The Global Existence and Averaging Theorem for the Strong Solution of the Stochastic Boussinesq Equations with the Low Froude Number

Abstract

Considering the stochastic Boussinesq equations with additive noise, we prove the global existence for the strong solution in the case of strong stable stratification. The averaging theorem is also established.

A The Proof of Lemma 5.1

Proof

First we proof the operator ${\mathcal {B}}^\varepsilon$ and ${\mathcal {B}}^0$ satisfy “cancelation property”. From Sect. 2, we know that the nonzero eigenvalue of ${\mathbf {P}}{\mathcal {A}}$ is pure imaginary, hence ${\mathcal {L}}^*(t)={\mathcal {L}}(-t)$. Then

$$\begin{aligned} \big ({\mathcal {B}}^{\varepsilon }(U,U),U\big )_{L^2}&=\big ({\mathcal {L}}\left( t/\varepsilon \right) {\mathbf {P}}\left( {\mathcal {L}}\left( -t/\varepsilon \right) U\cdot \nabla {\mathcal {L}}\left( -t/\varepsilon \right) U\right) ,U\big )_{L^2}\nonumber \\&=\big (\left( {\mathcal {L}}\left( -t/\varepsilon \right) U\cdot \nabla {\mathcal {L}}\left( -t/\varepsilon \right) U\right) ,{\mathcal {L}}\left( -t/\varepsilon \right) U\big )_{L^2}\nonumber \\&=\frac{1}{2}\left( \mathrm {div}\left( {\mathcal {L}}\left( -t/\varepsilon \right) U\right) ,\left| {\mathcal {L}}\left( -t/\varepsilon \right) U\right| ^2\right) _{L^2}\nonumber \\&=0. \end{aligned}$$

Now, we begin to consider the operator ${\mathcal {B}}$. By the definition of ${\mathcal {B}}$, we mention that

$$\begin{aligned} \big ({\mathcal {B}}(U,U),U\big )_{L^2}&=\sum _{n}\sum _{\begin{array}{c} \lambda _{k,m,n}^{a,b,c}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}\sum _{l=1}^4\sum _{j=1}^3 U(k)\varphi _k^{a,j}m_j U(m)\varphi _m^{b,l}{\bar{U}}(n){\bar{\varphi }}_n^{c,l}\nonumber \\&=\sum _{n}\sum _{\begin{array}{c} \lambda _{k,m,n}^{a,b,c}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}\sum _{l=1}^4\sum _{j=1}^3 U(k)\varphi _k^{a,j}n_j U(m)\varphi _m^{b,l}{\bar{U}}(n){\bar{\varphi }}_n^{c,l}\nonumber \\&=-\sum _{n}\sum _{\begin{array}{c} \lambda _{k,m,n}^{a,b,c}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}\sum _{l=1}^4\sum _{j=1}^3 U(-k)\varphi _{-k}^{a,j}n_j U(n)\varphi _n^{c,l}{\bar{U}}(m){\bar{\varphi }}_m^{b,l}\nonumber \\&=-\sum _{n}\sum _{\begin{array}{c} \lambda _{-k,n,m}^{a,c,b}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}\sum _{l=1}^4\sum _{j=1}^3 U(k)\varphi _k^{a,j}m_j U(m)\varphi _m^{b,l}{\bar{U}}(n){\bar{\varphi }}_n^{c,l}. \end{aligned}$$

We note that

$$\begin{aligned} \sum _{\begin{array}{c} \lambda _{-k,n,m}^{a,c,b}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}U(k)\varphi _k^{a,j}m_j U(m)\varphi _m^{b,l}{\bar{U}}(n){\bar{\varphi }}_n^{c,l}= \sum _{\begin{array}{c} \lambda _{k,m,n}^{a,b,c}=0\\ k+m=n\\ a,b,c\in \{0,\pm \} \end{array}}U(k)\varphi _k^{a,j}m_j U(m)\varphi _m^{b,l}{\bar{U}}(n){\bar{\varphi }}_n^{c,l}. \end{aligned}$$

For example,

$$\begin{aligned} \lambda _{-k,n,m}^{a,c,b}{\mathbf {1}}_{(b,c)= (+,-)}&=\lambda ^a(-k)+\lambda ^-(n)-\lambda ^+(m)\\&=\lambda ^a(k)+\lambda ^-(m)-\lambda ^+(n)\\&=\lambda _{k,m,n}^{a,b,c}{\mathbf {1}}_{(b,c)= (-,+)}. \end{aligned}$$

The other cases can be verify similarly. Hence

$$\begin{aligned} \big ({\mathcal {B}}^0(U,U),U\big )_{L^2}=-\big ({\mathcal {B}}^0(U,U),U\big )_{L^2}=0. \end{aligned}$$

Combine the It’o formula for $\Vert W\Vert _{L^2}^p$ and the equation for W of (4.1), for $p\ge 2$, we find

$$\begin{aligned}&\mathrm {d}\Vert W^\varepsilon \Vert _{L^2}^p+\Vert W\Vert _{L^2}^{p-2}\Vert \nabla W^\varepsilon \Vert _{L^2}^2\mathrm {d}t\nonumber \\&\quad =\frac{p}{2}\Vert W^\varepsilon \Vert _{L^2}^{p-2}\Vert {\mathcal {Q}}^\varepsilon \Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t+\frac{p(p-2)}{2}\Vert W^\varepsilon \Vert _{L^2}^2\Vert ({\mathcal {Q}}^\varepsilon ,W)_{L^2}\Vert _{L_2({\mathfrak {U}};{\mathbb {R}})}^{p-4}\mathrm {d}t\nonumber \\&\qquad +\Vert W\Vert _{L^2}^{p-2}(W,{\mathcal {Q}}^\varepsilon \mathrm {d}{\mathcal {W}})\nonumber \\&\quad =:H_1\mathrm {d}t+H_2\mathrm {d}{\mathcal {W}}. \end{aligned}$$

(A1)

We bounded $H_1$ as

$$\begin{aligned} \int _0^T H_1\mathrm {d}t&\le \frac{p(p-1)}{2}\int _0^T\Vert W^\varepsilon \Vert _{L^2}^{p-2}\Vert {\mathcal {Q}}^\varepsilon \Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\nonumber \\&\le C\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^{p-2}\int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\nonumber \\&\le \frac{1}{4}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^p+C\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^{p/2}. \end{aligned}$$

(A2)

Applying the Burkholder–Davis–Gundy inequality (2.4), we have

$$\begin{aligned} {\mathbb {E}}\sup \limits _{t\in [0,T]}\int _0^t H_2\mathrm {d}{\mathcal {W}}&\le C{\mathbb {E}}\left( \int _0^T\Vert W^\varepsilon \Vert _{L^2}^{2p-2}\Vert {\mathcal {Q}}^\varepsilon \Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^{1/2}\nonumber \\&\le C{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^{p-1}\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^{1/2}\nonumber \\&\le \frac{1}{4}{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^p+C{\mathbb {E}}\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^{p/2}. \end{aligned}$$

(A3)

Insert (A2) and (A3) into (A1), we get

$$\begin{aligned}&{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^p+{\mathbb {E}}\int _0^T\Vert W^\varepsilon \Vert _{L^2}^{p-2}\Vert \nabla W^\varepsilon \Vert _{L^2}^2\mathrm {d}t\\&\quad \le C{\mathbb {E}}\Vert U_0\Vert _{L^2}^p+C{\mathbb {E}}\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^{p/2}. \end{aligned}$$

Taking $p=4$ and $p=16$, respectively, we obtain

$$\begin{aligned} {\mathbb {E}}(\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^4+\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^{16})\le C. \end{aligned}$$

(A4)

Using (A1) for $p=2$, we obtain

$$\begin{aligned} {\mathbb {E}}\left( \int _0^T\Vert \nabla W^\varepsilon \Vert _{L^2}^2\mathrm {d}r\right) ^8&\le C{\mathbb {E}}\left( \int _0^T\Vert {\mathcal {Q}}^\varepsilon \Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^8+C{\mathbb {E}}\left( \int _0^T(W^\varepsilon ,{\mathcal {Q}}^\varepsilon \mathrm {d}{\mathcal {W}})_{L^2}\mathrm {d}t\right) ^8\nonumber \\&\le C{\mathbb {E}}\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^8+C{\mathbb {E}}\left( \int _0^T\Vert W^\varepsilon \Vert _{L^2}^2\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^4\nonumber \\&\le C+C{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^8\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^4\nonumber \\&\le C+C{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^{16}{\mathbb {E}}\left( \int _0^T\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^2\mathrm {d}t\right) ^8\nonumber \\ \quad&\le C, \end{aligned}$$

(A5)

where the Burkholder–Davis–Gundy inequality (2.4) is used in second line. For each $\phi \in C^\infty$, we apply It’o formula for $(W^\varepsilon ,\phi ){L^2}$ and obtain

$$\begin{aligned} \mathrm {d}(W^\varepsilon ,\phi )_{L^2}&=(W^\varepsilon ,\Delta \phi )_{L^2}\mathrm {d}t+({\mathcal {B}}^0(W^\varepsilon ,W^\varepsilon ),\phi )_{L^2}\mathrm {d}t+(\phi ,{\mathcal {Q}}^\varepsilon \mathrm {d}{\mathcal {W}})_{L^2}\nonumber \\ :&=J_1\mathrm {d}t+J_2\mathrm {d}{\mathcal {W}}. \end{aligned}$$

(A6)

We bounded $J_1$ as

$$\begin{aligned}&\int _{t_1}^{t_2}\left( (W^\varepsilon ,\Delta \phi )_{L^2}+({\mathcal {B}}^0(W^\varepsilon ,W^\varepsilon ),\phi )_{L^2}\right) \mathrm {d}r\nonumber \\&\quad \le C\int _{t_1}^{t_2}(\Vert W^\varepsilon \Vert _{L^2}\Vert \Delta \phi \Vert _{L^2}+\Vert \nabla \phi \Vert _{L^2}\Vert W^\varepsilon \Vert _{L^2}^{1/2} \Vert W^\varepsilon \Vert _{H^1}^{3/2})\mathrm {d}r\nonumber \\&\quad \le C\int _{t_1}^{t_2}\Vert W^\varepsilon \Vert _{L^2}^2\mathrm {d}r\int _{t_1}^{t_2}(1+\Vert W^\varepsilon \Vert _{H^1}^2)\mathrm {d}r\nonumber \\&\quad \le C(t_2-t_1)\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^2\int _{t_1}^{t_2}(1+\Vert W^\varepsilon \Vert _{H^1}^2)\mathrm {d}r. \end{aligned}$$

(A7)

Using the Burkholder–Davis–Gundy inequality (2.4) again, we obtain

$$\begin{aligned} {\mathbb {E}}\left( \int _{t_1}^{t_2}(\phi ,{\mathcal {Q}}^\varepsilon \mathrm {d}{\mathcal {W}})_{L^2}\right) ^4&\le C{\mathbb {E}}\left( \int _{t_1}^{t_2}\Vert \phi \Vert _{L^2}^2\Vert {\mathcal {Q}}^\varepsilon \Vert _{L_2({\mathfrak {U}};L^2)}^2\right) ^2\nonumber \\&\le C(t_2-t_1)^2\Vert \phi \Vert _{L^2}^4\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^4. \end{aligned}$$

(A8)

Hence, combine (A6)–(A8), and taking advantage of the bound (A4) and (A5), we have

$$\begin{aligned}&{\mathbb {E}}[|(W^\varepsilon (t_2),\phi )_{L^2}-(W^\varepsilon (t_1),\phi )_{L^2}|^4]\nonumber \\&\quad \le C(t_2-t_1)^4{\mathbb {E}}\left[ \sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^8\left( \int _{t_1}^{t_2}(1+\Vert W^\varepsilon \Vert _{H^1}^2)\mathrm {d}r\right) ^4\right] + C(t-s)^2\Vert \phi \Vert _{L^2}^4\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^4\nonumber \\&\quad \le C(t_2-t_1)^4{\mathbb {E}}\sup \limits _{t\in [0,T]}\Vert W^\varepsilon \Vert _{L^2}^{16}{\mathbb {E}}\left( \int _{t_1}^{t_2}(1+\Vert W^\varepsilon \Vert _{H^1}^2)\mathrm {d}r\right) ^8+ C(t-s)^2\Vert \phi \Vert _{L^2}^4\Vert {\mathcal {Q}}\Vert _{L_2({\mathfrak {U}};L^2)}^4\nonumber \\&\quad \le C(t_2-t_1)^2. \end{aligned}$$

The proof for $U^\varepsilon$ is as same as $W^\varepsilon$, we omit it. $\square$