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Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

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Abstract

This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray–Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time \(T>0\)) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

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Acknowledgments

The authors thank Professor Changxing Miao for discussions. Jiu was supported by NNSF of China under Grants No. 11171229 and No. 11231006. In addition, Jiu and Wu were partially supported by a Grant from NNSF of China under No.11228102. Wu was partially supported by NSF Grant DMS1209153 and by the AT&T Foundation at Oklahoma State University. Wu thanks the School of Mathematical Sciences, Capital Normal University for its hospitality.

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Correspondence to Jiahong Wu.

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Communicated by Peter Constantin.

Appendix 1: Proof of Proposition 4.3

Appendix 1: Proof of Proposition 4.3

This appendix provides a detailed proof of Proposition 4.3. The proof draws ideas from Dabkowski (2011), but the framework is generalized to deal with the general form of the SQG type equation and may be useful for future work. In addition, some technical details are simplified here, for example, the proof of (5.7).

Proof of Proposition 4.3

According to Definition 4.1, it suffices to verify that \(\psi \) possesses the following properties:

  1. (1)

    For any \(f_0\in Lip(1)\), namely \(f_0\) Lipschitz with Lipschitz constant \(1\),

    $$\begin{aligned} \left| \int _{\mathbb {T}^2} f_0(x)\, \psi (x, t-s)\,\mathrm{d}x\right| \le r\, e^{(\sigma ^{-1}-1)\delta \, s\, r^{-\beta }}; \end{aligned}$$
    (5.1)
  2. (2)

    For \(A\) and \(p\) as defined in Definition 4.1, and \(\frac{1}{p} +\frac{1}{q} =1\),

    $$\begin{aligned} \Vert \psi (x,t-s)\Vert _{L^p} \le A^{\frac{1}{p}} r^{-\frac{2}{q}}\, e^{-(1+\frac{2}{q\sigma })\delta \, s\, r^{-\beta }}. \end{aligned}$$
    (5.2)

The proof of (5.1) is involved. The idea is to consider the evolution of \(f_0\) according to an equation close to (1.6) so that we can use the condition \(\psi (x,t) \in \mathcal {U}(r)\). Let \(u_r\) to be the mollification of \(u\) (\(u\) is defined in (1.6)), namely

$$\begin{aligned} u_r = \rho _r *u, \end{aligned}$$

where \(\rho _r\) represents the standard mollifier, namely

$$\begin{aligned} \rho \ge 0, \,\, \rho \in C_0^\infty , \,\, \text{ supp } \rho \subset B(0,1), \,\,\int _{\mathbb {T}^2} \rho (x)\,dx =1, \,\, \rho _r(x) =r^{-2} \rho (r^{-1}x).\nonumber \\ \end{aligned}$$
(5.3)

Here \(r\) is assumed to be small. Assume \(f=f(x,\tau )\) with \(\tau \in [t-s,t]\) solves the linear equation

$$\begin{aligned} \partial _\tau f + u_r \cdot \nabla f + \Lambda ^\beta f = 0, \qquad f(x,t-s) =f_0(x). \end{aligned}$$
(5.4)

It is then easily checked using (4.3) and (5.4) that

$$\begin{aligned} \int f_0(x)\,\psi (x, t-s)\,\mathrm{d}x&= \int f(x,t)\,\psi (x, t)\,\mathrm{d}x \nonumber \\&+ \int _{t-s}^t \int (u(x,\tau )-u_r(x,\tau ))\cdot \nabla f(x,\tau )\,\psi (x,\tau )\,\mathrm{d}x\mathrm{d}\tau . \nonumber \\ \end{aligned}$$
(5.5)

To bound the first term on the right, we show that

$$\begin{aligned} f(\cdot ,t) \in Lip(C_1\,e^{C\, s\, r^{\alpha +\sigma -1}}). \end{aligned}$$
(5.6)

This can be achieved by simple energy estimates. In fact, \(\nabla f\) satisfies

$$\begin{aligned} \partial _\tau (\nabla f) + u_r \cdot \nabla (\nabla f) + \Lambda ^\beta (\nabla f) = -(\nabla u_r) (\nabla f). \end{aligned}$$

To bound \(\Vert \nabla f\Vert _{L^\infty }\), we first bound \(\Vert \nabla f\Vert _{L^\gamma }\) for large \(\gamma \) and then let \(\gamma \rightarrow \infty \). Multiplying each side by \(\nabla f |\nabla f|^{\gamma -2}\), integrating in space and applying \(\nabla \cdot u_r=0\), we find

$$\begin{aligned} \frac{1}{\gamma } \frac{\mathrm{d}}{\mathrm{d}t} \Vert \nabla f\Vert _{L^\gamma }^\gamma \le \Vert \nabla u_r\Vert _{L^\infty } \,\Vert \nabla f\Vert _{L^\gamma }^\gamma , \end{aligned}$$

where we have applied the following inequality involving fractional Laplacian operator (see Córdoba and Córdoba 2004),

$$\begin{aligned} \int |\nabla f|^{\gamma -2} \nabla f \cdot \Lambda ^\beta (\nabla f)\,\mathrm{d}x \ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \nabla f(\cdot ,\tau )\Vert _{L^\gamma (\mathbb {T}^2)} \le \Vert \nabla f_0\Vert _{L^\gamma (\mathbb {T}^2)} \, e^{\int _{t-s}^\tau \Vert \nabla u_r(\cdot ,\zeta )\Vert _{L^\infty (\mathbb {T}^2)} \mathrm{d}\zeta }. \end{aligned}$$

Due to \(\Vert \nabla f_0\Vert _{L^\gamma (\mathbb {T}^2)} \le (4 \pi ^2)^{\frac{1}{\gamma }}\,\Vert \nabla f_0\Vert _{L^\infty }\) and \(f_0\in Lip(1)\), we obtain by letting \(\gamma \rightarrow \infty \),

$$\begin{aligned} \Vert \nabla f(\cdot ,t)\Vert _{L^\infty } \le e^{\int _{t-s}^t \Vert \nabla u_r(\cdot ,\zeta )\Vert _{L^\infty (\mathbb {T}^2)} \mathrm{d}\zeta }. \end{aligned}$$
(5.7)

Recall that

$$\begin{aligned} \nabla u_r = \nabla (\rho _r*(\widetilde{u} +v)) = \rho _r*\nabla \widetilde{u} + \nabla \rho _r *v. \end{aligned}$$

Due to (1.7), namely \(\widetilde{u}\in Lip(M)\),

$$\begin{aligned} \Vert \rho _r*\nabla \widetilde{u}\Vert _{L^\infty } \le \Vert \rho _r\Vert _{L^1} \Vert \nabla \widetilde{u}\Vert _{L^\infty } \equiv M <\infty . \end{aligned}$$
(5.8)

It is not difficult to verify that \(\nabla \rho _r(x) = r^{-3} (\nabla \rho )(r^{-1} x) \in r^{-1} \mathcal {U}(2r)\) using the fact that \(\Vert \nabla \rho _r\Vert _{L^1} \le r^{-1}\), \(\Vert \nabla \rho _r\Vert _{L^\infty } \le C\,r^{-3}\) and \(\nabla \rho _r\) has mean zero. Thus, by (4.2),

$$\begin{aligned} |\nabla \rho _r *\theta | =\Big |\int \theta (y)\, \nabla \rho _r (x-y)\,\mathrm{d}y\Big | \le C r^{-1}\, (2r)^\sigma =C\,r^{\sigma -1}, \end{aligned}$$
(5.9)

where we have used the fact that \(\nabla \rho _r (x-y) =\nabla \rho _r (y-x) \in r^{-1} \mathcal {U}(2r)\) (the translation of a test function is still a test function). Since

$$\begin{aligned} v =-\nabla ^\perp \Lambda ^{-2} \Lambda ^{-\alpha } \partial _{1} \theta , \end{aligned}$$

we have \(\Vert v\Vert _{C^\alpha } \le \Vert \theta \Vert _{L^\infty }\) by using the fact that Riesz transforms \(\nabla ^\perp \Lambda ^{-2}\partial _{1}\) are bounded in Hölder space. By 5.9 and Lemma 4.2 (relating a Hölder function to the power of \(r\) when acting on a test function),

$$\begin{aligned} |\nabla \rho _r *v| \le C\,r^{\alpha +\sigma -1}. \end{aligned}$$
(5.10)

Inserting (5.8) and (5.10) in (5.7) leads to (5.6). By (5.6) and the fact that \(\psi (x,t)\in \mathcal {U}(r)\),

$$\begin{aligned} \Big |\int f(x,t)\,\psi (x, t)\,\mathrm{d}x \Big | \le C_1\, r\, e^{C\, s\, r^{\alpha +\sigma -1}}. \end{aligned}$$
(5.11)

Next we bound the second term on the right of (5.5). Applying Hölder’s inequality in space and integrating in time, we obtain

$$\begin{aligned}&\Big |\displaystyle \int _{t-s}^t \int (u-u_r)\cdot \nabla f\psi (x,\tau )\,\mathrm{d}x\mathrm{d}\tau \Big | \nonumber \\&\quad \le s \,\sup \limits _{\tau \in [t-s,t]} \Vert (u-u_r)(\cdot , \tau )\Vert _{L^q} \Vert \nabla f\Vert _{L^\infty } \Vert \psi (\cdot ,\tau )\Vert _{L^p}. \end{aligned}$$
(5.12)

Recall that \(\Vert \nabla f\Vert _{L^\infty }\) is bounded according to (5.6) and also \(\frac{1}{p}+\frac{1}{q}=1\). Since \(\psi (\cdot ,t)\in \mathcal {U}(r)\) and \(\Vert \psi (\cdot ,\tau )\Vert _{L^p}\) increases in time (due to the evolution equation (4.3)), we have

$$\begin{aligned} \Vert \psi (\cdot ,\tau )\Vert _{L^p} \le \Vert \psi (\cdot ,t)\Vert _{L^p} \le A^{\frac{1}{p}} r^{-\frac{2}{q}}. \qquad \qquad \end{aligned}$$
(5.13)

It then suffices to bound \(\Vert (u-u_r)(\cdot , \tau )\Vert _{L^q}\). We claim that

$$\begin{aligned} \Vert (u-u_r)(\cdot , \tau )\Vert _{L^q} \le C\, r^{\sigma +\alpha }. \end{aligned}$$
(5.14)

Since \(\widetilde{u}\in Lip(M)\),

$$\begin{aligned} \Vert (u-u_r)(\cdot , \tau )\Vert _{L^q}&\le \Vert (\widetilde{u}-\widetilde{u}_r)(\cdot , \tau )\Vert _{L^q} + \Vert (v-v_r)(\cdot , \tau )\Vert _{L^q}\\&\le C\,r\, \Vert \nabla \widetilde{u}\Vert _{L^\infty } + \Vert (v-v_r)(\cdot , \tau )\Vert _{L^q}\\&\le C\,r\, M + \Vert (v-v_r)(\cdot , \tau )\Vert _{L^q}. \end{aligned}$$

Recall \(v =-\nabla ^\perp \Lambda ^{-2} \Lambda ^{-\alpha } \partial _{1} \theta \). By the boundedness of Riesz transforms on \(L^q\) (\(1<q<\infty \)) and a simple analysis of Fourier series (Stein 1970),

$$\begin{aligned} \Vert (v-v_r)(\cdot , \tau )\Vert _{L^q(\mathbb {T}^2)} \le C\, \Vert \Lambda ^{-\alpha }(\theta -\theta _r)\Vert _{L^q(\mathbb {T}^2)} \le C r^{\alpha }\, \Vert \theta -\theta _r\Vert _{L^q(\mathbb {T}^2)}.\qquad \end{aligned}$$
(5.15)

To bound \(\Vert \theta -\theta _r\Vert _{L^q(\mathbb {T}^2)}\), we cover \(\mathbb {T}^2\) by \(B_{r}\), disks of radius \(r\) and the number of such disks is of order \(r^{-2}\). First we bound \(\Vert \theta -\theta _r\Vert _{L^q(B_{r})}\). For any constant \(c\),

$$\begin{aligned} \Vert \theta -\theta _r\Vert _{L^q(B_{r})} \le \Vert \theta -c\Vert _{L^q(B_{r})} + \Vert c-\theta _r\Vert _{L^q(B_{r})} \le 2 \Vert \theta -c\Vert _{L^q(B_{3r})}. \end{aligned}$$
(5.16)

To further the estimate, we choose \(c\) such that \(\text{ sign }(\theta -c)\, |\theta -c|^{q-1}\) has mean zero on \(B_{3r}\), namely

$$\begin{aligned} \int _{B_{3r}} \text{ sign }(\theta -c)\, |\theta -c|^{q-1} \,\mathrm{d}x =0. \end{aligned}$$

Then,

$$\begin{aligned} \Vert \theta -c\Vert ^q_{L^q(B_{3r})}&= \int _{B_{3r}} (\theta -c)\, \text{ sign }(\theta -c)\, |\theta -c|^{q-1}\,\mathrm{d}x \nonumber \\&= \int _{B_{3r}} \theta \,\text{ sign }(\theta -c)\, |\theta -c|^{q-1}\,\mathrm{d}x \nonumber \\&= r^{\frac{2}{q}}\, \Vert \theta -c\Vert _{L^{p(q-1)}(B_{3r})}^{q-1} \int _{\mathbb {T}^2} \theta (x,\tau )\, \phi (x,\tau )\,\mathrm{d}x, \end{aligned}$$
(5.17)

where we have set

$$\begin{aligned} \phi (x,\tau ) = r^{-\frac{2}{q}}\, \Vert \theta -c\Vert _{L^{p(q-1)}(B_{3r})}^{-(q-1)} \chi _{B_{3r}} \text{ sign }(\theta -c)\, |\theta -c|^{q-1}. \end{aligned}$$

Since \(\phi \) has mean zero, it is easily checked that \(\phi \) satisfies, for any \(f\in Lip(1)\),

$$\begin{aligned} \Vert \phi \Vert _{L^p(\mathbb {T}^2)} \le A^{\frac{1}{p}}\, (3r)^{-\frac{2}{q}}, \qquad \left| \int _{\mathbb {T}^2} f(x) \, \phi (x,\tau )\,\mathrm{d}x\right| \le C\,(3r) \end{aligned}$$

for some constants \(A\) and \(C\). That is, \(\phi \in \mathcal {U}(3r)\). Therefore, by (4.2) with \(e^\delta \le 3\),

$$\begin{aligned} \left| \int _{\mathbb {T}^2} \theta (x,\tau )\, \phi (x,\tau )\,\mathrm{d}x\right| \le (3r)^\sigma . \end{aligned}$$
(5.18)

Inserting (5.18) in (5.17) and realizing that \(p(q-1)=q\), we have

$$\begin{aligned} \Vert \theta -c\Vert ^q_{L^q(B_{3r})} \le C\, r^{\frac{2}{q} +\sigma }\,\Vert \theta -c\Vert ^{q-1}_{L^q(B_{3r})} \end{aligned}$$

or

$$\begin{aligned} \Vert \theta -c\Vert _{L^q(B_{3r})} \le C\, r^{\frac{2}{q} +\sigma }. \end{aligned}$$
(5.19)

Since the number of disks needed to cover \(\mathbb {T}^2\) is of the order \(r^{-2}\), we combine (5.15), (5.16) and (5.19) to obtain

$$\begin{aligned} \Vert v-v_r\Vert _{L^q(\mathbb {T}^2)} \le C\,r^{-\frac{2}{q}}\, r^{\frac{2}{q} +\sigma +\alpha } = C\,r^{\alpha +\sigma }. \end{aligned}$$
(5.20)

Therefore, by inserting (5.6), (5.13) and (5.20) in (5.12), we have

$$\begin{aligned} \Big |\!\int _{t-s}^t\!\int \!(u(x,\tau )-u_r(x,\tau ))\cdot \nabla f(x,\tau )\,\psi (x,\tau )\,\mathrm{d}x\mathrm{d}\tau \Big |\! \le C\, A^{\frac{1}{p}}\,r^{\alpha +\sigma -\frac{2}{q}}\,s\,e^{C s\,r^{\alpha +\sigma -1}}.\nonumber \\ \end{aligned}$$
(5.21)

Combining the bounds in (5.11) and (5.21) and recalling that

$$\begin{aligned} s\le r^{\beta }, \qquad \alpha +\beta +\sigma -\frac{2}{q} >1, \end{aligned}$$

we have, by taking \(C\,r_0^{\alpha +\beta +\sigma -1} \le C\, r_0^{\frac{2}{q}} \le (\sigma ^{-1}-1)\delta \),

$$\begin{aligned} \left| \int _{\mathbb {T}^2} f_0(x)\, \psi (x, t-s)\,\mathrm{d}x\right| \le C\,r \,e^{C s\,r^{\alpha +\sigma -1}} \le r\, e^{(\sigma ^{-1}-1)\delta \, s\, r^{-\beta }}. \end{aligned}$$

We have thus completed the proof of (5.1).

Next we prove (5.2). To bound \(\Vert \psi (\cdot ,\tau )\Vert _{L^p}\) for \(\tau \in [t-s,t]\), we multiply (4.3) by \(\Psi \equiv \psi \,|\psi |^{p-2}\), integrate in space and apply \(\nabla \cdot u=0\) to obtain

$$\begin{aligned} \frac{1}{p} \frac{\mathrm{d}}{\mathrm{d}t} \Vert \psi (\cdot ,\tau )\Vert _{L^p}^p = \int _{\mathbb {T}^2} \Psi \Lambda ^\beta \psi \,\mathrm{d}x. \end{aligned}$$
(5.22)

Applying the integral representation for \(\Lambda ^\beta \psi \) (see Córdoba and Córdoba 2004), we have

$$\begin{aligned} \int \Psi \Lambda ^\beta \psi \,\mathrm{d}x&= C_\beta \int _{\mathbb {T}^2} \Psi (x,\tau ) \sum _{n\in \mathbb {Z}^2} p.v.\int _{\mathbb {T}^2} \frac{\psi (x,\tau )-\psi (y,\tau )}{|x-y-n|^{2+\beta }}\,\mathrm{d}y\,\mathrm{d}x \\&= C_\beta \sum _{n\in \mathbb {Z}^2} p.v. \int _{\mathbb {T}^2\times \mathbb {T}^2} \frac{\Psi (x,\tau )(\psi (x,\tau )-\psi (y,\tau ))}{|x-y-n|^{2+\beta }}\,\mathrm{d}y\mathrm{d}x\\&= \! \frac{C_\beta }{2}\!\sum _{n\in \mathbb {Z}^2} p.v.\!\!\int _{\mathbb {T}^2\times \mathbb {T}^2} \!\frac{(\Psi (x,\tau )-\Psi (y,\tau ))(\psi (x,\tau )-\psi (y,\tau ))}{|x-y-n|^{2+\beta }}\,\mathrm{d}y\mathrm{d}x, \end{aligned}$$

where \(p.v.\) denotes principal value. Since the integrand on the right-hand side is non-negative, we have by keeping only the term with \(n=(0,0)\),

$$\begin{aligned} \int \Psi \Lambda ^\beta \psi \,\mathrm{d}x&\ge \frac{C_\beta }{2} p.v. \int _{\mathbb {T}^2\times \mathbb {T}^2} \frac{(\Psi (x,\tau )-\Psi (y,\tau ))(\psi (x,\tau )-\psi (y,\tau ))}{|x-y|^{2+\beta }}\,\mathrm{d}x\mathrm{d}y\\&\ge \frac{C_\beta }{2} \int _{|x-y|\le r}\frac{(\Psi (x,\tau )-\Psi (y,\tau ))(\psi (x,\tau )-\psi (y,\tau ))}{|x-y|^{2+\beta }}\,\mathrm{d}x\mathrm{d}y. \end{aligned}$$

For \(|x-y|\le r\),

$$\begin{aligned} |x-y|^{-2-\beta } \ge r^{-\beta } r^{-2} \ge r^{-\beta }\,\rho _r(x-y), \end{aligned}$$

where \(\rho _r\) is the standard mollifier defined in (5.3). Since \(\Psi \psi = |\psi |^p\),

$$\begin{aligned}&\int \!\!\Psi \Lambda ^\beta \psi \,\mathrm{d}x\nonumber \\&\quad \ge \frac{C_\beta }{2}\!\int _{\mathbb {T}^2\times \mathbb {T}^2}(\Psi (x,\tau )\!-\!\Psi (y,\tau ))(\psi (x,\tau )\!-\!\psi (y,\tau ))\, r^{-\beta } \rho _r(x\!-\!y)\,\mathrm{d}x\mathrm{d}y \nonumber \\&\quad = C_\beta \int _{\mathbb {T}^2\times \mathbb {T}^2} |\psi (x)|^p \,r^{-\beta } \rho _r(x-y) \mathrm{d}x\mathrm{d}y \nonumber \\&\qquad - \,\,C_\beta \int _{\mathbb {T}^2\times \mathbb {T}^2} \Psi (y,\tau )\,\psi (x,\tau )\,r^{-\beta } \rho _r(x-y) \mathrm{d}x\mathrm{d}y. \end{aligned}$$
(5.23)

Using the fact that \(\Vert \rho _r\Vert _{L^1}=1\), we have

$$\begin{aligned}&\int _{\mathbb {T}^2\times \mathbb {T}^2} |\psi (x, \tau )|^p\, r^{-\beta } \rho _r(x-y)\mathrm{d}x\mathrm{d}y\nonumber \\&\quad =r^{-\beta }\,\int _{\mathbb {T}^2} |\psi (x, \tau )|^p \int _{\mathbb {T}^2} \rho _r(x-y) \mathrm{d}y \mathrm{d}x \nonumber \\&\quad = r^{-\beta } \Vert \psi (\tau )\Vert _{L^p}^p. \end{aligned}$$
(5.24)

Furthermore,

$$\begin{aligned}&\left| \int _{\mathbb {T}^2\times \mathbb {T}^2} \Psi (y,\tau )\,\psi (x,\tau )\,r^{-\beta } \rho _r(x-y) \mathrm{d}x\mathrm{d}y\right| \\&\quad =r^{-\beta }\left| \int _{\mathbb {T}^2} \psi (x,\tau ) \int _{\mathbb {T}^2} \rho _r(x-y)\Psi (y,\tau ) \mathrm{d}y\,\mathrm{d}x\right| \\&\quad =r^{-\beta }\left| \int _{\mathbb {T}^2} \psi (x,\tau )\,(\rho _r*\Psi )(x)\,\mathrm{d}x\right| . \end{aligned}$$

This term is bounded by invoking (5.1). By taking suitable \(\delta \) and \(\sigma \) such that, for \(s\le r^{\beta }\),

$$\begin{aligned} e^{\delta ((\sigma ^{-1}-1) s r^{-\beta }} \le 2. \end{aligned}$$

If we set \(F= \rho _r*\Psi \) and \(f=F/\Vert \nabla F\Vert _{L^\infty }\), then \(f\in Lip(1)\) and we obtain by applying (5.1)

$$\begin{aligned} \left| \int _{\mathbb {T}^2\times \mathbb {T}^2} \Psi (y,\tau )\,\psi (x,\tau )\,r^{-\beta } \rho _r(x-y) dxdy\right|&\le r^{-\beta }\, 2r\, \Vert \nabla F\Vert _{L^\infty }\\&\le 2 r^{1-\beta } \Vert \nabla \rho _r*\Psi \Vert _{L^\infty }\\&\le 2 r^{1-\beta } \Vert \nabla \rho _r\Vert _{L^p} \, \Vert \Psi \Vert _{L^q}\\&\le C\, r^{1-\beta }\, r^{-1-\frac{2}{q}} \Vert \psi \Vert _{L^p}^{p-1}. \end{aligned}$$

If \(\Vert \psi \Vert _{L^p} \le \frac{1}{2} A^{\frac{1}{p}} r^{-\frac{2}{q}}\), then (5.2) is already proven. If \(\Vert \psi \Vert _{L^p} \ge \frac{1}{2} A^{\frac{1}{p}} r^{-\frac{2}{q}}\), then

$$\begin{aligned} \left| \int _{\mathbb {T}^2\times \mathbb {T}^2} \Psi (y,\tau )\,\psi (x,\tau )\,r^{-\beta } \rho _r(x-y) dxdy\right| \le C\, A^{-\frac{1}{p}}\, r^{-\beta } \Vert \psi \Vert _{L^p}^{p}. \end{aligned}$$
(5.25)

Inserting (5.24) and (5.25) in (5.23), we obtain

$$\begin{aligned} \int \Psi (x,\tau ) \Lambda ^\beta \psi (x,\tau ) \,dx \ge (1-C\, A^{-\frac{1}{p}})\, r^{-\beta } \Vert \psi \Vert _{L^p}^{p}. \end{aligned}$$

It then follows from integrating (5.22) that

$$\begin{aligned} \Vert \psi (\cdot ,t-s)\Vert _{L^p} \le e^{-s(1-C\, A^{-\frac{1}{p}})\, r^{-\beta }}\, \Vert \psi (\cdot ,t)\Vert _{L^p}. \end{aligned}$$

Since \(\psi (x,t) \in \mathcal {U}(r)\), in particular, \(\Vert \psi (\cdot ,t)\Vert _{L^p} \le A^{\frac{1}{p}} r^{-\frac{2}{q}}\), we have

$$\begin{aligned} \Vert \psi (\cdot ,t-s)\Vert _{L^p} \le A^{\frac{1}{p}} r^{-\frac{2}{q}} e^{-(1-C\, A^{-\frac{1}{p}}) \, s\,r^{-\beta }} \le A^{\frac{1}{p}} r^{-\frac{2}{q}}\, e^{-(1+\frac{2}{q\sigma })\delta s r^{-\beta }} \end{aligned}$$

when \((1+\frac{2}{q\sigma })\delta \le (1-C\, A^{-\frac{1}{p}})\). This proves (5.2). We thus have completed the proof of Proposition 4.3. \(\square \)

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Jiu, Q., Wu, J. & Yang, W. Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation. J Nonlinear Sci 25, 37–58 (2015). https://doi.org/10.1007/s00332-014-9220-y

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