On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates

Abstract

We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all $L^1\cap L^\infty$ localized initial data produces global solutions of the forced SQG, which converge to the steady states in $L^p({\mathbf {R}}^2), 1<p\le 2$ as time goes to infinity. This establishes that the steady states serve as one point attracting set. Moreover, by employing the method of scaling variables, we compute the sharp relaxation rates, by requiring slightly more localized initial data.

Appendix A. Proof of Lemma 2.2

The proof of this lemma is based by some modifications in the proof of relation (4.8), [27]. Recall, that for $s\in (0,2)$

$$\begin{aligned}{}[|\nabla |^{s}, g] f (x)= & {} |\nabla |^{s} (g f)- g \ |\nabla |^{s}f= c_s \int \frac{f(x) g(x)- f(y) g(y)}{|x- y|^{2+ s}} dy- g(x) c_s\int \frac{f(x)- f(y)}{|x- y|^{2+ s}} dy \\= & {} c_s \int \frac{ f(y)( g(x)- g(y) )}{|x- y|^{2+ s}} dy. \end{aligned}$$

Introduce a smooth partition of unity, that is a function $\psi \in C^\infty _0({\mathbf {R}})$, $supp\ \psi \subset (\frac{1}{2},2)$, so that

$$\begin{aligned} \sum _{k=-\infty }^\infty \psi (2^{-k} |\eta |) = 1, \eta \in {\mathbf {R}}^2, \eta \ne 0. \end{aligned}$$

Introduce another $C^\infty _0$ function $\Psi (z)=|z|^\sigma \psi (z)$, so that we can decompose

$$\begin{aligned} |\eta |^{\sigma } = \sum _{k=-\infty }^\infty |\eta |^{\sigma } \psi (2^{-k} |\eta |) = \sum _{k=-\infty }^\infty 2^{k \sigma } \Psi (2^{-k} |\eta |). \end{aligned}$$

We can then write

$$\begin{aligned} F(\eta )&{:}{=}&[\Lambda ^{s},|\eta |^\sigma ] f= \sum _k 2^{\sigma k} [\Lambda ^s, \Psi (2^{- k} \cdot )] f(\eta ) = \sum _k 2^{\sigma k} \int \frac{f(y) (\Psi (2^{-k}\eta )- \Psi (2^{-k}y))}{|\eta - y|^{2+ s}} dy. \end{aligned}$$

Introducing

$$\begin{aligned} F_k{:}{=} \int \frac{|f(y)| |\Psi (2^{-k}\eta )- \Psi (2^{-k}y)|}{|\eta - y|^{2+ s}} dy, \end{aligned}$$

we need to control

$$\begin{aligned} \Vert F\Vert _{L^2}^2= & {} \sum _l \int _{|\eta |\sim 2^l} |F(\eta )|^2 d\eta =\sum _l \int _{|\eta |\sim 2^l} \left| \sum _k 2^{s k} F_k(\eta )\right| ^2 d\eta \\= & {} \sum _l \int _{|\eta |\sim 2^l} \left| \sum _{k>l+10} 2^{s k} F_k(\eta )\right| ^2 d\eta +\sum _l \int _{|\eta |\sim 2^l} \left| \sum _{k=l-10}^{l+10} 2^{s k} F_k(\eta )\right| ^2 d\eta \\&+ \sum _l \int _{|\eta |\sim 2^l} \left| \sum _{k<l-10} 2^{s k} F_k(\eta )\right| ^2 d\eta =:K_1+K_2+K_3. \end{aligned}$$

We first consider the cases $k>l+10$. One can estimate easily $F_k$ point-wise. More specifically, since in the denominator of the expression for $F_k$, we have $|\eta -y|\ge \frac{1}{2} |\eta |\ge 2^{k-3}$,

$$\begin{aligned} |F_k(\eta )|\le 2^{-k(2+\sigma )} \int |f(y)||\Psi (2^{-k}y)| dy\le C 2^{-k(1+\sigma )} \Vert f\Vert _{L^2(|y| \sim 2^k)}, \end{aligned}$$

whence

$$\begin{aligned}&K_1 \le \sum _l 2^{2 l} \sum _{k_1>l+10} \sum _{k_2>l+10} 2^{k_1(s- 1-\sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k_1})} 2^{k_2(s- 1- \sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k_2})}\\&\quad \le \sum _{k_1} \sum _{k_2} 2^{2\min (k_1,k_2)}2^{k_1(s- 1- \sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k_1})} 2^{k_2(s- 1- \sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k_2})} \\&\quad \le C \sum _k 2^{2 k(s-\sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k})}^2 \le C \Vert |\eta |^{s- \sigma } f\Vert ^2. \end{aligned}$$

where we have used $\sum _{l: l<\min (k_1,k_2)-10} 2^{2l} \le C 2^{2\min (k_1,k_2)}$.

For the case $k<l-10$, we perform similar argument, since

$$\begin{aligned} |F_k(\eta )|\le C 2^{-l(2+\sigma )} 2^k \Vert f\Vert _{L^2(|y| \sim 2^k)}. \end{aligned}$$

So,

$$\begin{aligned}&K_3 \le C \sum _l 2^{2l} 2^{-2 l (2+ \sigma )} \sum _{k_1<l-10} \sum _{k_2<l-10} 2^{(s+ 1) k_1} \Vert f\Vert _{L^2(|y| \sim 2^{k_1})} 2^{(s+ 1) k_2} \Vert f\Vert _{L^2(|y| \sim 2^{k_2})} \\&\quad \le C \sum _{k_1} \sum _{k_2} 2^{(s+ 1) k_1} \Vert f\Vert _{L^2(|y| \sim 2^{k_1})} 2^{(s+ 1) k_2} \Vert f\Vert _{L^2(|y| \sim 2^{k_2})} 2^{-2 (1+ \sigma )\max (k_1,k_2)} \\&\quad \le C \sum _k 2^{2 k(s- \sigma )} \Vert f\Vert _{L^2(|y| \sim 2^{k})}^2 \le C \Vert |\eta |^{(s- \sigma )} f\Vert ^2. \end{aligned}$$

Finally, for the case $|l-k|\le 10$, we use

$$\begin{aligned} |\Psi (2^{- k} \eta )- \Psi (2^{- k} y)| \le 2^{- k} |\eta - y| |\nabla \Psi (2^{- k} (\eta - y))| \le C 2^{- k} |\eta - y|, \end{aligned}$$

so that

$$\begin{aligned} |F_k(\eta )|\le C 2^{-k} \int _{|y| \sim 2^k} \frac{|f(y)|}{|\eta - y|^{1+ \sigma }} dy = C 2^{-k} |f| \chi _{|y| \sim 2^k} * \frac{1}{|\cdot |^{1+ \sigma }}. \end{aligned}$$

Thus, by Hölder’s

$$\begin{aligned}&K_2 \le C \sum _k \int _{|\eta |\sim 2^k} 2^{s k} \left| |f| \chi _{|y| \sim 2^k} * \frac{1}{|\cdot |^{1+ \sigma }}\right| ^2 d\eta \le C \sum _k 2^{s k} \Vert |f| \chi _{|y| \sim 2^k} * \frac{1}{|\cdot |^{1+ \sigma }}\Vert _{L^2(|\eta |\sim 2^k)}^2 \\&\quad \le C \sum _k 2^{2 k(s-\sigma )} \Vert |f| \chi _{|y| \sim 2^k} * \frac{1}{|\cdot |^{1+ \sigma }}\Vert _{L^{\frac{2}{\sigma }}(|\eta |\sim 2^k)}^2 \le C \sum _k 2^{2 k(s-\sigma )} \Vert f\Vert _{L^2(|\eta | \sim 2^k)}^2 \le C \Vert |\eta |^{s-\sigma } f\Vert ^2. \end{aligned}$$

where we have used the Hausdorf–Young’s inequality

$$\begin{aligned} \Vert f \chi _{|y| \sim 2^k} * \frac{1}{|\cdot |^{1+ \frac{\alpha }{2}}}\Vert _{L^{\frac{2}{\sigma }}} \le C \Vert \frac{1}{|\cdot |^{1+ \sigma }} \Vert _{L^{\frac{2}{1+ \sigma }, \infty }} \ \Vert f\Vert _{L^2(|\eta | \sim 2^k)}\le C \Vert f\Vert _{L^2(|\eta | \sim 2^k)}. \end{aligned}$$