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1 Introduction↩︎

In this paper, we consider the dissipative surface quasi-geostrophic equation. \[\label{SQG} \begin{cases} \partial_t \theta + \Lambda^\alpha \theta + ({\boldsymbol{u}} \cdot \nabla) \theta = 0, &\quad \; t>0,x\in{\mathbb{R}}^2, \\ {\boldsymbol{u}} = \nabla ^\perp \Lambda^{-1}\theta, &\quad \;t>0,x\in{\mathbb{R}}^2,\\ \theta(0,x) = \theta_0(x), &\quad \;x\in{\mathbb{R}}^2,\\ \end{cases}\tag{1}\] where \(\Lambda^{\alpha} = (-\Delta)^{\frac{\alpha}{2}}\), \(1\leq \alpha\leq 2\) and \(\nabla^{\perp} = (-\partial_2, \partial_1)\). The fractional Laplacian is defined as follows using the Fourier transform: \[\Lambda^\alpha \theta (x) = \mathcal{F}^{-1}[|\xi|^\alpha \widehat{\theta}(\xi)](x).\] The real-valued unknown function \(\theta(t,x)\) denotes the potential temperature and \({\boldsymbol{u}}(t,x)\) denotes the velocity of a fluid. The surface quasi-geostrophic equation is derived from the quasi-geostrophic approximation of the three-dimensional Navier-Stokes equations and the assumption that the interior potential vorticity and the buoyancy frequency are constant. It is an evolution equation for the temperature at the boundary in the vertical direction and is known to be an important model in geophysical dynamics. For a detailed derivation of the equation and its physical background, we refer the reader to the papers Co_2002?, He_Pi_Ga_Sw_1995?, Pe_1979?. From a mathematical point of view, the surface quasi-geostrophic equation has similar structure to the three-dimensional Euler and Navier-Stokes equations (see Co_Ma_Ta_1994?, Co_Ma_Ta_1994_2?). Also, when \(\alpha = 2\), the scale-critical spaces of 1 are the same as those of the two-dimensional incompressible Navier-Stokes equations.

We briefly discuss the scale-critical spaces of 1 . If \(\theta\) is a solution of 1 , then for \(\lambda>0\), \[\theta_{\lambda}(t,x) := \lambda^{\alpha-1}\theta(\lambda^{\alpha}t, \lambda x)\] is also a solution of 1 . Namely, 1 is invariant under this scaling. Then, if a function space \(X\) satisfies \[{\|\theta_{\lambda}(0,\cdot)\|}_{X} = {\|\theta(0,\cdot)\|}_{X},\] then \(X\) is called a scale-critical space of 1 . It is known that scale-critical spaces play an important role in the study of well-posedness for initial value problems of partial differential equations (see, e.g.,Ko_Ta_2001?, Bo_Pa_2008?). When \(1\leq \alpha \leq 2\), the scale-critical space in the framework of Lebesgue spaces is \(L^{\frac{2}{\alpha-1}}(\mathbb{R}^{2})\). In particular, when \(\alpha=1\), the scale-critical space is \(L^{\infty}\). On the other hand, in the Besov space setting, the homogeneous Besov norm \({\|\cdot\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{d})}\) satisfies the following scaling property: \[c\lambda^{s-\frac{d}{p}}{\|f\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{d})} \leq {\|f(\lambda \cdot)\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{d})} \leq C\lambda^{s-\frac{d}{p}}{\|f\|}_{\dot{B}_{p,q}^{s}(\mathbb{R}^{d})}.\] Thus, \(\dot{B}_{p,q}^{1 - \alpha +\frac{2}{p}}(\mathbb{R}^{2})\) is a scale critical space of 1 . In this paper, we also refer to the non-homogeneous Besov space \(B_{p,q}^{1 - \alpha +\frac{2}{p}}(\mathbb{R}^{2})\) as a scale-critical space of 1 .

We begin by recalling several known results. Here, we consider \(0< \alpha \leq 2\). From the balance between the nonlinearity and the dissipation, the cases \(\alpha>1\), \(\alpha = 1\), \(\alpha<1\) are called the sub-critical case, critical case and super-critical case respectively. In the sub-critical case, the global-in-time regularity for large initial data was obtained by Constantin and Wu Co_Wu_1999?. Also, Carrillo and Ferreira Ca_Fe_2008? proved the global well-posedness of 1 in the scale-critical Lebesgue spaces \(L^{\frac{2}{\alpha-1}}(\mathbb{R}^2)\). In the critical case, Constantin, Cordoba and Wu Co_Co_Wu_2001? proved the global existence and regularity for small data. Later, large initial data case is solved by Kiselev, Nazarov and Volberg Ki_Na_Vo_2007? and Caffarelli and Vasseur Ca_Va_2010?. On the other hand, in the super-critical case, global-in-time regularity for large initial data still remains an open problem. For small initial data, global regularity is known (see e.g. Co_Vi_2016?).

Concerning the uniqueness of the solution, in the sub-critical case, the uniqueness in the scale-critical Lebesgue space \(C([0,T]; L^{\frac{2}{\alpha-1}})\) was established by Ferreira Fe_2011? (\(1<\alpha<2\)) and Iwabuchi and Ueda I_U_2024? (\(\alpha=2\)). Recently, for \(0<\alpha\leq 2\), Iwabuchi and Okazaki I_O_2026? proved that the uniqueness of solutions holds in scale-critical non-homogeneous Besov spaces \(C([0,T]; B_{p,q}^{1-\alpha+\frac{2}{p}})\) with \(2/(\alpha-1)\leq p\leq \infty\), \(1\leq q < \infty\) and \(1-\alpha+2/p>-1/2\). Note that \(-1/2\) is characterized as the least regularity for which the nonlinear term in 1 is well defined. As for non-uniqueness, if \(0<\alpha<3/2\), then Buckmaster, Shkoller and Vicol Bu_Sh_Vi_2019? proved the non-uniqueness of weak solutions in \(L_{\text{loc}}^2(0,T; \dot{H}^{-\frac{1}{2}}(\mathbb{T}^2))\).

We also refer to some literature on the Navier-Stokes equations. For the two-dimensional incompressible Navier-Stokes equations, it is a classical result that the weak solution \({\boldsymbol{u}}\in L^\infty(0,T; L^2)\cap L^2(0,T; \dot{H}^{1})\) is unique if \({\boldsymbol{u}}_0\in L^2\) (see, e.g., Te_1995? and the references therein). The uniqueness in the scale-critical Lebesgue space \(C([0,T]; L^{2})\) does not seem to be resolved. Regarding the non-uniqueness result, Cheskidov and Luo Ch_Lu_2023? established that the non-uniqueness in \(C([0,T]; L^{p}(\mathbb{T}^{2}))\) with \(1\leq p <2\). In the case when the space dimension is three, the uniqueness of the solution in the scale-critical space \(C([0,T];L^{3})\) was proved by Meyer Me_1997?, Furioli, Lemarié-Rieusset and Terraneo Fu_Le_Te_1997? and Monniaux Mo_1999?. Also, Lions and Masmoudi Li_Ma_2001? proved that the uniqueness in \(L^{3}\) using a dual problem. The non-uniqueness in \(C([0,T]; L^{2}(\mathbb{T}^{3}))\) was shown by Buckmaster and Vicol Bu_Vi_2019?. Recently, if \(d \geq 3\), then Fujii Fu_2026? established that the non-uniqueness of the \(d\)-dimensional incompressible Navier-Stokes equations in scale-critical homogeneous Besov space \(C([0,T]; B_{p,q}^{-1 + \frac{d}{p}}(\mathbb{R}^{d}))\), where \(B_{p,q}^{-1 + \frac{d}{p}}(\mathbb{R}^{d})\) is strictly larger than \(L^{d}(\mathbb{R}^{d})\). For solutions that are not continuous in time, if \(d\geq 4\), then the uniqueness of the \(n\)-dimensional incompressible Navier-Stokes equations in \(L^{\infty}(0,T; L^{d})\) is proved in Li_Ma_2001?.

In this work, when the sub-critical case, we prove that solutions of 1 are unique in scale-critical Lebesgue space \(L^{\infty}(0,T; L^{\frac{2}{\alpha-1}})\) without smallness assumptions. In addition, we include the critical case \(\alpha = 1\) and establish the uniqueness of solutions of 1 in the scale-critical non-homogeneous Besov spaces \(B_{p,q}^{1 - \alpha + \frac{2}{p}}\).

We first present the setting for the main result in the Lebesgue space setting. In this paper, the solution of 1 is defined as follows.

We next define the weak solution of 1 .

Actually, it is known that 2 and 3 are equivalent.

Proposition 1 can be proved using the same method as the corresponding equivalence proposition for the Navier-Stokes equations in Fabes, Jones and Riviére Fa_Jo_Ri_1972? (see also Li_Ma_2001?). For the convenience of the reader, we provide a proof of Proposition 1 in Appendix 4.

We state our main results in Lebesgue spaces.

As in the previous work I_O_2026?, we can extend Theorem 2 to Besov spaces with negative regularity index. To state this result, let us recall the definition of non-homogeneous Besov spaces. We refer to the book by Bahouri, Chemin and Danchin Ba_Ch_Da_2011?. Let us first introduce Littlewood-Paley decomposition.

Based on this, we define the non-homogeneous Besov spaces.

When attempting to define a solution of 1 in the sense of Definition 1 in Besov spaces with negative regularity index, we need to ensure that the product \((\nabla^{\perp}\Lambda^{-1}\theta)\theta\) is well-defined. Indeed, it is known that if \(s<0\), then there exist some \(f,g \in B_{p,q}^{s}\) such that \[fg\notin \mathcal{S}'.\] At this point, we consider the sum of two products \[(\nabla^{\perp}\Lambda^{-1}f)g + (\nabla^{\perp}\Lambda^{-1}g)f.\] It can be written in teams of first-order derivatives (see I_U_2024?). As a result, we can define \((\nabla^{\perp}\Lambda^{-1}f)g + (\nabla^{\perp}\Lambda^{-1}g)f\) in \(\mathcal{S}'\) for any \(f,g\in B_{p,q}^{s}\) with \(-1/2<s<0\), \(2\leq p \leq \infty\) and \(1\leq q\leq \infty\) (see I_O_2026?).

Thus, we can define a solution of 1 (as well as a weak solution) in scale-critical homogeneous Besov spaces provided that the product \((\nabla^{\perp}\Lambda^{-1}\theta)\theta\) is well-defined.

Then, in the sub-critical case, the following results holds.

If \(p< 4/(\alpha-1)\), then we can relax the assumption on the interpolation index \(q\).

In the critical case, the uniqueness of solutions of 1 holds in the scale-critical Besov spaces with finite integrability index and positive regularity index.

We now present an outline of the proof of our main results. Our proof is motivated by the method employed in Lions and Masmoudi Li_Ma_2001? to establish the uniqueness of the Navier-Stokes equations in \(L^{\infty}(0,T; L^{d}(\mathbb{R}^{d}))\) (\(d \geq 4\)). In Li_Ma_2001?, their approach relies on the energy equality that holds for the Navier-Stokes equations, \[\partial_t{\boldsymbol{u}} - \Delta{\boldsymbol{u}} + ({\boldsymbol{u}}\cdot \nabla){\boldsymbol{u}} + \nabla p = 0, \quad \nabla\cdot {\boldsymbol{u}}=0.\] If \({\boldsymbol{u}}, {\boldsymbol{v}}\) are solutions of the Navier-Stokes equations, then \({\boldsymbol{u}} - {\boldsymbol{v}}\) satisfies \[\partial_t({\boldsymbol{u}}-{\boldsymbol{v}}) - \Delta({\boldsymbol{u}}-{\boldsymbol{v}}) + ({\boldsymbol{u}}\cdot \nabla)({\boldsymbol{u}}-{\boldsymbol{v}}) + \big(({\boldsymbol{u}}-{\boldsymbol{v}})\cdot \nabla\big){\boldsymbol{v}} + \nabla p = 0.\] Thus, by the energy method, \[\frac{1}{2}\frac{\rm d}{{\rm d}t}{\|({\boldsymbol{u}}- {\boldsymbol{v}})(t)\|}_{L^2}^2 + {\|\nabla({\boldsymbol{u}}-{\boldsymbol{v}})(t)\|}_{L^2}^2 \leq -\int_{\mathbb{R}^{d}}({\boldsymbol{u}}- {\boldsymbol{v}})\big(({\boldsymbol{u}}-{\boldsymbol{v}})\cdot \nabla\big){\boldsymbol{v}}~{\rm d}x\] holds, which implies \({\boldsymbol{u}}\) does not appear in the estimate. Hence, Lions and Masmoudi Li_Ma_2001? assumed that \({\boldsymbol{u}}\in L^{\infty}(0,T; L^{d}(\mathbb{R}^{d}))\) and \({\boldsymbol{v}}\in C([0,T]; L^{d}(\mathbb{R}^{d}))\), and that \({\boldsymbol{u}}\), v have same initial data. And they proved the uniqueness by using a decomposition for solutions that are continuous in time.

The crucial point in the above argument is the justification of the energy inequality, that is, it is necessary to verify that \[\label{0215-2} {\boldsymbol{u}} - {\boldsymbol{v}}\in L^{\infty}(0,T; L^{2})\cap L^2(0,T; \dot{H}^{1})\tag{4}\] holds. Indeed, Li_Ma_2001? established that 4 holds for \({\boldsymbol{u}}, {\boldsymbol{v}}\in L^{\infty}(0,T; L^{d}(\mathbb{R}^{d}))\) with \(d\geq 4\) by using the smoothing effect of the Stokes operator combined with an iteration argument.

We briefly explain why dimension constraints \(d \geq 4\) arise. From the well-known Sobolev embedding, the following holds, \[H^{1}(\mathbb{R}^{d})\hookrightarrow W^{2-\frac{d}{2}, n}(\mathbb{R}^{d}),\] Thus, \({\boldsymbol{u}} - {\boldsymbol{v}}\in L^{2}(0,T; W^{2-\frac{d}{2}, d}(\mathbb{R}^{d}))\) is a necessary condition for 4 . Note that if \(d\geq 4\), then \(2-d/2\leq 0\). On the other hand, we consider the following Duhamel term for the Navier-Stokes equations \[\label{0215-3} \int_{0}^{t}e^{(t-s)\Delta}\mathbb{P}\big(\nabla\cdot ({\boldsymbol{u}}\otimes{\boldsymbol{u}})\big)~{\rm d}s,\tag{5}\] where \(\mathbb{P}\) is the Helmholtz projection. Since the smoothting effect of the Stokes operator, we can show that if \(d\geq4\), then for any \({\boldsymbol{u}}\in L^{\infty}(0,T; L^{d}(\mathbb{R}^{d}))\), we have \[\int_{0}^{t}e^{(t-s)\Delta}\mathbb{P}\big(\nabla\cdot ({\boldsymbol{u}}\otimes{\boldsymbol{u}})\big)~{\rm d}s \in L^{2}(0,T; W^{2-\frac{d}{2}, d}(\mathbb{R}^{d})),\] which implies \({\boldsymbol{u}} - {\boldsymbol{v}}\in L^{2}(0,T; W^{2-\frac{d}{2}, d}(\mathbb{R}^{d}))\). In contrast, when \(d = 2,3\), the Duhamel term 5 belonging to \(L^{2}(0,T; W^{2-\frac{d}{2}, d}(\mathbb{R}^{d}))\) is not generally guaranteed. From the above, it seems that the following condition \[\label{0215-5} H^{1}(\mathbb{R}^{d})\hookrightarrow W^{2-\frac{d}{2}, d}(\mathbb{R}^{d}), \;2-\frac{d}{2}\leq 0.\tag{6}\] can be regarded as one of the criteria for determining whether the uniqueness can be established using the present method. In fact, if we consider the fractional Navier-Stokes equations \[\partial_t{\boldsymbol{u}} + \Lambda^{\alpha}{\boldsymbol{u}} + ({\boldsymbol{u}}\cdot \nabla){\boldsymbol{u}} + \nabla p = 0, \quad \nabla\cdot {\boldsymbol{u}}=0,\] then the scale-critical Lebesgue space is \(L^{\frac{d}{\alpha-1}}(\mathbb{R}^{d})\), and the energy inequality \[\frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|{\boldsymbol{u}}(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}}{\boldsymbol{u}}(t)\|}_{L^2}^2 \leq 0\] is justified under the condition \[{\boldsymbol{u}} \in L^{\infty}(0,T; L^{2})\cap L^{2}(0,T; \dot{H}^{\frac{\alpha}{2}}).\] In this case, the condition corresponding to 6 is \[H^{\frac{\alpha}{2}}(\mathbb{R}^{d})\hookrightarrow W^{\frac{3}{2}\alpha - 1 - \frac{d}{2}, \frac{d}{\alpha-1}}(\mathbb{R}^{d}), \;\frac{3}{2}\alpha - 1-\frac{d}{2}\leq 0.\] Then, under the this condition, the uniqueness of solutions of fractional Navier-Stokes equations in \(L^{\infty}(0,T; L^{\frac{d}{\alpha-1}}(\mathbb{R}^{d}))\) can be proved using a method similar to that of Li_Ma_2001?. Regarding the range of \(\alpha\), for instance, it is \(\alpha\leq 4/3\) when \(d = 2\), and \(\alpha\leq 5/3\) when \(d = 3\).

In light of this, we discuss the energy equality in the surface quasi-geostrophic equation and the exponent \(\alpha = 5/3\) in our theorem. Since \({\boldsymbol{u}} = \nabla^{\perp}\Lambda^{-1}\theta\) satisfies the imcompressibility condition, the energy inequality also holds for 1 , \[\label{0215-4} \frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|\theta(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}}\theta(t)\|}_{L^2}^2 \leq 0.\tag{7}\] In this case, the condition necessary to justify energy equality 7 is \[\theta \in L^{\infty}(0,T; L^{2})\cap L^{2}(0,T; \dot{H}^{\frac{\alpha}{2}}),\] this is the same as in the case of two-dimensional fractional Navier-Stokes equations. On the other hand, since \(\nabla^{\perp}\cdot \nabla f=0\), we can express the nonlinear term in 1 as follows, \[({\boldsymbol{u}}\cdot \nabla)\theta = \left((\nabla^{\perp}\Lambda^{-1}\theta)\cdot \nabla\right)\theta = \nabla^{\perp}\cdot \left((\Lambda^{-1}\theta)\nabla \theta\right).\] From this, the following equality hold (see, e.g., Ma_2008?), \[\label{0215-6} \frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|\Lambda^{-\frac{1}{2}}\theta(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}-\frac{1}{2}}\theta(t)\|}_{L^2}^2 \leq 0.\tag{8}\] Then the condition required to justify 8 is \[\theta \in L^{\infty}(0,T; \dot{H}^{-\frac{1}{2}})\cap L^{2}(0,T; \dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}),\] and the condition corresponding to 6 is \[H^{\frac{\alpha}{2}-\frac{1}{2}}(\mathbb{R}^{2})\hookrightarrow W^{\frac{3}{2}\alpha - \frac{5}{2}, \frac{2}{\alpha-1}}(\mathbb{R}^{2}), \;\frac{3}{2}\alpha - \frac{5}{2}\leq 0\] Clearly, \[\frac{3}{2}\alpha - \frac{5}{2}\leq 0 \text{ if and only if } \alpha\leq 5/3,\] and the range of \(\alpha\) is wider than that of two-dimensional fractional Navier-Stokes equations.

The paper is organized as follows. In Section 2, we recall several lemmas and justify the energy inequality 8 for the solution defined in this paper. In Section 3, we prove the main theorems using the energy method together with bilinear estimates exploiting the structure of the nonlinear term in 1 . Finally, in Appendix 4, we show the equivalence of the two solutions defined in this paper by using the Littlewood-Paley decomposition.

2 Preliminarys↩︎

In this section, we introduce several lemmas which are elemental properties of the fractional heat semigroup, homogeneous Sobolev spaces and homogeneous Besov spaces. Using these lemmas, we show that the difference of two solutions of 1 belongs to \(L^{\infty}(0,T; \dot{H}^{-\frac{1}{2}})\cap L^{2}(0,T; \dot{H}^{\frac{\alpha}{2}-\frac{1}{2}})\).

2.1 Basic lemmas↩︎

The following lemma gives an \(L^{p} - L^{q}\) estimate for the fractional heat semigroup.

We define the homogeneous Sobolev spaces via Fourier multipliers (see Tr_1983?).

We often use the equivalent norm of \(\dot{H}^{s}\), which is given by the Littlewood-Paley decomposition. \[\label{0406-1} c{\|f\|}_{\dot{H}^{s}} \leq {\left\|\left\{2^{sj}{\left\|\phi_{j}*f\right\|}_{L^{2}}\right\}_{j\in\mathbb{Z}}\right\|}_{l^{2}} \leq C{\|f\|}_{\dot{H}^{s}}.\tag{9}\]

The next lemma is the bilinear estimate in the homogeneous Sobolev spaces.

The following lemmas provide basic estimates for frequency-localized functions and fundamental properties of Besov spaces.

The following lemma describes the smoothing effect of the fractional heat semigroup in the non-homogeneous Besov spaces. The case \(\alpha = 2\) was established by Kozono, Ogawa and Taniuchi Ko_Og_Ta_2003?. When \(0<\alpha <2\), the proof is similar to that of Zhai Zh_2010? in the homogeneous Besov spaces.

We recall several lemmas on bilinear estimate in non-homogeneous Besov spaces.

The next lemma establishes an bilinear estimate in Besov spaces with negative regularity indices, based on the derivative structure of the nonlinear terms in 1 .

At the end of this subsection, we describe the frequency decomposition of functions used in the proof of the main theorem. The following lemmas follows immediately from the definition of Besov spaces.

By density, the following also holds in Lebesgue spaces.

2.2 Justification of the energy inequality in Lebesgue spaces↩︎

In this subsection, we established a crucial step toward Theorem 2. We justify the energy inequality for the difference between two solutions of the integral equation of 1 in the sense of Definition 1 with same initial data by estimating the Duhamel term and performing an iteration based on the structure of the integral equation.

We prove the following proposition.

To prove the proposition, we need the following lemmas concerning estimates of Duhamel terms.

Applying an argument similar to that in Lemma 23, we estimate Duhamel terms in \(L^{2}(0,T; \dot{H}^{\frac{\alpha}{2}-\frac{1}{2}})\). Then the following exponent appears in the estimate appears in the estimate corresponding to 17 , \[-\frac{1}{\alpha}\left(\frac{\alpha}{2} + \frac{1}{2} - 3 + 2\alpha\right) = -\frac{5}{2\alpha}\left(\alpha - 1\right).\] Clearly, \(- 5(\alpha - 1)/2\alpha > -1\) if and only if \(\alpha < 5/3\). That is, the following lemma holds.

In case \(\alpha = 5/3\), we need to work in Besov spaces.

The following lemma follows from \(\nabla\cdot \nabla^{\perp}f = 0\) and Lemma 6.

Using the above lemmas, we prove Proposition 22.

2.3 Justification of the energy inequality in Besov spaces↩︎

We next consider the case where the solution belongs to Besov spaces.

Under the assumptions of Theorem 3, the following proposition holds.

We state lemmas needed to prove Proposition 27.

Thanks to Lemma 14 and using a similar product estimate to Lemma 28, we obtain following lemma.

We now prove Proposition 27.

The following Proposition will be used in the proof of Theorem 4.

Proposition 30 can be proved by a similar argument to Proposition 27. We state the lemmas required for the proof and omit the proof of Proposition 30.

The following lemma can be proved by the same method as in Lemma 28.

In the proof of Proposition 30 [0417-2], since the interpolation index is infinite, we use the next lemma instead of Lemma 29.

In the case \(\alpha = 1\), the following proposition holds.

Proposition 33 follows by the same argument as above, based on estimate of the Duhamel term and an iteration argument. Accordingly, as in Proposition 30, we only state the lemma used in the proof.

The next lemma follows from the same argument as Lemma 32 and the fact that \(B_{p,\infty}^{\frac{2}{p}} \hookrightarrow L^{p}\).

3 Proof of theorems↩︎

3.1 Proof of Theorem 2↩︎

By the (global) existence result (Ca_Fe_2008?), there exists \(\bar{\theta} \in C([0,T] ; L^{\frac{2}{\alpha-1}})\) a solution of 1 with \(\bar{\theta}(0) = \theta_0\) and satisfies 2 . We prove that \[\theta^{(1)}(t) = \bar{\theta}(t) \text{ and } \theta^{(2)}(t) = \bar{\theta}(t) \text{ in }L^{\frac{2}{\alpha-1}} \text{ for a.e }t\in (0, T).\] This implies Theorem 2. It is sufficient to show that \(\theta^{(1)}(t) = \bar{\theta}(t)\). Let \[w := \theta^{(1)} - \bar{\theta}.\] Since \(\theta^{(1)}, \bar{\theta}\) fulfill 3 (see also Proposition 1), \(w\) satisfies \[\begin{cases} \partial_t w + \Lambda^{\alpha} w + ((\nabla^{\perp}\Lambda^{-1}\theta^{(1)})\cdot\nabla)\theta^{(1)} - ((\nabla^{\perp}\Lambda^{-1}\bar{\theta})\cdot\nabla)\bar{\theta} = 0,\\ w(0,x) = 0, \end{cases}\] in the sense of distributions, and we can write \[((\nabla^{\perp}\Lambda^{-1}\theta^{(1)})\cdot\nabla)\theta^{(1)} - ((\nabla^{\perp}\Lambda^{-1}\bar{\theta})\cdot\nabla)\bar{\theta} = ((\nabla^{\perp}\Lambda^{-1}w)\cdot\nabla)\theta^{(1)} + ((\nabla^{\perp}\Lambda^{-1}\bar{\theta})\cdot\nabla)w\] Moreover, using Proposition 22, \(w \in L^{\infty}(0,T ; \dot{H}^{-\frac{1}{2}})\cap L^2(0,T ; \dot{H}^{\frac{\alpha}{2}-\frac{1}{2}})\). Thus, we can justify the energy inequality 8 i.e., \(w\) satisfies \[\begin{align} \label{0212-4} &\frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|\Lambda^{-\frac{1}{2}}w(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}-\frac{1}{2}}w(t)\|}_{L^2}^2\notag\\ \leq& -\int_{\mathbb{R}^2}\Big(\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)\theta^{(1)}(t) + \big((\nabla^{\perp}\Lambda^{-1}\bar{\theta}(t))\cdot \nabla\big) w(t)\Big)(\Lambda^{-1}w(t))~{\rm d}x. \end{align}\tag{19}\] Note that thanks to \(\nabla^{\perp} \cdot \nabla f =0\), we obtain \[\begin{align} &\int_{\mathbb{R}^2}\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)\theta^{(1)}(t)(\Lambda^{-1}w(t))~{\rm d}x\\ =& - \int_{\mathbb{R}^2}\Big(\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)\theta^{(1)}(t)\Big)\Lambda^{-1}w(t)~{\rm d}x, \end{align}\] and we have \[\int_{\mathbb{R}^2}\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)\theta^{(1)}(t)(\Lambda^{-1}w(t))~{\rm d}x = 0.\] We can also write \[\begin{align} \label{0204-5} &- \int_{\mathbb{R}^2}\big((\nabla^{\perp}\Lambda^{-1}\bar{\theta}(t))\cdot \nabla\big)w(t)(\Lambda^{-1}w(t))~{\rm d}x\notag\\ =& \int_{\mathbb{R}^2}(\Lambda^{-1}\bar{\theta}(t))\Big(\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\Big)~{\rm d}x. \end{align}\tag{20}\] We now estimate the right-hand side of 20 . Let \(N\in\mathbb{N}\) be fixed later. We divide \(\bar{\theta}\) as follows, \[\bar{\theta} = \sum_{|j|>N} \phi_j*\bar{\theta} + \sum_{|j|\leq N} \phi_j*\bar{\theta} =: {\bar{\theta}}_{>N} + {\bar{\theta}}_{\leq N}.\] By Lemma 20, Lemma 21 and continuity of \(\bar{\theta}\) with respect to time, for any \(\varepsilon>0\), there exists \(N\in \mathbb{N}\) such that \[\sup_{t\in (0, T)}{\|{\bar{\theta}}_{>N}(t)\|}_{L^{\frac{2}{\alpha-1}}}\leq \varepsilon,\] and there exists constant \(C>0\) such that \[\sup_{t\in (0, T)}{\|{\bar{\theta}}_{\leq N}(t)\|}_{W^{\alpha - \frac{1}{2}, \frac{2}{\alpha-1}}}\leq C.\] By Hölder’s inequality, we have \[\begin{align} &\int_{\mathbb{R}^2}(\Lambda^{-1}\bar{\theta}(t))\Big(\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\Big)~{\rm d}x\\ \leq& \sup_{t\in (0, T)}{\|{\bar{\theta}}_{>N}(t)\|}_{L^{\frac{2}{\alpha-1}}}{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{\dot{W}^{-1, \frac{2}{3-\alpha}}}\\ &+\sup_{t\in (0, T)}{\|{\bar{\theta}}_{\leq N}(t)\|}_{W^{\alpha - \frac{1}{2}, \frac{2}{\alpha-1}}}{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{W^{-\alpha - \frac{1}{2}, \frac{2}{3-\alpha}}}\\ \leq& \varepsilon {\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{\dot{W}^{-1, \frac{2}{3-\alpha}}} + C{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{W^{-\alpha - \frac{1}{2}, \frac{2}{3-\alpha}}}. \end{align}\] Using \(\nabla\cdot \nabla^{\perp}=0\) and Sobolev embedding \(\dot{W}^{\frac{\alpha}{2}-\frac{1}{2}, p}(\mathbb{R}^2)\hookrightarrow L^{\frac{2}{3-\alpha}}(\mathbb{R}^2)\), where \[\frac{1}{p} = \frac{3-\alpha}{2} + \frac{1}{2}\left(\frac{\alpha}{2} - \frac{1}{2}\right) = \frac{5-\alpha}{4},\] we get \[{\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\|}_{\dot{W}^{-1, \frac{2}{3-\alpha}}} \leq C{\|(\nabla^{\perp}\Lambda^{-1}w)w\|}_{\dot{W}^{\frac{\alpha}{2}-\frac{1}{2}, \frac{4}{5-\alpha}}}.\] Now \(0<\alpha/2 - 1/2\) and \(1<4/(5-\alpha)\). Thus, by Lemma 7 and boundedness of Riesz transform in \(L^p\) (\(1<p<\infty\)), we obtain \[\begin{align} {\|(\nabla^{\perp}\Lambda^{-1}w)w\|}_{\dot{W}^{\frac{\alpha}{2}-\frac{1}{2}, \frac{4}{5-\alpha}}} &\leq C({\|\nabla^{\perp}\Lambda^{-1}w\|}_{L^{\frac{4}{3-\alpha}}}{\|w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}} + {\|\nabla^{\perp}\Lambda^{-1}w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}{\|w\|}_{L^{\frac{4}{3-\alpha}}})\\ &\leq C {\|w\|}_{L^{\frac{4}{3-\alpha}}}{\|w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}. \end{align}\] Since \(2<4/(3-\alpha)\), we have \[\dot{W}^{\frac{\alpha}{2} - \frac{1}{2}, p}(\mathbb{R}^2)\hookrightarrow L^{\frac{4}{3-\alpha}}(\mathbb{R}^2), \text{ where } \frac{1}{p} = \frac{3-\alpha}{4} + \frac{1}{2}\left(\frac{\alpha}{2} - \frac{1}{2}\right) = \frac{1}{2}.\] Hence we obtain \[{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{\dot{W}^{-1, \frac{2}{3-\alpha}}} \leq C {\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}^2.\] We now estimate \[{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{W^{-\alpha - \frac{1}{2}, \frac{2}{3-\alpha}}}.\] Using Lemma 11 12 , we have \[{\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\|}_{W^{-\alpha - \frac{1}{2} , \frac{2}{3-\alpha}}} \leq C {\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}.\] By Bony’ s decomposition, we get \[\begin{align} &{\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ \leq& {\left\|\sum_{l\geq 2}\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\phi_l*w) + \sum_{k\leq l - 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}} \\ &+ {\left\|\sum_{k\geq 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_{k}*w))\cdot \nabla\big)(\psi*w) + \sum_{l\leq k - 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ &+ {\left\|\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\psi*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ &+ {\left\|\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\phi_{1}*w) + \big((\nabla^{\perp}\Lambda^{-1}(\phi_{1}*w))\cdot \nabla\big)(\psi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ &+ {\left\|\sum_{|k-l|\leq 1}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}. \end{align}\] Using Lemma 8, product estimate (Lemma 15 16 ) and embedding (Lemma 10), we have \[\begin{align} &{\left\|\sum_{l\geq 2}\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\phi_l*w) + \sum_{k\leq l - 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ \leq& C {\left\|\sum_{l\geq 2}(\nabla^{\perp}\Lambda^{-1}(\psi*w))(\phi_l*w) + \sum_{k\leq l - 2}(\nabla^{\perp}\Lambda^{-1}(\phi_k*w))(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha + \frac{1}{2}}}\\ \leq& C {\|\nabla^{\perp}\Lambda^{-1}w\|}_{B_{\frac{2}{2-\alpha}, 2}^{-\alpha + \frac{1}{2}}}{\|w\|}_{B_{2,2}^{0}}\\ \leq& C {\|w\|}_{B_{2, 2}^{-\frac{1}{2}}}{\|w\|}_{L^{2}}, \end{align}\] and \[\begin{align} &{\left\|\sum_{k\geq 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_{k}*w))\cdot \nabla\big)(\psi*w) + \sum_{l\leq k - 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ \leq& C {\|w\|}_{B_{2, 2}^{-\frac{1}{2}}}{\|w\|}_{L^{2}}. \end{align}\] Since it can be written as a finite sum, we obtain \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\psi*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}} \leq C {\|w\|}_{B_{2, 2}^{-\frac{1}{2}}}{\|w\|}_{L^{2}},\] and \[\begin{align} &{\left\|\big((\nabla^{\perp}\Lambda^{-1}(\psi*w))\cdot \nabla\big)(\phi_{1}*w) + \big((\nabla^{\perp}\Lambda^{-1}(\phi_{1}*w))\cdot \nabla\big)(\psi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ \leq& C {\|w\|}_{B_{2, 2}^{-\frac{1}{2}}}{\|w\|}_{L^{2}}. \end{align}\] Note that \(B_{2,2}^{s} = H^s\) (Lemma 12). Using Lemma 10, we get \[\begin{align} &{\left\|\sum_{|k-l|\leq 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{\frac{2}{3-\alpha}, 1}^{-\alpha - \frac{1}{2}}}\\ \leq& C {\left\|\sum_{|k-l|\leq 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{1, 1}^{-\frac{3}{2}}}. \end{align}\] Since we can write \[\begin{align} &\sum_{|k-l|\leq 1}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\\ =& \frac{1}{2}\sum_{|k-l|\leq 1}\Big(\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w) + \big((\nabla^{\perp}\Lambda^{-1}(\phi_l*w))\cdot \nabla\big)(\phi_k*w)\Big), \end{align}\] by Lemma 17, we get \[{\left\|\sum_{|k-l|\leq 2}\big((\nabla^{\perp}\Lambda^{-1}(\phi_k*w))\cdot \nabla\big)(\phi_l*w)\right\|}_{B_{1, 1}^{-\frac{3}{2}}} \leq C {\|w\|}_{L^{2}}{\|w\|}_{H^{- \frac{1}{2}}}.\] Hence, we have \[\frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|\Lambda^{-\frac{1}{2}}w(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}-\frac{1}{2}}w(t)\|}_{L^2}^2 \leq C\varepsilon{\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}^2 + C {\|w(t)\|}_{L^{2}}{\|w(t)\|}_{H^{- \frac{1}{2}}}.\] Thanks to \(\dot{H}^{-s} \hookrightarrow H^{-s}\) \((s>0)\) and interpolation, we obtain \[{\|w(t)\|}_{L^{2}}{\|w(t)\|}_{H^{- \frac{1}{2}}} \leq C {\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}}^{\theta}{\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2} - \frac{1}{2}}}^{1 - \theta}{\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}},\] where \(\theta\in \mathbb{R}\) satisfies \[-\frac{\theta}{2} + (1-\theta)\left(\frac{\alpha}{2} - \frac{1}{2}\right) = 0.\] Note that \(0<\theta<1\). Moreover, we have \[{\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}}^{\theta}{\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2} - \frac{1}{2}}}^{1 - \theta}{\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}} \leq \frac{C}{\varepsilon}{\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}}^{2} + \varepsilon{\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2} - \frac{1}{2}}}^{2}.\] Thus, we obtain \[\begin{align} \frac{1}{2}\frac{{\rm d}}{{\rm d}t}{\|\Lambda^{-\frac{1}{2}}w(t)\|}_{L^2}^2 + {\|\Lambda^{\frac{\alpha}{2}-\frac{1}{2}}w(t)\|}_{L^2}^2 \leq C \varepsilon {\|w(t)\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}^2 + \frac{C}{\varepsilon}{\|w(t)\|}_{\dot{H}^{- \frac{1}{2}}}^{2}. \end{align}\] We take \(\varepsilon>0\) sufficiently small such that \[C\varepsilon \leq \frac{1}{2},\] then we obtain \[\frac{{\rm d}}{{\rm d}t}{\|\Lambda^{-\frac{1}{2}}w(t)\|}_{L^2}^2 \leq \frac{C}{\varepsilon}{\|\Lambda^{-\frac{1}{2}}w(t)\|}_{L^2}^2.\] Using Gronwall’s inequality, we have \[w(t)=0 \text{ in } \dot{H}^{-\frac{1}{2}} \text{ for a.e. } t\in (0,T),\] also, \[w(t)=0 \text{ in } L^{\frac{2}{\alpha-1}} \text{ for a.e. } t\in (0,T).\]

3.2 Proof of Theorem 3↩︎

Similariy to proof of Theorem 2, let \(\bar{\theta}\in C([0,T]; B_{\frac{4}{\alpha-1}, 2}^{-\frac{1}{2}(\alpha - 1)})\) be a solution of the integral equation of 1 in the sense of Definition 1 with \(\theta(0)=\theta_0\). Then by Proposition 27, \(w = \theta^{(1)} - \bar{\theta}\) belongs to \(L^{\infty}(0,T; \dot{H}^{-\frac{1}{2}})\cap L^{2}(0,T; \dot{H}^{\frac{\alpha}{2} - \frac{1}{2}})\) and satisfies 19 . Moreover, by duality in Besov spaces (Lemma 13), Lemma 18 and Lemma 19, we have \[\begin{align} &\int_{\mathbb{R}^2}(\Lambda^{-1}\bar{\theta}(t))\Big(\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\Big)~{\rm d}x\\ \leq& \sup_{t\in (0, T)}{\|{\bar{\theta}}_{>N}(t)\|}_{B_{\frac{4}{\alpha-1}, 2}^{-\frac{1}{2}(\alpha - 1)}}{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{B_{\frac{4}{5-\alpha}, 2}^{\frac{1}{2}(\alpha - 1) - 1}}\\ &+\sup_{t\in (0, T)}{\|{\bar{\theta}}_{\leq N}(t)\|}_{B_{\frac{4}{\alpha-1}, 2}^{\frac{\alpha}{2}}}{\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\|}_{B_{\frac{4}{5-\alpha}, 2}^{-\frac{\alpha}{2} - 1}}\\ \leq& \varepsilon {\left\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\right\|}_{B_{\frac{4}{5-\alpha}, 2}^{\frac{1}{2}(\alpha - 1) - 1}} + C{\left\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\right\|}_{B_{\frac{4}{5-\alpha}, 2}^{-\frac{\alpha}{2} - 1}}. \end{align}\] Since \(\nabla\cdot\nabla^{\perp}f=0\) and \(\alpha>1\), using Lemma 15 15 and Lemma 16, we get \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{4}{5-\alpha}, 2}^{\frac{1}{2}(\alpha - 1) - 1}} \leq C{\|w\|}_{L^{\frac{4}{3-\alpha}}}{\|w\|}_{B_{2,2}^{\frac{\alpha}{2}-\frac{1}{2}}}.\] Thanks to \(B_{2,2}^{s} = H^{s}\), \(H^{s} = L^{2}\cap \dot{H}^{s}\) \((s>0)\) and interpolation, we obtain \[{\|w\|}_{B_{2,2}^{\frac{\alpha}{2}-\frac{1}{2}}} \leq C\left({\|w\|}_{L^{2}} + {\|w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}\right) \leq C\left({\|w\|}_{\dot{H}^{-\frac{1}{2}}} + {\|w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}\right).\] Note that \(3\leq 4/(3-\alpha)\), by Sobolev embedding, we have \[{\|w\|}_{L^{\frac{4}{3-\alpha}}} \leq C {\|w\|}_{\dot{H}^{\frac{\alpha}{2}- \frac{1}{2}}}.\] Thus, we get \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{4}{5-\alpha}, 2}^{\frac{1}{2}(\alpha - 1) - 1}} \leq C\left({\|w\|}_{\dot{H}^{-\frac{1}{2}}}^{2} + {\|w\|}_{\dot{H}^{\frac{\alpha}{2}-\frac{1}{2}}}^{2}\right).\] Also, using Lemma 17 instead of Lemma 16, we obtain \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w(t))\cdot \nabla\big)w(t)\right\|}_{B_{\frac{4}{5-\alpha}, 2}^{-\frac{\alpha}{2} - 1}} \leq C {\|w\|}_{L^{2}}{\|w\|}_{B_{2,2}^{-\frac{1}{2}}}.\] Therefore, by the same argument as in Theorem 2, we can prove that the uniqueness of the solution of 1 .

3.3 Proof of Theorem 4↩︎

In this case, we consider \[\theta_{0} \in B_{p,p}^{-\frac{2}{3} + \frac{2}{p}} \text{ if } \alpha = \frac{5}{3} \text{ and } 3\leq p < 6,\] or \[\theta_{0} \in B_{p,q}^{1 - \alpha + \frac{2}{p}} \text{ if } 1 < \alpha < \frac{5}{2}, \frac{2}{\alpha - 1} \leq p < \frac{4}{\alpha - 1} \text{ and } 1\leq q < \infty.\] The proof is carried out using the same method as before. The energy inequality is justified by Proposition 30. We need to estimate the following term \[\label{0409-2} {\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{-2 + \alpha - \frac{2}{p}}} \quad\text{ and }\quad {\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{- \frac{2}{p} - \frac{3}{2}}}.\tag{21}\] Thanks to \(-1+\alpha-2/p<\alpha/2-1/2\), by product estimate (Lemma 15 16 and Lemma 16 or Lemma 17) and embedding (Lemma 10), we obtain \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{-2 + \alpha - \frac{2}{p}}} \leq C {\|w\|}_{B_{\frac{2p}{p-2}, \frac{2q}{q-2}}^{\frac{\alpha}{2} - \frac{1}{2} - \frac{2}{p}}}{\|w\|}_{B_{2, 2}^{\frac{\alpha}{2} - \frac{1}{2}}} \leq C {\|w\|}_{H^{\frac{\alpha}{2} - \frac{1}{2}}}^2,\] and \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{- \frac{2}{p} - \frac{3}{2}}} \leq C {\|w\|}_{B_{\frac{2p}{p-2}, \frac{2q}{q-2}}^{- \frac{2}{p}}}{\|w\|}_{B_{2, 2}^{-\frac{1}{2}}} \leq C {\|w\|}_{L^{2}}{\|w\|}_{H^{- \frac{1}{2}}}.\] Thus, we can show that \(w(t) = 0\) in \(B_{p,q}^{1-\alpha + \frac{2}{p}}\) for a.e. \(t\in (0,T)\).

3.4 Proof of Theorem 5↩︎

In Theorem 5, we consider the cases \(\alpha = 1\) and \(\theta_{0}\in B_{p,q}^{\frac{2}{p}}\) with \(2\leq p < \infty\), \(1\leq q < \infty\). In this case, local well-posedness is established by Chen, Miao and Zhang Ch_Mi_Zh_2007?. Furthermore, by the argument in Wang and Zhang Wa_Za_2011? using modulus of continuity, the solution exists globally in time (see also Ki_Na_Vo_2007?). Hence, the same method as before can be applied. By Proposition 33, the difference between two solutions with same initial data \(\theta_{0}\in B_{p,q}^{\frac{2}{p}}\) belongs to \(L^{\infty}(0,T; \dot{H}^{-\frac{1}{2}})\cap L^{2}(0,T; L^{2})\). Using the energy method, the proof reduces to the estimate of the following term. \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{-\frac{2}{p} - 1}} \quad\text{ and }\quad {\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{- \frac{2}{p} - \frac{3}{2}}}.\] These terms can be estimated by the same argument as that used for terms 21 . Thus, we get \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{-\frac{2}{p} - 1}} \leq C {\|w\|}_{L^{2}}^{2},\] and \[{\left\|\big((\nabla^{\perp}\Lambda^{-1}w)\cdot \nabla\big)w\right\|}_{B_{\frac{p}{p-1}, \frac{q}{q-1}}^{- \frac{2}{p} - \frac{3}{2}}} \leq C {\|w\|}_{L^{2}}{\|w\|}_{H^{- \frac{1}{2}}}.\]

Therefore, the uniqueness of the solutions of the integral equation of 1 in the sense of Definition 1 in \(L^{\infty}(0,T; B_{p,q}^{\frac{2}{p}})\) holds.

4 Proof of Proposition 1.↩︎

In this appendix, we prove Proposition 1 using Littlewood-Paley decomposition. The overall strategy of the argument follows the proof of the analogous result for Navier-Stokes equations due to Fa_Jo_Ri_1972?.

Acknowledgments. The author would like to thank Professor Tsukasa Iwabuchi for valuable discussions and continuous encouragement. The author was supported by the Grant-in-Aid for JSPS Fellows, Grant Number 26KJ0527.
Data availability statement. This manuscript has no associated data.
Conflict of Interest. The author declares that he has no conflict of interest.

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1. Mathematics Subject Classification: 35Q35; 35Q86

   Keywords: surface quasi-geostrophic equation, fractional dissipation, uniqueness

   * Taiki Okazaki – E-mail: okazaki.taiki.r5@dc.tohoku.ac.jp↩︎