Optimal Gevrey Regularity for Supercritical Quasi-Geostrophic Equations

Abstract

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity endowed with the same decay exponent as the linear part. This settles several open problems in Biswas (J Differ Equ 257(6):1753–1772, 2014), Biswas et al. (J Funct Anal 269(10):3083–3119, 2015).

1 Introduction

We consider the following two-dimensional dissipative surface quasi-geostrophic equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta +(u \cdot \nabla ) \theta +\nu D^{\gamma } \theta =0, \quad (t,x) \in (0,\infty ) \times {\mathbb {R}}^2; \\ u= R^{\perp } \theta = (-R_2 \theta , R_1 \theta ), \quad R_j =D^{-1} \partial _j; \\ \theta (0,x) = \theta _0 (x),\quad x \in {\mathbb {R}}^2, \end{array}\right. } \end{aligned}$$

(1.1)

where $\nu \ge 0$, $0<\gamma \le 2$, $D=(-\Delta )^{\frac{1}{2}}$, $D^{\gamma }= (-\Delta )^{\frac{\gamma }{2}}$, and more generally the fractional operator $D^s =(-\Delta )^{\frac{s}{2}}$ corresponds to the Fourier multiplier $|\xi |^s$, i.e. $\widehat{D^s f }(\xi ) = |\xi |^s {\widehat{f}} (\xi )$ whenever it is suitably defined under certain regularity assumptions on f. The scalar-valued unknown $\theta$ is the potential temperature, and $u=D^{-1} \nabla ^{\perp } \theta$ corresponds to the velocity field of a fluid which is incompressible. One can write $u=(-R_2\theta , R_1\theta )$ where $R_j$ is the $j^{{\text {th}} }$ Riesz transform in 2D. The dissipative quasi-geostrophic equation (1.1) can be derived from general quasi-geostrophic equations in the special case of constant potential vorticity and buoyancy frequency [24]. It models the evolution of the potential temperature $\theta$ of a geostrophic fluid with velocity u on the boundary of a rapidly rotating half space. As such it is often termed surface quasi-geostrophic equations in the literature. If $\theta$ is a smooth solution to (1.1), then it obeys the $L^p$-maximum principle, namely

$$\begin{aligned} \Vert \theta (t, \cdot ) \Vert _{L^p({\mathbb {R}}^2)} \le \Vert \theta _0 \Vert _{L^p({\mathbb {R}}^2)}, \qquad t\ge 0, \; \forall \, 1\le p \le \infty . \end{aligned}$$

(1.2)

Similar results hold when the domain ${\mathbb {R}}^2$ is replaced by the periodic torus ${\mathbb {T}}^2$. Moreover, if $\theta _0$ is smooth and in $\dot{H}^{-\frac{1}{2}}({\mathbb {R}}^2)$, then one can show that

$$\begin{aligned} \Vert \theta (t,\cdot ) \Vert _{\dot{H}^{-\frac{1}{2}}({\mathbb {R}}^2)} \le \Vert \theta _0 \Vert _{\dot{H}^{-\frac{1}{2}}({\mathbb {R}}^2)}, \quad t>0. \end{aligned}$$

(1.3)

More precisely, for the inviscid case $\nu =0$ one has conservation and for the dissipative case $\nu >0$ one has dissipation of the ${\dot{H}}^{-\frac{1}{2}}$-Hamiltonian. Indeed for $\nu =0$ by using the identity (below $P_{<J}$ is a smooth frequency projection to $\{|\xi |\le \text{ constant }\cdot 2^J\}$)

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert D^{-\frac{1}{2}} P_{<J} \theta \Vert _2^2 =-\int P_{<J} (\theta {\mathcal {R}}^{\perp } \theta ) \cdot P_{<J} {\mathcal {R}} \theta dx, \end{aligned}$$

one can prove the conservation of $\Vert D^{-\frac{1}{2}} \theta \Vert _2^2$ under the assumption $\theta \in L_{t,x}^3$. The two fundamental conservation laws (1.2) and (1.3) play important roles in the wellposedness theory for both weak and strong solutions. In [25] Resnick proved the global existence of a weak solution for $0< \gamma \le 2$ in $L_t^{\infty }L_x^2$ for any initial data $\theta _0\in L_x^2$. In [23] Marchand proved the existence of a global weak solution in $L_t^{\infty } H^{-\frac{1}{2}}_x$ for $\theta _0\in {\dot{H}}^{-\frac{1}{2}}_x({\mathbb {R}}^2)$ or $L_t^{\infty } L^p_x$ for $\theta _0 \in L^p_x({\mathbb {R}}^2)$, $p\ge \frac{4}{3}$, when $\nu >0$ and $0<\gamma \le 2$. It should be pointed out that in Marchand’s result, the non-dissipative case $\nu =0$ requires $p> 4/3$ since the embedding $L^{\frac{4}{3}} \hookrightarrow \dot{H}^{-\frac{1}{2}}$ is not compact. On the other hand for the diffusive case one has extra $L_t^2 \dot{H}^{\frac{\gamma }{2}-\frac{1}{2}}$ conservation by construction. In recent [10], non-uniqueness of stationary weak solutions were proved for $\nu \ge 0$ and $\gamma <\frac{3}{2}$. In somewhat positive direction, uniqueness of surface quasi-geostrophic patches for the non-dissipative case $\nu =0$ with moving boundary satisfying the arc-chord condition was obtained in [7].

The purpose of this work is to establish optimal Gevrey regularity in the whole supercritical regime $0<\gamma < 1$. We begin by explaining the meaning of super-criticality. For $\nu >0$, the equation (1.1) admits a certain scaling invariance, namely: if $\theta$ is a solution, then for $\lambda >0$

$$\begin{aligned} \theta _{\lambda }(t,x) = \lambda ^{1-\gamma } \theta (\lambda ^{\gamma } t, \lambda x) \end{aligned}$$

(1.4)

is also a solution. As such the critical space for (1.1) is $\dot{H}^{2-\gamma }({\mathbb {R}}^2)$ for $0\le \gamma \le 2$. In terms of the $L^{\infty }$ conservation law, (1.1) is $L^{\infty }$-subcritical for $\gamma >1$, $L^{\infty }$-critical for $\gamma =1$ and $L^{\infty }$-supercritical for $0<\gamma <1$. Whilst the wellposedness theory for (1.1) is relatively complete for the subcritical and critical regime $1\le \gamma \le 2$ (cf. [1, 11, 12, 15, 18] and the references therein), there are very few results in the supercritical regime $0<\gamma <1$ ( [13, 17, 21, 27]). In this connection we mention three representative works: (1) The work of Miura [22] which establishes for the first time the large data local wellposedness in the critical space $H^{2-\gamma }$; (2) The work of Dong [12] which via a new set of commutator estimates establishes optimal polynomial in time smoothing estimates for critical and supercritical quasi-geostrophic equations; (3) The work of Biswas, Martinez and Silva [3] which establishes short-time Gevrey regularity with an exponent strictly less than $\gamma$, namely:

$$\begin{aligned} \sup _{0<t<T} \Vert e^{\lambda t^{\frac{\alpha }{\gamma } } D^{\alpha } } \theta (t,\cdot ) \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,q}} \lesssim \Vert \theta _0\Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,q}}, \end{aligned}$$

(1.5)

where $2\le p<\infty$, $1\le q<\infty$ and $\alpha <\gamma$.

Inspired by these preceding works, we develop in this paper an optimal local regularity theory for the super-critical quasi-geostrophic equation. Set $\nu =1$ in (1.1). If we completely drop the nonlinear term and keep only the linear dissipation term, then the linear solution is given by

$$\begin{aligned} \theta _{\textrm{linear}}(t, x) = (e^{ -tD^{\gamma } } \theta _0)(t,x). \end{aligned}$$

(1.6)

Formally speaking, one has the identity $e^{tD^{\gamma }}( \theta _{\textrm{linear}}(t,\cdot ) ) =\theta _0$ for any $t>0$. This shows that the best smoothing estimate one can hope for is

$$\begin{aligned} \Vert e^{tD^{\gamma } } (\theta _{\textrm{linear} } (t,\cdot ) ) \Vert _X \lesssim \Vert \theta _0 \Vert _X, \end{aligned}$$

(1.7)

where X is a working Banach space. The purpose of this work, rough speaking, is to show that for the nonlinear local solution to (1.1) (say taking $\nu =1$ for simplicity of notation), we have

$$\begin{aligned} \Vert e^{(1-\epsilon _0) t D^{\gamma } } ( \theta (t,\cdot ) ) \Vert _X \lesssim _{\epsilon _0} \Vert \theta _0 \Vert _X, \end{aligned}$$

(1.8)

where $\epsilon _0>0$ can be taken any small number, and X can be a Sobolev or Besov space. In this sense this is the best possible regularity estimate for this and similar problems.

We now state in more detail the main results. To elucidate the main idea we first showcase the result on the prototypical $L^2$-type critical $H^{2-\gamma }$ space. The following offers a substantial improvement of Miura [22] and Dong [12]. To keep the paper self-contained, we give a bare-hand harmonic-analysis-free proof. The framework we develop here can probably be applied to many other problems.

Theorem 1.1

Let $\nu =1$, $0<\gamma < 1$ and $\theta _0 \in H^{2-\gamma }$. For any $0<\epsilon _0<1$, there exists $T=T(\gamma ,\theta _0, \epsilon _0)>0$ and a unique solution $\theta \in C_t^0\,H^{2-\gamma } \cap C_t^1\,H^{1-\gamma } \cap L_t^2\,H^{2-\frac{\gamma }{2}} ([0,T]\times {\mathbb {R}}^2)$ to (1.1) such that $f(t,\cdot ) =e^{\epsilon _0 tD^{\gamma } }\theta (t,\cdot ) \in C_t^0\,H^{2-\gamma } \cap L_t^2\, H^{2-\frac{\gamma }{2}} ([0,T]\times {\mathbb {R}}^2)$ and

$$\begin{aligned} \sup _{0<t<T} \Vert f(t,\cdot ) \Vert _{H^{2-\gamma } }^2 + \int _0^{T} \Vert f(t,\cdot ) \Vert _{H^{2-\frac{\gamma }{2}}}^2 dt \le C \Vert \theta _0\Vert _{H^{2-\gamma } }^2, \end{aligned}$$

where $C>0$ is a constant depending on $(\gamma , \epsilon _0)$.

Our next result is devoted to the Besov case. In particular, we resolve the problem left open in [3], namely one can push to the optimal threshold $\alpha =\gamma$. Moreover we cover the whole regime $1\le p<\infty$.

Theorem 1.2

Let $\nu =1$, $0<\gamma <1$, $1\le p<\infty$ and $1\le q<\infty$. Assume the initial data $\theta _0 \in B^{1+\frac{2}{p}-\gamma }_{p,q}({\mathbb {R}}^2)$. There exists $T=T(\gamma ,\theta _0,p,q)>0$ and a unique solution $\theta \in C_t^0([0,T], B^{1+\frac{2}{p}-\gamma }_{p,q})$ to (1.1) such that $f(t,\cdot ) =e^{\frac{1}{2} t D^{\gamma }} \theta (t,\cdot ) \in C_t^0([0,T], B^{1 +\frac{2}{p}-\gamma }_{p,q})$ and

$$\begin{aligned} \sup _{0<t<T} \Vert e^{\frac{1}{2} t D^{\gamma } } \theta (t,\cdot ) \Vert _{B^{1+\frac{2}{p}-\gamma }_{p,q} } \le C \Vert \theta _0\Vert _{B^{1+\frac{2}{p}-\gamma }_{p,q}}, \end{aligned}$$

where $C>0$ is a constant depending on $(\gamma , p,q)$.

The techniques introduced in this paper may apply to many other similar models such as Burgers equations, generalized SQG models, and Chemotaxi equations (cf. recent very interesting works [4, 8, 14, 28]). Also there are some promising evidences that a set of nontrivial multiplier estimates can be generalized from our work. All these will be explored elsewhere. The rest of this paper is organized as follows. In Sect. 2 we collect some preliminary materials along with the needed proofs. In Sect. 3 we give the nonlinear estimates for the $H^{2-\gamma }$ case. In Sect. 4 we give the proof of Theorem 1.1. In Sect. 5 we give the proof of Theorem 1.2.

2 Notation and Preliminaries

In this section we introduce some basic notation used in this paper and collect several useful lemmas.

We define the sign function ${\text {sgn}}(x)$ on ${\mathbb {R}}$ as:

$$\begin{aligned} {\text {sgn}}(x)= {\left\{ \begin{array}{ll} 1, \quad x>0, \\ -1, \quad x<0, \\ 0, \quad x=0. \end{array}\right. } \end{aligned}$$

For any two quantities X and Y, we denote $X \lesssim Y$ if $X \le C Y$ for some constant $C>0$. The dependence of the constant C on other parameters or constants are usually clear from the context and we will often suppress this dependence. We denote $X \lesssim _{Z_1,\ldots , Z_N} Y$ if the implied constant depends on the quantities $Z_1,\ldots , Z_N$. We denote $X\sim Y$ if $X\lesssim Y$ and $Y \lesssim X$.

For any quantity X, we will denote by $X+$ the quantity $X+\epsilon$ for some sufficiently small $\epsilon >0$. The smallness of such $\epsilon$ is usually clear from the context. The notation $X-$ is similarly defined. This notation is very convenient for various exponents in interpolation inequalities. For example instead of writing

$$\begin{aligned} \Vert f g \Vert _{L^1({\mathbb {R}})} \le \Vert f \Vert _{L^{\frac{2}{1+\epsilon } } ({\mathbb {R}})} \Vert g \Vert _{L^{\frac{2}{1-\epsilon } } ({\mathbb {R}})}, \end{aligned}$$

we shall write

$$\begin{aligned} \Vert f g \Vert _{L^1({\mathbb {R}})} \le \Vert f \Vert _{L^{2-}({\mathbb {R}})} \Vert g \Vert _{L^{2+}({\mathbb {R}})}. \end{aligned}$$

(2.1)

For any two quantities X and Y, we shall denote $X\ll Y$ if $X \le c Y$ for some sufficiently small constant c. The smallness of the constant c (and its dependence on other parameters) is usually clear from the context. The notation $X\gg Y$ is similarly defined. Note that our use of $\ll$ and $\gg$ here is different from the usual Vinogradov notation in number theory or asymptotic analysis.

We shall adopt the following notation for Fourier transform on ${\mathbb {R}}^n$:

$$\begin{aligned}&({\mathcal {F}} f )(\xi ) = {\hat{f}}(\xi ) = \int _{{\mathbb {R}}^n} f(x) e^{-i x \cdot \xi } dx, \\&({\mathcal {F}}^{-1} g)(x) = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} g(\xi ) e^{i x\cdot \xi } d\xi . \end{aligned}$$

Similar notation will be adopted for the Fourier transform of tempered distributions. For real-valued Schwartz functions $f:\; {\mathbb {R}}^n \rightarrow {\mathbb {R}}$, $g:\, {\mathbb {R}}^n \rightarrow {\mathbb {R}}$, the usual Plancherel takes the form (note that $\overline{{\hat{g}}(\xi )}={\hat{g}}(-\xi )$)

$$\begin{aligned} \int _{{\mathbb {R}}^n} f(x) g(x) dx = \frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} {\hat{f}}(\xi ) {\hat{g}}(-\xi ) d\xi . \end{aligned}$$

We shall denote for $s>0$ the fractional Laplacian $D^s = (-\Delta )^{s/2} = |\nabla |^s$ as the operator corresponding to the symbol $|\xi |^s$. For any $0\le r \in {\mathbb {R}}$, the Sobolev norm $\Vert f\Vert _{\dot{H}^r}$ is defined as

$$\begin{aligned} \Vert f \Vert _{\dot{H}^r} = \Vert D^r f \Vert _2=\Vert (-\Delta )^{r/2} f \Vert _2. \end{aligned}$$

We will need to use the Littlewood–Paley (LP) frequency projection operators. To fix the notation, let $\phi _0 \in C_c^\infty ({\mathbb {R}}^n )$ and satisfy

$$\begin{aligned} 0 \le \phi _0 \le 1,\quad \phi _0(\xi ) = 1\ {\text { for}}\ |\xi | \le 1,\quad \phi _0(\xi ) = 0\ {\text { for}}\ |\xi | \ge 7/6. \end{aligned}$$

Let $\phi (\xi ):= \phi _0(\xi ) - \phi _0(2\xi )$ which is supported in $\frac{1}{2} \le |\xi | \le \frac{7}{6}$. For any $f \in {\mathcal {S}}^{\prime }({\mathbb {R}}^n)$, $j \in {\mathbb {Z}}$, define

$$\begin{aligned}&\widehat{P_{\le j} f} (\xi ) = \phi _0(2^{-j} \xi ) {\hat{f}}(\xi ), \quad P_{>j} f=f-P_{\le j} f,\\&\widehat{P_j f} (\xi ) = \phi (2^{-j} \xi ) {\hat{f}}(\xi ), \qquad \xi \in {\mathbb {R}}^n. \end{aligned}$$

Sometimes for simplicity we write $f_j = P_j f$, $f_{\le j} = P_{\le j} f$, and $f_{[a,b]}=\sum _{a\le j \le b} f_j$. Note that by using the support property of $\phi$, we have $P_j P_{j^{\prime }} =0$ whenever $|j-j^{\prime }|>1$. For $f \in {\mathcal {S}}^{\prime }$ with $\lim _{j\rightarrow -\infty } P_{\le j} f =0$, one has the identity

$$\begin{aligned} f= \sum _{j \in {\mathbb {Z}}} f_j , \quad ( \hbox { in}\ {\mathcal {S}}^{\prime } ) \end{aligned}$$

and for general tempered distributions the convergence (for low frequencies) should be taken as modulo polynomials.

The Bony paraproduct for a pair of functions $f,g\in {\mathcal {S}}({\mathbb {R}}^n)$ take the form

$$\begin{aligned} f g = \sum _{i \in {\mathbb {Z}}} f_i g_{[i-1,i+1]} + \sum _{i \in {\mathbb {Z}}} f_i g_{\le i-2} + \sum _{i \in {\mathbb {Z}}} g_i f_{\le i-2}. \end{aligned}$$

For $s\in {\mathbb {R}}$, $1\le p,q \le \infty$, the Besov norm $\Vert \cdot \Vert _{B^s_{p,q}}$ is given by

$$\begin{aligned} \Vert f \Vert _{B^s_{p,q}} = {\left\{ \begin{array}{ll} \Vert P_{\le 0} f \Vert _p + ( \sum _{k=1}^{\infty } 2^{sqk} \Vert P_k f \Vert _p^q )^{1/q}, \quad \ q<\infty ; \\ \Vert P_{\le 0} f \Vert _p + \sup _{k\ge 1} 2^{sk} \Vert P_k f \Vert _p, \quad \ q=\infty . \end{array}\right. } \end{aligned}$$

The Besov space $B^s_{p,q}$ is then simply

$$\begin{aligned} B^s_{p,q} = \biggl \{ f: \; f\in {\mathcal {S}}^{\prime }, \, \Vert f\Vert _{B^s_{p,q}} <\infty \biggr \}. \end{aligned}$$

Note that Schwartz functions are dense in $B^s_{p,q}$ when $1\le p,q<\infty$.

In the following lemma we give refined heat flow estimate and frequency localized Bernstein inequalities for the fractional Laplacian $|\nabla |^{\gamma }$, $0<\gamma <2$. Note that for $\gamma >2$ and $p\ne 2$ there are counterexamples to the frequency Bernstein inequalities (cf. Li and Sire [20]).

Lemma 2.1

(Refined heat flow estimate and Bernstein inequality, case $0<\gamma <2$). Let the dimension $n\ge 1$. Let $0<\gamma <2$ and $1\le q\le \infty$. Then for any $f\in L^q({\mathbb {R}}^n)$, and any $j \in {\mathbb {Z}}$, we have

$$\begin{aligned} \Vert e^{-t |\nabla |^{\gamma }} P_j f \Vert _q \le e^{-c_1 t 2^{j\gamma } } \Vert P_j f \Vert _q, \quad \forall \, t\ge 0, \end{aligned}$$

(2.2)

where $c_1>0$ is a constant depending only on ($\gamma$, n). For $0<\gamma <2$, $1\le q<\infty$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^n} (|\nabla |^{\gamma } P_j f) |P_j f|^{q-2} P_j f dx \ge c_2 2^{j\gamma } \Vert P_j f \Vert _q^q, \quad \text {if}\, 1<q<\infty ; \end{aligned}$$

(2.3)

$$\begin{aligned}&\int _{{\mathbb {R}}^n} (|\nabla |^{\gamma } P_j f) {\text {sgn}} (P_j f ) dx \ge c_2 2^{j\gamma } \Vert P_j f \Vert _1, \quad \hbox { if}\ q=1, \end{aligned}$$

(2.4)

where $c_2>0$ depends only on ($\gamma$, n).

The $q=\infty$ formulation of (2.3) is as follows. Let $0<\gamma <2$. For any $f \in L^{\infty }({\mathbb {R}}^n)$, if $j \in {\mathbb {Z}}$ and $|(P_j f)(x_0)| = \Vert P_j f \Vert _{\infty }$, then we have

$$\begin{aligned} {\text {sgn}}( P_j f(x_0) ) \cdot (|\nabla |^{\gamma } P_j f)(x_0) \ge c_3 2^{j\gamma } \Vert P_j f \Vert _{\infty }, \end{aligned}$$

(2.5)

where $c_3>0$ depends only on ($\gamma$, n).

Remark

For $1<q<\infty$ the first two inequalities also hold for $\gamma =2$, one can see Propositions 2.5 and 2.7 below. On the other hand, the inequality (2.4) does not hold for $\gamma =2$. One can construct a counterexample in dimension $n=1$ as follows. Take $g(x) = \frac{1}{4} (3 \sin x -\sin 3x) = (\sin x)^3$ which only has zeros of third order. Take h(x) with ${\hat{h}}$ compactly supported in $|\xi | \ll 1$ and $h(x)>0$ for all x. Set

$$\begin{aligned} f(x) = g(x) h(x) \end{aligned}$$

which obviously has frequency localized to $|\xi | \sim 1$ and have same zeros as g(x). Easy to check that $\Vert f\Vert _1 \sim 1$ but

$$\begin{aligned} \int _{{\mathbb {R}}} f^{\prime \prime }(x) {\text {sgn}} (f(x) ) dx = 0. \end{aligned}$$

Remark 2.2

For $\gamma >0$ sufficiently small, one can give a direct proof for $1\le q <\infty$ as follows. WLOG consider $g=P_1 f$ with $\Vert g \Vert _q =1$, and let

$$\begin{aligned} I(\gamma ) = \int _{{\mathbb {R}}^n} (|\nabla |^{\gamma } g ) |g|^{q-2} g dx. \end{aligned}$$

One can then obtain

$$\begin{aligned} I(\gamma ) -I(0) = \int _{{\mathbb {R}}^n} (\int _0^{\gamma } T_{s } g ds) |g|^{q-2} g dx, \quad \widehat{T_{s} g}(\xi ) =s (|\xi |^{s} \log |\xi | ) {\hat{g}} (\xi ). \end{aligned}$$

Since g has Fourier support localized in $\{ |\xi | \sim 1 \}$, one can obtain uniformly in $0<s\le 1$,

$$\begin{aligned} \Vert T_s g\Vert _q \lesssim _{n} \Vert g \Vert _q =1. \end{aligned}$$

Note that $I(0)=1$. Thus for $\gamma <\gamma _0(n)$ sufficiently small one must have $\frac{1}{2} \le I(\gamma )\le \frac{3}{2}$.

Remark

The inequality (2.5) was obtained by Wang-Zhang [26] by an elegant contradiction argument under the assumption that $f\in C_0({\mathbb {R}}^n)$ (i.e. vanishing at infinity) and f is frequency localized to a dyadic annulus. Here we only assume $f \in L^{\infty }$ and is frequency localized. This will naturally include periodic functions and similar ones as special cases. Moreover we provide two different proofs. The second proof is self-contained and seems quite short.

Proof of Lemma 2.1

For the first inequality and (2.3), see [19] for a proof using an idea of perturbation of the Lévy semigroup. Since the constant $c_2>0$ depends only on $(\gamma ,n)$, the inequality (2.4) can be obtained from (2.3) by taking the limit $q\rightarrow 1$. (Note that since $f_j = P_j f \in L^1$ and has compact Fourier support, $f_j$ can be extended to be an entire function on ${\mathbb {C}}^n$ and its zeros must be isolated.)

Finally for (2.5) we give two proofs. With no loss we can assume $j=1$ and write $f=P_1 f$. By using translation we may also assume $x_0=0$. With no loss we assume $\Vert f\Vert _{\infty }= f(x_0)=1$.

The first proof is to use (2.2) which yields

$$\begin{aligned} (e^{-t |\nabla |^{\gamma }} f )(0) \le e^{-c t}, \end{aligned}$$

where $c>0$ depends only on ($\gamma$, n). Then since $f=P_1 f$ is smooth and

$$\begin{aligned} f - e^{-t |\nabla |^{\gamma } } f = \int _0^t e^{-s |\nabla |^{\gamma } } |\nabla |^{\gamma } f ds = t |\nabla |^{\gamma } f - \int _0^t \left( \int _0^s e^{-\tau |\nabla |^{\gamma } } { |\nabla |^{2\gamma } f } d \tau \right) ds. \end{aligned}$$

One can then divide by $t\rightarrow 0$ and obtain

$$\begin{aligned} (|\nabla |^{\gamma } f) (0) \ge c. \end{aligned}$$

The second proof is more direct. We note that $\int \psi (y) dy =0$ where $\psi$ corresponds to the projection operator $P_1$. Since $1= (P_1 f)(0)$, we obtain

$$\begin{aligned}&1 = \int \psi (y) ( f(y) -1) dy \le \sup _{y \ne 0}( |\psi (y) | \cdot |y|^{n+\gamma } ) \cdot \int _{{\mathbb {R}}^n} \frac{1- f(y)}{ |y|^{n+\gamma } } dy\\&\quad \lesssim _{\gamma ,n} \int _{{\mathbb {R}}^n} \frac{1-f(y) }{ |y|^{n+\gamma } } dy. \end{aligned}$$

Thus

$$\begin{aligned} ( |\nabla |^{\gamma } f)(0) \gtrsim _{\gamma ,n } 1. \end{aligned}$$

$\square$

In what follows we will give a different proof of (2.3) (and some stronger versions, see Propositions 2.5 and (2.7)) and some equivalent characterization. For the sake of understanding (and keeping track of constants) we provide some details.

Lemma 2.3

Let $0<s<1$. Then for any $g \in L^2({\mathbb {R}}^n)$ with ${{\hat{g}}}$ being compactly supported, we have

$$\begin{aligned} \frac{1}{2} C_{2s,n} \cdot \int _{{\mathbb {R}}^n \times {\mathbb {R}}^n} \frac{ |g(x)-g(y)|^2}{|x-y|^{n+2s} } dx dy = \Vert \,|\nabla |^s g \Vert ^2_2, \end{aligned}$$

where $C_{2s,n}$ is a constant corresponding to the fractional Laplacian $|\nabla |^{2s}$ having the asymptotics $C_{2\,s,n} \sim _{n} s(1-s)$ for $0<s<1$. As a result, if $g \in L^2({\mathbb {R}}^n)$ and $\Vert \, |\nabla |^s g \Vert _2<\infty$, then

$$\begin{aligned} s(1-s)\cdot \int _{{\mathbb {R}}^n \times {\mathbb {R}}^n} \frac{ |g(x)-g(y)|^2}{|x-y|^{n+2s} } dx dy \sim _n \Vert \,|\nabla |^s g \Vert ^2_2. \end{aligned}$$

Similarly if $g \in L^2({\mathbb {R}}^n)$ and $\int \frac{|g(x)-g(y)|^2}{|x-y|^{n+2\,s} } dx dy<\infty$, then

$$\begin{aligned} \Vert \, |\nabla |^s g \Vert _2^2 \sim _n s(1-s)\cdot \int _{{\mathbb {R}}^n \times {\mathbb {R}}^n} \frac{ |g(x)-g(y)|^2}{|x-y|^{n+2s} } dx dy. \end{aligned}$$

Proof

Note that

$$\begin{aligned} \Vert |\nabla |^s g \Vert _2^2&= \int _{{\mathbb {R}}^n } (|\nabla |^{2s} g)(x) g(x) dx = \int _{{\mathbb {R}}^n} \Bigl ( \lim _{\epsilon \rightarrow 0} C_{2s,n} \int _{|y-x|>\epsilon } \frac{g(x)-g(y)}{|x-y|^{n+2s} } dy \Bigr ) g(x) dx, \end{aligned}$$

where $C_{2s,n}\sim _n s (1-s)$. Now for each $0<\epsilon <1$, it is easy to check that (for the case $\frac{1}{2}\le s<1$ one needs to make use of the regularised quantity $g(x)-g(y) + \nabla g(x) \cdot (y-x)$)

$$\begin{aligned} | \int _{|y-x|>\epsilon } \frac{g(x)-g(y)}{ |x-y|^{n+2s} } dy | \lesssim \; |g(x)| + |{\mathcal {M}}{\mathcal {g}}(x)| + |{\mathcal {M}} ( \partial g )(x)| + |{\mathcal {M}} ( \partial ^2 g)(x)|, \end{aligned}$$

where ${\mathcal {M}}g$ is the usual maximal function. By Lebesgue Dominated Convergence, we then obtain

$$\begin{aligned} \Vert |\nabla |^s g \Vert _2^2&= C_{2s,n} \lim _{\epsilon \rightarrow 0} \int _{{\mathbb {R}}^n} \int _{|y-x|>\epsilon } \frac{ g(x)-g(y)}{ |x-y|^{n+2s} } dy g(x) dx. \end{aligned}$$

Now note that for each $\epsilon >0$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^n} \int _{|y-x|>\epsilon } \frac{|g(x)-g(y)|}{|x-y|^{n+2s}} dy |g(x) |dx \lesssim _{\epsilon ,n} \int _{{\mathbb {R}}^n} (|g(x) |+ |{\mathcal {M}} g(x) |) |g(x)| dx <\infty . \end{aligned}$$

Therefore by using Fubini, symmetrising in x and y and Lebesgue Monotone Convergence, we obtain

$$\begin{aligned} \Vert |\nabla |^s g\Vert _2^2 = \frac{1}{2} C_{2s,n} \lim _{\epsilon \rightarrow 0} \int _{ |x-y|>\epsilon } \frac{|g(x)-g(y)|^2}{|x-y|^{n+2s} } dx dy =\frac{1}{2} C_{2s,n} \int \frac{|g(x)-g(y)|^2}{|x-y|^{n+2s}} dx dy. \end{aligned}$$

Now if $g \in L^2({\mathbb {R}}^n)$ with $\Vert \, |\nabla |^s g \Vert _2<\infty$, then by Fatou’s Lemma, we get

$$\begin{aligned}&s(1-s) \int \frac{|g(x)-g(y)|^2}{|x-y|^{n+2s} } dx dy \\&\le s(1-s) \cdot \liminf _{J\rightarrow \infty } \int \frac{|P_{\le J} g (x) - P_{\le J} g (y) |^2}{|x-y|^{n+2s} } dx dy \lesssim _n \, s(1-s) \Vert \, |\nabla |^s g \Vert _2^2. \end{aligned}$$

On the other hand, note that

$$\begin{aligned}&| P_{\le J} g(x) - P_{\le J} g(y)| \\&\quad \le \int |g(x-z)-g(y-z)| 2^{nJ} |\phi (2^J z) |d z \\&\quad \lesssim _n \left( \int |g(x-z)-g(y-z)|^2 2^{nJ} |\phi (2^J z) | dz \right) ^{\frac{1}{2}}, \end{aligned}$$

where $\phi \in L^1$ is a smooth function used in the kernel $P_{\le J}$. The desired equivalence then easily follows. $\square$

Lemma 2.4

Let $1<q<\infty$. Then for any a, $b\in {\mathbb {R}}$, we have

$$\begin{aligned}&(a-b)(|a|^{q-2} a - |b|^{q-2} b) \sim _{q} (|a|^{\frac{q}{2}-1} a - |b|^{\frac{q}{2}-1} b)^2,\\&(a-b) (|a|^{q-2}a - |b|^{q-2}b) \ge \frac{4(q-1)}{q^2} ( |a|^{\frac{q}{2}-1} a - |b|^{\frac{q}{2}-1} b)^2. \end{aligned}$$

Proof

The first inequality is easy to check. To prove the second inequality, it suffices to show for any $0<x<1$,

$$\begin{aligned} \frac{1+x^q-x-x^{q-1} }{(1-x^{\frac{q}{2}} )^2} \ge \frac{4(q-1)}{q^2} = \frac{q^2-(q-2)^2}{q^2}. \end{aligned}$$

Set $t= x^{\frac{q}{2}} \in (0,1)$. The inequality is obvious for $q=2$. If $2< q<\infty$, then we need to show

$$\begin{aligned} \frac{ t^{\frac{1}{q}} - t^{1-\frac{1}{q}} }{1-t} < \frac{q-2}{q}. \end{aligned}$$

If $1<q<2$, then we need

$$\begin{aligned} \frac{t^{1-\frac{1}{q}} -t^{\frac{1}{q}} }{1-t} < \frac{2-q}{q}. \end{aligned}$$

Set $\eta = \min \{ \frac{1}{q}, 1-\frac{1}{q}\} \in (0,\frac{1}{2})$. It then suffices for us to show the inequality

$$\begin{aligned} f(\eta )= 1-2\eta - \frac{t^{\eta } - t^{1-\eta } }{1-t} \ge 0. \end{aligned}$$

Note that $f(0)=f(1/2)=0$ and $f^{\prime \prime }(\eta )= -(t^{\eta }-t^{1-\eta }) (\log t)^2/(1-t) <0$. Thus the desired inequality follows. $\square$

Proposition 2.5

Let $1<q<\infty$ and $0<\gamma \le 2$. Then for any $f \in L^q({\mathbb {R}}^n)$ and any $j\in {\mathbb {Z}}$, we have

$$\begin{aligned}&\int _{{\mathbb {R}}^n} (|\nabla |^{\gamma } P_j f ) | P_j f |^{q-2} P_j f dx \sim _{q} \Bigl \Vert |\nabla |^{\frac{\gamma }{2}} ( |P_j f |^{\frac{q}{2}-1} P_j f ) \Bigr \Vert _2^2, \nonumber \\&\quad \int _{{\mathbb {R}}^n} (|\nabla |^{\gamma } P_j f ) | P_j f |^{q-2} P_j f dx \nonumber \\&\quad \ge \frac{4(q-1)}{q^2} \Bigl \Vert |\nabla |^{\frac{\gamma }{2}} ( |P_j f |^{\frac{q}{2}-1} P_j f ) \Bigr \Vert _2^2 \ge \frac{4(q-1)}{q^2} \Bigl \Vert |\nabla |^{\frac{\gamma }{2}} ( |P_j f |^{\frac{q}{2}} ) \Bigr \Vert _2^2. \end{aligned}$$

(2.6)

Consequently if $\Vert P_j f \Vert _q =1$, then for any $0<s\le 1$,

$$\begin{aligned} \Vert \, |\nabla |^s ( |P_j f |^{\frac{q}{2}-1} P_j f) \Vert _2 \sim _{q,n} 2^{js}. \end{aligned}$$

(2.7)

Also for any $0<s\le 1$,

$$\begin{aligned} \Vert \, |\nabla |^s ( |P_j f |^{\frac{q}{2}} ) \Vert _2 \sim _{q,n} 2^{js}. \end{aligned}$$

(2.8)

Remark

In [16], by using a strong nonlocal pointwise inequality (see also Córdoba-Córdoba [6]), Ju Proved an inequality of the form: if $0\le \gamma \le 2$, $2\le q<\infty$, $\theta$, $|\nabla |^{\gamma } \theta \in L^q$, then

$$\begin{aligned} \int (|\nabla |^{\gamma } \theta ) |\theta |^{q-2} \theta dx \ge \frac{2}{q} \Vert |\nabla |^{\frac{\gamma }{2}} ( |\theta |^{\frac{q}{2}} ) \Vert _2^2. \end{aligned}$$

A close inspection of our proof below shows that the inequality (2.6) also works with $P_j f$ replaced by $\theta$. Note that the present form works for any $1<q<\infty$. Furthermore in the regime $q>2$, we have $\frac{4(q-1)}{q^2} > \frac{2}{q}$ and hence the constant here is slightly sharper.

Remark

The inequality (2.7) was already obtained by Chamorro and P. Lemarié-Rieusset in [9]. Remarkably modulo a q-dependent constant it is equivalent to the corresponding inequality for the more localized quantity $\int ( |\nabla |^{\gamma } P_j f ) |P_j f|^{q-2} P_j f dx$. The inequality (2.8) for $q>2$ was obtained by Chen, Miao and Zhang [5] by using Danchin’s inequality $\Vert \nabla ( |P_1 f|^{q/2} ) \Vert _2^2 \sim _{q,n} \Vert P_1 f \Vert _q^q$ together with a fractional Chain rule in Besov spaces. The key idea in [5] is to show $\Vert \nabla P_{[N_0, N_1]} ( |P_1 f |^{q/2} ) \Vert _2 \gtrsim 1$ and in order to control the high frequency piece one needs the assumption $q>2$ (so as to use $|\nabla |^{1+\epsilon _0}$-derivative for $\epsilon _0>0$ sufficiently small). Our approach here is different: namely we will not use Danchin’s inequality and prove directly $\Vert \, |\nabla |^{s_0} ( |P_1 f|^{q/2} )\Vert _2 \gtrsim 1$ for some $s_0$ sufficiently small (depending on (q, n)). Together with some further interpolation argument we are able to settle the full range $1<q<\infty$. One should note that in terms of lower bound the inequality (2.8) is stronger than (2.7).

Proof

With no loss we can assume $j=1$ and for simplicity write $P_1 f$ as f. Assume first $0<\gamma <2$. Then for some constant $C_{\gamma ,n} \sim _n \gamma (2-\gamma )$, we have (the rigorous justification of the computation below follows a smilar argument as in the proof of Lemma 2.3)

$$\begin{aligned}&\int (|\nabla |^{\gamma } f ) |f|^{q-2} f dx = C_{\gamma ,n} \int (\lim _{\epsilon \rightarrow 0} \int _{|y-x|>\epsilon } \frac{f(x) -f(y)}{ |x-y|^{n+\gamma } } dy ) |f|^{q-2} f (x) dx \nonumber \\&\quad = \frac{1}{2} C_{\gamma ,n} \int \frac{ (f(x)-f(y)) ( |f|^{q-2} f (x) - |f|^{q-2} f(y) )}{|x-y|^{n+\gamma } } dx dy \\&\quad \ge \frac{4(q-1)}{q^2} \cdot \frac{1}{2} C_{\gamma ,n} \int \frac{ (|f|^{\frac{q}{2}-1} f(x) -|f|^{\frac{q}{2}-1} f(y) )^2}{ |x-y|^{n+\gamma } } dx dy \\&\quad = \, \frac{4(q-1)}{q^2} \Vert \,|\nabla |^{\frac{\gamma }{2}} ( |f|^{\frac{q}{2}-1} f ) \Vert _2^2, \end{aligned}$$

where in the last two steps we have used Lemmas 2.4 and 2.3 respectively. One may then carefully take the limit $\gamma \rightarrow 2$ to get the result for $\gamma =2$ (when estimating $\Vert |\nabla |^{\frac{\gamma }{2}} ( |f|^{\frac{q}{2}-1} f)\Vert _2$, one needs to split into $|\xi | \le 1$ and $|\xi |>1$, and use Lebesgue Dominated Convergence and Lebesgue Monotone Convergence respectively). By the simple inequality $|\, |f|^{\frac{q}{2}-1} f(x) - |f|^{\frac{q}{2}-1} f(y)| \ge |\, |f|^{\frac{q}{2}}(x)- |f|^{\frac{q}{2}}(y) |$, we also obtain $\Vert |\nabla |^{\frac{\gamma }{2} } ( |f|^{\frac{q}{2}-1} f ) \Vert _2 \ge \Vert |\nabla |^{\frac{\gamma }{2} } (|f|^{\frac{q}{2}} ) \Vert _2$.

Next to show (2.7), we can use Remark 2.2 to obtain $\Vert |\nabla |^{s} g \Vert _2 \sim _{q, n} 1$ for any $0<s\le s_0(n)$ and $g = |f|^{\frac{q}{2}-1} f$. Since $\Vert g\Vert _2=1$ and $\Vert \nabla g \Vert _2 \lesssim _{q,n}1$, a simple interpolation argument then yields $\Vert |\nabla |^s g \Vert _2 \sim _{q,n} 1$ uniformly for $0<s\le 1$.

Finally to show (2.8), we first use the simple fact that $\Vert \nabla (|g|) \Vert _2 \le \Vert \nabla g \Vert _2$ to get

$$\begin{aligned} \Vert |\nabla | ( |f|^{\frac{q}{2}} ) \Vert _2 \lesssim _{q} 1. \end{aligned}$$

It then suffices for us to show $\Vert |\nabla |^s ( |f|^{\frac{q}{2}} ) \Vert _2 \gtrsim _{q,n} 1$ for $0<s\le s_0(q,n)$ sufficiently small. To this end we consider the quantity

$$\begin{aligned} I(s) = \int _{{\mathbb {R}}^n} |\nabla |^s (|f|) | f|^{q-1} dx. \end{aligned}$$

For $0<s<1$ this is certainly well defined since $\Vert |\nabla |^s (|f|) \Vert _q \lesssim \Vert f\Vert _q + \Vert \nabla ( |f| )\Vert _q \lesssim 1$. To circumvent the problem of differentiating under the integral, one can further consider the regularized expression (later $N \rightarrow \infty$)

$$\begin{aligned} I_N(s) = \int _{{\mathbb {R}}^n} |\nabla |^s P_{\le N} ( |f| ) | f|^{q-1} dx. \end{aligned}$$

Then

$$\begin{aligned} I_N(s)- I_N(0)= \int _{{\mathbb {R}}^n} \int _0^s ( T_{{\tilde{s}}} P_{\le N} (|f|) ) d{\tilde{s}} |f|^{q-1} dx, \quad \widehat{T_{{\tilde{s}}}}(\xi ) ={\tilde{s}} |\xi |^{{\tilde{s}}} \log |\xi |. \end{aligned}$$

Define $\widehat{T_{{\tilde{s}}}^{(1)} }(\xi ) = {\tilde{s}} |\xi |^{{\tilde{s}}} (\log |\xi |)\cdot \chi _{|\xi |<1/10}$ and $T_{{\tilde{s}}}^{(2)} =T_{{\tilde{s}}}- T_{{\tilde{s}}}^{(1)}$. It is not difficult to check that uniformly in $0<{\tilde{s}}\le \frac{1}{2}$,

$$\begin{aligned} \sup _{\xi \ne 0} \max _{|\alpha |\le [n/2]} ( |\xi |^{|\alpha |} |\partial _{\xi }^{\alpha } ( \widehat{ T_{{\tilde{s}}}^{(1)}} (\xi ) ) |) \lesssim _{n} 1. \end{aligned}$$

Thus by Hörmander we get $\Vert T_{{\tilde{s}}}^{(1)} P_{\le N} ( |f|) \Vert _q \lesssim _{n,q} \Vert f\Vert _q =1$. For $T_{{\tilde{s}}}^{(2)}$ one can use $\Vert \nabla f \Vert _q \lesssim 1$ to get an upper bound which is uniform in $0<{\tilde{s}}\le \frac{1}{2}$. Therefore $\Vert T_{{\tilde{s}}} P_{\le N} (|f|) \Vert _q \lesssim _{q,n} 1$ for $0<{\tilde{s}}\le \frac{1}{2}$. One can then obtain for $0<s\le s_0(q,n)$ sufficiently small that $\frac{1}{2} \le I(s) \le \frac{3}{2}$. Finally view I(s) as

$$\begin{aligned} I(s) = \lim _{N\rightarrow \infty } \int _{{\mathbb {R}}^n} |\nabla |^s ( Q_{\le N} (|f|) ) ( Q_{\le N} (|f|) )^{q-1} dx, \end{aligned}$$

where $\widehat{ Q_{\le N} } (\xi ) = {\hat{q}}(2^{-N} \xi )$, and ${\hat{q}} \in C_c^{\infty }$ satisfies $q(x)\ge 0$ for any $x \in {\mathbb {R}}^n$ (such q can be easily constructed by taking $q(x) = \phi (x)^2$ which corresponds to ${\hat{q}} = {\hat{\phi }} * {\hat{\phi }}$). By using the integral representation of the operator $|\nabla |^s$ and a symmetrization argument (similar to what was done before), we can obtain

$$\begin{aligned} I(s) \sim _{q,n} \Vert \, |\nabla |^{\frac{s}{2}} ( |f|^{\frac{q}{2}} ) \Vert _2^2 \end{aligned}$$

and the desired result follows. $\square$

Lemma 2.6

Let the dimension $n\ge 1$ and $0<\gamma \le 2$. Suppose $g \in L^2({\mathbb {R}}^n)$ and for some $N_0>0$, $\epsilon _0>0$

$$\begin{aligned} \Vert {\hat{g}} \Vert _{L^2(|\xi | \ge N_0) }^2 \ge \epsilon _0 \Vert {\hat{g}} \Vert _{L^2_{\xi }({\mathbb {R}}^n)}^2. \end{aligned}$$

Then there exists $t_0=t_0(\epsilon _0, N_0,\gamma )>0$ such that for all $0\le t \le t_0$, we have

$$\begin{aligned} \Vert e^{ -t |\nabla |^{\gamma } } g \Vert _{2} \le e^{ - \frac{1}{2} \epsilon _0 N_0^{\gamma } t } \Vert g \Vert _2. \end{aligned}$$

Consequently if ${\tilde{g}} \in L^2({\mathbb {R}}^n)$ satisfies $\Vert {\tilde{g}} \Vert _2 =C_1>0$, $\Vert |\nabla |^s {\tilde{g}} \Vert _2=C_2>0$ for some $s>0$, then for any $0<\gamma \le 2$, there exists $t_0=t_0(C_1,C_2,\gamma , n, s)>0$, $c_0= c_0(C_1,C_2, \gamma , n, s)>0$, such that

$$\begin{aligned} \Vert e^{-t |\nabla |^{\gamma } } {\tilde{g}} \Vert _2 \le e^{-c_0 t} \Vert {\tilde{g}} \Vert _2. \end{aligned}$$

Proof

With no loss we assume $\Vert {\hat{g}} \Vert _{L_{\xi }^2 } =1$. Then

$$\begin{aligned} \int _{{\mathbb {R}}^n} e^{ -2t |\xi |^{\gamma } } |{\hat{g}} (\xi ) |^2 d\xi&\le 1- \Vert {\hat{g}} \Vert _{L^2 (|\xi | \le N_0)}^2 + e^{-2 t N_0^{\gamma } } \Vert {\hat{g}} \Vert _{L^2(|\xi |> N_0)}^2 \\&\le 1-\epsilon _0 +e^{-2t N_0^{\gamma } } \epsilon _0 \le 1- \epsilon _0 + (1-\frac{3}{2} tN_0^{\gamma } ) \epsilon _0 \le e^{- \epsilon _0 N_0^{\gamma } t}, \end{aligned}$$

where in the last two steps we used the fact $e^{-x} \sim 1- x +\frac{x^2}{2}$ for $x \rightarrow 0+$. The inequality for ${\tilde{g}}$ follows from the observation that $\Vert |\xi |^s \hat{{\tilde{g}}}(\xi ) \Vert _{L^2_{\xi }(|\xi | \le N_0)} \ll 1$ for $N_0$ sufficiently small. $\square$

Proposition 2.7

Let the dimension $n\ge 1$, $0<\gamma \le 2$ and $1<q<\infty$. Then for any $f \in L^q({\mathbb {R}}^n)$, any $j\in {\mathbb {Z}}$ and any $t>0$, we have

$$\begin{aligned} \Vert e^{ - t|\nabla |^{\gamma } } P_j f \Vert _q \le e^{- c 2^{j\gamma } t} \Vert P_j f \Vert _q, \end{aligned}$$

where $c>0$ is a constant depending on $(\gamma ,n,q)$.

Proof

With no loss we assume $j=1$ and write $P_1 f$ simply as f. In view of the semigroup property of $e^{-t |\nabla |^{\gamma }}$ it suffices to prove the inequality for $0<t \ll _{\gamma ,q,n} 1$. Denote $e^{-t |\nabla |^{\gamma } } f = K*f$ and observe that K is a positive kernel with $\int K(z) dz =1$. Consider first $2\le q<\infty$. Clearly

$$\begin{aligned} | \int K(x-y) f(y) dy|^{\frac{q}{2}} \le \int K(x-y) |f(y)|^{\frac{q}{2}} dy. \end{aligned}$$

By Lemma 2.6 and Proposition 2.5, we then get

$$\begin{aligned} \Vert e^{-t |\nabla |^{\gamma } } f \Vert _q^q \le \Vert e^{-t |\nabla |^{\gamma } } ( |f|^{\frac{q}{2}} ) \Vert _2^2 \le e^{-ct} \Vert f \Vert _q^q. \end{aligned}$$

For the case $1<q<2$, we observe

$$\begin{aligned}&\Vert \int K(x-y) |f(y)| dy \Vert _{L_x^q} \\&\quad \le \Vert ( \int K(x-y) |f(y)|^{\frac{q}{2}} dy )^{\frac{2(q-1)}{q}} \cdot ( \int K(x-y) |f(y)|^q dy)^{\frac{2-q}{q}} \Vert _{L_x^q} \\&\quad \le \Vert ( \int K(x-y) |f(y)|^{\frac{q}{2}} dy)^{\frac{2(q-1)}{q}} \Vert _{L_x^{\frac{q}{q-1} }} \Vert (\int K(x-y) |f(y)|^q d y)^{\frac{2-q}{q}} \Vert _{L_x^{\frac{q}{2-q} }} \\&\quad \le \Vert e^{-t |\nabla |^{\gamma } } ( |f|^{\frac{q}{2}} ) \Vert _2^2 \cdot \Vert f \Vert _q^{2-q}. \end{aligned}$$

Thus this case is also OK. $\square$

For the next lemma we need to introduce some terminology. Consider a function $F:\, (0,\infty ) \rightarrow {\mathbb {R}}$. We shall say F is admissible if $F\in C^{\infty }$ and

$$\begin{aligned} | F^{(k)} (x) | \lesssim _{k, F} x^{-k}, \quad \forall \, k\ge 0, \; 0<x<\infty . \end{aligned}$$

It is easy to check that ${\tilde{F}}(x) = x F^{\prime }(x)$ is admissible if F is admissible. A simple example of admissible function is $F(x) = e^{-x}$ which will show up in the bilinear estimates later.

Lemma 2.8

Let $0<\gamma <1$ and $\sigma (\xi ,\eta )= |\xi |^{\gamma } +|\eta |^{\gamma } - |\xi +\eta |^{\gamma }$ for $\xi$, $\eta \in {\mathbb {R}}^n$, $n\ge 1$. Then for $0<|\xi |\ll 1$, $|\eta | \sim 1$, the following hold:

• (1) $|\partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } \sigma (\xi , \eta ) | \lesssim _{\alpha , \beta ,\gamma ,n} |\xi |^{\gamma -|\alpha |}$, for any $\alpha$, $\beta$.

• (2) $| \partial _{\xi }^{\alpha } ( \sigma ^{-m} ) | \lesssim _{\alpha ,\gamma , m,n} |\xi |^{-m\gamma -|\alpha |}$ for any $m\ge 1$ and any $\alpha$.

• (3) $|\partial _{\xi }^{\alpha } ( F(t \sigma ) ) | \lesssim _{\alpha ,\gamma ,n, F} |\xi |^{-|\alpha |}$ for any $\alpha$, $t>0$, and any admissible F.

• (4) $|\partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } (F(t\sigma ) ) | \lesssim _{\alpha ,\gamma ,\beta , n, F} |\xi |^{-|\alpha |}$, for any $\alpha$, $\beta$, and any $t>0$, F admissible.

Remark 2.9

The condition $|\xi | \ll 1$, $|\eta |\sim 1$ can be replaced by $0<|\xi | \lesssim 1$, $|\eta |\sim 1$, $|\xi +\eta | \sim 1$.

Remark

This lemma also highlights the importance of the assumption $0<\gamma <1$. For $0<\gamma <1$, note that the function $g(x) =1 +x^{\gamma } - (1+x)^{\gamma } \sim \min \{ x^{\gamma }, \, 1 \}$. By the triangle inequality, this implies $\sigma (\xi ,\eta ) \ge |\xi |^{\gamma } + |\eta |^{\gamma } - (|\xi |+|\eta |)^{\gamma } \gtrsim \min \{ |\xi |^{\gamma }, |\eta |^{\gamma } \}$ which does not vanish as long as $|\xi |>0$ and $|\eta |>0$. However for $\gamma =1$, the phase $\sigma (\xi ,\eta ) = |\xi |+ |\eta | - |\xi +\eta |$ no longer enjoys such a lower bound since $\sigma \equiv 0$ on the one-dimensional cone $\xi = \lambda \eta$, $\lambda \ge 0$.

Proof

With no loss we consider dimension $n=1$. The case $n>1$ is similar except some minor changes in numerology.

(1) Note that for $|\xi |\ll 1$, $|\eta | \sim 1$, we have

$$\begin{aligned} \sigma (\xi , \eta ) = |\xi |^{\gamma } - \gamma \int _0^1 |\eta + \theta \xi |^{\gamma -2} (\eta +\theta \xi ) d\theta \cdot \xi . \end{aligned}$$

Observe that

$$\begin{aligned} \partial _{\eta } \sigma (\xi , \eta ) = ({\text {OK}}) \cdot \xi , \end{aligned}$$

where we use the notation ${\text {OK}}$ to denote any term which satisfy

$$\begin{aligned} | \partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } ( {\text {OK}} ) | \lesssim 1, \quad \forall \, \alpha ,\beta . \end{aligned}$$

This notation will be used throughout this proof. Thus for any $\beta \ge 1$, $\alpha \ge 0$, we have

$$\begin{aligned} | \partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } \sigma (\xi , \eta ) | \lesssim |\xi |^{1-|\alpha |}. \end{aligned}$$

On the other hand, if $\beta =0$ and $\alpha \ge 1$, then clearly

$$\begin{aligned} | \partial _{\xi }^{\alpha } \sigma (\xi ,\eta ) | = |\partial _{\xi }^{\alpha } ( |\xi |^{\gamma } - |\xi +\eta |^{\gamma } )| \lesssim |\xi |^{ \gamma -|\alpha |}. \end{aligned}$$

(2) Observe that for $|\xi | \ll 1$ and $|\eta | \sim 1$, we always have $\sigma (\xi ,\eta ) \gtrsim |\xi |^{\gamma }$. One can then induct on $\alpha$.

(3) One can induct on $\alpha$. The statement clearly holds for $\alpha =0$. Assume the statement holds for $\alpha \le m$ and any admissible F. Then for $\alpha =m+1$, we have

$$\begin{aligned} \partial _{\xi }^{m+1} ( F(t\sigma ) ) = \partial _{\xi }^m ( {\tilde{F}}( t\sigma ) \cdot \sigma ^{-1} \cdot \partial _{\xi } \sigma ), \end{aligned}$$

where ${\tilde{F}}(x) = x F^{\prime }(x)$ is again admissible. The result then follows from the inductive assumption, Leibniz and the estimates obtained in (1) and (2).

(4) Observe that $\partial _{\eta }(\frac{1}{\sigma } ) = - \sigma ^{-2} \partial _{\eta } \sigma = \sigma ^{-2} \xi \cdot ({\text {OK}})$, and in general for $\beta \ge 0$,

$$\begin{aligned} \partial _{\eta }^{\beta } ( \frac{1}{\sigma } ) = \sum _{ 0\le m\le \beta } \sigma ^{-m-1} \xi ^m \cdot ({\text {OK}}). \end{aligned}$$

Note that for $\beta \ge 1$ the summand corresponding to $m=0$ is actually absent (this is allowed in our notation since we can take the term (${\text {OK}}$) to be zero). Similarly one can check for any admissible F and $t>0$,

$$\begin{aligned} \partial _{\eta }^{\beta } ( F(t\sigma ) ) = \sum _{0\le m \le \beta } F_m(t\sigma ) \cdot \left( \frac{\xi }{\sigma } \right) ^m \cdot ({\text {OK}}), \end{aligned}$$

where $F_m$ are admissible functions. This then reduce matters to the estimate in (3). The result is obvious. $\square$

3 Nonlinear Estimates: $H^{2-\gamma }$ Case for $0<\gamma <1$

Lemma 3.1

Set $A=D^{\gamma }$, $s=2-\gamma$ and recall $R^{\perp }=(-D^{-1} \partial _2, D^{-1} \partial _1)$. For any real-valued $f, \, g \in L^2({\mathbb {R}}^2)$ with ${\hat{f}}$ and ${\hat{g}}$ being compactly supported, it holds that

$$\begin{aligned} | \int _{{\mathbb {R}}^2} D^s (e^{-t A} R^{\perp } g \cdot \nabla e^{-tA} f) D^s e^{tA} f dx | \lesssim _{\gamma } \Vert g \Vert _{\dot{H}^s} \Vert f \Vert _{\dot{H}^{s+\frac{\gamma }{2}} }^2. \end{aligned}$$

(3.1)

If in addition ${\text {supp}}({\hat{g}}) \subset B(0,N_0)$ for some $N_0>0$, then

$$\begin{aligned}&| \int _{{\mathbb {R}}^2} D^s (e^{-t A}R^{\perp } g \cdot \nabla e^{-tA} f) D^s e^{tA} f dx | \nonumber \\&\quad \lesssim _{\gamma } N_0^{2s+2} \Vert f\Vert _2^2 \Vert g \Vert _2+ \Vert f \Vert _{\dot{H}^s} \cdot \Vert f \Vert _{\dot{H}^{s+\frac{\gamma }{2}}} \cdot \Vert g\Vert _2\cdot N_0^{2-\frac{\gamma }{2}}, \end{aligned}$$

(3.2)

$$\begin{aligned}&| \int _{{\mathbb {R}}^2} D^s (e^{-tA} R^{\perp } f \cdot \nabla e^{-tA} g) D^s e^{tA} f dx | \lesssim _{\gamma } \Vert f \Vert _{\dot{H}^s}^2 \cdot N_0^2 \Vert g\Vert _2 + N_0^{2s+2} \Vert f \Vert _2^2 \Vert g\Vert _2, \end{aligned}$$

(3.3)

$$\begin{aligned}&| \int _{{\mathbb {R}}^2} D^s (e^{-tA} R^{\perp } g \cdot \nabla e^{-tA} g) D^s e^{tA} f dx | \lesssim _{\gamma } N_0^{2s+2} \Vert g \Vert _2^2 \Vert {\hat{f}} \Vert _{L^2(|\xi |\le 2N_0)}. \end{aligned}$$

(3.4)

Remark 3.2

Note that if ${\hat{f}}$ is localized to $|\xi |\gtrsim N_0$, then the low-frequency term $N_0^{2s+2} \Vert f \Vert _2^2 \Vert g\Vert _2$ can be dropped in (3.2) and (3.3).

Proof

We first show (3.1). For simplicity of notation we shall write $R^{\perp } g$ as g. Note that in the final estimates the operator $R^{\perp }$ can be easily discarded since we are in the $L^2$ setting. On the Fourier side we express the LHS inside the absolute value as (up to a multiplicative constant)

$$\begin{aligned} \int |\xi |^{2s} e^{-t ( |\eta |^{\gamma } + |\xi -\eta |^{\gamma } - |\xi |^{\gamma } )} {\hat{g}}(\eta ) \cdot (\xi -\eta ) {\hat{f}} (\xi -\eta ) {\hat{f}} (-\xi ) d\xi d\eta . \end{aligned}$$

Observe that by a change of variable $\xi \rightarrow \eta -{\tilde{\xi }}$ (and dropping the tildes), we have

$$\begin{aligned}&\int ( \xi -\eta ) |\xi |^{2s} e^{- t (|\xi -\eta |^{\gamma } - |\xi |^{\gamma } )} {\hat{f}}(\xi -\eta ) {\hat{f}}(-\xi ) d\xi \\ =&\frac{1}{2} \int \left( (\xi -\eta ) |\xi |^{2s} e^{-t (|\xi -\eta |^{\gamma } - |\xi |^{\gamma } )} - \xi |\xi -\eta |^{2s} e^{-t ( |\xi |^{\gamma } - |\xi -\eta |^{\gamma })} \right) {\hat{f}}(\xi -\eta ) {\hat{f}}(-\xi ) d\xi . \end{aligned}$$

Denote

$$\begin{aligned}&{\tilde{\sigma }}(\xi ,\eta )= e^{-t |\eta |^{\gamma }} \biggl ( (\xi -\eta ) |\xi |^{2s} e^{-t (|\xi -\eta |^{\gamma } - |\xi |^{\gamma } )} - \xi |\xi -\eta |^{2s} e^{-t ( |\xi |^{\gamma } - |\xi -\eta |^{\gamma })} \biggr ), \\&N(g^1,g^2,g^3) = \int {\tilde{\sigma }}(\xi ,\eta ) \widehat{g^1}(\eta ) \widehat{g^2}(\xi -\eta ) \widehat{g^3}(-\xi ) d\xi d\eta . \end{aligned}$$

We just need to bound N(g, f, f). By frequency localization, we have

$$\begin{aligned} N(g,f,f)&= \sum _j \Bigl (N(g_{<j-9}, f_j, f) + N(g_{>j+9}, f_{j},f ) + N(g_{[j-9,j+9]}, f_j, f) \Bigr ). \end{aligned}$$

Rewriting $\sum _j N(g_{>j+9}, f_j, f) = \sum _j N(g_j, f_{<j-9}, f)$, we obtain

$$\begin{aligned}&N(g,f,f) = \sum _j \Bigl (N(g_{<j-9}, f_j, f) + N(g_{j}, f_{<j-9},f ) + N(g_{[j-9,j+9]}, f_j, f) \Bigr ) \\&\quad = \sum _j \Bigl (N(g_{<j-9}, f_j, f_{[j-2,j+2]}) + N(g_{j}, f_{<j-9},f_{[j-2,j+2]} ) + N(g_{[j-9,j+9]}, f_j, f_{\le j+11} ) \Bigr ) \\&\quad = \sum _j \Bigl ( N(g_{\ll j}, f_{\sim j}, f_{\sim j}) + N(g_{\sim j}, f_{\ll j}, f_{\sim j}) + N(g_{\sim j}, f_{\sim j}, f_{\lesssim j} ) \Bigr ), \end{aligned}$$

where $g_{\ll j}$ corresponds to $|\eta | \ll 2^j$, $g_{\sim j}$ means $|\eta | \sim 2^j$, and $g_{\lesssim j}$ means $|\eta |\lesssim 2^j$. These notations are quite handy since only the relative sizes of the frequency variables $\eta$, $\xi$ and $\xi -\eta$ will play some role in the estimates. Note that we should have written $g_{\ll j}$ as $g_{\{l: \, 2^l \ll 2^j\} }$ according to our convention of the notation $\ll$ but we ignore this slight inconsistency here for the simplicity of notation.

1. Estimate of $N(g_{\ll j}, f_{\sim j}, f_{\sim j})$. Note that $|\eta | \ll 2^j$, $|\xi -\eta | \sim |\xi | \sim 2^j$. It is not difficult to check that in this regime

$$\begin{aligned}&|{\tilde{\sigma }}(\xi ,\eta )| \le | (\xi -\eta )|\xi |^{2s} - \xi |\xi -\eta |^{2s} | \cdot e^{-t (|\eta |^{\gamma }+|\xi -\eta |^{\gamma }-|\xi |^{\gamma })} \\&\quad +|\xi | |\xi -\eta |^{2s} \cdot |e^{-t(|\eta |^{\gamma }+|\xi -\eta |^{\gamma }-|\xi |^{\gamma })} -e^{-t (|\xi |^{\gamma }+|\eta |^{\gamma } -|\xi -\eta |^{\gamma } )} | \nonumber \\&\lesssim 2^{2js} \cdot |\eta |+|\xi | |\xi -\eta |^{2s} |e^{-t(|\eta |^{\gamma }+|\xi -\eta |^{\gamma }-|\xi |^{\gamma })} -e^{-t (|\xi |^{\gamma }+|\eta |^{\gamma } -|\xi -\eta |^{\gamma } )} |. \end{aligned}$$

To bound the second term, we shall use Lemma 2.8. More precisely, denote ${\tilde{\xi }} = -2^{-j} \eta$, ${\tilde{\eta }} = 2^{-j} \xi$, $T=2^{j\gamma } t$, $F(x)=e^{-x}$. Clearly (recall in Lemma 2.8, $\sigma (\xi ,\eta )= |\xi |^{\gamma } +|\eta |^{\gamma } - |\xi +\eta |^{\gamma }$)

$$\begin{aligned}&e^{-t(|\eta |^{\gamma }+|\xi -\eta |^{\gamma }-|\xi |^{\gamma })} = e^{-T ( |{\tilde{\xi }}|^{\gamma } + |{\tilde{\xi }} + {\tilde{\eta }}|^{\gamma } - |{\tilde{\eta }}|^{\gamma } )} = F(T \sigma ( {\tilde{\xi }} , {\tilde{\eta }} -{\tilde{\xi }}) ), \nonumber \\&e^{-t (|\xi |^{\gamma }+|\eta |^{\gamma } -|\xi -\eta |^{\gamma } )} = F(T \sigma ( {\tilde{\xi }}, {\tilde{\eta }}) ). \end{aligned}$$

(3.5)

Consider for $0\le \theta \le 1$, the function $G(\theta ) = F(T \sigma ({\tilde{\xi }}, {\tilde{\eta }} -\theta {\tilde{\xi }}) )$. By Lemma 2.8, we have

$$\begin{aligned} |G(1)-G(0)| \le \int _0^1 | \partial _{{\tilde{\eta }}} ( F(T \sigma ({\tilde{\xi }}, {\tilde{\eta }}-\theta {\tilde{\xi }}) )) | d\theta \cdot |{\tilde{\xi }} | \lesssim |{\tilde{\xi }}|=2^{-j} |\eta |. \end{aligned}$$

Thus

$$\begin{aligned} |{\tilde{\sigma }}(\xi , \eta )| \lesssim 2^{2js} \cdot |\eta |. \end{aligned}$$

Then by taking ${\tilde{\epsilon }}=\gamma$ below (note that $0<\gamma <1$), we get (below “$*$" denotes the usual convolution)

$$\begin{aligned} |\sum _j N(g_{\ll j}, f_{\sim j}, f_{\sim j} ) |&\lesssim \sum _j 2^{2js} \cdot \Vert f_{\sim j} \Vert _2 \cdot ( \Vert \; | \widehat{Dg_{\ll j} }| * |\widehat{f_{\sim j} }|) \Vert _2 ) \nonumber \\&\lesssim \sum _j 2^{2js} \cdot \Vert f_{\sim j} \Vert _2 \cdot (\Vert D g_{\ll j} \Vert _{\dot{H}^{1-{\tilde{\epsilon }}} } \cdot \Vert f_{\sim j} \Vert _{\dot{H}^{{\tilde{\epsilon }}} } ) \nonumber \\&\lesssim \Vert g \Vert _{\dot{H}^s} \Vert f \Vert _{\dot{H}^{s+\frac{\gamma }{2}} }^2. \end{aligned}$$

Here in the second inequality above, we have used the simple fact that

$$\begin{aligned} \Vert |{\hat{A}}| * |{\hat{B}}| \Vert _{L_{\xi }^2} \lesssim \Vert {\hat{A}} \Vert _{L_{\xi }^1} \Vert {\hat{B}} \Vert _{L_{\xi }^2} \lesssim \Vert |\xi |^{\theta } {\hat{A}} \Vert _{L_{\xi }^2} \Vert |\xi |^{1-\theta }{\hat{B}} \Vert _{L_{\xi }^2}, \end{aligned}$$

if ${\text {supp}}({\hat{A}}) \subset \{ |\xi | \lesssim 1 \}$, ${\text {supp}}({\hat{B}}) \subset \{ |\xi | \sim 1 \}$, and $\theta <1$.

2. Estimate of $N(g_{\sim j}, f_{\ll j}, f_{\sim j})$. In this case $|\eta | \sim |\xi | \sim 2^j$, $|\xi -\eta | \ll 2^j$. Since $s=2-\gamma \in (1,2)$, in this regime we have

$$\begin{aligned} |{\tilde{\sigma }}(\xi ,\eta )| \lesssim |\xi -\eta | 2^{2js} + 2^{j} |\xi -\eta |^{2s} \lesssim 2^{2js} |\xi -\eta |. \end{aligned}$$

Then

$$\begin{aligned}&| \sum _j N(g_{\sim j}, f_{\ll j}, f_{\sim j} ) | \lesssim \sum _j 2^{2js} \Vert \, |\widehat{g_{\sim j} } | * |\widehat{D f_{\ll j}} | \, \Vert _2 \cdot \Vert f_{\sim j} \Vert _2 \nonumber \\&\quad \lesssim \sum _j 2^{2js} \Vert g_{\sim j} \Vert _{\dot{H}^{\frac{\gamma }{2} }} \cdot \Vert D f_{\ll j} \Vert _{\dot{H}^{1-\frac{\gamma }{2}}} \Vert f_{\sim j}\Vert _2 \lesssim \Vert g \Vert _{\dot{H}^s} \Vert f \Vert _{\dot{H}^{s+\frac{\gamma }{2} } }^2. \end{aligned}$$

3. Estimate of $N(g_{\sim j}, f_{\sim j}, f_{\lesssim j})$. In this case $|\eta | \sim |\xi -\eta | \sim 2^j$, $|\xi | \lesssim 2^j$, and

$$\begin{aligned} | \sum _j N(g_{\sim j}, f_{\sim j}, f_{\lesssim j } ) |&\lesssim \sum _j 2^{2js} \Vert |\widehat{g_{\sim j}}| * |\widehat{f_{\sim j} } | \Vert _{L^{\infty }_{\xi } } \Vert \widehat{D f_{\lesssim j} } \Vert _{L^1_{\xi }} \nonumber \\&\lesssim \sum _j 2^{2js} \Vert g_{\sim j} \Vert _2 \Vert f_{\sim j} \Vert _2 \cdot \Vert |\xi |^{1-\frac{\gamma }{2}} \widehat{Df_{\lesssim j } } \Vert _{L^2_{\xi }} \cdot 2^{j\frac{\gamma }{2} } \nonumber \\&\lesssim \Vert g \Vert _{\dot{H}^s} \Vert f\Vert _{\dot{H}^{s+\frac{\gamma }{2}} }^2. \end{aligned}$$

Now we turn to (3.2). Choose $J_0\in {\mathbb {Z}}$ such that $2^{J_0-1} \le N_0 <2^{J_0}$. Clearly by frequency localization,

$$\begin{aligned} N(g,f,f) = N(g, f_{\lesssim J_0}, f_{\lesssim J_0} ) +\sum _{j:\, 2^j \gg N_0} N(g, f_{\sim j}, f_{\sim j} ). \end{aligned}$$

For the first term we have

$$\begin{aligned} | N(g,f_{\lesssim J_0}, f_{\lesssim J_0} ) |&\lesssim N_0^{2s+1} \Vert |{\widehat{g}} | *|\widehat{f_{\lesssim J_0} }| \Vert _{L_{\xi }^{\infty }} \Vert \widehat{f_{\lesssim J_0} } \Vert _{L_{\xi }^1} \\&\lesssim N_0^{2s+2} \Vert g \Vert _2 \Vert f \Vert _2^2. \end{aligned}$$

For the second term we can use the estimate of $N(g_{\ll j}, f_{\sim j}, f_{\sim j})$ and take ${\tilde{\epsilon }}=\frac{\gamma }{2}$ to get

$$\begin{aligned} |\sum _{j: 2^j\gg N_0} N(g,f_{\sim j}, f_{\sim j} ) |&\lesssim \sum _{j:\, 2^j \gg N_0} 2^{2js} \cdot \Vert f_{\sim j} \Vert _2 \cdot ( \Vert D g \Vert _{\dot{H}^{1-{\tilde{\epsilon }}} } \Vert f_{\sim j} \Vert _{\dot{H}^{{\tilde{\epsilon }}} }) \nonumber \\&\lesssim \Vert f \Vert _{\dot{H}^s} \cdot \Vert f \Vert _{\dot{H}^{s+\frac{\gamma }{2}}} \cdot \Vert g\Vert _2\cdot N_0^{2-\frac{\gamma }{2}}. \end{aligned}$$

The estimates of (3.3) and (3.4) are much simpler. We omit the details. $\square$

4 Proof of Theorem 1.1

To simplify numerology we conduct the proof for the case $\epsilon _0=1/2$. Throughout this proof we shall denote $s=2-\gamma$.

Step 1. A priori estimate. Denote $A= \frac{1}{2} D^{\gamma }$ and $f= e^{t A } \theta$. It will be clear from Step 2 below that f is smooth and well-defined, and the following computations can be rigorously justified. Then f satisfies the equation

$$\begin{aligned} \partial _t f = -\frac{1}{2} D^{\gamma } f - e^{t A} ( R^{\perp } e^{-tA} f \cdot \nabla e^{-t A} f ). \end{aligned}$$

Take $J_0>0$ which will be made sufficiently large later. Set $N_0=2^{J_0}$. Then

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} ( \Vert D^s P_{>J_0} f \Vert _2^2 ) + \frac{1}{2} \Vert D^{s+\frac{\gamma }{2}} P_{>J_0} f \Vert _2^2 \\&= -\int D^s ( R^{\perp } e^{-t A} f \cdot \nabla e^{-tA} f ) D^s e^{t A} P_{>J_0}^2 f dx. \end{aligned}$$

Now for convenience of notation we denote

$$\begin{aligned} N(g_1,g_2,g_3) = \int D^s ( R^{\perp } e^{-tA} g_1 \cdot \nabla e^{-tA} g_2) D^s e^{tA} g_3 dx. \end{aligned}$$

Denote $f_h = P_{>J_0}^2 f$ and $f_l = f-f_h$. Then clearly

$$\begin{aligned} N(f, f, f_h) = N(f_h, f_h, f_h) + N(f_l, f_h,f_h) + N(f_h,f_l, f_h) + N(f_l,f_l, f_h). \end{aligned}$$

By Lemma 3.1 and noting that $\Vert f_l(t)\Vert _2 \lesssim e^{ N_0^{\gamma } t} \Vert \theta _0\Vert _2$, we get (see Remark 3.2)

$$\begin{aligned}&|N(f,f,f_h) | \\&\quad \lesssim \Vert f_h \Vert _{\dot{H}^s} \Vert f_h \Vert _{\dot{H}^{s+\frac{\gamma }{2}} }^2 + \Vert f_h \Vert _{\dot{H}^s} \Vert f_h \Vert _{\dot{H}^{s+\frac{\gamma }{2}} } \cdot N_0^{2-\frac{\gamma }{2}} e^{N_0^{\gamma } t} \Vert \theta _0\Vert _2 + \Vert f_h\Vert _{\dot{H}^s}^2 \cdot N_0^2 e^{N_0^{\gamma } t} \Vert \theta _0\Vert _2 \nonumber \\&\quad + N_0^{2s+2} e^{4N_0^{\gamma } t} \Vert \theta _0\Vert _2^3. \end{aligned}$$

This implies for $0<t\le N_0^{-\gamma }$,

$$\begin{aligned}&\frac{d}{dt} ( \Vert D^s P_{>J_0} f \Vert _2^2) + \biggl (\frac{1}{2}-c_1 \Vert D^s P_{>J_0} f\Vert _{2} \biggr ) \cdot \Vert D^{s+\frac{\gamma }{2}} P_{>J_0} f \Vert _2^2 \nonumber \\ \le&\; \underbrace{c_2 \cdot (N_0^2 \Vert \theta _0\Vert _2 +N_0^{4-\gamma } \Vert \theta _0 \Vert _2^2) }_{=:\beta } \cdot \Vert D^s P_{>J_0} f \Vert _{2}^2 +c_3 N_0^{2s+2} \Vert \theta _0\Vert _2^3, \end{aligned}$$

where $c_1, c_2, c_3>0$ are constants depending on $\gamma$.

Thus as long as $\sup _{0\le s \le t} c_1 \Vert D^s P_{>J_0} f(s) \Vert _2 <\frac{1}{10}$ and $t\le N_0^{-\gamma }$, we obtain

$$\begin{aligned} \sup _{0\le s \le t} \Vert D^s P_{>J_0} f(s) \Vert _2^2 \le e^{\beta t} \Vert D^s P_{>J_0} \theta _0 \Vert _2^2 +t e^{\beta t} c_3 N_0^{2s+2} \Vert \theta _0\Vert _2^3. \end{aligned}$$

(4.1)

In particular, for any prescribed small constant $\epsilon _0>0$, we can first choose $J_0$ sufficiently large such that

$$\begin{aligned} \Vert D^s P_{>J_0} \theta _0 \Vert _2 < \frac{1}{2} \epsilon _0. \end{aligned}$$

(4.2)

Then by using (4.1) and choosing $T_0= T_0(J_0, \theta _0, \epsilon _0)$ sufficiently small we can guarantee

$$\begin{aligned} \sup _{0\le s \le T_0} \Vert D^s P_{>J_0} f(s) \Vert _2 < \epsilon _0. \end{aligned}$$

(4.3)

Step 2. Approximation system. For $n=1,2,3,\cdots$, define $\theta ^{(n)}$ as solutions to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(n)} =- P_{<n} ( R^{\perp } P_{<n} \theta ^{(n)} \cdot \nabla P_{<n} \theta ^{(n)} ) - D^{\gamma } \theta ^{(n)}, \\ \theta ^{(n)} \bigr |_{t=0} =P_{<n} \theta _0. \end{array}\right. } \end{aligned}$$

The solvability of the above regularized system is not an issue thanks to frequency cut-offs. It is easy to check that $\theta ^{(n)}$ has frequency supported in $|\xi | \lesssim 2^n$ and $\Vert \theta ^{(n)} \Vert _2 \le \Vert \theta _0 \Vert _2$. In particular for any ${\tilde{s}} \ge 0$ we have

$$\begin{aligned} \Vert D^{{\tilde{s}} } P_{\le J_0} \theta ^{(n) } \Vert _2 \lesssim 2^{c J_0 {\tilde{s}}} \Vert \theta _0 \Vert _2, \end{aligned}$$

(4.4)

where $c>0$ is a constant.

For any integer $J_0$ to be fixed momentarily, it is not difficult to check that

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} (\Vert D^s P_{>J_0} e^{tA} \theta ^{(n)} \Vert _2^2) + \frac{1}{2} \Vert D^{s+\frac{\gamma }{2}} P_{>J_0} e^{tA} \theta ^{(n)} \Vert _2^2 \nonumber \\ =&\; - \int D^s ( R^{\perp }P_{<n} \theta ^{(n)} \cdot \nabla P_{<n} \theta ^{(n)} ) D^s e^{tA} P_{>J_0}^2 e^{tA} P_{<n} \theta ^{(n)} dx. \end{aligned}$$

Now fix $J_0$ sufficiently large such that

$$\begin{aligned} c_1 \Vert D^s P_{>J_0} \theta _0 \Vert _2 < 1/10. \end{aligned}$$

By using the nonlinear estimate derived in Step 1 (easy to check that these estimates hold for $\theta ^{(n)}$ with slight changes of the constants $c_i$ if necessary), one can then find $T_0=T_0(\gamma ,\theta _0)>0$ sufficiently small such that uniformly in n tending to infinity, we have

$$\begin{aligned} \sup _{0\le t \le T_0} \Vert e^{tA} D^s \theta ^{(n)} (t,\cdot ) \Vert _2 + \int _0^{T_0} \Vert e^{tA} D^{s+\frac{\gamma }{2}} \theta ^{(n)} (t,\cdot ) \Vert _2^2 dt \lesssim 1. \end{aligned}$$

By slightly shrinking $T_0$ further if necessary and repeating the argument for ${\tilde{A}}= \frac{4}{3} A = \frac{2}{3} D^{\gamma }$, we have uniformly in n tending to infinity,

$$\begin{aligned} \sup _{0\le t \le T_0} \Vert e^{\frac{4}{3} tA} D^s \theta ^{(n)} (t,\cdot ) \Vert _2 \lesssim 1. \end{aligned}$$

Furthermore for any prescribed small constant $\epsilon _0>0$, by using (4.3), we can choose $J_0$ and $T_0$ such that uniformly in n,

$$\begin{aligned} \sup _{0\le t \le T_0} \Vert D^s P_{>J_0} e^{tA} \theta ^{(n)} \Vert _2 <\epsilon _0. \end{aligned}$$

Note that this implies

$$\begin{aligned} \sup _{0\le t \le T_0} \Vert D^s P_{>J_0} \theta ^{(n)} \Vert _2 <\epsilon _0. \end{aligned}$$

(4.5)

The estimate (4.5) will be needed later.

Step 3. Strong contraction of $\theta ^{(n)}$ in $C_t^0 L_x^2$. Denote $\eta _{n+1}= \theta ^{(n+1)}-\theta ^{(n)}$. Then (below for simplicity of notation we write $-R^{\perp }$ as R)

$$\begin{aligned} \partial _t \eta _{n+1}&= P_{<n+1}( RP_{<n+1} \eta _{n+1} \cdot \nabla P_{<n+1} \eta _{n+1}) - D^{\gamma } \eta _{n+1} \nonumber \\&\quad + P_{<n+1} (R P_{<n+1} \theta ^{(n)} \cdot \nabla P_{<n+1} \theta ^{(n)} ) - P_{<n} (R P_{<n} \theta ^{(n)} \cdot \nabla P_{<n} \theta ^{(n)} ) \nonumber \\&\quad + P_{<n+1} ( RP_{<n+1} \theta ^{(n)} \cdot \nabla P_{<n+1} \eta _{n+1} ) \nonumber \\&\quad +P_{<n+1} ( R P_{<n+1} \eta _{n+1} \cdot \nabla P_{<n+1} \theta ^{(n)} ). \end{aligned}$$

(4.6)

By using the divergence-free property, we have

$$\begin{aligned}&\int (4.6) \cdot \eta _{n+1} dx \\&\quad = -\int \biggl ( P_{<n+1} (R P_{<n+1} \theta ^{(n)} P_{<n+1} \theta ^{(n)} ) - P_{<n} (R P_{<n} \theta ^{(n)} P_{<n} \theta ^{(n)} ) \biggr )\cdot \nabla \eta _{n+1} dx. \end{aligned}$$

Clearly

$$\begin{aligned}&\Vert P_{<n+1} (R P_{<n+1} \theta ^{(n)} P_{<n+1} \theta ^{(n)} ) - P_{<n} (R P_{<n} \theta ^{(n)} P_{<n} \theta ^{(n)} ) \Vert _2 \nonumber \\ \le&\; \Vert P_{<n+1} ( R(P_{<n+1}-P_{<n}) \theta ^{(n)} P_{<n+1} \theta ^{(n)} ) \Vert _2 + \Vert P_{<n+1} ( RP_{<n} \theta _n (P_{<n+1} -P_{<n}) \theta ^{(n)} ) \Vert _2 \nonumber \\&\quad + \Vert (P_{<n+1}-P_{<n}) (RP_{<n} \theta ^{(n)} P_{<n} \theta ^{(n)} ) \Vert _2 \nonumber \\ \lesssim&\; 2^{-n(2-\gamma )} \Vert \theta ^{(n)}\Vert _{H^{2-\gamma } }^2 + 2^{- n} \Vert \theta ^{(n)}\Vert _{H^{2-\gamma }}^2 \lesssim 2^{-n}, \end{aligned}$$

where we have used the uniform Sobolev estimates in Step 2. Note that

$$\begin{aligned} \Vert \nabla \eta _{n+1} \Vert _2 \lesssim \Vert \theta ^{(n)} \Vert _{H^{2-\gamma }} + \Vert \theta ^{(n+1)} \Vert _{H^{2-\gamma }} \lesssim 1. \end{aligned}$$

It follows that

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \eta _{n+1}\Vert _2^2 + \Vert D^{\frac{\gamma }{2}} \eta _{n+1} \Vert _2^2 \\&\qquad \le \int (RP_{<n+1} \eta _{n+1} \cdot \nabla P_{<n+1} \theta ^{(n)} ) P_{<n+1} \eta _{n+1} dx \;+{\text {const}}\cdot 2^{-n} \nonumber \\&\quad \le {\text {const}}\cdot 2^{-n} + \int (RP_{<n+1} \eta _{n+1} \cdot \nabla P_{>J_0} P_{<n+1} \theta ^{(n)} ) P_{<n+1} \eta _{n+1} dx \nonumber \\&\qquad + \int (RP_{<n+1} \eta _{n+1} \cdot \nabla P_{\le J_0} P_{<n+1} \theta ^{(n) }) P_{<n+1} \eta _{n+1} dx \nonumber \\&\lesssim 2^{-n}+ \Vert \eta _{n+1} \Vert _{(\frac{1}{2} -\frac{\gamma }{4})^{-1}}^2 \cdot \Vert \nabla P_{>J_0} P_{<n+1} \theta ^{(n)} \Vert _{\frac{2}{\gamma }} \nonumber \\&\quad + \Vert \eta _{n+1} \Vert _2^2 \cdot 2^{2J_0} \Vert \theta ^{(n)} \Vert _2 \nonumber \\&\lesssim 2^{-n}+ \Vert D^{\frac{\gamma }{2}} \eta _{n+1} \Vert _2^2 \cdot \Vert D^s P_{>J_0} \theta ^{(n)} \Vert _2 + 2^{2J_0} \Vert \eta _{n+1} \Vert _2^2. \end{aligned}$$

By using the nonlinear estimates in Step 2 and (4.5), one can choose $J_0$ sufficiently large (and slightly shrink $T_0$ further if necessary) such that the term $\Vert D^s P_{>J_0} \theta ^{(n)}\Vert _2$ becomes sufficiently small (to kill the implied constant pre-factors in the above inequality). This implies

$$\begin{aligned} \frac{d}{dt} \Vert \eta _{n+1} \Vert _2^2 \lesssim 2^{-n} + 2^{2J_0} \Vert \eta _{n+1} \Vert _2^2. \end{aligned}$$

(4.7)

Thus for some constants ${\tilde{c}}_1>0$, ${\tilde{c}}_2>0$, we have

$$\begin{aligned} \sup _{0\le t \le T_0} \Vert \eta _{n+1} \Vert _2^2&\le e^{{\tilde{c}}_1 \cdot 2^{2J_0} T_0} \Vert \eta _{n+1}(0) \Vert _2^2 + e^{{\tilde{c}}_1 \cdot 2^{2J_0} T_0} \cdot 2^{-n} {\tilde{c}}_2, \end{aligned}$$

(4.8)

The desired strong contraction of $\theta ^{(n)} \rightarrow \theta$ in $C_t^0 L_x^2$ follows easily.

Step 4. Higher norms. By using the estimates in previous steps, we have for any $0\le t\le T_0$,

$$\begin{aligned}&\Vert D^s e^{\frac{4}{3}t A} \theta \Vert _2^2 \le \limsup _{N\rightarrow \infty } \Vert D^s e^{\frac{4}{3} tA} P_{\le N} \theta \Vert _2^2 \\&\quad = \limsup _{N\rightarrow \infty } \lim _{n \rightarrow \infty } \Vert D^s e^{\frac{4}{3} t A} P_{\le N} \theta ^{(n)} (t) \Vert _2^2< B_1<\infty , \end{aligned}$$

where the constant $B_1>0$ is independent of t.

It follows easily that for any $0\le s^{\prime }<s$,

$$\begin{aligned} \Vert D^{s^{\prime }} e^{tA} (\theta ^{(n)}(t) - \theta (t) ) \Vert _{L_t^{\infty } L_x^2} \rightarrow 0, \quad \hbox { as}\ n\rightarrow \infty , \end{aligned}$$

This implies $f(t) =e^{tA} \theta (t) \in C_t^0 H_x^{s^{\prime }}$ for any $s^{\prime }<s$. To show $f \in C_t^0 H^s_x$ it suffices to consider the continuity at $t=0$ (for $t>0$ one can use the fact $e^{\frac{1}{3} t A } f \in L_t^{\infty } H^s_x$ which controls frequencies $|\xi | \gg t^{-1/\gamma }$, and for the part $|\xi | \lesssim t^{-1/\gamma }$ one uses $C_t^0 L_x^2$). Since we are in the Hilbert space setting with weak continuity in time, the strong continuity then follows from norm continuity at $t=0$ which is essentially done in Step 1.

5 Nonlinear Estimates for Besov Case: $0<\gamma <1$

For $\sigma =\sigma (\xi ,\eta )$ we denote the bilinear operator

$$\begin{aligned} T_{\sigma }(f,g) (x) = \int _{{\mathbb {R}}^d_{\eta }} \int _{{\mathbb {R}}^d_{\xi }} \sigma (\xi ,\eta ) {\hat{f}}(\xi ) {\hat{g}}(\eta ) e^{i x \cdot (\xi +\eta )} d\xi d\eta . \end{aligned}$$

Lemma 5.1

Suppose ${\text {supp}} (\sigma ) \subset \{ (\xi ,\eta ):\, |\xi |< 1, \; \frac{1}{C_1}< |\eta | <C_1\}$ for some constant $C_1>0$. Let $n_0 =2d+[d/2]+1$ and $\Omega _0= \{(\xi ,\eta ): \, 0<|\xi |<1, \; \frac{1}{C_1}<|\eta |<C_1 \}$. Suppose $\sigma \in C^{n_0}_{{\text {loc}}} (\Omega _0)$ and for some $A_1>0$

$$\begin{aligned} \sup _{ \begin{array}{c} |\alpha |\le [d/2]+1\\ |\beta |\le 2d \end{array} } \sup _{(\xi ,\eta ) \in \Omega _0} |\xi |^{|\alpha |} |\eta |^{|\beta |} |\partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } \sigma (\xi ,\eta )| \le A_1. \end{aligned}$$

Then for any $1<p_1<\infty$, $1\le p_2\le \infty$, $f,g \in \mathcal S({\mathbb {R}}^d)$,

$$\begin{aligned} \Vert T_{\sigma }(f,g) \Vert _r\lesssim _{d,C_1,A_1,p_1,p_2} \Vert f \Vert _{p_1} \Vert g\Vert _{p_2}, \end{aligned}$$

where $\frac{1}{r} = \frac{1}{p_1} + \frac{1}{p_2}$.

Similarly if ${\text {supp}} (\sigma ) \subset \{ (\xi ,\eta ):\, \frac{1}{{\tilde{C}}_1}<|\xi |<{\tilde{C}}_1, \; \frac{1}{{\tilde{C}}_2}< |\eta | <{\tilde{C}}_2\}=\Omega _1$ for some constants ${\tilde{C}}_1$, ${\tilde{C}}_2>0$. Suppose $\sigma \in C^{4d+1}_{{\text {loc}}} (\Omega _1)$ and for some ${\tilde{A}}_1>0$

$$\begin{aligned} \sup _{ |\alpha |+|\beta |\le 4d+ 1} \sup _{(\xi ,\eta ) \in \Omega _1} |\xi |^{|\alpha |} |\eta |^{|\beta |} |\partial _{\xi }^{\alpha } \partial _{\eta }^{\beta } \sigma (\xi ,\eta )| \le {\tilde{A}}_1. \end{aligned}$$

Then for any $1\le p_1 \le \infty$, $1\le p_2\le \infty$, $f,g \in \mathcal S({\mathbb {R}}^d)$,

$$\begin{aligned} \Vert T_{\sigma }(f,g) \Vert _r\lesssim _{d,{\tilde{C}}_1,{\tilde{C}}_2, {\tilde{A}}_1, p_1,p_2} \Vert f \Vert _{p_1} \Vert g\Vert _{p_2}, \end{aligned}$$

where $\frac{1}{r} = \frac{1}{p_1} + \frac{1}{p_2}$.

Proof

For the first case see Theorem 3.7 in [3]. The idea is to make a Fourier expansion in the $\eta$-variable:

$$\begin{aligned} \sigma (\xi , \eta ) = \sum _{ k \in {\mathbb {Z}}^d} L^{-d} \int _{[-\frac{L}{2},\frac{L}{2}]^d} \sigma (\xi , {\tilde{\eta }}) e^{- 2\pi i \frac{k \cdot {\tilde{\eta }}}{L} } d{\tilde{\eta }} \; e^{2\pi i \frac{k \cdot \eta }{L} } \chi (\eta ), \end{aligned}$$

where $L=8C_1$ and $\chi \in C_c^{\infty }( (-\frac{L}{4},\frac{L}{4})^d)$ is such that $\chi (\eta ) \equiv 1$ for $1/C_1<|\eta |<C_1$. A rough estimate on the number of derivatives required is $n_0=2d+[d/2]+1$. Note that $r>1/2$ and (by paying 2d derivatives) $2dr >d$ so that the resulting summation in k converges in $l^r$-norm. For the second case, one can make a Fourier expansion in $(\xi ,\eta )$. $\square$

Remark 5.2

For $t>0$, $0<\gamma <1$, $j \in {\mathbb {Z}}$, consider

$$\begin{aligned} \sigma _0(\xi ,\eta ) = e^{-t (|\xi |^{\gamma } + |\eta |^{\gamma } - |\xi +\eta |^{\gamma } )} \chi _{|\xi | \sim 2^j} \chi _{|\eta | \sim 2^j} \chi _{|\xi +\eta |\ll 2^j}. \end{aligned}$$

By using the estimates $\Vert {\mathcal {F}}^{-1} ( e^{t|\xi |^{\gamma } } \chi _{|\xi | \ll 1} ) \Vert _1 =\Vert {\mathcal {F}}^{-1} ( e^{|\xi |^{\gamma } } \chi _{|\xi | \ll t^{\frac{1}{\gamma } }} ) \Vert _1 \lesssim e^{ {\tilde{ct}}}$ (${\tilde{c}}\ll 1$), $\Vert {\mathcal {F}}^{-1} (e^{-t |\xi |^{\gamma } } \chi _{|\xi | \sim 1} ) \Vert _1 \lesssim e^{- Ct}$ ($C\sim 1$), we have for any $1\le r,p_1,p_2\le \infty$ with $\frac{1}{r}=\frac{1}{p_1}+\frac{1}{p_2}$,

$$\begin{aligned} \Vert T_{\sigma _0} (f,g) \Vert _r \lesssim _{\gamma ,d} e^{-c 2^{j\gamma } t} \Vert f \Vert _{p_1} \Vert g\Vert _{p_2} \lesssim \Vert f\Vert _{p_1} \Vert g\Vert _{p_2}, \end{aligned}$$

where $c>0$ is a small constant. Denote

$$\begin{aligned}&\sigma _1(\xi ,\eta ) = e^{-t(|\xi |^{\gamma } + |\eta |^{\gamma } - |\xi +\eta |^{\gamma } )} \chi _{|\xi |\ll 2^j} \chi _{|\eta | \sim 2^j}, \\&\sigma _2(\xi ,\eta )= e^{-t( |\xi |^{\gamma } +|\eta |^{\gamma } - |\xi +\eta |^{\gamma } )} \chi _{|\xi |\sim 2^j} \chi _{|\eta | \ll 2^j}, \\&\sigma _3(\xi ,\eta )= e^{-t( |\xi |^{\gamma } +|\eta |^{\gamma } - |\xi +\eta |^{\gamma } )} \chi _{|\xi |\sim 2^j} \chi _{|\eta |\sim 2^j} \chi _{|\xi +\eta | \sim 2^j}. \end{aligned}$$

By using Lemmas 5.1, 2.8 and some elementary computations, it is not difficult to check that for any $\frac{1}{2}<r<\infty$, $1<p_1,p_2<\infty$, with $\frac{1}{r}=\frac{1}{p_1}+\frac{1}{p_2}$,

$$\begin{aligned} \Vert T_{\sigma _l} (f,g) \Vert _{r} \lesssim _{\gamma ,p_1,p_2,d} \Vert f \Vert _{p_1} \Vert g \Vert _{p_2}, \quad \forall \, l=1,2,3. \end{aligned}$$

We shall need to use these inequalities (sometimes without explicit mentioning) below.

Fix $t>0$, $j\in {\mathbb {Z}}$, $0<\gamma <1$, and denote

$$\begin{aligned} B_j(f,g) =&[P_j e^{t D^{\gamma } }, e^{-t D^{\gamma }} R^{\perp } f ]\cdot \nabla e^{-t D^{\gamma } } g \nonumber \\ =&P_je^{t D^{\gamma }} ( e^{-tD^{\gamma } }R^{\perp } f \cdot \nabla e^{-t D^{\gamma }} g) - e^{-t D^{\gamma } }R^{\perp } f \cdot \nabla P_j g. \end{aligned}$$

For integer $J_0 \ge 10$ which will be made sufficiently large later, we decompose

$$\begin{aligned}&\;\;B_j(f, g) \nonumber \\&= B_j(f_{\le J_0+2}, g_{\le J_0+4} ) + B_j(f_{\le J_0+2}, g_{>J_0+4}) +B_j(f_{>J_0+2}, g_{\le J_0+2}) + B_j(f_{>J_0+2}, g_{>J_0+2} ) \nonumber \\&=B_j(f_{\le J_0+2}, g_{\le J_0+4} ) +B_j(f_{[J_0+2,J_0+4]}, g_{\le J_0+2}) \nonumber \\&\qquad + B_j(f_{>J_0+4}, g_{\le J_0+2})+ B_j(f_{\le J_0+2}, g_{>J_0+4}) + B_j(f_{>J_0+2}, g_{>J_0+2} ). \nonumber \end{aligned}$$

Lemma 5.3

$B_j(f_{\le J_0+2}, g_{\le J_0+4} )=0$ and $B_j(f_{[J_0+2,J_0+4]}, g_{\le J_0+2} )=0$ for $j>J_0+6$. For $j\le J_0+6$ and $1\le p<\infty$,

$$\begin{aligned}&\Vert B_j(f_{\le J_0+2}, g_{\le J_0+4} ) \Vert _p + \Vert B_j(f_{[J_0+2,J_0+4]}, g_{\le J_0+2} )\Vert _p \\&\quad \lesssim e^{c_1(1+t) } (\Vert P_{\le J_0+10} f \Vert _p^2+ \Vert P_{\le J_0+10} g \Vert _p^2), \end{aligned}$$

where $c_1>0$ depends on $(J_0,p,\gamma )$.

Proof

We only deal with $B_j(f_{\le J_0+2}, g_{\le J_0+4} )$ as the estimate for $B_j(f_{[J_0+2,J_0+4]}, g_{\le J_0+2})$ is similar and therefore omitted. Clearly for $j\le J_0+6$ (below the notation $\infty -$, $p+$ is defined in the same way as in (2.1)),

$$\begin{aligned}&\Vert e^{-t D^{\gamma }} R^{\perp } f_{\le J_0+2} \cdot \nabla P_j g_{\le J_0+4} \Vert _p \lesssim \Vert R^{\perp } f_{\le J_0+2} \Vert _{p+} 2^{J_0} \Vert P_ {\le J_0+10} g \Vert _{\infty -} \\&\quad \lesssim 2^{J_0(1+\frac{2}{p})} (\Vert P_{\le J_0+10} f\Vert _p^2 + \Vert P_{\le J_0+10} g \Vert _p^2). \end{aligned}$$

Here $p+$ is needed for $p=1$ so that the Riesz transform can be discarded. On the other hand for $j\le J_0+6$,

$$\begin{aligned} \Vert P_je^{t D^{\gamma }} ( e^{-tD^{\gamma } }R^{\perp } f_{\le J_0+2} \cdot \nabla e^{-t D^{\gamma }} g_{\le J_0+4}) \Vert _p \lesssim&e^{c_1 (1+t) } (\Vert P_{\le J_0+10} f \Vert _p^2 + \Vert P_{\le J_0+10} g \Vert _p^2). \end{aligned}$$

$\square$

Lemma 5.4

For $j\le J_0+6$ and $0<t\le 1$,

$$\begin{aligned} \Vert B_j(f_{\le J_0+2}, g_{>J_0+4}) \Vert _p \lesssim c_2 \Vert e^{-t D^{\gamma }} f_{\le J_0+2} \Vert _p \Vert g_{[J_0+5,J_0+10]} \Vert _p. \end{aligned}$$

For $j>J_0+6$, $t>0$ and $1\le p<\infty$,

$$\begin{aligned}&\Vert B_j (f_{\le J_0+2}, g_{>J_0+4} ) \Vert _p \\&\quad \lesssim c_2 \cdot 2^{j0+} \Vert P_{\le J_0+2} f \Vert _p \Vert g_{[j-2,j+2]} \Vert _p+ c_2 (t+t^2) \Vert P_{\le J_0+2} f \Vert _p \cdot 2^{j\gamma } \Vert g_{[j-2,j+2]}\Vert _p, \end{aligned}$$

where $c_2>0$ depends on $(p,J_0)$, and the notation $0+$ is defined in the paragraph preceding (2.1).

Proof

The first inequality for $j\le J_0+6$ is obvious. Consider now $j>J_0+6$. Observe that by frequency localization $B_j(f_{\le J_0+2}, g_{>J_0+4} ) = B_j ( f_{\le J_0+2}, P_{>J_0+4} g_{[j-2,j+2]} )$. We just need to consider $T_{\sigma }(R^{\perp } f_{\le J_0+2}, \nabla P_{>J_0+4} g_{[j-2,j+2]} )$ with $|\xi |\ll 2^j$, $|\eta | \sim 2^j$, and

$$\begin{aligned} \sigma (\xi ,\eta )&= [\phi (2^{-j} (\xi +\eta )) e^{-t(|\xi |^{\gamma } + |\eta |^{\gamma }- |\xi +\eta |^{\gamma } )} - \phi (2^{-j} \eta ) e^{-t |\xi |^{\gamma }}]\chi _{|\xi |\ll 2^j} \chi _{|\eta | \sim 2^j} \nonumber \\&= (\phi (2^{-j} (\xi +\eta )) - \phi (2^{-j} \eta ) )e^{-t(|\xi |^{\gamma } + |\eta |^{\gamma }- |\xi +\eta |^{\gamma } )} \chi _{|\xi |\ll 2^j} \chi _{|\eta | \sim 2^j} \nonumber \\&\qquad - \phi (2^{-j} \eta ) (e^{-t(|\xi |^{\gamma } +|\eta |^{\gamma }-|\xi +\eta |^{\gamma } )} - e^{-t|\xi |^{\gamma } }) \chi _{|\xi |\ll 2^j} \chi _{|\eta | \sim 2^j} \nonumber \\&=: \sigma _1(\xi ,\eta ) + \sigma _2(\xi ,\eta ). \end{aligned}$$

By Lemma 5.1, it is easy to check that for some $c_2>0$ depending on $(J_0,p)$,

$$\begin{aligned}&\Vert T_{\sigma _1} (R^{\perp }f_{\le J_0+2}, \nabla P_{>J_0+4} g_{[j-2,j+2]})\Vert _p \\&\quad \lesssim \Vert \partial R^{\perp }f_{\le J_0+2} \Vert _{\infty -} \Vert g_{[j-2,j+2]} \Vert _{p+} \lesssim c_2 \cdot 2^{j0+} \Vert P_{\le J_0+2} f\Vert _p \Vert g_{[j-2,j+2]}\Vert _p. \end{aligned}$$

On the other hand for $\sigma _2$, we introduce for $0\le \tau \le 1$

$$\begin{aligned} F(\tau ) = e^{-t (|\xi |^{\gamma } + \tau (|\eta |^{\gamma } - |\xi +\eta |^{\gamma } ) ) }, \quad F_1(\tau ) =|\eta +\tau \xi |^{\gamma }. \end{aligned}$$

Then observe

$$\begin{aligned} F(1) - F(0)&= F^{\prime }(0 ) + \int _0^1 F^{\prime \prime }(\tau ) (1-\tau ) d\tau \nonumber \\&=e^{-t |\xi |^{\gamma }} t (|\xi +\eta |^{\gamma } - |\eta |^{\gamma }) + \int _0^1 F(\tau ) t^2 (|\eta |^{\gamma } -|\xi +\eta |^{\gamma })^2 (1-\tau ) d\tau \nonumber \\&=\gamma t e^{-t|\xi |^{\gamma }} |\eta |^{\gamma -2} (\eta \cdot \xi ) + te^{- t|\xi |^{\gamma } } \int _0^1 F_1^{\prime \prime }(\tau ) (1-\tau ) d\tau \\&\qquad +\int _0^1 F(\tau ) t^2 (|\eta |^{\gamma } -|\xi +\eta |^{\gamma })^2 (1-\tau ) d\tau . \end{aligned}$$

To handle the last term above we make the following observation. Note that for $|\xi | \ll 2^j$, $|\eta |\sim 2^j$, one has $| |\eta |^{\gamma }-|\xi +\eta |^{\gamma } |= {\mathcal {O}}( |\eta |^{\gamma -1} \cdot |\xi | ) \le \frac{\alpha _0}{2} |\xi |^{\gamma }$, for some constant $0<\alpha _0<1$. In particular one may write

$$\begin{aligned} F(\tau ) = e^{-t(1-\alpha _0) |\xi |^{\gamma } } e^{-t ( \alpha _0 |\xi |^{\gamma } +\tau ( |\eta |^{\gamma } -|\xi +\eta |^{\gamma } )}. \end{aligned}$$

(5.1)

The symbol corresponding to $e^{-t( \alpha _0 |\xi |^{\gamma } +\tau ( |\eta |^{\gamma } -|\xi +\eta |^{\gamma } )}$ is clearly good for us whilst the term $e^{-t (1-\alpha _0) |\xi |^{\gamma } }$ can be used to extract additional decay (see below).

It is then clear that

$$\begin{aligned}&\Vert T_{\sigma _2} (R^{\perp } f_{\le J_0+2}, \nabla P_{>J_0+4} g_{[j-2,j+2]} ) \Vert _p \\&\quad \lesssim t \Vert e^{-t D^{\gamma } } \partial R^{\perp } f_{\le J_0+2} \Vert _{\infty } \cdot 2^{j\gamma } \Vert g_{[j-2,j+2]} \Vert _p \nonumber \\&\quad + 2^{j(\gamma -1)} \cdot t \Vert e^{-t D^{\gamma }} \partial ^2 R^{\perp } f_{\le J_0+2} \Vert _{\infty -} \Vert g_{[j-2.j+2]} \Vert _{p+} \nonumber \\&\quad + 2^{j(2\gamma -1)} \cdot t^2 \Vert \partial ^2 e^{-t(1-\alpha _0) D^{\gamma } }R^{\perp } f_{\le J_0+2} \Vert _{\infty -} \Vert g_{[j-2,j+2]} \Vert _{p+} \nonumber \\&\lesssim c_2 \cdot (t+t^2) \Vert P_{\le J_0+2} f \Vert _p \cdot 2^{j\gamma } \Vert g_{[j-2,j+2]} \Vert _p. \end{aligned}$$

$\square$

Lemma 5.5

Let $1\le p<\infty$. For $j\le J_0+6$, $0<t\le 1$, we have

$$\begin{aligned} \Vert B_j(f_{>J_0+4}, g_{\le J_0+2} ) \Vert _p \lesssim c_2 \Vert e^{-t D^{\gamma }} f_{>J_0+4} \Vert _p \Vert g_{\le J_0+2} \Vert _p. \end{aligned}$$

For $j>J_0+6$ and any $t>0$, we have

$$\begin{aligned} \Vert B_j (f_{> J_0+4}, g_{\le J_0+2} ) \Vert _p \lesssim c_2 \cdot 2^{j0+} \Vert P_{\le J_0+2} g \Vert _p \Vert f_{[j-2,j+2]} \Vert _p, \end{aligned}$$

In the above $c_2>0$ depends on $(\gamma , p,J_0)$.

Proof

The estimate for $j\le J_0+6$ is obvious. Observe that for $j>J_0+6$,

$$\begin{aligned} B_j(f_{>J_0+4}, g_{\le J_0+2} ) = P_j e^{t D^{\gamma } } ( e^{- t D^{\gamma } } R^{\perp } P_{>J_0+4} f_{[j-2,j+2]}\cdot \nabla e^{-t D^{\gamma } } P_{\le J_0+2} g). \end{aligned}$$

Thus by Lemma 5.1,

$$\begin{aligned}&\Vert B_j(f_{>J_0+4}, g_{\le J_0+2} ) \Vert _p \\&\quad \lesssim \Vert R^{\perp } f_{[j-2,j+2]} \Vert _{p+} \Vert \nabla P_{\le J_0+2} g \Vert _{\infty -} \lesssim c_2 2^{j0+} \Vert P_{\le J_0+2} g \Vert _p \Vert f_{[j-2,j+2]} \Vert _p. \end{aligned}$$

$\square$

Lemma 5.6

Denote $f^h= f_{>J_0+2}$, $g^h=g_{>J_0+2}$. Then for $j\ge J_0$, $1\le p<\infty$, $0<t\le 1$, we have

$$\begin{aligned} \Vert B_j (f^h, g^h)\Vert _p&\lesssim 2^{j\gamma } \Vert f^h \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty }} \Vert g^h_{[j-2,j+2]}\Vert _p + 2^{j\gamma } \Vert g^h \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty }} \Vert f^h_{[j-2,j+9]} \Vert _p \nonumber \\&\qquad \qquad +2^j \sum _{k\ge j+8} 2^{k\cdot \frac{2}{p}} \Vert f^h_k \Vert _p \Vert g^h_{[k-2,k+2]} \Vert _p. \end{aligned}$$

Proof

Write

$$\begin{aligned} B_j(f^h, g^h )&= B_j(f^h_{<j-2}, g^h) + B_j(f_{[j-2,j+9]}^h, g^h) + B_j(f^h_{>j+9}, g^h) \nonumber \\&= B_j(f^h_{<j-2}, g^h_{[j-2,j+2]}) + B_j(f^h_{[j-2,j+9]}, g^h_{<j-4})\\&\qquad +B_j(f^h_{[j-2,j+9]}, g^h_{[j-4,j+12]}) +\sum _{k\ge j+10} B_j(f^h_k, g^h) \nonumber \\&=: (1)+ (2) +(3) +(4). \end{aligned}$$

$\underline{{\hbox {Estimate of}}\, (1)}.$

Note that for given integer $J_1\ge 2$, the term $B_j(f^h_{[j-J_1,j-3]},g^h_{[j-2,j+2]})$ can be included in the estimate of (3).

It suffices for us to estimate $(1\text {A})= T_{\sigma } ( R^{\perp } f^h_{<j-J_1}, \nabla g^h_{[j-2,j+2]})$ with (here to ensure $|\xi | \ll 2^j$ we need to take $J_1$ sufficiently large)

$$\begin{aligned}&\sigma (\xi ,\eta ) = ( \phi (\frac{\xi +\eta }{2^j}) -\phi (\frac{\eta }{2^j}) ) e^{-t(|\xi |^{\gamma }} \\&\quad {+ |\eta |^{\gamma } - |\xi +\eta |^{\gamma } )} \chi _{|\xi |\ll 2^j} \chi _{|\eta | \sim 2^j} - \phi (\frac{\eta }{2^j}) ( e^{-t(|\xi |^{\gamma } +|\eta |^{\gamma }- |\xi +\eta |^{\gamma })} -e^{-t|\xi |^{\gamma } }) \chi _{|\xi | \ll 2^j} \chi _{|\eta | \sim 2^j}. \end{aligned}$$

By an argument similar to that in Lemma 5.4, we get

$$\begin{aligned}&\Vert (1\text {A}) \Vert _p \\&\lesssim \Vert \partial R^{\perp } f^h_{<j-2} \Vert _{\infty -} \Vert g^h_{[j-2,j+2]} \Vert _{p+} +t \Vert e^{-t D^{\gamma } } \partial R^{\perp } f_{<j-2}^h \Vert _{\infty } \\&\cdot 2^{j\gamma } \Vert g_j^h\Vert _p+ 2^{j(\gamma -1)} \cdot t \Vert e^{-t D^{\gamma }} \partial ^2 R^{\perp } f_{<j-2}^h \Vert _{\infty -} \Vert g_j^h \Vert _{p+} \nonumber \\&\quad + 2^{j(2\gamma -1)} \Vert D^{2-2\gamma } R^{\perp } f_{<j-2}^h \Vert _{\infty -} \Vert g_j^h \Vert _{p+} \nonumber \\&\lesssim 2^{j\gamma } \Vert f^h\Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty }} \Vert g^h_{[j-2,j+2]} \Vert _p. \end{aligned}$$

$\underline{{\hbox {Estimate of}} \,(2)}$. Clearly

$$\begin{aligned} (2)= P_j e^{t D^{\gamma }} ( R^{\perp } e^{-t D^{\gamma } } f^h_{[j-2,j+9]}\cdot \nabla e^{-t D^{\gamma }} g^h_{<j-4} ). \end{aligned}$$

Thus

$$\begin{aligned} \Vert (2) \Vert _p \lesssim \Vert f^h_{[j-2,j+9]} \Vert _{p+} \Vert \nabla g^h_{<j-4} \Vert _{\infty -} \lesssim \Vert g^h \Vert _{\dot{B}^{1-\gamma + \frac{2}{p}}_{p,\infty } } 2^{j\gamma }\Vert f^h_{[j-2,j+9]} \Vert _p. \end{aligned}$$

$\underline{{\hbox {Estimate of}} \,(3)}$. Clearly

$$\begin{aligned} \Vert (3) \Vert _p \lesssim 2^{j\gamma } \Vert g^h \Vert _{\dot{B}^{1-\gamma +\frac{2}{p}}_{p,\infty } } \Vert f^h_{[j-2,j+9]}\Vert _p. \end{aligned}$$

$\underline{{\hbox {Estimate of}} \,(4)}$. We first note that

$$\begin{aligned} \sum _{k\ge j+10} \Vert e^{-tD^{\gamma }} R^{\perp } f_k^h \cdot \nabla P_j g \Vert _p \lesssim 2^{j\gamma } \Vert f^h \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty }} \Vert g^h_{[j-2,j+2]} \Vert _p. \end{aligned}$$

(5.2)

On the other hand by using that $R^{\perp } f$ is divergence-free, we have

$$\begin{aligned}&\sum _{k\ge j+10} \left\| P_j e^{t D^{\gamma } } \nabla \cdot \biggl ( (e^{-t D^{\gamma } } R^{\perp } f^h_k ) (e^{-t D^{\gamma } } g^h_{[k-2,k+2]} ) \biggr ) \right\| _p \nonumber \\ \lesssim&\sum _{k\ge j+10} 2^j \Vert f^h_k \Vert _{2p} \Vert g^h_{[k-2,k+2]} \Vert _{2p} \nonumber \\ \lesssim&\; 2^j \sum _{k\ge j+8} 2^{k\cdot \frac{2}{p}} \Vert f^h_k \Vert _p \cdot \Vert g^h_{[k-2,k+2]} \Vert _p. \end{aligned}$$

$\square$

6 Proof of Theorem 1.2

Recall that the initial data $\theta _0 \in B_{p,q}^{1-\gamma +\frac{2}{p}}$, $1\le p<\infty$, $0<\gamma <1$ and $1\le q<\infty$.

Lemma 6.1

Let $\chi \in C_c^{\infty }({\mathbb {R}}^2)$ and $\theta _0 \in B_{p,q}^s({\mathbb {R}}^2)$ with $1\le p<\infty$, $1\le q<\infty$, $s>0$. Let $(\lambda _n)_{n=1}^{\infty }$ be a sequence of positive numbers such that $\inf _n \lambda _n >0$. Then

$$\begin{aligned} \lim _{J_0\rightarrow \infty } \sup _{n\ge 1} \Vert P_{>J_0} \bigl ( \,\chi (\lambda _n^{-1} x) P_{\le n+2} \theta _0 \,\bigr ) \Vert _{ B^s_{p,q}} =0. \end{aligned}$$

Proof

Write $f= \chi (\lambda _n^{-1} x)$, $g= P_{\le n+2} \theta _0$, then

$$\begin{aligned} fg = \sum _{j \in {\mathbb {Z}}} ( f_{j} g_{<j-2} + g_j f_{<j-2} + f_j {\tilde{g}}_j), \end{aligned}$$

where ${\tilde{g}}_j = g_{[j-2,j+2]}$. Clearly

$$\begin{aligned} (2^{js}\Vert f_j g_{<j-2} \Vert _p )_{l_j^q( j \ge J_0)} \lesssim \Vert g \Vert _p \Vert \partial ^{2[s]+2} f\Vert _{\infty } (2^{-j(2[s]+2-s)})_{l_j^q(j\ge J_0)} \rightarrow 0, \end{aligned}$$

uniformly in n as $J_0 \rightarrow \infty$. A similar estimate also shows that the diagonal piece $f_j {\tilde{g}}_j$ is OK. On the other hand

$$\begin{aligned} ( 2^{js} \Vert f_{<j-2} g_j \Vert _p ) _{l_j^q(j\ge J_0)} \lesssim \Vert f\Vert _{\infty } (2^{js} \Vert P_j \theta _0 \Vert _p)_{l_j^q(j\ge J_0)} \rightarrow 0, \end{aligned}$$

uniformly in n as $J_0\rightarrow \infty$. $\square$

We now complete the proof of Theorem 1.2. This will be carried out in several steps below.

Step 1. Definition of approximating solutions. Define $\theta ^{(0)} \equiv 0$. For $n \ge 0$, define the iterates $\theta ^{(n+1)}$ as solutions to the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(n+1)} = -R^{\perp }\theta ^{(n)} \cdot \nabla \theta ^{(n+1)} - D^{\gamma } \theta ^{(n+1)}, \quad (t,x) \in (0,\infty ) \times {\mathbb {R}}^2;\\ \theta ^{(n+1)}\Bigr |_{t=0} = \chi (\lambda _n^{-1} x) P_{\le n+2} \theta _0, \end{array}\right. } \end{aligned}$$

where $\chi \in C_c^{\infty }({\mathbb {R}}^2)$ satisfies $0\le \chi \le 1$ for all x, $\chi (x)\equiv 1$ for $|x|\le 1$, and $\chi (x)=0$ for $|x|\ge 2$. Here we introduce the spatial cut-off $\chi$ so that $\theta ^{(n+1)}\Bigr |_{t=0} \in H^k$ for all $k\ge 0$ when we only assume $\theta _0$ lies in $L^p$ type spaces. The scaling parameters $\lambda _n \ge 1$ are inductively chosen such that $\lambda _n >\max \{4\lambda _{n-1},2^n\}$ and

$$\begin{aligned} \Vert \theta _0 \Vert _{L^p(|x|>\frac{1}{100} \lambda _n)} <2^{-100n}. \end{aligned}$$

Easy to check that

$$\begin{aligned} \Vert \theta ^{(n+1)}(0) - \theta ^{(n)} (0) \Vert _p \lesssim 2^{-n(1-\gamma +\frac{2}{p})}, \end{aligned}$$

and by interpolation for $0<{\tilde{s}}<1-\gamma +\frac{2}{p}$, ${\tilde{s}}=0+$,

$$\begin{aligned} \Vert \theta ^{(n+1)}(0) - \theta ^{(n)}(0) \Vert _{B^{{\tilde{s}}}_{p,\infty }} \lesssim 2^{-n(1-\gamma +\frac{2}{p})+}. \end{aligned}$$

(6.1)

Also by Lemma 6.1, we have

$$\begin{aligned} \Vert \theta ^{(n+1)}(0) - \theta _0 \Vert _{B^{1-\gamma +\frac{2}{p}}_{p,q} } \rightarrow 0, \quad \hbox { as}\ n\rightarrow \infty . \end{aligned}$$

These estimates will be needed for the contraction estimate later.

Clearly we have the uniform boundedness of $L^p$ norm:

$$\begin{aligned} \sup _{n\ge 0} \sup _{0\le t <\infty } \Vert \theta ^{(n+1)} (t) \Vert _p \le \sup _{n\ge 0} \Vert P_{\le n+2} \theta _0 \Vert _p \lesssim \Vert \theta _0\Vert _p. \end{aligned}$$

This will often be used without explicit mentioning below.

Step 2. Denote $A=\frac{1}{2} D^{\gamma }$, $f^{(n+1)}(t)=e^{t A } \theta ^{(n+1)} (t)$. Then

$$\begin{aligned} \partial _t f^{(n+1)} = - A f^{(n+1)} - e^{t A} ( R^{\perp }e^{-tA} f^{(n)} \cdot \nabla e^{-tA} f^{(n+1)} ). \end{aligned}$$

One can view $f^{(n+1)}$ as the unique limit of the sequence of solutions $(f^{(n+1)}_m)_{m=1}^{\infty }$ solving the regularized system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t f^{(n+1)}_m = - A f^{(n+1)}_m - e^{tA} {P_{\le m}} ( R^{\perp } e^{-t A} f^{(n)} \cdot \nabla e^{-t A } P_{\le m} f^{(n+1)}_m), \\ f^{(n+1)}_m \Bigr |_{t=0} = P_{\le m} \Bigl ( \chi (\lambda _n^{-1} x) P_{\le n+2} \theta _0 \Bigr ). \end{array}\right. } \end{aligned}$$

By using the estimates in Sect. 2 (and the inductive assumption that $f^{(n)} \in C_t^0\,H^k$ for all $k\ge 0$), we can then obtain $f^{(n+1)} \in C_t^0 ([0,T], H^k)$ for all $T>0$, $k\ge 0$. Write

$$\begin{aligned} f^{(n+1)}(t) = e^{-t A} f^{(n+1)}(0) - \int _0^t e^{-(t-s) A} e^{sA} ( R^{\perp }e^{-sA} f^{(n)} \cdot \nabla e^{-sA} f^{(n+1)} ) ds. \end{aligned}$$

By using the fact that $f^{(n)}$, $f^{(n+1)} \in C_t^0\,H^k$, it is not difficult to check that

$$\begin{aligned} \sup _{0\le t \le T} \Vert \partial ^k f^{(n+1)}(t) \Vert _p <\infty , \quad \forall \, T>0,\, k\ge 0. \end{aligned}$$

It follows that for any $T>0$

$$\begin{aligned} \sup _{0\le t\le T} \Vert \partial _t f^{(n+1)}\Vert _p\le \sup _{0\le t \le T} \Vert D^{\gamma } f^{(n+1)} \Vert _p + \Vert e^{tA} ( R^{\perp } e^{-tA} f^{(n)} \cdot \nabla e^{-t A} f^{(n+1)} ) \Vert _p <\infty . \end{aligned}$$

This together with interpolation implies $f^{(n+1)} \in C_t^0([0,T], W^{k,p})$ for any $T>0$, $k\ge 0$. These estimates establish the (a priori) finiteness of the various Besov norms and associated time continuity needed in the following steps.

Step 3. Besov norm estimates. Denote $f_j^{(n+1)} = P_j f^{(n+1)}$. For any $\epsilon _0>0$, we show that there exists $J_1$ sufficiently large, and $T_1>0$ sufficiently small, such that

$$\begin{aligned} \sup _{n\ge 0} ( 2^{j(1-\gamma +\frac{2}{p})}\Vert f_j^{(n)} \Vert _p )_{l_j^q L_t^{\infty } (t\in [0,T_1],\, j\ge J_1) } <\epsilon _0. \end{aligned}$$

(6.2)

Clearly for each $j\in {\mathbb {Z}}$,

$$\begin{aligned} \partial _t f^{(n+1)}_j = -\frac{1}{2} D^{\gamma } f^{(n+1)}_j- P_j e^{tA} ( R^{\perp } e^{-t A } f^{(n)} \cdot \nabla e^{-t A} f^{(n+1)}). \end{aligned}$$

Then by using Lemma 2.1, we get for some constants ${\tilde{C}}_1>0$, ${\tilde{C}}_2>0$,

$$\begin{aligned} \partial _t ( \Vert f_j^{(n+1)} \Vert _p) + {\tilde{C}}_1 2^{j\gamma } \Vert f_j^{(n+1)} \Vert _p \le {\tilde{C}}_2 \Vert [P_j e^{tA}, e^{-t A} R^{\perp } f^{(n)} ] \cdot \nabla e^{-t A} f^{(n+1)}\Vert _p. \end{aligned}$$

Take an integer $J_0\ge 10$ which will be made sufficiently large later. By using the nonlinear estimates derived before (see Lemma 5.3–5.6), we then obtain

$$\begin{aligned} \Vert f_j^{(n+1)}(t) \Vert _p\le e^{-{\tilde{C}}_1 2^{j\gamma } t} \Vert f_j^{(n+1)}(0) \Vert _p + \int _0^t e^{-{\tilde{C}}_1 2^{j\gamma } (t-s) } N_j ds, \end{aligned}$$

where for some constants ${\tilde{C}}_3>0$, ${\tilde{C}}_4>0$, ${\tilde{C}}_5>0$,

$$\begin{aligned} N_j&= 1_{j\le J_0+6} \cdot {{\tilde{C}}_3 } \cdot ( \Vert P_{\le J_0+10} f^{(n)} \Vert _p^2 \\&\quad + \Vert P_{\le J_0+10} f^{(n+1) } \Vert _p^2 +\Vert \theta _0\Vert _p^2 ) + 1_{j>J_0+6} \cdot {\tilde{C}}_4 \cdot 2^{j0+} (\Vert P_{\le J_0+2} f^{(n)} \Vert _p \Vert f^{(n+1)}_{[j-2,j+2]} \Vert _p ) \nonumber \\&\quad + 1_{j>J_0+6} \cdot {\tilde{C}}_4 \cdot (2^{j0+}\Vert P_{\le J_0+2} f^{(n+1) } \Vert _p \Vert f^{(n)}_{[j-2,j+2]} \Vert _p \\&\quad + s\Vert P_{\le J_0+2} f^{(n)} \Vert _p \cdot 2^{j\gamma } \Vert f_{[j-2,j+2]}^{(n+1)} \Vert _p ) \nonumber \\&\quad + {\tilde{C}}_5 2^{j\gamma } (\Vert P_{>J_0+2} f^{(n)} \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty } } \Vert P_{>J_0+2} f^{(n+1)}_{[j-2,j+2]} \Vert _p \\&+ \Vert P_{>J_0+2} f^{(n+1)} \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty } } \Vert P_{>J_0+2} f^{(n)}_{[j-2,j+9]} \Vert _p ) \nonumber \\&\quad + {\tilde{C}}_5 2^j \sum _{k\ge j+8} 2^{k\frac{2}{p}} \Vert P_{>J_0+2} f^{(n)}_k \Vert _p \Vert P_{>J_0+2} f^{(n+1) }_{[k-2,k+2]}\Vert _p. \end{aligned}$$

Denote

$$\begin{aligned} \Vert f^{(n+1)} \Vert _{T,J_0} = (2^{j(1-\gamma +\frac{2}{p})} \Vert f_j^{(n+1) } \Vert _p)_{l_j^q L_t^{\infty } (t \in [0,T],\, j\ge J_0)}. \end{aligned}$$

One should note that by the estimates derived in Step 2, the above norm of $f^{(n+1)}$ is finite. Then for $0<T\le 1$,

$$\begin{aligned} \Vert f^{(n+1)}\Vert _{T,J_0}&\le ( 2^{j(1-\gamma +\frac{2}{p})}\Vert f_j^{(n+1)}(0) \Vert _p)_{l_j^q(j\ge J_0)} + C_{J_0}^{(1)} T \Vert \theta _0\Vert _p^2 \\&\quad +C_1 \cdot 2^{-\frac{1}{2} J_0 \gamma } \cdot e^{C_3 \cdot 2^{J_0\gamma } T} \Vert \theta _0\Vert _p \cdot ( \Vert f^{(n)} \Vert _{T,J_0} + \Vert f^{(n+1)} \Vert _{T,J_0} ) \nonumber \\&\quad + C_{J_0}^{(2)} \cdot T \Vert \theta _0\Vert _p \Vert f^{(n+1)}\Vert _{T,J_0} + C_2 \Vert f^{(n) } \Vert _{T, J_0} \Vert f^{(n+1) } \Vert _{T,J_0}, \end{aligned}$$

where $C_{J_0}^{(1)}$, $C_{J_0}^{(2)}>0$ are constants depending on $(J_0,\gamma , p,q)$, $C_1$, $C_2>0$ are constants depending only on $(\gamma ,p,q)$, and $C_3>0$ depends only on $\gamma$.

By Lemma 6.1, one can find $J_0$ sufficiently large such that

$$\begin{aligned}&\sup _{n\ge 0} (2^{j(1-\gamma +\frac{2}{p})} \Vert f_j^{(n+1)}(0) \Vert _p)_{l^q_j(j\ge J_0)}< \frac{1}{100 C_2}, \\&C_1 \cdot 2^{-\frac{1}{2} J_0 \gamma } \cdot 10 (1+ \Vert \theta _0\Vert _p )<\frac{1}{20}. \end{aligned}$$

Fix such $J_0$ and then choose $T=T_0\le 1$ such that

$$\begin{aligned}&C_{J_0}^{(1)} T_0 \Vert \theta _0 \Vert _p^2<\frac{1}{100 C_2}, \quad C_3 \cdot 2^{J_0\gamma }\cdot T_0<\frac{1}{100}, \quad C_{J_0}^{(2)} T_0 \Vert \theta _0\Vert _p <\frac{1}{20}. \end{aligned}$$

The inductive assumption is $\Vert f^{(n)}\Vert _{T_0,J_0} <\frac{1}{4C_2}$. Then clearly

$$\begin{aligned}&\Vert f^{(n+1)} \Vert _{T_0, J_0} \le \frac{1}{100 C_2} \\&\quad + \frac{1}{100 C_2} +\frac{1}{20} \Vert f^{(n+1) }\Vert _{T_0,J_0} + \frac{1}{20} \cdot \frac{1}{4C_2} + \frac{1}{20} \Vert f^{(n+1)} \Vert _{T_0,J_0} + \frac{1}{4} \Vert f^{(n+1)} \Vert _{T_0,J_0}. \end{aligned}$$

This easily implies $\Vert f^{(n+1)} \Vert _{T_0,J_0} < \frac{1}{4C_2}$ which completes the argument.

The statement (6.2) clearly follows by a slight modification of the above argument.

Step 4. Contraction in $B^{s_0}_{p,\infty }$ where $s_0>0$ is a sufficiently small number.

Remark

We chose the space $C_t^0 B^{0+}_{p,\infty }$ since it contains $L^p$ and its norm coincides with the usual Chemin-Lerner space ${\tilde{L}}_t^{\infty } B^{0+}_{p,\infty }$ (see (6.10)). This way one can make full use of the smoothing effect of the linear semigroup on each dyadic frequency block which is needed for this critical problem.

Set $\eta ^{(n+1)}= f^{(n+1)}-f^{(n)}$. Then

$$\begin{aligned} \partial _t \eta ^{(n+1)} = - A \eta ^{(n+1)} - e^{tA} (R^{\perp } e^{-tA} \eta ^{(n)} \cdot \nabla e^{-tA} f^{(n+1)} ) - e^{tA} (R^{\perp } e^{-t A} f^{(n-1)} \cdot \nabla e^{-tA} \eta ^{(n+1) } ). \end{aligned}$$

It is easy to check for $0 \le t \le T_0$, $J_1 \in {\mathbb {Z}}$ (below we work with $p+$ to avoid the end-point situation $p=1$)

$$\begin{aligned}&\partial _t \Vert P_{\le 2J_1} \eta ^{(n+1) } \Vert _p \\&\quad \lesssim _{J_1, T_0,p,\gamma } \Vert e^{-tA} R^{\perp }\eta ^{(n)} \Vert _{p+} \Vert f^{(n+1) } \Vert _{\infty -} + \Vert R^{\perp } f^{(n-1) } \Vert _{\infty -} \Vert e^{-tA} \eta ^{(n+1)} \Vert _{p+} \nonumber \\&\quad \lesssim _{J_1, T_0,p,\gamma } \Vert \eta ^{(n)} \Vert _p \Vert f^{(n+1)} \Vert _{\infty -} + \Vert f^{(n-1)} \Vert _{\infty -} \Vert \eta ^{(n+1)} \Vert _p \nonumber \\&\quad \le C_{T_0, J_1} \cdot (\Vert \eta ^{(n) } \Vert _p + \Vert \eta ^{(n+1)} \Vert _p), \end{aligned}$$

where $C_{T_0,J_1}$ is a constant depending only on $(\theta _0, J_1, T_0, \gamma , p, q)$. Here in the last inequality we used the estimates obtained in Step 3.

On the other hand for $j\ge J_1$, denoting $\eta ^{(n+1)}_j =P_j \eta ^{(n+1)}$, we have

$$\begin{aligned}&\partial _t \Vert \eta ^{(n+1)}_j \Vert _p + {\tilde{C}}_1 2^{j\gamma } \Vert \eta ^{(n+1)}_j \Vert _p \\&\lesssim \Vert P_j e^{tA} (R^{\perp } e^{-tA} \eta ^{(n)} \cdot \nabla e^{-tA} f^{(n+1)} ) \Vert _p {+} \Vert [P_j e^{tA}, R^{\perp } e^{-t A} f^{(n-1)}] \cdot \nabla e^{-tA} \eta ^{(n{+}1) } \Vert _p. \end{aligned}$$

We now need a simple lemma.

Lemma 6.2

Let $0<t\le 1$, $1\le p<\infty$, $J_1 \ge 10$. We have for any $j\ge J_1$,

$$\begin{aligned}&\Vert P_j e^{tA} ( R^{\perp } e^{-tA} \eta \cdot \nabla e^{-t A} f ) \Vert _p \lesssim 2^{j(1+\frac{2}{p}-s_0)} \Vert f_{[j-2,j+2]}\Vert _p \Vert \eta \Vert _{B^{s_0}_{p,\infty }} \nonumber \\&\quad + 2^{j\gamma } \Vert \eta _{[j-2,j+3]}\Vert _p \cdot \;(2^{j_1(1-\gamma +\frac{2}{p})} \Vert f_{j_1}\Vert _p)_{l_{j_1}^{\infty } (j_1\ge 2J_1)} \nonumber \\&\quad +2^{3J_1(1+\frac{2}{p})} \Vert P_{\le 3J_1} f\Vert _p \cdot 2^{j0+} \Vert \eta _{[j-2,j+3]} \Vert _p +2^j \Vert \eta \Vert _{B^{s_0}_{p,\infty }} \sum _{k\ge j+4} 2^{k(\frac{2}{p}-s_0)}\Vert f_{[k-2,k+2]}\Vert _p; \nonumber \\&\Vert P_j e^{tA} ( R^{\perp } e^{-tA} f \cdot \nabla e^{-tA} \eta ) - R^{\perp } e^{-tA} f \cdot \nabla P_j \eta \Vert _p \nonumber \\&\quad \lesssim 2^{j\gamma } \Vert f_{>J_1} \Vert _{\dot{B}^{1+\frac{2}{p} -\gamma }_{p,\infty } } \Vert \eta _{[j-2,j+3]} \Vert _p + C_{J_1} \cdot \Vert f_{\le J_1+10} \Vert _p \cdot 2^{j0+} \Vert \eta _{[j-2,j+3]} \Vert _p \nonumber \\&\qquad \qquad + C_{J_1} \cdot t\Vert f_{\le J_1+10} \Vert _p \cdot 2^{j\gamma } \Vert \eta _{[j-2,j+3]} \Vert _p\nonumber \\&\qquad + \Vert \eta \Vert _{B^{s_0}_{p,\infty }} \cdot 2^{j(1+\frac{2}{p}-s_0)} \Vert f_{[j-5,j+5]} \Vert _p+ 2^{j\gamma }\Vert P_{\ge j+6} f\Vert _{B^{1+\frac{2}{p}-\gamma }_{p,\infty }} \Vert \eta _j \Vert _p, \end{aligned}$$

where $C_{J_1}$ is a constant depending on $J_1$.

Proof of Lemma 6.2

For the first inequality we denote $\widetilde{N_j}(g,h)= P_j e^{tA} ( R^{\perp } e^{-tA} g \cdot \nabla e^{-tA} h)$. By frequency localization, we write

$$\begin{aligned} \widetilde{N_j}(\eta , f) = \widetilde{N_j}( \eta _{<j-2}, f_{[j-2,j+2]} )+ \widetilde{N_j}(\eta _{[j-2,j+3]}, f_{\le j+5}) +\sum _{k\ge j+4} \widetilde{N_j}(\eta _k, f_{[k-2,k+2]} ). \end{aligned}$$

Clearly

$$\begin{aligned}&\Vert \widetilde{N_j}(\eta _{<j-2}, f_{[j-2,j+2]} ) \Vert _p \\&\quad \lesssim 2^j \Vert f_{[j-2,j+2]}\Vert _{p+} \Vert \eta _{<j-2} \Vert _{\infty -} \lesssim 2^{j(1+\frac{2}{p}-s_0)} \Vert f_{[j-2,j+2]} \Vert _p \Vert \eta \Vert _{B^{s_0}_{p,\infty }}. \end{aligned}$$

For the second term we split f as $f= f_{>3J_1} +f_{\le 3J_1}$. Then (below we work again with $p+$ for the $\eta$-term which give rises to $2^{j0+}$; the reason for $p+$ is to avoid the end-point case $p=1$)

$$\begin{aligned}&\Vert \widetilde{N_j} (\eta _{[j-2,j+3]}, f_{\le j+5} ) \Vert _p \\&\quad \lesssim 2^{j\gamma } \Vert \eta _{[j-2,j+3]}\Vert _p \cdot \;(2^{j_1(1-\gamma +\frac{2}{p})} \Vert f_{j_1}\Vert _p)_{l_{j_1}^{\infty } (j_1\ge 2J_1)} +2^{3J_1(1+\frac{2}{p})} \Vert P_{\le 3J_1} f\Vert _p \cdot 2^{j0+} \Vert \eta _{[j-2,j+3]} \Vert _p. \end{aligned}$$

For the diagonal piece, we have

$$\begin{aligned}&\sum _{k\ge j+4} \Vert \widetilde{N_j} (\eta _k, f_{[k-2,k+2]})\Vert _p \\&\quad \lesssim 2^j \sum _{k\ge j+4} 2^{k \frac{2}{p}} \Vert \eta _k \Vert _p \Vert f_{[k-2,k+2]} \Vert _p \lesssim 2^j \Vert \eta \Vert _{B^{s_0}_{p,\infty }} \sum _{k\ge j+4} 2^{k(\frac{2}{p}-s_0)}\Vert f_{[k-2,k+2]}\Vert _p. \end{aligned}$$

For the second inequality, we denote

$$\begin{aligned} N_j(f,\eta ) =P_j e^{tA} (R^{\perp } e^{-tA} f \cdot \nabla e^{-tA} \eta ) - R^{\perp } e^{-tA} f \cdot \nabla P_j \eta . \end{aligned}$$

(6.3)

Observe that

$$\begin{aligned}&N_j(f,\eta _{<j-2}) \\&=P_j e^{tA} ( R^{\perp } e^{-tA} f \cdot \nabla e^{-tA} \eta _{<j-2} ) =P_je^{tA} (R^{\perp } e^{-tA} f_{[j-2,j+2]} \cdot \nabla e^{-tA} \eta _{<j-2} ). \end{aligned}$$

Thus

$$\begin{aligned} \Vert N_j(f,\eta _{<j-2} )\Vert _p \lesssim \Vert f_{[j-2,j+2]} \Vert _p \cdot 2^{j(1+\frac{2}{p}-s_0)} \Vert \eta \Vert _{B^{s_0}_{p,\infty } }. \end{aligned}$$

(6.4)

On the other hand,

$$\begin{aligned}&N_j(f, \eta _{>j+3}) =P_je^{tA} ( R^{\perp } e^{-tA} f \cdot \nabla e^{-tA} \eta _{>j+3} ) \nonumber \\&\quad =\sum _{k\ge j+4} P_j e^{tA} ( R^{\perp } e^{-tA} f_{[k-2,k+2]} \cdot \nabla e^{-tA} \eta _k ). \end{aligned}$$

(6.5)

Thus

$$\begin{aligned} \Vert N_j(f, \eta _{>j+3})\Vert _p \lesssim 2^j \Vert \eta \Vert _{B^{s_0}_{p,\infty }}\sum _{k\ge j+4} 2^{k(\frac{2}{p}-s_0)} \Vert f_{[k-2,k+2]} \Vert _p. \end{aligned}$$

(6.6)

It remains to estimate $N_j(f, \eta _{[j-2,j+3]})$. We first note that

$$\begin{aligned} \Vert N_j(f_{\ge j+6}, \eta _{[j-2,j+3]} )\Vert _p&= \Vert R^{\perp } e^{-tA} f_{\ge j+6} \cdot \nabla P_j \eta \Vert _p \nonumber \\&\lesssim 2^{j\gamma }\Vert P_{\ge j+6} f \Vert _{B^{1+\frac{2}{p}-\gamma }_{p,\infty } } \Vert \eta _j\Vert _p. \end{aligned}$$

(6.7)

On the other hand,

$$\begin{aligned} \Vert N_j( f_{[j-5,j+5]}, \eta _{[j-2,j+3]} ) \Vert _p \lesssim 2^{j(1+\frac{2}{p})} \Vert f_{[j-5, j+5]} \Vert _p \Vert \eta _{[j-2,j+3]} \Vert _p. \end{aligned}$$

(6.8)

Finally to deal with the piece $N_j(f_{\le j-6}, \eta _{[j-2,j+3]} )$, we appeal to similar estimates in Lemmas 5.4 and 5.6. We obtain

$$\begin{aligned} \Vert N_j(f_{[j-5,j+5]}, \eta _{[j-2,j+3]} )\Vert _p&\lesssim C_{J_1} \cdot (2^{j0+} +t \cdot 2^{j\gamma } ) \Vert P_{\le J_1+10} f\Vert _p \Vert \eta _{[j-2,j+3]} \Vert _p \nonumber \\&\qquad +2^{j\gamma } \Vert P_{>J_1} f_{<j-6} \Vert _{\dot{B}^{1+\frac{2}{p}-\gamma }_{p,\infty } } \Vert \eta _{[j-2,j+3]} \Vert _p. \end{aligned}$$

(6.9)

The desired result follows. $\square$

It is clear that for any $T>0$,

$$\begin{aligned} \Vert \eta ^{(n+1)} \Vert _{ C_t^0 B^{s_0}_{p,\infty } ([0,T])}&\sim \Vert P_{\le 1} \eta ^{(n+1) } \Vert _{L_t^{\infty } L_x^p([0,T])} + (2^{js_0} \Vert \eta _j^{(n+1)} \Vert _p )_{l_t^{\infty }l_j^{\infty } (t\in [0,T],\, j\ge 2)} \nonumber \\&= \Vert P_{\le 1} \eta ^{(n+1) } \Vert _{L_t^{\infty } L_x^p([0,T])} + (2^{js_0} \Vert \eta _j^{(n+1)} \Vert _p )_{l_j^{\infty }l_t^{\infty } (t\in [0,T],\, j\ge 2)}, \end{aligned}$$

(6.10)

where the implied constant (in the notation “$\sim$”) depends only on $(s_0,p)$.

By this simple observation, using Lemma 6.2, (6.2), (6.1), and choosing first $J_1$ sufficiently large and then $T_1$ sufficiently small, we obtain

$$\begin{aligned} \Vert \eta ^{(n+1)} \Vert _{C_t^0 B^{s_0}_{p,\infty }([0,T_1])} \le \frac{1}{2} \Vert \eta ^{(n)} \Vert _{C_t^0 B^{s_0}_{p,\infty } ([0,T_1])} + C_{\theta _0} \cdot 2^{-n \sigma _0}, \end{aligned}$$

where $C_{\theta _0}$ is a constant depending on $(\theta _0,s_0,p,\gamma ,q)$ and $\sigma _0>0$ depends on $(s_0,\gamma ,p)$. This clearly yields the desired contraction in the Banach space $C_t^0([0,T_1], B^{s_0}_{p,\infty })$.

Step 5. Time continuity in $B^{1-\gamma +\frac{2}{p}}_{p,q}$. By the previous step and interpolation, we get $f^{(n)}$ converges strongly to the limit f also in $C_t^0 B^{s^{\prime }}_{p,1}([0,T_1])$ for any $0<s^{\prime }<1-\gamma +\frac{2}{p}$. We still have to show $f \in C_t^0 B^{1-\gamma +\frac{2}{p}}_{p,q}([0,T_1])$. Since $f^{(n)} \rightarrow f$ in $C_t^0 L_x^p$ we only need to consider the high frequency part. Denote $s=1-\gamma +\frac{2}{p}$. By using the estimates in Step 3 and strong convergence in each dyadic frequency block, we have for any $M\ge 10$,

$$\begin{aligned} \sum _{1\le j\le M} 2^{jsq} \Vert f_j \Vert _{L_t^{\infty } L_x^p([0,T_1])}^q = \lim _{n\rightarrow \infty } \sum _{1\le j\le M} 2^{jsq} \Vert f_j^{(n)} \Vert ^q_{L_t^{\infty } L_x^p([0,T_1])}<A_1<\infty , \end{aligned}$$

where $A_1>0$ is a constant independent of M. Thus $\Vert (f_j )\Vert _{l_j^q L_t^{\infty } L_x^p( j\ge 1, \, t \in [0,T_1])} <\infty$. Since $P_{\le M} f \in C_t^0 B^s_{p,q}$ for any M, and

$$\begin{aligned} \Vert P_{\le M} f - P_{\le M^{\prime }} f \Vert _{C_t^0 B^s_{p,q} } \lesssim \biggl (\sum _{M-2\le j \le M^{\prime }+2} 2^{jsq} \Vert f_j \Vert _{L_t^{\infty } L_x^p}^q \biggr )^{\frac{1}{q}} \rightarrow 0, \quad \hbox { as}\ M^{\prime }>M \rightarrow \infty , \end{aligned}$$

we obtain $f \in C_t^0 B^s_{p,q}$.

Remark

An alternative argument to show time continuity is to use directly (6.2) to get time continuity at $t=0$. For $t>0$ one can proceed similarly as the last part of Sect. 3 and show $e^{\epsilon _0 A t} f \in L_t^{\infty } B^{1-\gamma +\frac{2}{p}}_{p,q}$ for some $\epsilon _0>0$ small and use it to “damp" the high frequencies.

Step 6. Set $\theta (t) = e^{-t A} f(t,\cdot )$. Clearly $\theta \in C_t^0 B^{1-\gamma +\frac{2}{p}}_{p,q}$. Recall $\theta _n = e^{-t A} f_n(t,\cdot )$. In view of strong convergence of $f_n$ to f, we have $\theta _n \rightarrow \theta$ strongly in $C_t^0 B^{s^{\prime }}_{p,1}$ for any $0<s^{\prime } <1-\gamma +\frac{2}{p}$. Since for any $0\le t_0 <t \le T_0$ we have

$$\begin{aligned} \theta ^{(n+1)} (t) = e^{-(t-t_0) D^{\gamma }} \theta ^{(n+1)}(t_0) - \int _{t_0}^t \nabla \cdot e^{ -(t-s) D^{\gamma }} ( R^{\perp } \theta ^{(n)} \theta ^{(n+1)} )(s) ds. \end{aligned}$$

Taking the limit $n\rightarrow \infty$ yields

$$\begin{aligned} \theta (t) = e^{-(t-t_0) D^{\gamma }} \theta (t_0) - \int _{t_0}^t \nabla \cdot e^{ -(t-s) D^{\gamma }} ( R^{\perp } \theta (s) \theta (s) ) ds. \end{aligned}$$

(6.11)

It should be mentioned that the above equality holds in the sense of $L^p_x$ and even stronger topology. It is easy to check the absolute convergence of the integral on the RHS since

$$\begin{aligned}&\Vert \nabla \cdot e^{-(t-s) D^{\gamma }} (R^{\perp } \theta (s) \theta (s) ) \Vert _p \\&\lesssim (t-s)^{-1+} \Vert D^{(1-\gamma )+} ( R^{\perp } \theta (s) \theta (s) ) \Vert _p \lesssim (t-s)^{-1+} \Vert \theta (s) \Vert ^2_{ B^{1-\gamma +\frac{2}{p}}_{p,\infty }}. \end{aligned}$$

Thus $\theta$ is the desired local solution. One can regard (6.11) (together with some regularity assumptions) as a variant of the usual mild solution. Note that $\theta _j=P_j \theta$ is smooth, and one can easy deduce from the integral formulation (6.11) the point-wise identity:

$$\begin{aligned} \partial _t \theta _j = - D^{\gamma } \theta _j - \nabla \cdot P_j ( \theta R^{\perp }\theta ). \end{aligned}$$

From this one can proceed with the localized energy estimates and easily check the uniqueness of solution in $C_t^0 B^{1-\gamma +\frac{2}{p}}_{p,\infty }$. We omit the details.

Remark

Much better uniqueness results can be obtained by exploiting the specific form of $R^{\perp }\theta$ in connection with the $H^{-1/2}$ conservation law for non-dissipative SQG. Since this is not the focus of this work, we will not dwell on this issue here.