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.
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Notes
Here, we denote for conciseness \({{\mathscr {K}}}_n={{\mathscr {K}}}_{{\tilde{\theta }}_{n}}\).
Note however that not all this eigenvalues are isolated, hence they are in the essential spectrum.
Note that its derivation relies on the fact that \(\Vert \nabla \Theta \Vert _{L^\frac{2}{\alpha }}=\Vert \nabla {\tilde{\theta }}\Vert _{L^\frac{2}{\alpha }} <\epsilon _0(p)\).
Which applies since \({\tilde{\theta }}\) is small enough as in the Corollary 5.2.
(which we multiply by a large constant M and we take C large so that \(\frac{2}{\alpha }-1+C>\frac{m+2}{\alpha }-1\).
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Stefanov is partially supported by NSF-DMS under No. 1908626.
Appendix A. Proof of Lemma 2.2
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 \(s\in (0,2)\)
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
Introduce another \(C^\infty _0\) function \(\Psi (z)=|z|^\sigma \psi (z)\), so that we can decompose
We can then write
Introducing
we need to control
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 \(|\eta -y|\ge \frac{1}{2} |\eta |\ge 2^{k-3}\),
whence
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
So,
Finally, for the case \(|l-k|\le 10\), we use
so that
Thus, by Hölder’s
where we have used the Hausdorf–Young’s inequality
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Hadadifard, F., Stefanov, A.G. On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates. J. Math. Fluid Mech. 23, 24 (2021). https://doi.org/10.1007/s00021-021-00559-1
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DOI: https://doi.org/10.1007/s00021-021-00559-1