Global Regularity for a 2D Tropical Climate Model with Fractional Dissipation

Abstract

This paper examines the global regularity problem for a 2D tropical climate model with fractional dissipation. The inviscid version of this model was derived by Frierson, Majda and Pauluis for large-scale dynamics of precipitation fronts in the tropical atmosphere. The model considered here has some very special features. This nonlinear system involves interactions between a divergence-free vector field and a non-divergence-free vector field. In addition, the fractional dissipation not only models long-range interactions but also allows simultaneous investigations of a family of system. Our study leads to the global regularity of solutions when the indices of the fractional Laplacian are in two very broad ranges. In order to establish the global-in-time bounds, we introduce an efficient way to control the gradient of the non-divergence-free vector field and make sharp estimates by controlling the regularity of related quantities simultaneously.

Appendix A: Local Well-Posedness Theory on (1.1)

For the sake of completeness, this Appendix presents the local existence and uniqueness result for (1.1) with initial data $(u_{0}, v_{0}, \theta _{0}) \in H^{s}({\mathbb {R}}^{2})$ with $s>2$. More precisely, in this Appendix, we prove the following local well-posedness result.

Proposition A.1

Let $(u_{0}, v_{0}, \theta _{0}) \in H^{s}({\mathbb {R}}^{2})$ with $s>2$ and $\nabla \cdot u_{0}=0$. Then, there exists a positive time T depending on $\Vert u_{0}\Vert _{H^{s}}$, $\Vert v_{0}\Vert _{H^{s}}$ and $\Vert \theta _{0}\Vert _{H^{s}}$ such that (1.1) admits a unique solution $(u, v, \theta )\in C([0, T]; H^{s}({\mathbb {R}}^{2}))$.

We remark that we only consider the case $s>2$. Actually, $s>2$ can be weakened to $s>f(\alpha ,\,\gamma )$ with the function $f(\alpha ,\,\gamma )\le 2$. However, to make the idea clear, we assume this condition throughout this Appendix. The proof of Proposition A.1 can be performed by the method similar to Chapter 3 in Majda and Bertozzi (2002). To prove Proposition A.1, the main step is to approximate (1.1) in order to easily produce a family of global smooth solutions. In order to do this, we may, for instance, make use of the Friedrichs method. Now we define the spectral cutoff as follows

$$\begin{aligned} \widehat{{\mathcal {J}}_{N}f}(\xi )=\chi _{B(0,N)}(\xi ){\widehat{f}}(\xi ), \end{aligned}$$

where $N>0, \,B(0,N)=\{\xi \in {\mathbb {R}}^{2}| \, |\xi |\le N\}$ and $\chi _{B(0,N)}$ is the characteristic function on B(0, N). Also we define

$$\begin{aligned} L^{2}_{N}\triangleq \{f\in L^{2}({\mathbb {R}}^{2})|\, \text{ supp } \,{\widehat{f}}\subset B(0,N)\}. \end{aligned}$$

Proof of Proposition A.1

The first step is to consider the following approximate system of (1.1),

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u^{N}+{\mathcal {P}}\mathcal {J}_{N}(({\mathcal {J}}_{N}u^{N}\cdot \nabla ) {\mathcal {J}}_{N}u^{N})+ \nu \Lambda ^{2\alpha }{\mathcal {J}}_{N}u^{N} +{\mathcal {P}}{\mathcal {J}}_{N}\nabla \cdot ( {\mathcal {J}}_{N}v^{N}\otimes {\mathcal {J}}_{N}v^{N})=0, \\&\partial _{t}v^{N}+{\mathcal {J}}_{N} ({\mathcal {J}}_{N}u^{N} \cdot \nabla ){\mathcal {J}}_{N}v^{N}- \mu \Delta {\mathcal {J}}_{N}v^{N}+\nabla {\mathcal {J}}_{N}\theta ^{N}+ {\mathcal {J}}_{N} ({\mathcal {J}}_{N}v^{N}\cdot \nabla ){\mathcal {J}}_{N}u^{N}=0,\\&\partial _{t}\theta ^{N}+{\mathcal {J}}_{N} ({\mathcal {J}}_{N}u^{N} \cdot \nabla ) {\mathcal {J}}_{N}\theta ^{N}+ \eta \Lambda ^{2\gamma }{\mathcal {J}}_{N}\theta ^{N}+\nabla \cdot {\mathcal {J}}_{N}v^{N}=0,\\&\nabla \cdot u^{N}=0,\\&u^{N}(x, 0)={\mathcal {J}}_{N}u_{0}(x), \quad v^{N}(x,0)={\mathcal {J}}_{N}v_{0}(x), \quad \theta ^{N}(x,0)={\mathcal {J}}_{N}\theta _{0}(x), \end{aligned}\right. \end{aligned}$$

(A.1)

where ${\mathcal {P}}$ denotes the standard projection onto divergence-free vector fields. Taking advantage of the Cauchy–Lipschitz theorem (Picard’s Theorem, see Majda and Bertozzi 2002), we can find that for any fixed N, there exists a unique local solution $(u^{N},v^{N},\theta ^{N})$ on $[0,\,T_{N})$ in the functional setting $L^{2}_{N}$ with $T_{N}=T(N, u_{0}, v_{0},\theta _{0})$. Due to ${\mathcal {J}}_{N}^{2}={\mathcal {J}}_{N},\,\mathcal {P}^{2}={\mathcal {P}}$ and ${\mathcal {P}}{\mathcal {J}}_{N}=\mathcal {J}_{N}{\mathcal {P}}$, we find that $({\mathcal {J}}_{N}u^{N},\,\mathcal {J}_{N}v^{N},\,\mathcal {J}_{N}\theta ^{N})$ is also a solution to (A.1) with the same initial datum. Thanks to the uniqueness, we thus find

$$\begin{aligned} {\mathcal {J}}_{N}u^{N}=u^{N},\ \ \ {\mathcal {J}}_{N}v^{N}=v^{N}, \ \ \ {\mathcal {J}}_{N}\theta ^{N}=\theta ^{N}. \end{aligned}$$

Consequently, approximate system (A.1) reduces to

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u^{N}+{\mathcal {P}}\mathcal {J}_{N}((u^{N}\cdot \nabla )u^{N})+ \nu \Lambda ^{2\alpha }u^{N} +{\mathcal {P}}{\mathcal {J}}_{N}\nabla \cdot (v^{N}\otimes v^{N})=0, \\&\partial _{t}v^{N}+{\mathcal {J}}_{N} (u^{N} \cdot \nabla )v^{N}- \mu \Delta v^{N}+\nabla \theta ^{N}+ {\mathcal {J}}_{N} ( v^{N}\cdot \nabla ) u^{N}=0,\\&\partial _{t}\theta ^{N}+{\mathcal {J}}_{N} ( u^{N} \cdot \nabla ) \theta ^{N}+ \eta \Lambda ^{2\gamma } \theta ^{N}+\nabla \cdot v^{N}=0,\\&\nabla \cdot u^{N}=0,\\&u^{N}(x, 0)={\mathcal {J}}_{N}u_{0}(x), \quad v^{N}(x,0)={\mathcal {J}}_{N}v_{0}(x), \quad \theta ^{N}(x,0)={\mathcal {J}}_{N}\theta _{0}(x). \end{aligned}\right. \end{aligned}$$

(A.2)

A basic energy estimate implies $(u^{N},v^{N},\theta ^{N})$ of (A.2) satisfies

$$\begin{aligned}&\Vert u^{N}(t)\Vert _{L^{2}}^{2}+\Vert v^{N}(t)\Vert _{L^{2}}^{2}+\Vert \theta ^{N}(t)\Vert _{L^{2}}^{2}+ \int _{0}^{t}{ (\Vert \Lambda ^{\alpha } u^{N}\Vert _{L^{2}}^{2}+\Vert \nabla v^{N}\Vert _{L^{2}}^{2}+\Vert \Lambda ^{\gamma }\theta ^{N}\Vert _{L^{2}}^{2})(\tau )\,\hbox {d}\tau } \\&\quad \le \Vert u_{0}\Vert _{L^{2}}^{2}+\Vert v_{0}\Vert _{L^{2}}^{2}+\Vert \theta _{0}\Vert _{L^{2}}^{2}. \end{aligned}$$

Therefore, the local solution can be extended into a global one, by the standard Picard extension theorem (see, e.g., Majda and Bertozzi 2002). Moreover, by direct $H^s$-estimates, we have

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}\left( \Vert u^{N}(t)\Vert _{H^{s}}^{2} +\Vert v^{N}(t)\Vert _{H^{s}}^{2}+\Vert \theta ^{N}(t)\Vert _{H^{s}}^{2} \right) +\Vert \Lambda ^{\alpha }u^{N}\Vert _{H^{s}}^{2} +\Vert \nabla v^{N} \Vert _{H^{s}}^{2}+\Vert \Lambda ^{\gamma }\theta ^{N}\Vert _{H^{s}}^{2}\nonumber \\&\quad \le C(\Vert \nabla u^{N}\Vert _{L^{\infty }}+\Vert \nabla v^{N}\Vert _{L^{\infty }}+\Vert \nabla \theta ^{N}\Vert _{L^{\infty }}+\Vert v^{N}\Vert _{L^{\infty }}^{2}) (\Vert u^{N}\Vert _{H^{s}}^{2}+\Vert v^{N}\Vert _{H^{s}}^{2}+\Vert \theta ^{N}\Vert _{H^{s}}^{2})\nonumber \\&\quad \le C(\Vert u^{N}\Vert _{H^{s}}+\Vert v^{N}\Vert _{H^{s}}+\Vert \theta ^{N}\Vert _{H^{s}}+\Vert v^{N}\Vert _{H^{s}}^{2}) (\Vert u^{N}\Vert _{H^{s}}^{2}+\Vert v^{N}\Vert _{H^{s}}^{2}+\Vert \theta ^{N}\Vert _{H^{s}}^{2}) \nonumber \\&\quad \le C (\Vert u^{N}\Vert _{H^{s}}^{2}+\Vert v^{N}\Vert _{H^{s}}^{2}+\Vert \theta ^{N}\Vert _{H^{s}}^{2})^{2}, \end{aligned}$$

(A.3)

where here and in what follows we use the fact that, in 2D case

$$\begin{aligned} \Vert \nabla f\Vert _{L^{\infty }}\le C\Vert f\Vert _{H^{s}},\quad s>2. \end{aligned}$$

Notice that in (A.3) we assume that $\Vert u^{N}\Vert _{H^{s}}+\Vert v^{N}\Vert _{H^{s}}+\Vert \theta ^{N}\Vert _{H^{s}}\ge 1$, otherwise we replace $\Vert u^{N}\Vert _{H^{s}}+\Vert v^{N}\Vert _{H^{s}}+\Vert \theta ^{N}\Vert _{H^{s}}$ by $1+\Vert u^{N}\Vert _{H^{s}}+\Vert v^{N}\Vert _{H^{s}}+\Vert \theta ^{N}\Vert _{H^{s}}$. For the convenience of notation, we denote

$$\begin{aligned} X(t)\triangleq {\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert v^{N}(t)\Vert _{H^{s}}^{2}+\Vert \theta ^{N}(t)\Vert _{H^{s}}^{2}}. \end{aligned}$$

Consequently, (A.3) becomes

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}X(t) \le {\widetilde{C}}X(t)^{2}, \end{aligned}$$

where ${\widetilde{C}}>0$ is an absolute constant. Standard calculations show that for all N

$$\begin{aligned} \sup _{0\le t\le T}( {\Vert u^{N}(t)\Vert _{H^{s}}^{2}+\Vert v^{N}(t)\Vert _{H^{s}}^{2}+\Vert \theta ^{N}(t)\Vert _{H^{s}}^{2}}) \le \frac{ {\Vert u_{0}\Vert _{H^{s}}^{2}+\Vert v_{0}\Vert _{H^{s}}^{2}+\Vert \theta _{0}\Vert _{H^{s}}^{2}}}{1-{\widetilde{C}}T({\Vert u_{0}\Vert _{H^{s}}^{2} +\Vert v_{0}\Vert _{H^{s}}^{2}+\Vert \theta _{0}\Vert _{H^{s}}^{2}})}. \end{aligned}$$

Therefore, the family $(u^{N},v^{N},\theta ^{N})$ is uniformly bounded in $C([0, T]; H^{s})$ with $s>2$, provided that

$$\begin{aligned} T<\frac{1}{{\widetilde{C}}(\Vert u_{0}\Vert _{{H}^{s}}^{2}+\Vert b_{0}\Vert _{H^{s}}^{2}+ \Vert \theta _{0}\Vert _{H^{s}}^{2})}. \end{aligned}$$

Thus, it is not hard to see that

$$\begin{aligned} \partial _{t}u^{N},\,\,\partial _{t}v^{N},\,\,\partial _{t}\theta ^{N}\in L_{t}^{\infty } ([0, T]);\,H_{x}^{-\sigma }({\mathbb {R}}^{2})\quad \text{ for } \text{ some } \,\, \sigma \ge 2. \end{aligned}$$

Since the embedding $L^{2}\hookrightarrow H^{-\sigma }$ is locally compact, the well-known Aubin–Lions argument allows us to conclude that, up to extraction, subsequence $(u^{N},v^{N},\theta ^{N})_{N\in {\mathbb {N}}}$ satisfies

$$\begin{aligned} \Vert u^{N}-u^{N'}\Vert _{L^{2}},\quad \Vert v^{N}-v^{N'}\Vert _{L^{2}},\quad \Vert \theta ^{N}-\theta ^{N'}\Vert _{L^{2}}\rightarrow 0,\quad as\quad N,\,\,N'\rightarrow \infty . \end{aligned}$$

Thanks to the interpolation ($\Vert f\Vert _{H^{s'}}\le C \Vert f\Vert _{L^{2}}^{1-\frac{s'}{s}}\Vert f\Vert _{H^{s}}^{\frac{s'}{s}}$ for any $s'<s$), we deduce that

$$\begin{aligned} \Vert u^{N}-u^{N'}\Vert _{H^{s'}},\quad \Vert v^{N}-v^{N'}\Vert _{H^{s'}},\quad \Vert \theta ^{N}-\theta ^{N'}\Vert _{H^{s'}}\rightarrow 0,\quad as\quad N,\,\,N'\rightarrow \infty , \end{aligned}$$

which imply that we have strong convergence limit $(u, v,\theta )\in C([0, T]; H^{s'})$ for any $s'<s$. Therefore, this is enough for us to show that up to extraction, sequence $(u^{N},v^{N},\theta ^{N})_{N\in {\mathbb {N}}}$ has a limit $(u,\,v,\,\theta )$ satisfying

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u+{\mathcal {P}}((u\cdot \nabla )u)+ \nu \Lambda ^{2\alpha }u +{\mathcal {P}}\nabla \cdot (v\otimes v)=0, \\&\partial _{t}v+(u\cdot \nabla )v- \mu \Delta v+\nabla \theta +(v\cdot \nabla ) u=0,\\&\partial _{t}\theta +(u\cdot \nabla )\theta + \eta \Lambda ^{2\gamma } \theta +\nabla \cdot v=0,\\&\nabla \cdot u=0,\\&u(x, 0)=u_{0}(x), \quad v(x,0)=v_{0}(x), \quad \theta (x,0)=\theta _{0}(x). \end{aligned}\right. \end{aligned}$$

(A.4)

Moreover, one may show that $(u, v, \theta )\in L^{\infty }([0, T]; H^{s}({\mathbb {R}}^{2}))$. Finally, we claim that $(u, v, \theta )\in C([0, T]; H^{s}({\mathbb {R}}^{2}))$; namely, $(u,v,\theta )$ is strongly continuous in $H^{s}({\mathbb {R}}^{2})$ in time. It suffices to consider $u\in C([0, T]; H^{s}({\mathbb {R}}^{2})$ as the same fashion can be applied to v and $\theta$ to obtain the desired result. From the above argument, we first have

$$\begin{aligned} \sup _{0\le t\le T}(\Vert u\Vert _{H^{s}}+\Vert v\Vert _{H^{s}}+\Vert \theta \Vert _{H^{s}})<\infty . \end{aligned}$$

By the equivalent norm, it yields

$$\begin{aligned} \Vert u(t_{1})-u(t_{2})\Vert _{H^{s}}=\left\{ \left( \sum _{k<N}+ \sum _{k\ge N}\right) (2^{ks}\Vert \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})\Vert _{L^{2}})^{2} \right\} ^{\frac{1}{2}}, \end{aligned}$$

(A.5)

where the Fourier localization operator $\Delta _{k}$ is defined through the Littlewood–Paley decomposition (see Chapter 2 in Bahouri et al. 2011 for details). Let $\varepsilon >0$ be arbitrarily small. Due to $u\in L^{\infty }([0, T]; H^{s}({\mathbb {R}}^{2}))$, there exists an integer $M=M(\varepsilon )>0$ such that

$$\begin{aligned} \Big \{\sum _{k\ge M} (2^{ks}\Vert \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})\Vert _{L^{2}})^{2} \Big \}^{\frac{1}{2}}<\frac{\varepsilon }{2}. \end{aligned}$$

(A.6)

Recalling system (A.4)$_{1}$, we obtain

$$\begin{aligned} \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2}) =&\int _{t_{1}}^{t_{2}}{\frac{d}{\hbox {d}\tau } \Delta _{k}u(\tau )\,\hbox {d}\tau } \\ =&-\int _{t_{1}}^{t_{2}}{ \Delta _{k}{\mathcal {P}}[\nabla \cdot (v\otimes v)+(u\cdot \nabla ) u+\nu \Lambda ^{2\alpha } u](\tau )\,\hbox {d}\tau }. \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\sum _{k<M} 2^{2ks}\Vert \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})\Vert _{L^{2}}^{2}\\&\quad = \sum _{k<M} 2^{2ks}\Big (\Big \Vert \int _{t_{1}}^{t_{2}}{ \Delta _{k}{\mathcal {P}}[\nabla \cdot (v\otimes v)+(u\cdot \nabla ) u+\nu \Lambda ^{2\alpha } u](\tau )\,\hbox {d}\tau }\Big \Vert _{L^{2}}\Big )^{2}\\&\quad \le \sum _{k<M} 2^{2ks}\Big (\int _{t_{1}}^{t_{2}}{ \Vert \Delta _{k}[\nabla \cdot (v\otimes v)+(u\cdot \nabla ) u+\nu \Lambda ^{2\alpha } u]\Vert _{L^{2}}(\tau )\,\hbox {d}\tau }\Big )^{2} \\&\quad \le \sum _{k<M} 2^{2ks}\Big (\int _{t_{1}}^{t_{2}}{ [\Vert \Delta _{k}(\nabla \cdot (v\otimes v)) \Vert _{L^{2}}+ \Vert \Vert \Delta _{k}(u\cdot \nabla u)\Vert _{L^{2}}+\nu \Vert \Vert \Delta _{k}\Lambda ^{2\alpha }u\Vert _{L^{2}}](\tau )\,\hbox {d}\tau }\Big )^{2} \\&\quad = \sum _{k<M} 2^{4k}\Big (\int _{t_{1}}^{t_{2}}{ [2^{k(s-2)}\Vert \Delta _{k}\nabla \cdot (v\otimes v) \Vert _{L^{2}}+ 2^{k(s-2)}\Vert \Delta _{k}\nabla \cdot (u\otimes u)\Vert _{L^{2}}](\tau )\,\hbox {d}\tau }\Big )^{2} \\&\qquad + \sum _{k<M} 2^{4k}\Big (\int _{t_{1}}^{t_{2}}{ \nu 2^{k(s-2+2\alpha )}\Vert \Delta _{k}u(\tau )\Vert _{L^{2}}\,\hbox {d}\tau }\Big )^{2} \\&\quad \le C\sum _{k<M} 2^{4k}\Big (\Vert vv\Vert _{L_{t}^{\infty }H^{s}}^{2}|t_{1}-t_{2}|^{2} +\Vert uu\Vert _{L_{t}^{\infty }H^{s}}^{2}|t_{1}-t_{2}|^{2}+ \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2}|t_{1}-t_{2}|^{2}\Big )\\&\quad \le C\sum _{k<M} 2^{4k}|t_{1}-t_{2}|^{2}\Big (\Vert v\Vert _{L_{t}^{\infty }L^{\infty }}^{2}\Vert v\Vert _{L_{t}^{\infty }H^{s}}^{2}+ \Vert u\Vert _{L_{t}^{\infty }L^{\infty }}^{2}\Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2} + \nu \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2}\Big )\\&\quad \le C 2^{4M}|t_{1}-t_{2}|^{2}\Big (\Vert v\Vert _{L_{t}^{\infty }H^{s}}^{4}+\Vert u\Vert _{L_{t}^{\infty } H^{s}}^{4} +\nu \Vert u\Vert _{L_{t}^{\infty }H^{s}}^{2}\Big ). \end{aligned}$$

Thus, the following holds true

$$\begin{aligned} \Big \{\sum _{k< M} (2^{ks}\Vert \Delta _{k}u(t_{1})-\Delta _{k}u(t_{2})\Vert _{L^{2}})^{2} \Big \}^{\frac{1}{2}}<\frac{\varepsilon }{2} \end{aligned}$$

(A.7)

provided $|t_{1}-t_{2}|$ small enough. Combining (A.5), (A.6) with (A.7) implies $u\in C([0, T]; H^{s}({\mathbb {R}}^{2})$. The uniqueness can be easily obtained since $(u, v, \theta )$ are all in Lipschitz space. Therefore, the proof of Proposition A.1 is completed. $\square$