{L^2(\Omega )}^{2}+\int _{\partial \Omega } u(M-M_\mathrm{w})u\,dS+\frac{1}{2}\int _{\Omega }|\nabla \times u|^2\,dx\ge \frac{1}{4C_\Omega }\Vert u\Vert _{H^1(\Omega )}^2.\nonumber \\ \end{aligned}$$

(A.17)

Then we get, by summing up $K\times$(A.9) and (A.16), that

$$\begin{aligned} \frac{d}{dt}E_1(u)+\frac{1}{2}\Vert \partial _tu\Vert _{L^2(\Omega )}^2\le CE_1(u)\bigl (1+E_1^2(u)\bigr ), \end{aligned}$$

(A.18)

from which, we deduce by a comparison argument that for any $T>0$ and $R>0$, there exists a continuous function $C_{T,p,R}$ from $[0,+\infty )$ to $[0,+\infty )$ with $C_{T,1,R}(0)=0$, such that there exists $T_1$ in (0, T) and such that for any $u_0$ in $H^1(\Omega )$, with $\Vert u_0\Vert _{H^1(\Omega )} \le R$, divergence free and tangent to $\partial \Omega$, the unique strong solution u in $C([0,T_1];H^1(\Omega ))\cap L^2([0,T_1];H^2(\Omega ))$ to (A.1) satisfies (A.2) holds true for $p=1.$

$\bullet$ $\underline{\hbox {Higher}\,\, \hbox {energy}\,\, \hbox {estimates}}$

Inductively, we assume that (A.2) holds for $p\le \ell -1,$ we are going to show that (A.2) holds for $p=\ell .$ Without loss of generality, we may assume that $\ell$ is an even integer. The odd integer case can be proved along the same line. Indeed we first get, by applying $\partial _t^{\ell /2}$ to (A.1), that

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^{1+\frac{\ell }{2}} u+\partial _t^{\frac{\ell }{2}}(u\cdot \nabla u)- \Delta \partial _t^{\frac{\ell }{2}}u +\nabla \partial _t^{\frac{\ell }{2}}p=0, \\ \mathrm {div} \,\partial _t^{\frac{\ell }{2}}u=0 \quad \text{ in } (0,T_1)\times \Omega ,\\ \partial _t^{\frac{\ell }{2}}u\cdot {\mathbf {n}}=0 \quad \text{ and } \quad {\mathcal {N}}(\partial _t^{\frac{\ell }{2}}u)=0\quad \text{ on } \ (0,T_1)\times \partial \Omega , \end{array}\right. } \end{aligned}$$

(A.19)

from which, we get, by a similar derivation of (A.4) that

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl (t^{\ell -1}\Vert \partial _t^{\frac{\ell }{2}}u(t)\Vert _{L^2(\Omega )}^2\bigr )+t^{\ell -1}\Vert \nabla \times \partial _t^{\frac{\ell }{2}}u\Vert _{L^2(\Omega )}^2 =\frac{\ell -1}{2}t^{\ell -2}\Vert \partial _t^{\frac{\ell }{2}}u\Vert _{L^2(\Omega )}^2\\&\quad +t^{\ell -1}\int _{\partial \Omega }\bigl [\partial _t^{\frac{\ell }{2}}u\times (\nabla \times \partial _t^{\frac{\ell }{2}}u)\bigr ]\cdot \mathbf{n}\,dS -t^{\ell -1}\bigl (\partial _t^{\frac{\ell }{2}}(u\cdot \nabla u) | \partial _t^{\frac{\ell }{2}}u\bigr )_{L^2(\Omega )}. \end{aligned} \end{aligned}$$

(A.20)

Similarly to (A.7), we have

$$\begin{aligned} t^{\ell -1}\bigl |\int _{\partial \Omega }\bigl [\partial _t^{\frac{\ell }{2}}u\times (\nabla \times \partial _t^{\frac{\ell }{2}}u)\bigr ]\cdot \mathbf{n}\,dS\bigr | \le \lambda \bigl \Vert t^{\frac{\ell -1}{2}}\nabla \partial ^{\frac{\ell }{2}}_tu\bigr \Vert _{L^2(\Omega )}^2+C_\lambda \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^2(\Omega )}^2. \end{aligned}$$

On the other hand, due to $u\cdot \mathbf{n}|_{\partial \Omega }=0$ and $\mathrm {div} \,u=0,$ we get, by using integration by parts, that

$$\begin{aligned} \begin{aligned} \bigl (\partial _t^{\frac{\ell }{2}}(u\cdot \nabla u) | \partial _t^{\frac{\ell }{2}}u\bigr )_{L^2(\Omega )}=&\bigl (\partial _t^{\frac{\ell }{2}}( u\cdot \nabla u)-u\cdot \nabla \partial _t^{\frac{\ell }{2}}u | \partial ^{\frac{\ell }{2}}_tu \bigr )_{L^2(\Omega )}\\ =&-\sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ \ell _1\ge 1 \end{array}}C_{\frac{\ell }{2}}^{\ell _1}\bigl (\partial _t^{\ell _1}u\otimes \partial _t^{\ell _2}u | \nabla \partial ^{\frac{\ell }{2}}_tu \bigr )_{L^2(\Omega )}, \end{aligned} \end{aligned}$$

from which we infer

$$\begin{aligned} \begin{aligned}&t^{\ell -1}\bigl |\bigl (\partial _t^{\frac{\ell }{2}}(u\cdot \nabla u) | \partial _t^{\frac{\ell }{2}}u\bigr )_{L^2(\Omega )}\bigr | \\&\quad \lesssim \sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ \ell _1\ge 1 \end{array}} t^{\ell -1}\Vert \partial _t^{\ell _1}u\Vert _{L^3(\Omega )}\Vert \partial _t^{\ell _2}u\Vert _{L^6(\Omega )}\Vert \nabla \partial ^{\frac{\ell }{2}}_tu\Vert _{L^2(\Omega )}\\&\quad \lesssim \sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ \ell _1\ge 1 \end{array}} t^{\ell -1}\Vert \partial _t^{\ell _1}u\Vert _{L^2(\Omega )}^{\frac{1}{2}}\Vert \partial _t^{\ell _1}u\Vert _{H^1(\Omega )}^{\frac{1}{2}}\Vert \partial _t^{\ell _2}u\Vert _{H^1(\Omega )}\Vert \nabla \partial ^{\frac{\ell }{2}}_tu\Vert _{L^2(\Omega )}\\&\quad \le \lambda \bigl \Vert t^{\frac{\ell -1}{2}} \partial _t^{\frac{\ell }{2}}u\bigr \Vert _{H^1(\Omega )}^2+C_\lambda \Vert u\Vert _{H^1(\Omega )}^{4}\bigl \Vert t^{\frac{\ell -1}{2}} \partial ^{\frac{\ell }{2}}_tu\bigr \Vert _{L^2(\Omega )}^2\\&\qquad +C_\lambda \sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ 1\le \ell _1\le \frac{\ell }{2}-1 \end{array}} \bigl \Vert t^{\ell _1-\frac{1}{2}}\partial _t^{\ell _1}u\bigr \Vert _{H^1(\Omega )}^2\bigl \Vert |t^{\ell _2}\partial _t^{\ell _2}u\bigr \Vert _{H^1(\Omega )}^2. \end{aligned} \end{aligned}$$

By substituting the above estimates into (A.20) and using Korn’s type inequality (A.8), we find

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{\frac{\ell }{2}}u(t)\bigr \Vert _{L^2(\Omega )}^2+\frac{1}{C_\Omega }\bigl \Vert t^{\frac{\ell -1}{2}} \partial _t^{\frac{\ell }{2}}u\bigr \Vert _{H^1(\Omega )}^2 \\&\quad \le \frac{\ell -1}{2}\bigl \Vert t^{\frac{\ell }{2}-1}\partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^2(\Omega )}^2+C_\lambda \bigl (1+ \Vert u\Vert _{H^1(\Omega )}^{4}\bigr )\bigl \Vert t^{\frac{\ell -1}{2}} \partial ^{\frac{\ell }{2}}_tu\bigr \Vert _{L^2(\Omega )}^2\\&\qquad +2\lambda \bigl \Vert t^{\frac{\ell -1}{2}} \partial _t^{\frac{\ell }{2}}u\bigr \Vert _{H^1(\Omega )}^2+ C_\lambda \sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ 1\le \ell _1\le \frac{\ell }{2}-1 \end{array}} \bigl \Vert t^{\ell _1-\frac{1}{2}}\partial _t^{\ell _1}u\bigr \Vert _{H^1(\Omega )}^2\bigl \Vert |t^{\ell _2}\partial _t^{\ell _2}u\bigr \Vert _{H^1(\Omega )}^2. \end{aligned} \end{aligned}$$

By taking $\lambda =\frac{1}{4C_\Omega }$ in the above inequality and then applying Gronwall’s inequality to the resulting inequality, we achieve

$$\begin{aligned} \begin{aligned}&\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^\infty _t(L^2(\Omega ))}^2+\frac{1}{C_\Omega }\bigl \Vert t^{\frac{\ell -1}{2}} \partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^2_t(H^1(\Omega ))}^2\le C\exp \left( C\bigl (1+t\Vert u\Vert _{L^\infty _{t}(H^1(\Omega ))}^4\bigr )\right) \\&\quad \times \Bigl (\bigl \Vert t^{\frac{\ell }{2}-1}\partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^2_t(L^2(\Omega ))}^2+ \sum _{\begin{array}{c} \ell _1+\ell _2=\frac{\ell }{2}\\ 1\le \ell _1\le \frac{\ell }{2}-1 \end{array}} \bigl \Vert t^{\ell _1-\frac{1}{2}}\partial _t^{\ell _1}u\bigr \Vert _{L^2_t(H^1(\Omega ))}^2\bigl \Vert |t^{\ell _2}\partial _t^{\ell _2}u\bigr \Vert _{L^\infty _t(H^1(\Omega ))}^2\Bigr ), \end{aligned} \end{aligned}$$

from which, with the inductive assumption, we deduce that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^\infty _{T_1}(L^2(\Omega ))}^2+\frac{1}{C_\Omega }\bigl \Vert t^{\frac{\ell -1}{2}} \partial _t^{\frac{\ell }{2}}u\bigr \Vert _{L^2_{T_1}(H^1(\Omega ))}^2\le C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned} \end{aligned}$$

(A.21)

On the other hand, for any non-negative integer $j\le \frac{\ell }{2}-1,$ we infer from the inductive assumption that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j}(\Omega ))} =&\bigl \Vert t^{\frac{\ell -1}{2}}\nabla ^2\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))} +\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -1-2j}(\Omega ))}\\ \le&\bigl \Vert t^{\frac{\ell -1}{2}}\nabla ^2\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}+ C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned} \end{aligned}$$

Moreover in view of (A.1), we write

$$\begin{aligned} - \Delta \partial _t^{j}u +\nabla \partial _t^{j}p=-\partial _t^{j+1} u-\partial _t^{j}(u\cdot \nabla u), \end{aligned}$$

from which, with Lemma A.3, we infer

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\nabla ^2\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}\lesssim&\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j+1}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}\\&+ \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}(u\cdot \nabla u)\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}. \end{aligned} \end{aligned}$$

As a result, we get that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j}(\Omega ))} \le&C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )})+\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j+1}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}\\&+ \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}(u\cdot \nabla u)\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}, \quad \forall \ j\le \frac{\ell }{2}-1. \end{aligned} \end{aligned}$$

(A.22)

However, it follows from Moser type inequality and the inductive assumption that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}\nabla (u\otimes u)\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))} \lesssim&\sum _{j_1+j_2=j}\bigl \Vert t^{j_1+\frac{1}{2}}\partial _t^{j_1}u\bigr \Vert _{L^\infty _{T_1}(H^{2}(\Omega ))}\\&\times \bigl \Vert t^{\frac{\ell -2}{2}-j+j_2}\partial _t^{j_2}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j-1}(\Omega ))}\\ \le&C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned} \end{aligned}$$

Substituting the above estimates into (A.22) gives rise to

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j}(\Omega ))} \le&C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )})+\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j+1}u\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2-2j}(\Omega ))}. \end{aligned} \end{aligned}$$

We deduce from this inequality and from (A.21), by an iterative argument, that

$$\begin{aligned} \sum _{0\le j \le \frac{\ell }{2}}\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^ju\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j}(\Omega ))} \le C_{\ell ,T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned}$$

(A.23)

Exactly along the same line to the proof of (A.23), for any non-negative integer $j\le \frac{\ell }{2}-1,$ we infer from the inductive assumption that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^2_{T_1}(H^{\ell +1-2j}(\Omega ))} \le&\bigl \Vert t^{\frac{\ell -1}{2}}\nabla ^2\partial _t^{j}u\bigr \Vert _{L^2_{T_1}(H^{\ell -1-2j}(\Omega ))}+ C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned} \end{aligned}$$

On the other hand it follows from Lemma A.3 that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\nabla ^2\partial _t^{j}u\bigr \Vert _{L^2_{T_1}(H^{\ell -1-2j}(\Omega ))}\lesssim&\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j+1}u\bigr \Vert _{L^2_{T_1}(H^{\ell -1-2j}(\Omega ))}\\&+ \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}(u\otimes u)\bigr \Vert _{L^\infty _{T_1}(H^{\ell -2j}(\Omega ))}. \end{aligned} \end{aligned}$$

For, any $j\le \frac{\ell }{2}-1,$ it follows from Moser type inequality and the inductive assumption that

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}(u\otimes u)\bigr \Vert _{L^2_{T_1}(H^{\ell -2j}(\Omega ))} \lesssim&\sum _{j_1+j_2=j}\bigl \Vert t^{j_1+\frac{1}{2}}\partial _t^{j_1}u\bigr \Vert _{L^\infty _{T_1}(H^{2}(\Omega ))}\\&\times \bigl \Vert t^{\frac{\ell -2}{2}-j+j_2}\partial _t^{j_2}u\bigr \Vert _{L^2_{T_1}(H^{\ell -2j}(\Omega ))}\\ \le&C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned} \end{aligned}$$

As a result, for any $j\le \frac{\ell -1}{2},$ we arrive at

$$\begin{aligned} \begin{aligned} \bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j}u\bigr \Vert _{L^2_{T_1}(H^{\ell +1-2j}(\Omega ))} \le&C_{\ell , T_1}(\Vert u_0\Vert _{H^1(\Omega )})+\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^{j+1}u\bigr \Vert _{L^2_{T_1}(H^{\ell -1-2j}(\Omega ))}, \end{aligned} \end{aligned}$$

from which, with (A.21), we deduce by an iterative argument that

$$\begin{aligned} \sum _{0\le j \le \frac{\ell }{2}}\bigl \Vert t^{\frac{\ell -1}{2}}\partial _t^ju\bigr \Vert _{L^2_{T_1}(H^{\ell +1-2j}(\Omega ))} \le C_{\ell ,T_1}(\Vert u_0\Vert _{H^1(\Omega )}). \end{aligned}$$

(A.24)

By combining (A.23) and (A.24), we obtain that (A.2) holds for $p=\ell .$ This finishes the proof of (A.2) and therefore the proof of Theorem 2.1. $\square$