A class of large solution of 2D tropical climate model without thermal diffusion

TL;DR

This paper constructs global smooth solutions for a 2D tropical climate model without thermal diffusion, even with arbitrarily large initial data, advancing understanding of such models' solution behavior.

Contribution

It introduces a novel method to establish global solutions for the 2D tropical climate model without thermal diffusion with large initial data.

Findings

Global smooth solutions exist despite large initial data

Construction of solutions relies on special initial data

Advances understanding of 2D tropical climate models without thermal diffusion

Abstract

In this paper, we consider the Cauchy problem of 2D tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data whose $L^{\infty}$ norm can be arbitrarily large.

Equations

$$\displaystyle\left\{\begin{array}[]{ll}\partial_{t}u+u\cdot\nabla u+\mu u+\nabla p+\mathord{{\rm div}\ }(v\otimes v)=0,&x\in\mathbb{R}^{2},t>0,\\ \partial_{t}v+u\cdot\nabla v+v\cdot\nabla u+\nabla\theta-\nu\Delta v=0,&x\in\mathbb{R}^{2},t>0,\\ \partial_{t}\theta+u\cdot\nabla\theta+\mathord{{\rm div}\ }v=0,&x\in\mathbb{R}^{2},t\geq 0,\\ \mathord{{\rm div}\ }u=0,&x\in\mathbb{R}^{2},t\geq 0,\\ (u,v,\theta)|_{t=0}=(u_{0},v_{0},\theta_{0}),&x\in\mathbb{R}^{2}.\end{array}\right.$$

$$A(t)\triangleq ||(w,c,\theta )(t)|{|}_{H^3}^2,B(t)\triangleq ||(w,\nabla c)(t)|{|}_{H^3}^2+||\nabla \theta (t)|{|}_{H^2}^2\text{mbox}and{A}_0\triangleq ||w_0,c_0,{\theta }_0|{|}_{H^3}^2.$$

$$u_0=U_0+w_0\text{mbox}and{v}_0=V_0+c_0$$

$$U_0=\left(\begin{array}{c}{\partial }_2{a}_0\\ -{\partial }_1{a}_0\end{array}\right)\text{mbox}and{V}_0=\left(\begin{array}{c}{m}_0\\ m_0\end{array}\right)$$

$$\displaystyle\mathrm{supp}\ \hat{a}_{0}(\xi),\mathrm{supp}\ \hat{m}_{0}(\xi)\subset\mathcal{C}\triangleq\Big{\{}\xi\in\mathbb{R}^{2}:\ |\xi_{1}+\xi_{2}|\leq\varepsilon,\ 1\leq|\xi|\leq 2\Big{\}},\quad\varepsilon>0.$$

$$\displaystyle\Big{(}A_{0}+\varepsilon\big{(}||a_{0}||_{L^{2}}||\hat{a}_{0}||_{L^{1}}+||m_{0}||_{L^{2}}(1+||\hat{m}_{0}||_{L^{1}})\big{)}\Big{)}\exp\Big{\{}C\varepsilon\big{(}||a_{0}||_{L^{2}}||\hat{a}_{0}||_{L^{1}}+||m_{0}||_{L^{2}}(1+||\hat{m}_{0}||_{L^{1}})\big{)}$$

$$\displaystyle\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+C\big{(}||\hat{a}_{0},\hat{m}_{0}||_{L^{1}}+||\hat{a}_{0},\hat{m}_{0}||^{2}_{L^{1}}\big{)}\Big{\}}\leq\varepsilon_{0},$$

$$\mathrm{supp}\hat{\chi }\in \mathcal{C},\hat{\chi }(\xi )\in [0,1]\text{mbox}and\hat{\chi }(\xi )=1\text{mbox}for\xi \in \tilde{\mathcal{C}},$$

$$\displaystyle\mathcal{\widetilde{C}}\triangleq\Big{\{}\xi\in\mathbb{R}^{2}:\ |\xi_{1}+\xi_{2}|\leq\frac{\varepsilon}{2},\ \frac{4}{3}\leq|\xi|\leq\frac{5}{3}\Big{\}}.$$

$$\displaystyle C\varepsilon^{\frac{1}{2}}\Big{(}\log\log\frac{1}{\varepsilon}\Big{)}\exp\Big{\{}C\log\log\frac{1}{\varepsilon}\Big{\}}.$$

$$\displaystyle||\hat{a}_{0}||_{L^{1}}\approx\Big{(}\log\log\frac{1}{\varepsilon}\Big{)}^{\frac{1}{2}}\quad\mbox{and}\quad||{a}_{0}||_{L^{2}}\approx\varepsilon^{-\frac{1}{2}}\Big{(}\log\log\frac{1}{\varepsilon}\Big{)}^{\frac{1}{2}}.$$

$$\displaystyle||u_{0}||_{L^{\infty}}\gtrsim||\Delta a_{0}||_{L^{\infty}}\gtrsim||\hat{a}_{0}||_{L^{1}}\gtrsim\Big{(}\log\log\frac{1}{\varepsilon}\Big{)}^{\frac{1}{2}}$$

$$\displaystyle||v_{0}||_{L^{\infty}}\approx||m_{0}||_{L^{\infty}}\gtrsim||\hat{m}_{0}||_{L^{1}}\gtrsim\Big{(}\log\log\frac{1}{\varepsilon}\Big{)}^{\frac{1}{2}}.$$

$$\displaystyle\left\{\begin{array}[]{ll}\partial_{t}a+a=0,\\ \partial_{t}m-\Delta m=0,\\ (a,m)|_{t=0}=(a_{0},m_{0}).\end{array}\right.$$

$$U=\left(\begin{array}{c}{\partial }_{2}a\\ -{\partial }_{1}a\end{array}\right)\text{mbox}andV=\left(\begin{array}{c}m\\ m\end{array}\right),$$

$$\displaystyle\left\{\begin{array}[]{ll}\partial_{t}U+U=0,\\ \partial_{t}V-\Delta V=0,\\ \mathord{{\rm div}\ }U=0,\\ (U,V)|_{t=0}=(U_{0},V_{0}).\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}\partial_{t}w+w\cdot\nabla w+U\cdot\nabla w+c\cdot\nabla c+c\mathord{{\rm div}\ }c+w+\nabla p=f\\ \qquad\qquad-c\cdot\nabla V-c\mathord{{\rm div}\ }V-V\cdot\nabla c-V\mathord{{\rm div}\ }c-w\cdot\nabla U,\\ \partial_{t}c+w\cdot\nabla c+U\cdot\nabla c-\Delta c+\nabla\theta=g-c\cdot\nabla w-V\cdot\nabla w-c\cdot\nabla U-w\cdot\nabla V,\\ \partial_{t}\theta+w\cdot\nabla\theta+U\cdot\nabla\theta+\mathord{{\rm div}\ }c=h,\\ \mathord{{\rm div}\ }w=0,\\ (w,c,\theta)|_{t=0}=(w_{0},c_{0},\theta_{0}).\end{array}\right.$$

$$f=-U\cdot \nabla U-V\cdot \nabla V-V\mathrm{div}V,g=-U\cdot \nabla V-V\cdot \nabla U,h=-\mathrm{div}V.$$

$$\hat{f}(\xi )\triangleq {\int }_{{\mathbb{R}}^2}{e}^{-ix\cdot \xi }f(x)dx.$$

$$||f||_{H^{s}(\mathbb{R}^{2})}\triangleq\Big{(}\sum_{|\alpha|\leq s}||D^{\alpha}f||^{2}_{L^{2}(\mathbb{R}^{2})}\Big{)}^{\frac{1}{2}}=||f||_{\dot{H}^{s}(\mathbb{R}^{2})}+||f||_{L^{2}(\mathbb{R}^{2})}$$

$$||f|{|}_{W^{s,\infty }({\mathbb{R}}^2)}\triangleq \sum _{0\le i\le s}||{\nabla }^{i}f|{|}_{L^{\infty }({\mathbb{R}}^2)}.$$

$$\sum _{0<|\alpha |\le 3}||[D^{\alpha },\mathbf{f}]\mathbf{g}|{|}_{L^2}\le C(||\mathbf{g}|{|}_{H^2}||\nabla \mathbf{f}|{|}_{L^{\infty }}+||\mathbf{g}|{|}_{L^{\infty }}||\mathbf{f}|{|}_{H^3}),$$

$$\sum _{0<|\alpha |\le 3}||[D^{\alpha },\mathbf{f}]\mathbf{g}|{|}_{L^2}\le C(||\nabla \mathbf{f}|{|}_{L^{\infty }}+||{\nabla }^3\mathbf{f}|{|}_{L^{\infty }})||\mathbf{g}|{|}_{H^2}.$$

$$||\mathbf{fg}|{|}_{H^m}\le C||\mathbf{f}|{|}_{H^m}||\mathbf{g}|{|}_{H^m},$$

$$\displaystyle||\mathbf{f}\mathbf{g}||_{H^{m}}\leq C\big{(}||\mathbf{f}||_{L^{\infty}}+||\nabla^{m}\mathbf{f}||_{L^{\infty}}\big{)}||\mathbf{g}||_{H^{m}}.$$

$$\displaystyle E(t)\triangleq||f,g,h||_{H^{3}}\leq Ce^{-t}\varepsilon\big{(}||a_{0}||_{L^{2}}||\hat{a}_{0}||_{L^{1}}+||m_{0}||_{L^{2}}||\hat{m}_{0}||_{L^{1}}+||m_{0}||_{L^{2}}\big{)}$$

$$||U,V|{|}_{W^{M,\infty }}\le C{e}^{-t}||{\hat{a}}_0,{\hat{m}}_0|{|}_{L^1}.$$

$$||h|{|}_{H^M}=||({\partial }_1+{\partial }_2)m|{|}_{H^M}\le C{e}^{-t}\epsilon ||m_0|{|}_{L^2}.$$

$$f^1$$

$$={\partial }_{1}a{\partial }_2{\partial }_{2}a-{\partial }_{2}a{\partial }_1{\partial }_{2}a-2m({\partial }_1+{\partial }_2)m$$

$$=({\partial }_1+{\partial }_2)a{\partial }_2{\partial }_{2}a-{\partial }_{2}a{\partial }_2({\partial }_1+{\partial }_2)a-2m({\partial }_1+{\partial }_2)m,$$

$$f^2$$

$$=-{\partial }_{1}a{\partial }_2{\partial }_{1}a+{\partial }_{2}a{\partial }_1{\partial }_{1}a-2m({\partial }_1+{\partial }_2)m$$

$$=-({\partial }_1+{\partial }_2)a{\partial }_1{\partial }_{2}a+{\partial }_{2}a{\partial }_1({\partial }_1+{\partial }_2)a-2m({\partial }_1+{\partial }_2)m,$$

Full text

A class of large solution of 2D tropical climate model without thermal diffusion

Jinlu Li1111E-mail: lijinlu@gnnu.cn, Yanghai Yu2222E-mail: yuyanghai214@sina.com(Corresponding author)

1*School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China

2School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241002, China*

Abstract: In this paper, we consider the Cauchy problem of 2D tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data whose $L^{\infty}$ norm can be arbitrarily large.

Keywords: Tropical Climate Model; Global Large Solution.

MSC (2010): 35D35; 76D03

1 Introduction

This paper focuses on the following 2D tropical climate model (TCM) given by

[TABLE]

here $\mu,\nu$ are non-negative parameters, $u=u(t,x)$ and $v=v(t,x)$ stand for the barotropic mode and the first baroclinic mode of the vector velocity, respectively. $p=p(t,x)$ and $\theta=\theta(t,x)$ denote the scalar pressure and scalar temperature, respectively.

In the completely inviscid case, namely, $\mu=\nu=0$, (1.6) was first derived by Frierson–Majda–Pauluis [4] by performing a Galerkin truncation to the hydrostatic Boussinesq equations, of which the first baroclinic mode had been originally used in some studies of tropical atmosphere. More relevant background on the tropical climate model can be found in [5, 11, 12] and the references therein. The mathematical studies on (1.6) have attracted considerable attention recently from various authors and have motivated a large number of research papers concerning the local well-posedness [13], global small solutions, global regularity and so on. Li–Titi [6] introduced a new quantity to bypass the obstacle caused by the absence of thermal diffusion and proved the global well-posedness of the 2D TCM (1.6) with two Laplacian terms $\Delta u$ and $\Delta v$. From the mathematical point of view, adding some fractional dissipation terms does bring the regularity for the system (1.6) and also significantly change the model properties and physics. Concerning the global regularity of TCM with fractional dissipation terms, we refer to [1, 2, 3, 16, 17] and the references therein. However, adding only the damping term $u$ does not appear to be enough to establish the global well-posedness of TCM (1.6). Wan [15] proved the global strong solution to TCM (1.6) with additional damping term $v$ under suitable smallness assumptions on the initial data. Later, Ma–Wan [14] removed the damping term $v$ and obtained the global strong solution to TCM (1.6) with different smallness assumptions on the initial data. Naturally, we ask that whether or not there exists a global solution to TCM (1.6) with some classes of large initial value. Motivated by the ideas that used in [7, 8, 9], we give a positive answer in this paper.

For the sake of implicity, we will limit ourselves to $\mu=\nu=1$ since their values do not play any role in our analysis and introduce the following notations

[TABLE]

Our main result is stated as follows.

Theorem 1.1

Assume that the initial data fulfills $\mathord{{\rm div}\ }w_{0}=0$ and

[TABLE]

where

[TABLE]

with

[TABLE]

There exists a sufficiently small positive constant $\varepsilon_{0}$, and a universal constant $C$ such that if

[TABLE]

then the system (1.6) has a unique global solution.

Remark 1.1

By choosing a class of special initial data, we can show that general initial value whose $L^{\infty}$ norms can be arbitrarily large generates a global unique solution to the system (1.6). Here, we construct an example to verify this.

Let $(w_{0},c_{0},\theta_{0})=(0,0,0)$ and a_{0}=m_{0}=\frac{1}{\varepsilon}\big{(}\log\log\frac{1}{\varepsilon}\big{)}^{\frac{1}{2}}\chi, where the smooth function $\chi$ satisfying

[TABLE]

where

[TABLE]

Then, direct calculations show that the left side of (1.1) becomes

[TABLE]

In fact, one has

[TABLE]

Therefore, choosing $\varepsilon$ small enough, we deduce that the system (1.6) has a global solution.

Moreover, we also have

[TABLE]

and

[TABLE]

2 Renormalized System

Let $(a,m)=(e^{-t}a_{0},e^{t\Delta}m_{0})$ be the solutions of the following system

[TABLE]

Setting

[TABLE]

we can deduce from (2.12) that

[TABLE]

Denoting $w=u-U$ and $c=v-V$, the system (1.6) can be written as follows

[TABLE]

where

[TABLE]

3 Useful Tools

Firstly, we introduce some notations and conventions which will be used throughout this paper.

• •

We will use the notation $||f_{1},\cdots,f_{n}||_{X}\triangleq||f_{1}||_{X}+\cdots+||f_{n}||_{X}$ for some Banach space $X$.

• •

Let $\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{N}^{2}$ be a multi-index and $D^{\alpha}=\partial^{|\alpha|}/\partial^{\alpha_{1}}_{x_{1}}\partial^{\alpha_{2}}_{x_{2}}$ with $|\alpha|=\alpha_{1}+\alpha_{2}$.

• •

$\langle f,g\rangle$ denotes the inner product in $L^{2}(\mathbb{R}^{2})$, namely, $\langle f,g\rangle\triangleq\int_{\mathbb{R}^{2}}fg\ \mathrm{d}x$.

• •

The Fourier transform of $f$ with respect to the space variable is given by

[TABLE]

• •

For $s\in\mathbb{N}$, the norm of the integer order Sobolev space $H^{s}(\mathbb{R}^{2})$ and $W^{s,\infty}(\mathbb{R}^{2})$ are defined by

[TABLE]

and

[TABLE]

Next, we present some estimates which will be used in the proof of Theorem 1.1.

Lemma 3.1

[10]** (Commutator estimates) There hold that

[TABLE]

Lemma 3.2

[10]** (Product estimates) For $m\in\mathbb{Z}^{+}$ and $m\geq 2$, we have

[TABLE]

Lemma 3.3

Under the assumptions of Theorem 1.1, for all $M\in\mathbb{Z}^{+}\cup\{0\}$, the following estimates hold

[TABLE]

and

[TABLE]

Proof of Lemma 3.3 Recalling that $h=-(\partial_{1}+\partial_{2})m$ and $m=e^{t\Delta}m_{0}$, then using the condition $\mathrm{supp}\ \hat{m}_{0}(\xi)\subset\mathcal{C}$, one has

[TABLE]

Direct calculations show that

[TABLE]

For all $M\in\mathbb{Z}^{+}\cup\{0\}$, using the fact $||\mathbf{f}||_{L^{\infty}}\leq C||\hat{\mathbf{f}}||_{L^{1}}$, we deduce

[TABLE]

and

[TABLE]

where we have used the conditions $\mathrm{supp}\ \hat{a}_{0}(\xi)\subset\mathcal{C}$ and $\mathrm{supp}\ \hat{m}_{0}(\xi)\subset\mathcal{C}$.

By the product estimate (3.4), then we have

[TABLE]

An argument similar to that used above, we also have

[TABLE]

Combining (3.7), (3) and (3.11) yields the desired result (3.5). (3.6) is just a direct consequence of (3.8) and (3.9). Thus, we end the proof of Lemma 3.3. $\Box$

4 Proof of Theorem 1.1

By standard energy method, we can obtain that there exists a unique solution $(u,v,\theta)$ to (1.6) on some time interval $[0,T^{*})$, where $T^{*}$ is the maximal time of existence of solution $(u,v,\theta)$. It remains to prove $T^{*}=\infty$.

Step 1: The Estimate of $w$.

Applying $D^{\ell}$ to the both sides of Eq.$\eqref{c-tcm}_{1}$, taking the inner product with $D^{\ell}w$ and summing the resulting over $|\alpha|\leq 3$, we get

[TABLE]

where

[TABLE]

Next, we need to estimate the above terms one by one.

For the terms $I_{1}$ and $I_{2}$, notice that the embedding $H^{2}(\mathbb{R}^{2})\hookrightarrow L^{\infty}(\mathbb{R}^{2})$, using the commutate estimate (3.1) and (3.2), respectively, one has

[TABLE]

By the product estimate (3.3), one has

[TABLE]

Using Hölder’s inequality gives

[TABLE]

For the last term $I_{5}$, by product estimate (3.4), one has

[TABLE]

Gathering the above estimates $I_{1}-I_{5}$ to (4.12) together yields

[TABLE]

Step 2: The Estimates of $c$ and $\theta$.

Applying $D^{\ell}$ to the both sides of Eqs.$\eqref{c-tcm}_{2}$ and $\eqref{c-tcm}_{3}$, taking the inner product with $D^{\ell}c$ and $D^{\ell}\theta$, respectively, then summing the resulting over $|\alpha|\leq 3$, we get

[TABLE]

where

[TABLE]

Next, we need to estimate the above terms one by one.

Using the commutate estimate (3.1) and (3.2), respectively, we obtain

[TABLE]

Due to the fact $c\cdot\nabla w=\mathord{{\rm div}\ }(w\otimes c)-w\mathord{{\rm div}\ }c$, integrating by parts, one has

[TABLE]

Similarly, we deduce

[TABLE]

For the term $J_{5}$, by the product estimate (3.4), one has

[TABLE]

Using Hölder’s inequality gives

[TABLE]

Gathering the above estimates $J_{1}-J_{6}$ to (4.19) together yields

[TABLE]

Step 3: The Estimate of Crossing Term $\sum_{0\leq|\ell|\leq 2}\langle D^{\ell}c,D^{\ell}\nabla\theta\rangle$.

Applying $D^{\ell}$ to $\eqref{c-tcm}_{2}$ and $\eqref{c-tcm}_{3}$, taking the scalar product of them with $D^{\ell}\nabla\theta$ and $D^{\ell}\mathord{{\rm div}\ }c$, respectively, adding them together and then summing the resulting over $|\alpha|\leq 2$, we get

[TABLE]

where

[TABLE]

By Leibniz’s formula and Hölder’s inequality, we have

[TABLE]

Gathering the above estimates $K_{1}-K_{6}$ to (4.27) together yields

[TABLE]

Step 4: Closure of The A Priori Estimates.

By simple computations, we deduce easily that for some suitable positive constant $\gamma$

[TABLE]

and

[TABLE]

Multiplying the inequality (4.27) by $\gamma$, combining (4.18) and (4), then integrating in time yields

[TABLE]

Let us define

[TABLE]

where $\eta$ is a small enough positive constant which will be determined later on.

Assume that $\Gamma<T^{*}$. Choosing $\eta$ small enough such that the first term of RHS of (4) is absorbed, then we infer from (4) that for all $t\in[0,\Gamma]$

[TABLE]

Then by Gronwall’s inequality and (1.1), (3.5)–(3.6), we have for all $t\in[0,\Gamma]$

[TABLE]

Choosing $\eta=2C\varepsilon_{0}$, thus we can get

[TABLE]

So if $\Gamma<T^{*}$, due to the continuity of the solutions, we can obtain that there exists $0<\epsilon\ll 1$ such that

[TABLE]

which is contradiction with the definition of $\Gamma$.

Thus, we can conclude $\Gamma=T^{*}$ and

[TABLE]

which implies that $T^{*}=+\infty$. This completes the proof of Theorem 1.1. $\Box$

Acknowledgments

J. Li is supported by NSFC (No.11801090). Y. Yu is supported by NSF of Anhui Province (No.1908085QA05).