Stability of the Couette Flow for a 2D Boussinesq System Without Thermal Diffusivity

Abstract

In this paper, we prove the stability of the Couette flow for a 2D Navier–Stokes Boussinesq system without thermal diffusivity for the initial perturbation in Gevrey-$\frac{1}{s}$, ($1/3<s\leqq 1$). The synergism of density mixing, vorticity mixing and velocity diffusion leads to the stability.

Appendices

Appendix A. Paraproduct Tools

In this section, we introduce the tools and notations that we should use in the Fourier analysis and paraproduct. We first introduce the dyadic partition of unity that we should use throughout the paper. Let $\varkappa (\xi )$ be a real radial bump function which satisfies $\varkappa (\xi )=1$ for $|\xi |\leqq 1/2$ and $\varkappa (\xi )=0$ for $|\xi |\geqq 3/4$. We define $\varphi (\xi )=\varkappa (\xi /2)-\varkappa (\xi )$ supported on the annulus $\{\frac{1}{2}\leqq |\xi |\leqq \frac{3}{2}\}$. By construction, we have the following partition of unity:

$$\begin{aligned} 1=\varkappa (\xi )+\sum _{k\in {\mathbb {Z}}}\varphi (\xi /2^k)=\varkappa (\xi )+\sum _{\mathrm{M\in 2^{{\mathbb {N}}}}} \varphi (\mathrm {M}^{-1}\xi ) \end{aligned}$$

and we define the cut-off $\varphi _\mathrm {M}=\varphi (\mathrm {M}^{-1}\xi )$, each supported in in the annulus $\frac{\mathrm {M}}{2}\leqq |\xi |\leqq \frac{3\mathrm {M}}{2}$

For $\mathrm {f}\in L^2({\mathbb {R}})$, we define

$$\begin{aligned} \begin{aligned} \mathrm {f}_\mathrm {M}&=\varphi _\mathrm {M}(|\partial _v|)\mathrm {f},\\ \mathrm {f}_{\frac{1}{2}}&=\varkappa (|\partial _v|)\mathrm {f},\\ \mathrm {f}_{<\mathrm {M}}&=\mathrm {f}_{\frac{1}{2}}+\sum _{\mathrm{K\in 2^{\mathbb {N}}}: \mathrm {K}<\mathrm {M}}\mathrm {f}_{<\mathrm {K}} \end{aligned} \end{aligned}$$

Hence, we have the decomposition

$$\begin{aligned} \mathrm {f}=\mathrm {f}_{\frac{1}{2}}+\sum _{\mathrm{M\in 2^{\mathbb {N}}}}\mathrm {f}_{\mathrm{M}}. \end{aligned}$$

We also have the almost orthogonality property

$$\begin{aligned} \Vert \mathrm {f}\Vert _2^2\approx \sum _{\mathrm{M\in {\mathbf {D}}}}\Vert \mathrm {f_\mathrm {M}}\Vert _2^2 \end{aligned}$$

and the approximate projection property

$$\begin{aligned} \Vert \mathrm {f}_{\mathrm{M}}\Vert _2\approx \Vert (\mathrm {f}_{\mathrm{M}})_\mathrm {M}\Vert _2. \end{aligned}$$

More generally if $\mathrm {f} ={\sum _{k}\mathrm {D}_{k}}$ for any $\mathrm {D}_k$ with $\frac{1}{\mathrm {C}}2^\mathrm{{k}}\subset \mathrm {supp } \,\mathrm {D}_k\subset \mathrm {C}2^k$ it follows that

$$\begin{aligned} \Vert \mathrm {f}\Vert _2^2\approx _c \sum _k\Vert \mathrm {D}_k\Vert _2^2 \end{aligned}$$

During much of the proof we are also working with Littlewood–Paley decompositions defined in the (z, v) variables, with the notation conventions being analogous. Our convention is to use $\mathrm {N}$ to denote Littlewood–Paley projections in (z, v) and $\mathrm {M}$ to denote projections only in the v direction.

For any $1\leqq p\leqq q\leqq \infty$, there exists a constant independent of $\mathrm {M}$ such that

$$\begin{aligned} {\Vert \partial _x^\alpha \mathrm f_\mathrm {M}\Vert } _q+{\Vert \partial _x^\alpha \mathrm {f}_{<\mathrm {M}/8} \Vert }_q\leqq C\mathrm {M}^{d(1/p-1/q)+|\alpha |}{\Vert \mathrm f\Vert }_p. \end{aligned}$$

Appendix B. Important Inequalities

If $|x-y|\leqq |x|/\mathrm {K}$, then it holds that

$$\begin{aligned} |x^s-y^s|\leqq \frac{s}{(\mathrm {K}-1)^{s-1}}|x-y|^s,\qquad 0<s<1. \end{aligned}$$

(B.1)

In many occasions, we use the following inequality, which is a result of (B.1):

$$\begin{aligned} e^{\lambda |x|^s}\leqq e^{\lambda |y|^s}e^{c\lambda |x-y|^s} \end{aligned}$$

for $|x-y|<|x|/\mathrm {K}$, with $\mathrm {K}>1$, $s\in (0,1)$ and $c=c(s)\in (0,1)$.

If $|y|\leqq |x|\leqq \mathrm {K}|y|$, then it holds that

$$\begin{aligned} |x+y|^s\leqq \Big (\frac{\mathrm {K}}{\mathrm {K}+1}\Big )^{1-s}(x^s+y^s). \end{aligned}$$

Lemma B.1

($L^p-L^q$ decay of the heat kernel) Let u be the solution of the heat equation

$$\begin{aligned} u_t-\Delta u=0,\quad u(t=0)=\varphi (x),\quad x\in {\mathbb {R}}^d,\, t\geqq 0. \end{aligned}$$

Let $S(t)=e^{t\Delta }$ being the heat operator. Then it holds that for any $1\leqq q\leqq p\leqq \infty$

$$\begin{aligned} \Big \Vert \partial _t^j\partial _x^\alpha u\Big \Vert _{p}=\Big \Vert \partial _t^j\partial _x^\alpha S(t)\varphi \Big \Vert _{p}\leqq C t^{-\frac{d}{2}(\frac{1}{q}-\frac{1}{p})-j-\frac{|\alpha |}{2}} \Vert \varphi \Vert _{L^q} \end{aligned}$$

where j is a positive integer and $\alpha =(\alpha _1, \dots ,\alpha _d)$, where $\alpha _i, \, 1\leqq i\leqq d$ is a positive integer and $|\alpha |=\alpha _1+\dots +\alpha _d$ and $\partial _x^\alpha =\partial _{x_1}^{\alpha _1}\partial _{x_2}^{\alpha _2}\dots \partial _{x_d}^{\alpha _d}$.

We define the Gevrey in the physical space as (see e. g. [53])

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {G}}}^{\lambda ;s}}\approx \Big [\sum _{n=0}^\infty \Big (\frac{\lambda ^n}{(n!)^{\frac{1}{s}}}\Vert \mathrm {D}^n f\Vert _2\Big ) ^2\Big ]^{\frac{1}{2}} \end{aligned}$$

(B.2a)

We can also extend (B.2a) define the more general $\ell ^qL^p$ based spaces (see [69]) as

$$\begin{aligned} \Vert f\Vert _{\ell ^q\mathrm {L}^p;\lambda }\approx \Big [\sum _{n=0}^\infty \Big (\frac{\lambda ^n}{(n!)^{\frac{1}{s}}}\Vert \mathrm {D}^n f\Vert _p\Big ) ^q\Big ]^{\frac{1}{q}} \end{aligned}$$

(B.2b)

Then it holds that (see [8, Appendix A.]) for $\lambda >\lambda '$ and for $p,\,q\in [1,\infty ]$, we have

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{\ell ^p\mathrm {L}^q;\lambda '}\leqq&\, \Vert f\Vert _{\ell ^1\mathrm {L}^q;\lambda }\lesssim _{\lambda -\lambda '}\Vert f\Vert _{\ell ^p\mathrm {L}^q;\lambda }\\ \Vert f\Vert _{\ell ^2\mathrm {L}^\infty ;\lambda '}\lesssim&\, \Vert f\Vert _{\ell ^2\mathrm {L}^2;\lambda }. \end{aligned} \end{aligned}$$

(B.3)

Next, we show the following lemma, which useful to prove the scattering result in Section 2.5.

Lemma B.2

The following inequality holds:

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{\ell ^1\mathrm {L}^{2};\lambda } \lesssim&\, \Vert f\Vert _{\ell ^1\mathrm {L}^2;\lambda }\Vert g\Vert _{\ell ^1\mathrm {L}^\infty ;\lambda }. \end{aligned} \end{aligned}$$

Proof

We have, by using (B.2b) together with Leibniz’s, rule, that

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{\ell ^1\mathrm {L}^{2};\lambda }=&\,\sum _{n=0}^\infty \frac{\lambda ^n}{(n!)^{1/s}}\Vert D^n(fg)\Vert _{\mathrm{L^2}} \\ \lesssim&\,\sum _{n=0}^\infty \sum _{k=0}^n \Big (\frac{n!}{k!(n-k)!}\Big )^{\frac{1}{s}} \frac{\lambda ^k \lambda ^{n-k}}{(n!)^{1/s}} \Vert D^{k} f\Vert _{\mathrm{L^2}}\Vert D^{n-k} g\Vert _{\mathrm{L^\infty }}\\ \lesssim&\, \Vert f\Vert _{\ell ^1\mathrm {L}^2;\lambda }\Vert g\Vert _{\ell ^1\mathrm {L}^\infty ;\lambda }, \end{aligned} \end{aligned}$$

which gives the lemma. $\quad \square$