Data Assimilation to the Primitive Equations with $L^p$-$L^q$-based Maximal Regularity Approach

Abstract

In this paper, we show a mathematical justification of the data assimilation of nudging type in $L^p$-$L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations constructed by the data assimilation converges to the true solution with exponential order in the Besov space $B^{2/q}_{q,p}(\Omega )$ for $1/p + 1/q \le 1$ on the periodic layer domain $\Omega = \mathbb {T}^2 \times (-h, 0)$.

Appendices

Appendix A: $L^2_t H^3_x$ Estimate for the Solution to the Primitive Equations

In this appendix, we show the $L^2_t H^3_x$ a priori estimate

$$\begin{aligned} \Vert \Delta v (t) \Vert _{L^2(\Omega )^2} + \int _0^t \Vert \nabla \Delta v (t) \Vert _{L^2(\Omega )^6} ds < \infty \end{aligned}$$

(A1)

for all $t>0$ if $f \in L^2_t H^1_x ((0, \infty ) \times \Omega )$. Although Giga et   al. [5] proved $L^\infty _t H^2_x$-estimates and the lower order estimates, they did not show $L^2_t H^3_x$-a priori estimates. Let $\psi \in H^1_t L^2_{\overline{\sigma }, x}((0, \infty ) \times \Omega ) \cap L^2_t H^2_{\sigma , x} ((0, \infty )\times \Omega )$ be a solution of the hydrostatic Stokes equations

$$\begin{aligned}&\partial _t \psi + A_2 \psi = g \in L^2_t H^1_{\overline{\sigma }, x} ((0, \infty ) \times \Omega ), \end{aligned}$$

(A2)

$$\begin{aligned}&\psi (0) = \psi _0 \in H^2_{\overline{\sigma }}(\Omega ), \end{aligned}$$

(A3)

with boundary conditions (2). Note that we have the local well-posedness for the primitive equation from [5]. The local solution v is embedded into $C(0,T_0; H^2_{\overline{\sigma }}(\Omega )^2)$ for some small $T_0>0$. A priori estimates are used to extend the local solution globally. Therefore, without loss of generality, we can assume (A3). Assume that $\psi$ satisfies

$$\begin{aligned} \int _0^t \Vert \nabla \partial _t \psi (s) \Vert _{L^2 (\Omega )^6} ds < \infty \end{aligned}$$

(A4)

for all $t > 0$. We denote $D^h_j \psi (x) = (\psi (x + h e_j) - \psi (x))/h$ for $j = 1, 2, 3$ and $h > 0$. Then $D^h_j \psi$ satisfies (A2) with an external force $D^h_j g$ and initial data $D^h_j \psi _0$. We find that

$$\begin{aligned}&\int ^t_0 \Vert \Delta D^h_j \psi \Vert _{L^2(\Omega )^2}^2 ds \le C \int ^t_0 \Vert A_2 D^h_j \psi \Vert _{L^2(\Omega )^2}^2 ds \\&\quad \le C \int ^t_0 \Vert D^h_j \partial _t \psi \Vert _{L^2(\Omega )^2}^2 ds + C \int ^t_0 \Vert D^h_j g \Vert _{L^2(\Omega )^2}^2 ds\\&\quad \le C \int ^t_0 \Vert \partial _j \partial _t \psi \Vert _{L^2(\Omega )^2}^2 ds + C \int ^t_0 \Vert \partial _j g \Vert _{L^2(\Omega )^2}^2 ds. \end{aligned}$$

Since the right-hand side is uniform with respect to $h>0$, we take a $\limsup$ to get

$$\begin{aligned} \int ^t_0 \Vert \partial _j \psi \Vert _{H^2(\Omega )^2}^2 ds \le C \int ^t_0 \Vert \partial _j \partial _t \psi \Vert _{L^2(\Omega )^2}^2 ds + C \int ^t_0 \Vert \partial _j g \Vert _{L^2(\Omega )^2}^2 ds. \end{aligned}$$

The reader is referred to Section 3.1.5 and 4.2.7 of the book [16] by Sohr. Therefore, we see that

$$\begin{aligned}&\int _0^t \Vert \partial _j v \Vert _{H^2(\Omega )^2}^2 ds \\&\quad \le C \int _0^t \Vert \partial _t \partial _j v \Vert _{L^2(\Omega )^2}^2 ds \\&\quad + C \int _0^t \Vert \partial _j \left( v \cdot \nabla _H v + w \partial _3 v \right) \Vert _{L^2(\Omega )^2}^2 ds \\&\quad + C \int _0^t \Vert \partial _j f \Vert _{L^2(\Omega )^2}^2 dx. \end{aligned}$$

The boundedness of the first term was shown in [5]. The Sobolev inequalities and the interpolation inequalities yield

$$\begin{aligned}&\int _\Omega \left| \nabla v \cdot \nabla _H v \right| ^2 dx \le C \Vert \nabla v \Vert _{L^4(\Omega )^6}^2 \le C \Vert \nabla v \Vert _{L^2(\Omega )^6}^{3/2} \Vert \nabla v \Vert _{H^1(\Omega )^6}^{1/2}, \\&\int _\Omega \left| v \cdot \nabla \nabla _H v \right| ^2 dx \le C \Vert v \Vert _{L^\infty (\Omega )^2} \Vert {\nabla \nabla _H} v \Vert _{L^2(\Omega )^{12}} \le C \Vert v \Vert _{H^1(\Omega )^2}^{\frac{1}{2}} \Vert v \Vert _{H^2(\Omega )^2}^\frac{1}{2} \Vert v \Vert _{\dot{H}^2(\Omega )^2}. \end{aligned}$$

In the same way, we see that

$$\begin{aligned} \int _\Omega \left| \nabla w \partial _3 v \right| ^2 dx&\le C \Vert \nabla v \Vert _{L^2(\Omega )^6} \Vert \nabla v \Vert _{H^1(\Omega )^6} \Vert \partial _3 v \Vert _{L^2(\Omega )^2} \Vert \partial _3 v\Vert _{H^1(\Omega )^2}\\&\le C \Vert v \Vert _{H^1(\Omega )^2}^2 \Vert \nabla v \Vert _{H^1(\Omega )^6}^2,\\ \int _\Omega \left| w \nabla \partial _3 v \right| ^2 dx&\le C \Vert \nabla _H v \Vert _{L^2(\Omega )^4} \Vert \nabla _H v \Vert _{H^1(\Omega )^4} \Vert \partial _3 v \Vert _{L^2(\Omega )^2} \Vert \partial _3 v \Vert _{H^1(\Omega )^2}\\&\le C \Vert v \Vert _{H^1(\Omega )^2}^2 \Vert \nabla v \Vert _{H^1(\Omega )^6}^2. \end{aligned}$$

Using the Young inequality, we obtain (A1).

Appendix B: Remarks for Semi-Group Based Approach

The main result of this paper is a mathematical justification of the DA in $L^p$-$L^q$ based maximal regularity settings. However, the analytic semi-group based approach as used by Hieber and Kashiwabara [4] can be applicable. As in [4], we set

$$\begin{aligned}&{\mathcal {S}}_{q, \eta , T} = \left\{ v \in C([0, T]; H^{2/q, q}_{\overline{\sigma }}(\Omega )) \cap C((0, T]; H^{1 + 1/q, q}_{\overline{\sigma }}(\Omega )) \, ; \, \Vert v \Vert _{{\mathcal {S}}_{q, \eta , T}}< \infty \right\} , \\&\Vert v \Vert _{{\mathcal {S}}_{q, \eta , T}} := \sup _{0< t< T} e^{\eta t} \Vert v(t) \Vert _{H^{2/q, q}_{\overline{\sigma }}(\Omega )} + \sup _{0< t< T} e^{\eta t} t^{1/2 - 1/2q} \Vert v(t) \Vert _{H^{1 + 1/q, q}_{\overline{\sigma }}(\Omega )}, \\&\widetilde{{\mathcal {S}}}_{q, \eta , T} = \left\{ v \in C((0, T]; H^{1 + 1/q, q}_{\overline{\sigma }}(\Omega )^2) \, ; \, \Vert v \Vert _{\widetilde{{\mathcal {S}}}_{q, \eta , T}}< \infty \right\} , \\&\Vert v \Vert _{\widetilde{{\mathcal {S}}}_{q, \eta , T}} := \sup _{0< t < T} e^{\eta t} t^{1/2 - 1/2q} \Vert v(t) \Vert _{H^{1 + 1/q, q}_{\overline{\sigma }}(\Omega )}. \end{aligned}$$

We also set $X_q = \left\{ \varphi \in H^{2/q, q}_{\overline{\sigma }}(\Omega )^2 \, ; \, \varphi \vert _{\Gamma _b = 0} \right\}$. Using the same kind of method based on analytic semi-group theory as [4, 10], we can show

Theorem B.1

Let $1< p < \infty$ and $0< \alpha < 1$. Let $v_0 \in X_q$ be an initial datum. Let $f \in H^1_{loc} (0, \infty ; L^q(\Omega )^2)$ satisfy

$$\begin{aligned} \sup _{t>0} e^{\gamma _0 t} \Vert f(s) \Vert _{L^q(\Omega )^2} + \sup _{0< s< t} e^{\gamma _0 s} s^{1 - \frac{1}{q}} \Vert f (s) \Vert _{L^q(\Omega )^2}&< \infty ,\\ \Vert f \Vert _{H^1_{loc}(0, \infty ; L^2(\Omega )^2 \cap L^q(\Omega )^2)} + \Vert e^{\gamma _0 t} f \Vert _{L^2(0, \infty ; H^1(\Omega )^2)}&< \infty , \end{aligned}$$

for some constant $\gamma _0 > 0$. Assume that $v \in C(0, \infty ; X_q)$ is the solution to (1), which is obtained by Hieber et al. [10] with zero temperature and salinity, satisfying

$$\begin{aligned} \Vert v \Vert _{{\mathcal {S}}_{q, \gamma _1, \infty }} < \infty \end{aligned}$$

(B5)

for some constant $C_0>0$ and $\gamma _1 < \gamma _0$. Then there exist $\mu _0, \delta _0>0$ such that if $\mu \ge \mu _0$ and $\delta \le \delta _0$, there exists a unique solution $V \in C(0, \infty ;X_{1/q, p, q})$ to (4) such that

$$\begin{aligned} \Vert {V} \Vert _{{\mathcal {S}}_{q, {\mu _*}, \infty }} < \infty \end{aligned}$$

for some constants $\gamma _0< \mu _*< \gamma _1$ and $C>0$. Moreover, V satisfies

$$\begin{aligned} \Vert \partial _t V (t) \Vert _{L^q(\Omega )^2} + \Vert V (t) \Vert _{D(\tilde{A}_{q, \mu })} = O(e^{- \mu _*t /2}). \end{aligned}$$

Remark 5

The assumption on the regularity for the initial data in Theorem B.1 is stronger than Theorem 1 since $H^{2/q, q}(\Omega ) \hookrightarrow B^{2/q}_{q, p}(\Omega )$ for $p > 2$.

We show the local well-posedness in $C(0,T_*; L^q(\Omega )^2)$-framework with the analytic semi-group $e^{- t \tilde{A}_{\mu , q}}$ and the Fujita-Kato principle. The mild solution is given by

$$\begin{aligned} \begin{aligned} V (t)&= e^{- t \tilde{A}_{\mu , q}} V_0 \\&\quad +\,\int _0^t e^{- (t - s) \tilde{A}_{\mu , q}} P \left( - U (s) \cdot \nabla V (s) + u (s) \cdot \nabla V (s) \right. \\&\left. + U (s) \cdot \nabla v (s) \right) ds \\&\quad +\,\int _0^t e^{- (t - s) \tilde{A}_{\mu , q}} P F(s) ds \\&=: N_1(V, v) + N_2(F) := N(V, v). \end{aligned} \end{aligned}$$

(B6)

Lemma B.2

Let $T>0$, $0 \le \tilde{\mu } \le \mu _*$, and $V_0, v_0 \in H^{2/q, q}(\Omega )^2$. Let $F \in C(0, T; L^q(\Omega )^2)$ satisfy

$$\begin{aligned} \sup _{0< s< T} e^{\gamma _0 s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2} < \infty \end{aligned}$$

for some constant $\gamma _0 > 0$. Let v satisfy the same assumptions as Theorem B.1. Then, there exists $T_0 > 0$ such that if $T \le T_0$, the integral Eq. (B6) admit the unique solution $V \in C(0,T; H^{2/q, q}_{\overline{\sigma }}(\Omega )^2)$ such that

$$\begin{aligned} \Vert V \Vert _{{{\mathcal {S}}_{q, \tilde{\mu },T}}} \le C. \end{aligned}$$

(B7)

Moreover, there exists $\varepsilon _0>0$ such that if

$$\begin{aligned} \Vert v_0 \Vert _{H^{2/q, q}(\Omega )^2}, \sup _{0< s < T} e^{\gamma _0 s} s^{1 - \frac{1}{q}} \Vert f(s) \Vert _{L^q(\Omega )^2} \le \varepsilon _0, \end{aligned}$$

it can be taken $T = \infty$.

Proof

The proof is based on Proposition 5.2 in [4]. We prove that

$$\begin{aligned} N : {{\mathcal {S}}_{q, \tilde{\mu },T}} {\times {{\mathcal {S}}_{q, \tilde{\mu },T}}} \rightarrow {{\mathcal {S}}_{q, \tilde{\mu },T}} \end{aligned}$$

is a contraction mapping if T is small or $v_0$ and F are small. We find from Proposition 6 that

$$\begin{aligned} \begin{aligned}&e^{\tilde{\mu }t} \Vert N_1(V, v) \Vert _{H^{2/q, q}(\Omega )^2} \\&\quad \le \int _0^t e^{\tilde{\mu }t} e^{- \mu _*(t - s)} (t - s)^{ - \frac{1}{q}} \\&\quad \times \left( \Vert U (s) \cdot \nabla V (s) \Vert _{L^q(\Omega )^2} + \Vert u (s) \cdot \nabla V (s) \Vert _{L^q(\Omega )^2} + \Vert U (s) \cdot \nabla v (s) \Vert _{L^q(\Omega )^2} \right) ds \\&\quad \le C \int _0^t e^{\tilde{\mu } (t - s)} e^{ - \mu _*(t - s)} (t - s)^{ - \frac{1}{q}} s^{- 1 - 1/q} ds {\left( \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}^2 + \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}} \Vert v \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}} \right) }\\&\quad \le {C \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}^2 + C \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}} \Vert v \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}.} \end{aligned} \end{aligned}$$

(B8)

We also find from Propositions 4 and 6 that

$$\begin{aligned}&e^{\tilde{\mu }t} \Vert N_1(V, v) \Vert _{H^{1 + 1/q, q}(\Omega )^2} \\&\quad \le \int _0^t e^{\tilde{\mu }t} e^{- \mu _*(t - s)} (t - s)^{ - \frac{1}{2} - \frac{1}{2q}} \\&\quad \times \left( \Vert U (s) \cdot \nabla V (s) \Vert _{L^q(\Omega )^2} + \Vert u (s) \cdot \nabla V (s) \Vert _{L^q(\Omega )^2} + \Vert U (s) \cdot \nabla v (s) \Vert _{L^q(\Omega )^2} \right) ds \\&\quad \le C t^{- \frac{1}{2} + \frac{1}{2q}} \Vert V \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}}^2 + C \Vert V \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}} \Vert v \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}}. \end{aligned}$$

Similarly we have

$$\begin{aligned}&e^{\tilde{\mu }t} \Vert N_1(V_1, v) - N_1(V_2, v) \Vert _{H^{1 + 1/q, q}(\Omega )^2} \\&\quad \le C \Vert V_1 - V_2 \Vert _{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}} \left( \Vert V_1 \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}} + \Vert V_2 \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}} + \Vert v \Vert _{{\widetilde{{\mathcal {S}}}_{q, \tilde{\mu }, T}}} \right) . \end{aligned}$$

Since $e^{- t \tilde{A}_{\mu , q}}$ is analytic for t, we can observe that

$$\begin{aligned}&e^{\tilde{\mu }t} \Vert e^{- t \tilde{A}_{\mu , q}} V_0 \Vert _{H^{2/q, q}(\Omega )^2} \le C \Vert V_0 \Vert _{H^{2/q, q}(\Omega )^2}, \\&e^{\tilde{\mu }t} \Vert e^{- t \tilde{A}_{\mu , q}} V_0 \Vert _{H^{1 + 1/q, q}(\Omega )^2} \le C t^{- \frac{1}{2} + \frac{1}{2q}} \Vert V_0 \Vert _{H^{2/q, q}(\Omega )^2}. \end{aligned}$$

Moreover, since $D(\tilde{A}_{\mu , q})$ is densely embedded into $H^{s, q}(\Omega )^2$ for $s \in [0, 2)$, we see that

$$\begin{aligned} \Vert e^{- t \tilde{A}_{\mu , q}} V_0 \Vert _{H^{2/q, q}(\Omega )^2} \rightarrow 0 \quad \text {as} \quad t \rightarrow 0. \end{aligned}$$

We find from Proposition 4 that

$$\begin{aligned}&e^{\tilde{\mu }t} \left\| \int _0^t e^{- (t - s) \tilde{A}_{\mu , q}} F (s) ds \right\| _{H^{2/q, q}(\Omega )^2} \\&\quad \le C \int _0^t e^{\tilde{\mu }t} e^{- \mu _*(t - s)} (t - s)^{- \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2} ds \\&\quad \le C \sup _{0< s < t} e^{\tilde{\mu }s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2}, \end{aligned}$$

and

$$\begin{aligned}&e^{\tilde{\mu }t} \left\| \int _0^t e^{- (t - s) \tilde{A}_{\mu , q}} F (s) ds \right\| _{H^{1 + 1/q, q}(\Omega )^2} \\&\quad \le \int _0^t e^{\tilde{\mu }t} e^{- \mu _*(t - s)} (t - s)^{ - \frac{1}{2} - \frac{1}{2q}} \Vert F (s) \Vert _{L^q(\Omega )^2} ds \\&\quad \le C t^{- \frac{1}{2} + \frac{1}{2q}} \sup _{0< s < t} e^{\tilde{\mu }s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2}. \end{aligned}$$

Combining the above estimates, we obtain the quadratic inequality

$$\begin{aligned}&\Vert N(V, v) \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} \\&\quad \le C_2 \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}}^2 + C_1 \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} \Vert v \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} \\&\quad + C_0 \left( \sup _{0< t< T} e^{\tilde{\mu }t} t^{\frac{1}{2} - \frac{1}{2q}} \Vert e^{- t \tilde{A}_{\mu , q}} V_0 \Vert _{H^{1 + 1/q, q}(\Omega )^2} + C \sup _{0< s < t} e^{\tilde{\mu }s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2} \right) . \end{aligned}$$

We assume that $\Vert v \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} \le \frac{1}{2C_1}$ is sufficiently small.

The quadratic estimate implies that, if we take $T>0$ so small or $\Vert V_0 \Vert _{H^{2/q, q}(\Omega )^2}$ and $\sup _{0< s < t} e^{\tilde{\mu }s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2}$ so small that

$$\begin{aligned} R&:= 16 C_2 C_0 ( \sup _{0< t< T} e^{\tilde{\mu }t} t^{\frac{1}{2} - \frac{1}{2q}} \Vert e^{- t \tilde{A}_{\mu , q}} V_0 \Vert _{H^{1 + 1/q, q}(\Omega )^2} \\&+ C \sup _{0< s < t} e^{\tilde{\mu }s} s^{1 - \frac{1}{q}} \Vert F (s) \Vert _{L^q(\Omega )^2} ) \le \frac{1}{2}, \end{aligned}$$

then

$$\begin{aligned} \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} \le \frac{ 1 - \sqrt{ 1 - R } }{2 C_2} =: R_*. \end{aligned}$$

We find that $N (\cdot , v)$ is a self-mapping on $\left\{ V \in \widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T} \, ; \, \Vert V \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}} < R_* \right\}$. We take R and $\Vert v \Vert _{\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}}$ sufficiently sufficiently small again. Then $N (\cdot , v)$ is a contraction mapping. Banach’s fixed point theorem implies that there exists a unique mild solution V to (B6) in $\widetilde{{\mathcal {S}}}_{\tilde{\mu }, q, T}$. The estimate (B8) implies that $N (V, v) \in {\mathcal {S}}_{\tilde{\mu }, q, T}$, so $V \in {\mathcal {S}}_{\tilde{\mu }, q, T}$. $\square$

We improve the regularity of the solution V to (B6) for regular F.

Proposition B.3

Let $0 \le \tilde{\mu } \le \mu _*$, $T > 0$, and $1< q < \infty$. Let $f \in C^\alpha (0, T; L^q(\Omega )^2)$ such that

$$\begin{aligned} \begin{aligned}&\Vert f (t) \Vert _{L^q(\Omega )^2} \le C t^{- \beta } e^{-\gamma t}, \\&\Vert f (t + \tau ) - f (t) \Vert _{L^q(\Omega )^2} \le C \tau ^{\alpha } t^{- \beta } e^{-\gamma t}, \end{aligned} \end{aligned}$$

(B9)

for $0< \alpha < 1$ and $\beta , \gamma > 0$. Set $\phi$ such that

$$\begin{aligned} \phi (t) = \int _{0}^t e^{- (t - s)\tilde{A}_{\mu , q}} P f ds. \end{aligned}$$

Then $\phi$ satisfies

$$\begin{aligned} \Vert \phi (t) \Vert _{L^q(\Omega )^2} \le C t^{1 - \beta } ( e^{- \gamma t} + e^{- \mu _*t/2} ) \end{aligned}$$

and

$$\begin{aligned}&\Vert \partial _t \phi (t) \Vert _{L^q(\Omega )^2} + \Vert \tilde{A}_{\mu , q} \phi (t) \Vert _{L^q(\Omega )^2} \\&\le C ( t^{\alpha - \beta }e^{- \gamma t/2} + t^{- \beta } e^{- (\mu _*/2 + \gamma ) t} + t^{- \beta } e^{- \mu _*t /2} ) \end{aligned}$$

for some constant $C > 0$. In particular, if $\mu _*\le \gamma$, it follows that

$$\begin{aligned} \Vert \partial _t \phi (t) \Vert _{L^q(\Omega )^2} + \Vert \tilde{A}_{\mu , q} \phi (t) \Vert _{L^q(\Omega )^2} = O(t^{-\beta } e^{{-}\mu _*/2}). \end{aligned}$$

Proof

Since the proof is quite parallel as Lemma 5.6 in [4], we omit the details here. $\square$

Proof of Theorem B.1

By Proposition 7 for $p = q = 2$, we first deduce that there exists an $L^2$-solution $V \in C(0, \infty ; H^1(\Omega )^2) \cap C(0, T; D(\tilde{A}_{2, \mu }))$ such that

$$\begin{aligned} \Vert V(t) \Vert _{H^1(\Omega )^2} \le C e^{- \mu _*t} \end{aligned}$$

for some constant $C>0$. Moreover, we repeat the same argument as in the proof of Theorem 1 for $p = q = 2$ to see that there exist small $\varepsilon > 0$ and large $T_*> 0$ such that $\Vert V (T_*) \Vert _{D(\tilde{A}_{2, \mu })} \le \varepsilon$. Applying Lemma B.2, we have $V \in {\mathcal {S}}_{q, {\mu _*}, \infty }$. By the assumptions for f and Proposition 6, we see that

$$\begin{aligned} \begin{aligned}&\Vert F \Vert _{L^2(\Omega )} = O (e^{- \gamma _0 t}), \\&\Vert U(t) \cdot \nabla V(t) \Vert _{L^2(\Omega )} = O(e^{- 2 \mu _*t}), \\&\Vert u(t) \cdot \nabla V(t) \Vert _{L^2(\Omega )} + \Vert U(t) \cdot \nabla v(t) \Vert _{L^2(\Omega )} = O(e^{- (\gamma _1 + \mu _*) t}), \end{aligned} \end{aligned}$$

(B10)

as $t \rightarrow \infty$. We use Proposition B.3 to conclude that

$$\begin{aligned} \Vert \partial _t V (t) \Vert _{L^2(\Omega )^2} + \Vert V (t) \Vert _{D(\tilde{A}_{2, \mu })} = O(e^{- \mu _*t /2}). \end{aligned}$$

(B11)

Note that if we take $\mu$ sufficiently large, the decay rate $e^{- \mu _*t /2}$ is larger than that of $\Vert \partial _t v (t) \Vert _{L^2(\Omega )^2} + \Vert v(t) \Vert _{D(A_2)}$. Proposition B.2 implies that there exists a unique solution in ${\mathcal {S}}_{q, \mu _*, T_*}^\prime$ for small $T_*^\prime > 0$ and all $1< q < \infty$. We consider the case $q > 2$. In this case, we have $V(t) \in H^{1 + 1/q,q}(\Omega )^2 \hookrightarrow H^1(\Omega )^2$ for $0< t < T_*^\prime$. We can extend V as $H^1$-solution with initial data $V(T_*^\prime )$ such that

$$\begin{aligned} \begin{aligned} V \in {\mathcal {S}}_{q, \mu _*, T_*^\prime } \cap C(T_*^\prime /2, \infty ; H^1_{\overline{\sigma }}(\Omega )) \cap C(T_*^\prime /2, \infty ; H^2(\Omega )^2), \\ \Vert V (t) \Vert _{H^2(\Omega )^2 } = O(e^{- \mu _*t /2}) \quad \text {as} \rightarrow \infty . \end{aligned} \end{aligned}$$

(B12)

Since $H^2(\Omega )^2 \hookrightarrow H^{2/q, q}(\Omega )^2$, we see that $V \in {\mathcal {S}}_{q, \mu _*, \infty }$. By Proposition 6, we have the same order decay estimate as (B10) even for $L^q$ cases and conclude that

$$\begin{aligned} \Vert \partial _t V (t) \Vert _{L^q(\Omega )^2} + \Vert V (t) \Vert _{D(\tilde{A}_{q, \mu })} = O(e^{- \mu _*t /2}) \end{aligned}$$

for $q > 2$. We consider the case $q < 2$. Since $f \in L^2(0, \infty ;L^2(\Omega )^2)$ we find from the same type of bootstrapping argument as in [10] that V is also an $L^2$-solution satisfying (B12). Since $D(\tilde{A}_{\mu _*, 2}) \hookrightarrow X_q$, we can extend the solution V globally in $X_q$ and find that V satisfies (B11). We proved the theorem. $\square$