The Galerkin approximation to the coupled Eqs. (
1)–(
5) reads as follows
$$\begin{aligned} \begin{aligned}&\frac{\partial \mathbf{u}_m^a}{\partial t} +P_m\left[ (\mathbf{u}^a_m\cdot \nabla )\mathbf{u}^a_m\right] +\frac{1}{Ro^a}\mathbf{u}^{a\bot }_m +\frac{1}{Ro^a}\nabla \theta ^a_m = \frac{1}{Re^a}\triangle \mathbf{u}^a_m,\\&\frac{\partial \tilde{Fr}^a\theta ^a_m}{\partial t} +\frac{1}{Ro^a}{div}(\mathbf{u}^a_m) = - \gamma \theta ^o_m +\frac{1}{Pe^a}\triangle \theta ^a_m,\\&\frac{\partial \mathbf{u}_m^o}{\partial t} +P_m[(\mathbf{u}_m^o\cdot \nabla )\mathbf{u}_m^o] +\frac{1}{Ro^o}\mathbf{u}_m^{o\bot } +\frac{1}{Ro^o}\nabla p_m^o = \sigma \mathbf{u}^a_m +\frac{1}{Re^o}\triangle \mathbf{u}^o_m,\\&\frac{\partial \theta _m^o}{\partial t} +P_m\left[ (\mathbf{u}_m^o\cdot \nabla )\theta _m^o\right] =\frac{1}{Pe^o}\triangle \theta _m^o, \end{aligned} \end{aligned}$$
(28)
here the Galerkin projections
\(\mathbf{u}_m^a,\theta _m^a\) for the atmospheric component are given as truncation of the expansions (
18)
$$\begin{aligned} \begin{aligned}&\mathbf{u}_m^a(x,t):=P_m\mathbf{u}^a(x,t):=\sum _{n\in \{\mathbb {Z}^2{\setminus } \{ 0\}, |n|\le m\}} \hat{\mathbf{u}}^a_n(t) w_n(x)\\ \text { and }&\ \theta _m^a(x,t):=P_m\theta ^a(x,t):=\sum _{n\in \{\mathbb {Z}^2{\setminus } \{ 0\}, |n|\le m\}} \hat{\theta }_n(t) w_n(x), \end{aligned} \end{aligned}$$
(29)
with
\(w_n\) given by (
4). The Galerkin system (
28) has periodic boundary conditions and its initial condition is given by
$$\begin{aligned} \mathbf{u}^a_m(t_0)=\mathbf{u}^a_0,\quad \theta ^a_m(t_0)=\theta ^a_0,\quad U^o_m(t_0)=\mathbf{u}^o_0,\quad \theta ^o_m(t_0)=\theta ^o_0. \end{aligned}$$
(30)
For the oceanic component an analogous expansion holds. The corresponding operator equation for the state vector
\(\psi _m:=(\mathbf{u}_m^a,\theta _m^a,\mathbf{u}_m^o,\theta _m^o)\) is obtained by replacing
\(\psi \) in (
6) by
\(\psi _m\), with initial condition
\(\psi (t_0)=(\mathbf{u}^a_m(t_0), \theta ^a_m(t_0), \mathbf{u}^o_m(t_0), \theta ^o_m(t_0))\).
Step 2:\(H^s\)-
Estimate Applying the derivative operator
\(\mathcal {D}^\alpha \) to the Galerkin system (
28) yields for the atmospheric component
$$\begin{aligned} \begin{aligned}&\frac{\partial \mathcal {D}^\alpha \mathbf{u}^a_m}{\partial t}+(\mathbf{u}^a_m\cdot \nabla )\mathcal {D}^\alpha \mathbf{u}^a_m +\frac{1}{Ro^a}\mathcal {D}^\alpha \mathbf{u}^{a\bot }_m +\frac{1}{Ro^a}\nabla \mathcal {D}^\alpha \theta ^a_m = \frac{1}{Re^a}\triangle \mathcal {D}^\alpha \mathbf{u}^a_m +\mathcal {G}^a_\mathbf{u}(\alpha ),\\&\tilde{Fr}^a\frac{\partial \mathcal {D}^\alpha \theta ^a_m}{\partial t} +\frac{1}{Ro^a}{div}(\mathcal {D}^\alpha \mathbf{u}^a_m) = -\mathcal {D}^\alpha (\gamma \theta ^o_m) +\mathcal {G}^a_\theta (\alpha ) +\frac{1}{Pe^a}\triangle \mathcal {D}^\alpha \theta ^a_m, \end{aligned} \end{aligned}$$
(31)
where the nonlinear terms on the right-hand side are defined by
$$\begin{aligned} \begin{aligned}&\mathcal {G}^a_\mathbf{u}(\alpha ):=\mathbf{u}^a_m\cdot \nabla \mathcal {D}^\alpha \mathbf{u}^a_m-\mathcal {D}^\alpha [\mathbf{u}^a_m\cdot \nabla \mathbf{u}^a_m],\\&\mathcal {G}^a_\theta (\alpha ):=\mathbf{u}^a_m\cdot \mathcal {D}^\alpha \nabla \theta ^a_m-\mathcal {D}^\alpha [\mathbf{u}^a_m\cdot \nabla \theta ^a_m]. \end{aligned} \end{aligned}$$
Taking the
\(L^2\)-inner product of (
31) with
\((\mathcal {D}^\alpha \mathbf{u}^a_m,\mathcal {D}^\alpha \theta ^a_m)\), integrating by parts and adding the two equations yields
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\big \{||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\tilde{Fr}^a||\mathcal {D}^\alpha \theta ^a_m||_{L^2}^2\big \} +\frac{1}{Re^a}||\mathcal {D}^\alpha \nabla \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\frac{1}{Pe^a}||\mathcal {D}^\alpha \nabla \theta ^a_m||_{L^2}^2\\&\quad = -\int _{\varOmega }\mathbf{u}^a_m\cdot \nabla \frac{|\mathcal {D}^\alpha \mathbf{u}^a_m|^2}{2}\, dx -\int _{\varOmega }\frac{1}{Ro^a}\nabla \mathcal {D}^\alpha \theta ^a_m\cdot \mathcal {D}^\alpha \mathbf{u}^a_m\, dx\\&\qquad -\int _{\varOmega }\frac{1}{Ro^a}{div}(\mathcal {D}^\alpha \mathbf{u}^a_m)\cdot \mathcal {D}^\alpha \theta ^a_m\,dx\\&\qquad -\int _\varOmega \mathcal {D}^\alpha (\gamma \theta ^a_m)\cdot \mathcal {D}^\alpha \theta ^o_m\,dx +\int _\varOmega \mathcal {G}^a_\mathbf{u}(\alpha )\cdot \mathcal {D}^\alpha \mathbf{u}^a_m\, dx +\int _\varOmega \mathcal {G}^a_\theta (\alpha )\cdot \mathcal {D}^\alpha \theta ^a_m\, dx, \end{aligned} \end{aligned}$$
(32)
where the Coriolis term vanishes due to
\(\big \langle \mathcal {D}^\alpha \mathbf{u}^{a\bot }_m\mathcal {D}^\alpha \mathbf{u}^a_m\big \rangle _{L^2}=0 \). For the coupling term we derive with the inequalities of Cauchy–Schwarz, Poincaré and with Lemma
2$$\begin{aligned} \begin{aligned}&\left| \int _\varOmega \mathcal {D}^\alpha (\gamma \theta ^a_m)\cdot \mathcal {D}^\alpha \theta ^o_m\,dx\right| \le \int _\varOmega |\gamma \mathcal {D}^\alpha \theta ^a_m\cdot \mathcal {D}^\alpha \theta ^o_m|\, dx +\int _\varOmega |\theta ^a_m\mathcal {D}^\alpha \gamma \cdot \mathcal {D}^\alpha \theta ^o_m|\,dx\\&\quad \le K_sC_P||\gamma ||_{H^{s_\theta ^a}}||\mathcal {D}^\alpha \theta ^o_m||_{L^2}||\mathcal {D}^\alpha \theta ^a_m||_{L^2}. \end{aligned} \end{aligned}$$
(33)
After integrating the second term on the right hand side of (
32) by parts and invoking the periodic boundary condition, this term cancels with the third term on the right hand side and we obtain with (
33) and the Young inequality the following estimate
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\big \{||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\tilde{Fr}^a||\mathcal {D}^\alpha \theta ^a_m||_{L^2}^2\big \} +\frac{1}{Re^a}||\mathcal {D}^\alpha \nabla \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\frac{1}{Pe^a}||\mathcal {D}^\alpha \nabla \theta ^a_m||_{L^2}^2\\&\quad \le |\int _{\varOmega }div(\mathbf{u}^a_m)\frac{|\mathcal {D}^\alpha \mathbf{u}^a_m|^2}{2}\, dx| +\frac{1}{2Pe^a}||\mathcal {D}^\alpha \theta ^a_m||_{L^2}^2 +\frac{K_s^2C_P^2Pe^a||\gamma ||_{H^{s_\theta ^a}}^2}{2}||\mathcal {D}^\alpha \theta ^o_m||_{L^2}^2\\&\qquad +||\mathcal {G}^a_\mathbf{u}(\alpha )||_{\mathbf{L}^2}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2} +||\mathcal {G}^a_\theta (\alpha )||_{L^2}||\mathcal {D}^\alpha \theta ^a_m||_{L^2}. \end{aligned} \end{aligned}$$
(34)
For the nonlinear forcing terms we obtain with Lemma
1 for all
\(\alpha \in \mathbb {Z}^2_+\) with
\(|\alpha |\le s_{u^a}\)$$\begin{aligned} \begin{aligned} ||\mathcal {G}^a_\mathbf{u}(\alpha )||_{\mathbf{L}^2}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2} \le C_s||\nabla \mathbf{u}^a_m||_{\mathbf{L}^\infty }||\mathbf{u}^a_m||_{\mathbf{H}^{s_{u^a}}}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2} , \end{aligned} \end{aligned}$$
(35)
and
$$\begin{aligned} \begin{aligned}&||\mathcal {G}^a_\theta (\alpha )||_{L^2}||\mathcal {D}^\alpha \theta ^a_m||_{L^2}\\&\quad \le C_s(||\nabla \mathbf{u}^a_m||_{\mathbf{L}^\infty }||\mathcal {D}^{s-1}\nabla \theta ^a_m||_{L^2} +||\nabla \theta ^a_m||_{L^\infty }||\mathcal {D}^s\mathbf{u}^a_m||_{\mathbf{L}^2})||\mathcal {D}^\alpha \theta ^a_m||_{L^2}. \end{aligned} \end{aligned}$$
(36)
From (
35), (
36) we obtain for (
34)
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\big \{||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\tilde{Fr}^a||\mathcal {D}^\alpha \theta ^a_m||_{L^2}^2\big \} +\frac{1}{Re^a}||\mathcal {D}^\alpha \nabla \mathbf{u}^a_m||_{\mathbf{L}^2}^2+\frac{1}{Pe^a}||\mathcal {D}^\alpha \nabla \theta ^a_m||_{L^2}^2\\&\quad \le 2||{div}(u^a_m)||_{\mathbf{L}^\infty }||\mathcal {D}^\alpha u^a_m||_{\mathbf{L}^2}^2 +K_s^2C_P^2Pe^a||\gamma ||_{H^{s_\theta ^a}}^2||\mathcal {D}^\alpha \theta ^o_m||_{L^2}^2\\&\qquad + 2C_s\big ( ||\nabla \mathbf{u}^a_m||_{\mathbf{L}^\infty }+||\nabla \theta ^a_m||_{L^\infty }\big ) \big ( ||\mathcal {D}^s\mathbf{u}^a_m||_{\mathbf{L}^2}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2} +||\mathcal {D}^s\theta ^a_m||_{\mathbf{L}^2}||\mathcal {D}^\alpha \theta ^a_m||_{\mathbf{L}^2}\\&\qquad +||\mathcal {D}^s\mathbf{u}^a_m||_{\mathbf{L}^2}||\mathcal {D}^\alpha \theta ^a_m||_{\mathbf{L}^2} \big ). \end{aligned} \end{aligned}$$
(37)
We sum over all derivatives
\(\mathcal {D}^\alpha \) such that the degree
\(|\alpha |\) of any derivative is less or equal to the degree of the corresponding
\(\mathbf{s}\)-component. This implies with the Young inequality the following upper bound for the atmospheric state
\(\psi ^a=(\mathbf{u}^a,\theta ^a)\)$$\begin{aligned} \begin{aligned} \frac{d}{dt}||\psi ^a_m||_{\mathcal {H}^{\mathbf{s}}}^2 +\frac{1}{Re^a}||\nabla \mathbf{u}^a_m||_{\mathbf{H}^{s_u^a}}^2+\frac{1}{Pe^a}||\nabla \theta ^a_m||_{H^{s_\theta ^a}}^2&\le C||div\psi ^a_m||_{\mathcal {L}^\infty } ||\psi ^a_m||_{\mathcal {H}^{\mathbf{s}}}^2\\&\quad +CK_s^2C_P^2Pe^a||\gamma ||_{H^{s_\theta ^a}}^2||\theta ^o_m||_{H^s}^2. \end{aligned} \end{aligned}$$
(38)
For the oceanic state we proceed analogously to the atmosphere. After applying
\(\mathcal {D}^\alpha \) to (
28) we take the
\(L^2\)-inner product with
\((\mathcal {D}^\alpha \mathbf{u}^o_m,\mathcal {D}^\alpha \theta ^o_m)\), integrate by parts and add the two equations. This yields
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\big \{||\mathcal {D}^\alpha \mathbf{u}^o_m||_{\mathbf{L}^2}^2+||\mathcal {D}^\alpha \theta ^o_m||_{L^2}^2\big \} +\frac{1}{Re^o}||\mathcal {D}^\alpha \nabla \mathbf{u}^o_m||_{\mathbf{L}^2}^2+\frac{1}{Pe^o}||\mathcal {D}^\alpha \nabla \theta ^o_m||_{L^2}^2\\&\quad = -\int _{\varOmega }\mathbf{u}^o_m\cdot \nabla \frac{|\mathcal {D}^\alpha \mathbf{u}^o_m|^2}{2}\, dx -\int _{\varOmega }\mathbf{u}^o_m\cdot \nabla \frac{|\mathcal {D}^\alpha \theta ^o_m|^2}{2}\,dx +\int _{\varOmega }\sigma \mathcal {D}^\alpha \mathbf{u}^a_m\cdot \mathcal {D}^\alpha \mathbf{u}^o_m\, dx \\&\qquad +\int _\varOmega \mathcal {G}^o_\mathbf{u}(\alpha )\cdot \mathcal {D}^\alpha \mathbf{u}^a_m\, dx +\int _\varOmega \mathcal {G}^a_\theta (\alpha )\cdot \mathcal {D}^\alpha \theta ^a_m\, dx\\&\quad \le \int _\varOmega \mathcal {G}^o_\mathbf{u}(\alpha )\cdot \mathcal {D}^\alpha \mathbf{u}^a_m\, dx +\int _\varOmega \mathcal {G}^a_\theta (\alpha )\cdot \mathcal {D}^\alpha \theta ^a_m\, dx +K_sC_P||\sigma ||_{H^{s_u^o}}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{L^2}||\mathcal {D}^\alpha \mathbf{u}^o_m||_{\mathbf{L}^2}, \end{aligned} \end{aligned}$$
(39)
where the pressure and the two gradient terms in the second line vanish after integration by part due to incompressibility and the periodic boundary conditions. The coupling term in the velocity equations has been treated analogously to (
33) with the temperature variable
\(\theta ^a_m\) replaced by the velocity variable
\(\mathbf{u}^o_m\)$$\begin{aligned} \begin{aligned}&\left| \int _\varOmega \mathcal {D}^\alpha (\sigma \mathbf{u}^a_m)\cdot \mathcal {D}^\alpha \mathbf{u}^o_m\,dx\right| \le \int _\varOmega |\sigma \mathcal {D}^\alpha \mathbf{u}^a_m\cdot \mathcal {D}^\alpha \mathbf{u}^o_m|\, dx +\int _\varOmega |\mathbf{u}^a_m\mathcal {D}^\alpha \sigma \cdot \mathcal {D}^\alpha \mathbf{u}^o_m|\,dx\\&\le K_sC_P||\sigma ||_{H^{s_u^o}}||\mathcal {D}^\alpha \mathbf{u}^o_m||_{\mathbf{L}^2}||\mathcal {D}^\alpha \mathbf{u}^a_m||_{\mathbf{L}^2}. \end{aligned} \end{aligned}$$
(40)
With the estimates (
35) and (
36) for
\(\mathcal {G}^o_\mathbf{u}(\alpha )\) and
\(\mathcal {G}^o_\theta (\alpha )\) respectively we arrive analogously to the atmospheric estimate at the following inequality
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}||\psi ^o_m||_{\mathcal {H}^{\mathbf{s}}}^2 +\frac{1}{Re^o}||\nabla \mathbf{u}^o_m||_{\mathbf{H}^{s_u^o}}^2+\frac{1}{Pe^o}||\nabla \theta ^o_m||_{H^{s_\theta ^o}}^2\\&\quad \le C ||div\psi ^o_m||_{\mathcal {L}^\infty }||\psi ^o_m||_{\mathcal {H}^{\mathbf{s}}}^2 +CK_s^2C_P^2 Re^o||\sigma ||_{H^{s_u^o}}^2||\mathbf{u}^a_m||_{\mathbf{H}^{s_u^a}}^2. \end{aligned} \end{aligned}$$
(41)
Adding (
38) and (
41) results in
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 +\frac{1}{R}(||\nabla \mathbf{u}_m^a||_{\mathbf{H}^{s_u^a}}^2+||\nabla \mathbf{u}_m^o||_{\mathbf{H}^{s_u^o}}^2)+\frac{1}{P}(||\nabla \theta _m^a||_{H^{s_\theta ^a}}^2+||\nabla \theta _m^o||_{H^{s_\theta ^o}}^2)\\&\quad \le C||div\psi _m||_{\mathcal {L}^\infty }||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 +CK_s^2C_P^2(||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o)||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2, \end{aligned} \end{aligned}$$
(42)
where
\(\frac{1}{R}:=\min \{\frac{1}{Re^a},\frac{1}{Re^o}\}\),
\(\frac{1}{P}:=\min \{\frac{1}{Pe^a}, \frac{1}{Pe^o}\}\). The inequality is still true if we neglect the positive gradient-terms on the left hand side
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 \le C||div\psi _m||_{\mathcal {L}^\infty } ||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 +CK_s^2C_P^2(||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o)||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2. \end{aligned} \end{aligned}$$
(43)
We now make use of our assumption that all components of the state vector belong to a Sobolev space
\(\mathcal {H}^{\mathbf{s}}\) and that all component of
\(\mathbf{s}\) are greater or equal to 3. This allows to apply Lemma
2 (with
\(s=2,\ k=1\)) to the divergence term in (
42) and it follows
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 \le C_s ||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^3 +CK_s^2C_P^2\left( ||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o\right) ||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2, \end{aligned} \end{aligned}$$
(44)
where the constant
\(C_s\) from Lemma
2 does depend on
\(\mathbf{s}\) but not on
m. Hence
$$\begin{aligned} \begin{aligned}&\frac{d}{dt}||\psi _m||_{\mathcal {H}^{\mathbf{s}}} \le C_s ||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2 +K_s\left( ||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o\right) ||\psi _m||_{\mathcal {H}^{\mathbf{s}}}, \end{aligned} \end{aligned}$$
and with the Young inequality
$$\begin{aligned} \begin{aligned} \frac{d }{dt}||\psi _m||_{\mathcal {H}^{\mathbf{s}}}&\le (C_s+1)||\psi _m||_{\mathcal {H}^{\mathbf{s}}}^2+ M, \end{aligned} \end{aligned}$$
(45)
with
\(M:=CK_s^2C_P^2(||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o)\). Integrating this from
\(t_0\) to
\(t_1\) yields
2$$\begin{aligned} \begin{aligned} \arctan \sqrt{\frac{(C_s+1)}{M}}||\psi _m(t_1)||_{\mathcal {H}^{\mathbf{s}}} - \arctan \sqrt{\frac{(C_s+1)}{M}}||\psi _m(t_0)||_{\mathcal {H}^{\mathbf{s}}} \le \sqrt{(C_s+1)M}. \end{aligned} \end{aligned}$$
(46)
We chose now
\(t_1>t_0\) such that the following condition is satisfied
$$\begin{aligned} \begin{aligned} \sqrt{C_sM}t_1\le \frac{\pi }{2}-\arctan \sqrt{\frac{C_s}{M}}||\psi _0||_{\mathcal {H}^{\mathbf{s}}}. \end{aligned} \end{aligned}$$
(47)
A
\(t_1\) with this property exist, because the right-hand-side of (
47) is positive. From (
46) follows
$$\begin{aligned} \begin{aligned} \sqrt{\frac{(C_s+1)}{M}}||\psi _m(t_1)||_{\mathcal {H}^{\mathbf{s}}} \le \tan \bigg \{\sqrt{\frac{(C_s+1)}{M}}t_1 +\arctan \sqrt{\frac{(C_s+1)}{M}}||\psi _m(t_0)||_{\mathcal {H}^{\mathbf{s}}}\bigg \}. \end{aligned}\end{aligned}$$
(48)
With (
30) follows
$$\begin{aligned} \begin{aligned} \sqrt{\frac{(C_s+1)}{M}}||\psi _m(t_1)||_{\mathcal {H}^{\mathbf{s}}} \le \tan \bigg \{\sqrt{\frac{(C_s+1)}{M}}t_1 +\arctan \sqrt{\frac{(C_s+1)}{M}}||\psi _0||_{\mathcal {H}^{\mathbf{s}}}\bigg \}. \end{aligned} \end{aligned}$$
(49)
Since the upper bound is independent on
m, we have thus proven that the sequence
\((\psi _m)_m\) is uniformly bounded in
\(L^\infty ([t_0,t_1],\mathcal {H}^{\mathbf{s}})\), where the endpoint
\(t_1\) satisfies (
47). It follows that for each
m the solutions
\(\psi _m\) of the Galerkin system do have a joint interval of existence
\([t_0,t_1]\). The boundedness of
\((\psi _m)_m\) establishes the existence of a subsequence
\((\psi _k)_k\) that converges weakly in
\(L^2(T;\mathcal {H}^{\mathbf{s}})\) to a
\(\psi \in L^2(T;\mathcal {H}^{\mathbf{s}})\). According to (
49) it holds that
\(\psi \in L^\infty (T;\mathcal {H}^{\mathbf{s}})\).
The boundedness of
\((\psi _m)_m\) in
\(L^2(T;\mathcal {H}^{\mathbf{s}})\) and of
\((\frac{d\psi _m}{dt})_m\) in
\(L^2(T,\mathcal {L}^2)\) implies with the Aubin compactness theorem (cf. [
10], Lemma 8.2) that a subsequence
\((\psi _{k})_k\) of
\((\psi _m)_m\) exists that converges strongly in
\(L^2(T,\mathcal {L}^2)\) and weakly in
\(L^2(T;\mathcal {H}^{\mathbf{s}})\) to the limit
\(\psi \in L^2(T,\mathcal {H}^{\mathbf{s}})\). We consider the
\((\psi _k-\psi )\), and denote an arbitrary component of this difference by
\((f_k^{(i)}-f^{(i)})\), with
\(i=1\ldots 4\). From Lemma
3 applied to the components
\((f_k^{(i)}-f^{(i)})\) of the difference
\((\psi _k-\psi )\) follows that for all
\(s_i' < s_i\) and
\(t\in T\)$$\begin{aligned} ||f_k^{(i)}(t)-f^{(i)}(t)||_{\mathcal {H}^{{ s_i}'}}\le C_s ||f_k^{(i)}(t)-f^{(i)}(t)||_{\mathcal {L}^2}^{1-s'_i/s_i} ||f_k^{(i)}(t)-f^{(i)}(t)||_{\mathcal {H}^{{s_i}}}^{s_i'/s_i}. \end{aligned}$$
(50)
From (
50) we derive with the convergence of
\((\psi _k)_k\) in
\(L^2(T,\mathcal {L}^2)\) and with the boundedness in
\( L^2(T,\mathcal {H}^{\mathbf{s}})\) the (strong) convergence of
\((\psi _k)_k\) in
\(C(T;\mathcal {H}^{\mathbf{s}'})\) for all
\(\mathbf{s}'<\mathbf{s}\). We now show that
\(\psi \in C(T;\mathcal {H}^\mathbf{s})\). The strong convergence in
\(C(T;\mathcal {H}^{\mathbf{s}'})\) and the density of
\(\mathcal {H}^{-\mathbf{s}'}\) in
\(\mathcal {H}^{-\mathbf{s}}\) for
\(\mathbf{s}'<s\) imply for all
\(\phi \in \mathcal {H}^{-\mathbf{s}'}\) that
$$\begin{aligned} \lim _{k\rightarrow \infty }\big \langle \psi _k(\cdot ,t),\phi \big \rangle _{\mathcal {L}^2}=\big \langle \psi (\cdot ,t),\phi \big \rangle _{\mathcal {L}^2}. \end{aligned}$$
(51)
This proves continuity in the weak sense, i.e.
\(\psi \in C_w(T;\mathcal {H}^\mathbf{s})\). The weak continuity implies for
\(\tau \in [t_0,t_1]\)$$\begin{aligned} \lim _{\tau \rightarrow t_0+}\inf ||\psi (\cdot ,\tau )||_{\mathcal {H}^{\mathbf{s}}}\ge ||\psi _0||_{\mathcal {H}^{\mathbf{s}}}. \end{aligned}$$
(52)
From (
49) follows
$$\begin{aligned} \lim _{\tau \rightarrow t_0+}\sup ||\psi (\cdot ,\tau )||_{\mathcal {H}^{\mathbf{s}}}\le ||\psi _0||_{\mathcal {H}^{\mathbf{s}}}. \end{aligned}$$
(53)
From (
52) and (
53) we obtain continuity of the
\(\mathcal {H}^{\mathbf{s}}\)-norm of the solution at initial time
$$\begin{aligned} \lim _{\tau \rightarrow t_0+}||\psi (\cdot ,\tau )||_{\mathcal {H}^{\mathbf{s}}}= ||\psi _0||_{\mathcal {H}^{\mathbf{s}}}. \end{aligned}$$
(54)
From (
42) we get after integration over
\(T=[t_0,t_1]\)$$\begin{aligned} \begin{aligned}&\int _{t_0}^{t_1}\frac{1}{R}(||\nabla \mathbf{u}_m^a(s)||_{\mathbf{H}^s}^2+||\nabla \mathbf{u}_m^o(s)||_{\mathbf{H}^s}^2) +\frac{1}{P}(||\nabla \theta _m^a(s)||_{H^s}^2||\nabla \theta _m^o(s)||_{H^s}^2)\, ds\\&\quad \le ||\psi _m(t_0)||_{\mathcal {H}^{\mathbf{s}}}^2 + C_s\int _{t_0}^{t_1}||\psi _m(s)||_{\mathbf{H}^s}^2\, ds +K_s^2 (||\gamma ||_{H^{s_\theta ^a}}^2Pe^a+||\sigma ||_{H^{s_u^o}}^2Re^o)^2(t_1-t_0). \end{aligned} \end{aligned}$$
(55)
From (
49) follows that the right hand side of (
55) is bounded independent from
m. This implies that
\(\psi \in L^2(T,\mathcal {H}^{s+1})\). Consequently there exists a set
\(E\subseteq T\) of Lebesgue-measure zero such that for all
\(\tau \in T{\setminus } E\) it holds that
\(\psi (\cdot ,\tau )\in \mathcal {H}^{s+1}\). This implies that for all
\(\delta >0\) there exists a
\(t_0^* < \delta \) such that
\(\psi (\cdot ,t_0^*)\in \mathcal {H}^{s+1}\). If we use
\(\psi _{t_0^*}:=\psi (\cdot ,t_0^*)\) as initial condition we can repeat all the arguments of our proof to establish the existence of a solution
\(\tilde{\psi }\in C([t_0^*,t_1^*],\mathcal {H}^{\mathbf{s}^*})\), with
\(s^*<s+1\). The two solutions
\(\psi ,\tilde{\psi }\) coincide on their joint interval of existence
\([t_0,t_1]\cap [t_0^*,t_1^*]\). We obviously have for the two endpoints
\(t_1\le t_1^*\) and hence
\(\psi ,\tilde{\psi }\) coincide on
\([t_0^*,t_1]\). Since
\(\delta >0\) was arbitrary we have
\(\psi \in C((t_0,t_1], \mathcal {H}^{\mathbf{s}})\) and combined with the continuity at
\(t_0\) (see (
54)) it follows that
\(\psi \in C([t_0,t_1], \mathcal {H}^{\mathbf{s}})\).