Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects

Abstract

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter κ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ regardless of the size of initial data.

Keywords

Boussinesq equations Convection problem Long-time solvability Dispersive effects Besov spaces Borderline regularity

1. Introduction

We begin by considering the two-dimensional (2D) inviscid Boussinesq system $\begin{matrix} (1.1) & \{\begin{array}{l} \partial_{t} u + (u \cdot \nabla) u + \nabla p = κ θ e_{2}, \\ \partial_{t} θ + (u \cdot \nabla) θ = 0, \\ div u = 0, \\ u (x, 0) = u_{0} (x), θ (x, 0) = θ_{0} (x), \end{array} \end{matrix}$ where $(x, t) \in R^{2} \times (0, \infty)$ , u is the fluid velocity, θ denotes the temperature (or the density in geophysical flows), p stands for the pressure, $κ > 0$ is a gravitational constant which will be associated with the stratification of the fluid (as described below), and the vector $e_{2} = (0, 1)$ indicates the positive vertical direction.

The 2D Boussinesq equations arise as a model in lower dimensions for the 3D hydrodynamics equations by approximating the exact density of the fluid by a constant representative value [28]. In particular, these equations serve to model large scale atmospheric and oceanic flows that are responsible for cold fronts and the jet stream (see [25,27]).

Applying the “curl” to the first equation in (1.1), and recalling that the vorticity $ω = curl (u) = \nabla^{⊥} \cdot u$ , we arrive at the equivalent vorticity formulation $\begin{matrix} (1.2) & \{\begin{array}{l} \partial_{t} ω + (u \cdot \nabla) ω = κ \partial_{1} θ, \\ \partial_{t} θ + (u \cdot \nabla) θ = 0, \\ u = \nabla^{⊥} {(- Δ)}^{- 1} ω, \nabla^{⊥} = (- \partial_{2}, \partial_{1}), \\ ω (x, 0) = ω_{0} (x), θ (x, 0) = θ_{0} (x), in R^{2} . \end{array} \end{matrix}$

Given that the atmosphere is physically observed to be mainly stable around the hydrostatic balance between the pressure gradient and gravitational effects [27], we wish to consider the initial temperature close to a physically nontrivial, stably stratified, stationary solution; namely, the hydrostatic balance $θ_{0} (x) = ρ_{0} (x) - κ x_{2}$ , and then look for the solution of (1.2) with the temperature in the form $θ (t, x) = ρ (t, x) - κ x_{2}$ . These modifications lead us to work with the following new system (see [31] for more details) $\begin{matrix} (1.3) & \{\begin{array}{l} \partial_{t} ω + (u \cdot \nabla) ω = κ \partial_{1} ρ, \\ \partial_{t} ρ + (u \cdot \nabla) ρ = κ u_{2}, \\ u = \nabla^{⊥} {(- Δ)}^{- 1} ω, \\ ω (x, 0) = ω_{0} (x), ρ (x, 0) = ρ_{0} (x), in R^{2} . \end{array} \end{matrix}$

The parameter κ can then be interpreted as a gravitational constant with $κ = N^{2}$ , where $N > 0$ is the buoyancy or Brunt-Väisälä frequency, representing the strength of stable stratification. System (1.3) thus exhibits a dispersive nature due to the stable stratification terms, as will be developed below, and can be seen as a Rayleigh-Bénard convection model where hot fluid sits on top of cooler fluid.

From the mathematical point of view, the inviscid 2D Boussinesq equations (1.1) are also important because they retain essential structural features of the 3D Euler equations which derive from the vortex stretch mechanism; in fact, they are equivalent to the 3D axisymmetric Euler equations away from the symmetry axis. Additionally, this connection has prompted the analysis of this system in the presence of nonlocal dissipative mechanisms for the velocity or the temperature (or both), represented by a fractional Laplacian operator. Global regularity has been shown in various scenarios of these viscous systems, but there are still a number of open problems; see for instance [10,29,35], and references therein. In particular, the challenging problem of stabilization around the hydrostatic balance with partial viscosity or partial dissipation has been shown to be feasible by specifically exploiting the wave structure provided by the coupling of the velocity with the temperature (cf. [7,32]).

On the other hand, the regularity problem for the inviscid system (1.1) is in general more difficult, as it still appears to be unknown whether the classical application of the Beale–Kato–Majda (BKM) regularity criterion, based on the control of $‖ ω ‖_{L^{\infty}}$ , is enough for the global regularity (cf. [37]). Thus, most results for this system are local in time, with various works dealing with the local well-posedness, in subcritical Besov and Sobolev spaces, by using the embedding of $H^{s} (R^{2})$ into $L^{\infty}$ , $s > 1$ . The borderline case $s = 1$ has added complications since $‖ \nabla u ‖_{L^{\infty}}$ is not bounded by $‖ ω ‖_{H^{1}}$ , but this can be circumvented in the critical Besov space to show local well-posedness, where the embedding into $L^{\infty}$ does hold (cf. [24]).

Moreover, system (1.1) has been shown to be ill-posed in a borderline or critical regularity setting, where the notion of criticality is defined as the lower threshold where local well-posedness of strong solutions holds (cf. [9,18]). In particular, the approach in [18] hinges on showing that “the non-linearity does not serve as a stabilizing mechanism” [18], and is based on the condition that the equation be locally well-posed in the critical Besov space which imbeds in $L^{\infty}$ , along with the application of a nontrivial commutator estimate; so that the ill-posedness is linear and critical. Furthermore, it is shown that system (1.3) is ill-posed in the Yudovich class due to norm inflation of the vorticity. Thus, these results provide the backdrop for the ensuing proof of finite time blow-up for strong solutions of system (1.1) (cf. [17]).

However, there is a stark contrast for the inviscid system (1.3) in the presence of dispersive effects. The absence of dissipation presents the possibility of instabilities, but in [19] Elgindi and Widmayer are able to derive a sharp dispersive estimate and prove the long-time existence and non-linear stability of (1.3) about the stationary configuration, see also [34] for related results. The approach consists in showing that, in the case of $κ = 1$ , the dispersive estimate reveals the explicit time decay rate of the linearized system, while not affecting the energy estimates. The nonlinear stability then follows by improving the dependence of the local time of existence on the size of the initial data. As this structure is preserved at each level of the iterative scheme one can then obtain uniform estimates for the approximation, as we show below. Deriving the corresponding Strichartz estimates one can then control for the size of the initial data by making κ sufficiently large.

This dispersive estimate is also used for the analogous result in the case of the dispersive inviscid surface quasigeostrophic (SQG) equation, where dispersive effects are associated with strong rotation; indeed, both proofs have the same structure. We refer the reader to [1,3,16,23], and references therein, where the authors showed that high speeds of rotation tend to smooth out 3D Navier–Stokes and Euler flows.

In this direction, the long-time solvability of the system in the limit of strong stratification is important to reveal the structure of solutions and their main dynamics. For instance, in the case of the 3D Boussinesq system, Widmayer [36] shows that as the dispersive parameter grows to infinity, the limiting system is a stratified system of 2D Euler equations with stratified density.

This problem was analyzed for the SQG equations and the Boussinesq system (1.3) by Wan and Chen [35] to show the long-time solvability of strong solutions for large enough $| κ |$ . The term solvability here refers to the pair of existence-uniqueness of solutions in an appropriate sense. The approach in [35] is based on generalizing the dispersive estimate of [19] and deriving the corresponding Strichartz estimates, to then show long-time solvability via a blow-up criterion of BKM-type in Sobolev spaces $({\dot{H}}^{s} \cap {\dot{H}}^{- 1}) \times H^{s + 1}$ , with $s > 3$ . Using an alternative argument, Takada [31] then improved this result with a weaker smoothness condition on the initial data, as only belonging to $H^{s}$ ( $s > 2$ ), also showing the asymptotics of solutions as $κ \to \infty$ . Nevertheless, the results of [31] and [35] do not reach the case $s = 2$ that appears as a borderline value for the long-time solvability of (1.3) in Sobolev spaces $H_{p}^{s}$ , and Besov spaces $B_{p, q}^{s}$ , with $p = 2$ .

Thus, bearing in mind the lack of control on the vorticity in the context of local well-posedness in the borderline Besov space, we are motivated to show how the solvability can be extended in this context to arbitrary times through the strong linear dispersive stabilization. For this we exploit the paraproduct estimates which derive from the embedding of ${\dot{B}}_{2, 1}^{1}$ into $L^{\infty}$ , following the theme in Vishik’s result of long-time uniqueness for the 2D Euler equations in the borderline regularity case [33]. In particular, since we need to show local estimates which are uniform in κ, in contrast to [24], we must develop specific commutator estimates.

Then, in view of the identifications $B_{2, 2}^{s} \equiv H^{s}$ and ${\dot{B}}_{2, 2}^{s} \equiv {\dot{H}}^{s}$ for $p = 2$ and $q = 2$ (see [8, Theorem 6.4.4]), we are able to recover the results in [35] and also cover the case $s = 2$ , with similar power laws for the time decay rate. Our main results read as follows:

Theorem 1.1.
Let s and q be real numbers such that $s > 2$ with $1 ⩽ q ⩽ \infty$ or $s = 2$ with $q = 1$ .
(Local uniform solvability) Let $ω_{0} \in {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})$ and $ρ_{0} \in B_{2, q}^{s} (R^{2})$ . There exists $T > 0$ (depending of $‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}$ and $‖ ρ_{0} ‖_{B_{2, q}^{s}}$ ) such that ( 1.3 ) has a unique solution $(ω, ρ)$ with $ω \in C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T]; {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $ρ \in C ([0, T]; B_{2, q}^{s} (R^{2})) \cap C^{1} ([0, T]; B_{2, q}^{s - 1} (R^{2}))$ , for all $κ \in R$ .

(Long-time solvability) Let $T \in (0, \infty)$ , $ω_{0} \in {\dot{B}}_{2, q}^{s} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})$ , and $ρ_{0} \in B_{2, q}^{s + 1} (R^{2})$ . There exists $κ_{0} = κ_{0} (T, ‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}, ‖ ρ_{0} ‖_{B_{2, q}^{s + 1}}) > 0$ such that if $| κ | ⩾ κ_{0}$ then ( 1.3 ) has a unique solution $(ω, ρ)$ such that $ω \in C ([0, T]; {\dot{B}}_{2, q}^{s} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $ρ \in C ([0, T]; B_{2, q}^{s + 1} (R^{2})) \cap C^{1} ([0, T]; B_{2, q}^{s} (R^{2}))$ .

An analogous theorem for the SQG system was proved in our previous work [2], and although the general argument is structured in a similar vein, there are significant technical differences with the proof provided here for Theorem 1.1, which derive from the how the stabilization due to the coupling of velocity and temperature works at different levels.

In order to prove Theorem 1.1, and show how the regularization of the temperature feeds into further stabilization, we construct approximate solutions ${(ω_{n}, ρ_{n})}_{n \in N}$ via a Picard iteration scheme, and show a priori estimates uniform with respect to the dispersive parameter κ, so as to obtain a solution as the limit of $(ω_{n}, ρ_{n})$ in the Besov spaces with the borderline regularity for (1.3) (see Section 4 and the proof of Theorem 1.1).

To do this in the framework of borderline regularity, we employ an intersection of spaces for the vorticity ω. In fact, ${\dot{H}}^{- 1}$ is important to control the influence of κ on the existence time T, and then obtain a uniform time w.r.t κ, via a cancellation effect involving the ${\dot{H}}^{- 1}$ -inner product (see, e.g., (4.2)), while the homogeneous Besov space ${\dot{B}}_{2, q}^{s - 1} (R^{2})$ provides the necessary control on the regularity, particularly for the borderline case. More precisely, we need to show that ${(ω_{n}, ρ_{n})}_{n \in N}$ is bounded in $L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}) \times L^{\infty} (0, T; B_{2, q}^{s})$ and Cauchy in $L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}) \times L^{\infty} (0, T; B_{2, q}^{s - 1})$ , both uniformly w.r.t. κ. It is worth noting that this difficulty does not appear in the context of inviscid SQG and Euler equations (with or without dispersive effects) when analyzing local solvability and borderline regularity in Besov spaces (see [1,2,26,33]).

We also note that to prove the uniform solvability in this context, it is central to use commutator estimates in the framework of homogeneous Besov spaces, which we present in a form which we could not find elsewhere in the literature (see Section 3). Subsequently, for large values of $| κ |$ and $s ⩾ 2$ , we obtain the long-time solvability by showing a blow-up criterion and handling globally the integral $\int_{0}^{t} ‖ (ω \pm Λ ρ) (τ) ‖_{{\dot{B}}_{\infty, 1}^{0}} d τ$ , using Strichartz estimates as presented in [23,35].

Lastly, we remark that the stability problem for the more general setting where the viscosity and thermal diffusivity are non-zero, which is also physically important [21,27], has received considerable attention. In particular, for the case of system (1.1) with positive viscosity but zero thermal diffusivity, the global well-posedness and inviscid limit have been shown through refined energy methods, see [12,14,20,22], whereas the stabilization around the hydrostatic balance has been recently studied in the case of bounded or strip domains with Dirichlet or periodic boundary conditions, see [10,15,32]. For the latter problem, the viscous term introduces a wave structure in the linearized system for which explicit decay rates can be shown, but so that the convergence to the full nonlinear equations is considerably more involved. Given that the energy estimates for this system has a parallel structure to the inviscid system, it is possible that the iterative scheme could be useful for controlling the decay rate and obtaining the asymptotic limit of large dispersive forcing.

The plan of the manuscript is as follows. In Section 2 we present some preliminaries about Besov spaces and Strichartz estimates, among others. Section 3 is devoted to the commutator estimates. In Section 4, we analyze the approximation scheme ${(ω_{n}, ρ_{n})}_{n \in N}$ and obtain the local-in-time solvability of (1.3) uniformly with respect to the parameter κ. The proof of Theorem 1.1 (ii) is carried out in Section 5.
2. Preliminaries

The purpose of this section is to provide some basic definitions and properties about Besov spaces as well as some estimates useful for our ends, such as product, embeddings, Strichartz estimates, among others. We refer the reader to the book [8] for more details on Besov spaces and their properties.

First, we denote the Schwartz space on $R^{2}$ by $S (R^{2})$ and its dual by $S^{'} (R^{2})$ (tempered distributions). For $f \in S^{'} (R^{2})$ , $\hat{f}$ stands for the Fourier transform of f. Select a radial function $ψ_{0} \in S (R^{2})$ satisfying $0 ⩽ {\hat{ψ}}_{0} (ξ) ⩽ 1$ , $supp {\hat{ψ}}_{0} \subset {ξ \in R^{2} : \frac{5}{8} ⩽ | ξ | ⩽ \frac{7}{4}}$ and $\begin{matrix} \sum_{j \in Z} {\hat{ψ}}_{j} (ξ) = 1 for all ξ \in R^{2} ∖ {0}, \end{matrix}$ where $ψ_{j} (x) : = 2^{2 j} ψ_{0} (2^{j} x)$ . For each $k \in Z$ , we consider $S_{k}, {\dot{S}}_{k} \in S$ defined in Fourier variables as $\begin{matrix} \hat{S_{k}} (ξ) = 1 - \sum_{j ⩾ k + 1} {\hat{ψ}}_{j} (ξ) and \hat{{\dot{S}}_{k}} (ξ) = \sum_{j ⩽ k} {\hat{ψ}}_{j} (ξ) . \end{matrix}$ We observe that $\begin{matrix} supp {\hat{ψ}}_{j} \cap supp {\hat{ψ}}_{j^{'}} = \emptyset if | j - j^{'} | ⩾ 2 . \end{matrix}$ For $f \in S^{'} (R^{2})$ and $j \in Z$ , the Littlewood–Paley operator $Δ_{j}$ is the convolution $Δ_{j} f : = ψ_{j} * f$ which works as a filter on the support of $ψ_{j}$ . We also consider the family of operators ${{\tilde{Δ}}_{k}}_{k \in {0} \cup N}$ defined as ${\tilde{Δ}}_{0} = S_{0} * f$ and ${\tilde{Δ}}_{k} = Δ_{k}$ for every integer $k ⩾ 1$ .

Let $P = P (R^{2})$ denote the set of polynomials and consider $s \in R$ and $p, q \in [1, \infty]$ . The homogeneous Besov space ${\dot{B}}_{p, q}^{s} (R^{2})$ is the set of all $f \in S^{'} (R^{2}) / P$ such that $\begin{matrix} ‖ f ‖_{{\dot{B}}_{p, q}^{s}} : = {‖ {2^{s j} ‖ Δ_{j} f ‖_{L^{p}}} ‖}_{l^{q} (Z)} < \infty . \end{matrix}$ The nonhomogeneous version of ${\dot{B}}_{p, q}^{s} (R^{2})$ , namely the Besov space $B_{p, q}^{s} = B_{p, q}^{s} (R^{2})$ , is the space of all $f \in S^{'} (R^{n})$ such that the norm $‖ f ‖_{B_{p, q}^{s}} < \infty$ , where $\begin{matrix} ‖ f ‖_{B_{p, q}^{s}} = \{\begin{array}{ll} {(\sum_{k = 0}^{\infty} 2^{k s q} ‖ {\tilde{Δ}}_{k} f ‖_{p}^{q})}^{\frac{1}{q}}, & if q < \infty, \\ {sup}_{k \in {0} \cup N} {2^{k s} ‖ {\tilde{Δ}}_{k} f ‖_{p}}, & if q = \infty . \end{array} \end{matrix}$ The pairs $({\dot{B}}_{p, q}^{s}, ‖ \cdot ‖_{{\dot{B}}_{p, q}^{s}})$ and $(B_{p, q}^{s}, ‖ \cdot ‖_{B_{p, q}^{s}})$ are Banach spaces. Also, for $s \in R$ and $p, q \in [1, \infty]$ , it follows that $\begin{matrix} ‖ f ‖_{B_{p, q}^{s}} ⩽ C (‖ f ‖_{L^{p}} + ‖ f ‖_{{\dot{B}}_{p, q}^{s}}) . \end{matrix}$ For $s > 0$ , we have the equivalence of norms $\begin{matrix} ‖ f ‖_{B_{p, q}^{s}} \sim ‖ f ‖_{L^{p}} + ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{matrix}$ In the case $s = 0$ , we recall the inclusion $B_{p, 1}^{0} ↪ L^{p}$ , for all $p \in [1, \infty]$ .

Lemma 2.1 (Bernstein’s Lemma).

Let $1 ⩽ p ⩽ \infty$ and $f \in L^{p}$ be such that $supp \hat{f} \subset {ξ \in R^{2} : 2^{j - 2} ⩽ | ξ | ⩽ 2^{j}}$ . Then, we have the estimates $\begin{matrix} C^{- 1} 2^{j k} ‖ f ‖_{L^{p}} ⩽ {‖ D^{k} f ‖}_{L^{p}} ⩽ C 2^{j k} ‖ f ‖_{L^{p}}, \end{matrix}$ where $C = C (k)$ is a positive constant.

Remark 2.2.
Using the above lemma, one can prove the equivalence $\begin{matrix} ‖ f ‖_{{\dot{B}}_{p, q}^{s + k}} \sim {‖ D^{k} f ‖}_{{\dot{B}}_{p, q}^{s}} . \end{matrix}$ Moreover, considering $1 ⩽ p, q ⩽ \infty$ and $s ⩾ n / p$ , with $q = 1$ if $s = n / p$ , we have that (see, e.g., [8, Sections 6.5 and 6.8]) $\begin{matrix} (2.1) & ‖ f ‖_{L^{\infty}} ⩽ C ‖ f ‖_{B_{p, q}^{s}} . \end{matrix}$ Then, $\begin{matrix} (2.2) & ‖ \nabla f ‖_{L^{\infty}} ⩽ C ‖ \nabla f ‖_{B_{p, q}^{s - 1}} ⩽ C ‖ f ‖_{B_{p, q}^{s}}, \end{matrix}$ where $1 ⩽ p, q ⩽ \infty$ and $s ⩾ n / p + 1$ with $q = 1$ in the case $s = n / p + 1$ .

The uniform estimates we develop rely on inequalities (2.1) and (2.2) in their respective regularity ranges $s ⩾ n / p$ and $s ⩾ n / p + 1$ . For the case $q = 1$ and the corresponding s, these inequalities hold in homogeneous Besov spaces. However, as our results are aimed for more general $q ⩾ 1$ and s, and deal with the full inviscid case, we restrict to the case of nonhomogeneous spaces, where they are guaranteed to be valid, and leave open the question of working with homogeneous Besov spaces for the component ρ.

Some Leibniz-type rules in Besov spaces are the subject of the lemma below (see [11]).
Lemma 2.3.
Let $s > 0$ , $1 ⩽ p_{1}, p_{2} ⩽ \infty$ , $1 ⩽ r_{1}, r_{2} ⩽ \infty$ and $1 ⩽ p, q ⩽ \infty$ be such that $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{r_{1}} + \frac{1}{r_{2}}$ . Then, we have the estimates $\begin{aligned} ‖ f g ‖_{{\dot{B}}_{p, q}^{s}} ⩽ C (‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s}} + ‖ f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}}), \\ ‖ f g ‖_{B_{p, q}^{s}} ⩽ C (‖ g ‖_{L^{p_{2}}} ‖ f ‖_{B_{p_{1}, q}^{s}} + ‖ f ‖_{L^{r_{2}}} ‖ g ‖_{B_{r_{1}, q}^{s}}), \end{aligned}$ where $C > 0$ is a universal constant.

We will employ the Strichartz estimates of [35], linked to the dispersive term $κ (\partial_{1} ρ + Λ u_{2})$ obtained from (1.3), which will allow us to obtain long-time solvability for (1.3). In particular, we will use the following results found in [2,23] and [35]. Lemma 2.4.
Let $κ \in R$ , $4 ⩽ γ ⩽ \infty$ and $2 ⩽ r ⩽ \infty$ be such that $\begin{matrix} (2.3) & \frac{1}{γ} + \frac{1}{2 r} ⩽ \frac{1}{4} . \end{matrix}$ Then, there holds $\begin{matrix} {‖ G_{\pm} (κ t) f ‖}_{L^{γ} (0, \infty; L^{r})} ⩽ C | κ |^{- \frac{1}{γ}} ‖ f ‖_{L^{2}}, \end{matrix}$ for all $f \in L^{2} (R^{2})$ , where $\begin{matrix} G_{\pm} (t) f (x) : = \int_{R^{2}} e^{i ξ \cdot x \pm i t \frac{ξ_{1}}{| ξ |}} \hat{ϕ} (ξ) \hat{f} (ξ) d ξ \end{matrix}$ and $\hat{ϕ}$ is a compactly supported smooth function in $R^{2}$ .
Lemma 2.5.
Let $s, t, κ \in R$ , $1 ⩽ q ⩽ \infty$ , $4 ⩽ γ ⩽ q$ , and $2 ⩽ r ⩽ \infty$ . Assume also ( 2.3 ). Then, $\begin{matrix} {‖ e^{\pm t κ R_{1}} f ‖}_{L^{γ} (0, \infty; {\dot{B}}_{r, q}^{s})} ⩽ C | κ |^{- \frac{1}{γ}} ‖ f ‖_{{\dot{B}}_{2, q}^{s + 1 - \frac{2}{r}}}, \end{matrix}$ for all $f \in {\dot{B}}_{2, q}^{s + 1 - \frac{2}{r}} (R^{2})$ .

3. Commutator estimates

The present section is devoted to commutator estimates in ${\dot{B}}_{p, q}^{s}$ and $B_{p, q}^{s}$ that will be useful to obtain convergence of our approximate solutions. We state and prove some of them, as we have not been able to locate them in the literature with the needed hypotheses and conclusions for our purposes.

Recall the commutator operator $\begin{matrix} [f \cdot \nabla, Δ_{j}] g = f \cdot \nabla (Δ_{j} g) - Δ_{j} (f \cdot \nabla g) . \end{matrix}$ Using the Hölder inequality in a slightly different way than used in [11,30,38], it is possible to obtain the following estimates for the commutator:

Lemma 3.1.
Let $1 < p < \infty$ and $1 ⩽ q, p_{1}, p_{2}, r_{1}, r_{2} ⩽ \infty$ be such that $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{r_{1}} + \frac{1}{r_{2}}$ .
Let $s > 0$ , $f \in {\dot{B}}_{p_{1}, q}^{s} (R^{n})$ and $g \in {\dot{B}}_{r_{1}, q}^{s} (R^{n})$ . Assume further that $\nabla f \in L^{r_{2}} (R^{n})$ , $\nabla \cdot f = 0$ and $\nabla g \in L^{p_{2}} (R^{n})$ . Then, we have the estimate $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ [f \cdot \nabla, Δ_{j}] g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}} + ‖ \nabla g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s}}), \end{matrix}$ where $C > 0$ is a universal constant.

Let $s > - 1$ , $f \in {\dot{B}}_{p_{1}, q}^{s + 1} (R^{n})$ and $g \in {\dot{B}}_{r_{1}, q}^{s} (R^{n}) \cap L^{p_{2}} (R^{n})$ . Assume further that $\nabla f \in L^{r_{2}} (R^{n})$ and $\nabla \cdot f = 0$ . Then, we have the estimate $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ [f \cdot \nabla, Δ_{j}] g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}} + ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s + 1}}), \end{matrix}$ where $C > 0$ is a universal constant.

Remark 3.2.
With the help of Lemma 6.3 in [38] (see also [4]), the properties of the operator $Λ^{- 1}$ , considering the same hypotheses of Lemma 3.1 and using the same arguments in the proof of the same result, we obtain the following commutator estimate $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ [f \cdot \nabla, Λ^{- 1} Δ_{j}] g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s - 1}} + ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s}}) . \end{matrix}$

Recall that the Bony formula for the paraproduct of f and g is given by $\begin{matrix} (3.1) & f g = T_{f} g + T_{g} f + R (f, g), \end{matrix}$ where $\begin{matrix} (3.2) & T_{f} g : = \sum_{j \in Z} S_{j - 2} f Δ_{j} g and R (f, g) : = \sum_{| j - j^{'} | ⩽ 1} Δ_{j} f Δ_{j^{'}} g . \end{matrix}$ In the sequel we state and prove the following commutator-type estimates: Lemma 3.3.
Let $1 < p < \infty$ and $1 ⩽ q, p_{1}, p_{2}, r_{1}, r_{2} ⩽ \infty$ be such that $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} = \frac{1}{r_{1}} + \frac{1}{r_{2}}$ .
Let $s > 0$ , $f \in {\dot{B}}_{p_{1}, q}^{s} (R^{n})$ with $\nabla f \in L^{r_{2}} (R^{n})$ and $\nabla \cdot f = 0$ , and $g \in {\dot{B}}_{r_{1}, q}^{s} (R^{n})$ with $\nabla g \in L^{p_{2}} (R^{n})$ . Then, there exists a universal constant $C > 0$ such that $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}} + ‖ \nabla g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s}}) . \end{matrix}$

Let $s > - 1$ , $f \in {\dot{B}}_{p_{1}, q}^{s + 1} (R^{n})$ with $\nabla f \in L^{r_{2}} (R^{n})$ and $\nabla \cdot f = 0$ , and $g \in {\dot{B}}_{r_{1}, q}^{s} (R^{n}) \cap L^{p_{2}} (R^{n})$ . Then, there exists a universal constant $C > 0$ such that $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}} + ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s + 1}}) . \end{matrix}$

Proof.
The proof of part (i), it follows from the calculations obtained by Chae in [11]. We show the part (ii). We follow closely the argument in [33] (see also [5,13]). By Bony’s paraproduct formula (3.1), we can write $\begin{aligned} ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g \\ = - \sum_{i = 1}^{n} Δ_{j} T_{\partial_{i} g_{k}} f_{i} + \sum_{i = 1}^{n} [T_{f_{i}} \partial_{i}, Δ_{j}] g_{k} - \sum_{i = 1}^{n} Δ_{j} T_{f_{i} - S_{j - 2} f_{i}} (\partial_{i} Δ_{j} g_{k}) \\ - \sum_{i = 1}^{n} {Δ_{j} R (f_{i}, \partial_{i} g_{k}) - R (S_{j - 2} f_{i}, \partial_{i} Δ_{j} g_{k})} \\ : = I + II + III + IV . \end{aligned}$ For I, in view of (3.2), it follows that $\begin{matrix} I = - \sum_{j^{'}} \sum_{i = 1}^{n} Δ_{j} {\dot{S}}_{j^{'} - 2} (\partial_{i} g_{k}) Δ_{j^{'}} f_{i} . \end{matrix}$ We observe that $supp F ({\dot{S}}_{j^{'} - 2} (\partial_{i} g_{k}) Δ_{j^{'}} f_{i}) \subset {ξ : 2^{j^{'} - 3} ⩽ | ξ | ⩽ 2^{j^{'} + 1}}$ and $Δ_{j} {\dot{S}}_{j^{'} - 2} (\partial_{i} g_{k}) Δ_{j^{'}} f_{i} = 0$ , if $| j - j^{'} | ⩾ 4$ . Then, $\begin{matrix} I = - \sum_{| j - j^{'} | ⩽ 3} \sum_{i = 1}^{n} Δ_{j} {\dot{S}}_{j^{'} - 2} (\partial_{i} g_{k}) Δ_{j^{'}} f_{i} . \end{matrix}$ Using integration by parts, we arrive at $\begin{aligned} I & = - \sum_{| j - j^{'} | ⩽ 3} \sum_{i = 1}^{n} 2^{j n} \int_{R^{n}} ψ_{0} (2^{j} (x - y)) ({\dot{S}}_{j^{'} - 2} \partial_{i} g_{k}) (y) (Δ_{j^{'}} f_{i}) (y) d y \\ = - \sum_{| j - j^{'} | ⩽ 3} \sum_{i = 1}^{n} 2^{j} 2^{j n} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) ({\dot{S}}_{j^{'} - 2} g_{k}) (y) (Δ_{j^{'}} f_{i}) (y) d y \\ = - \sum_{| j - j^{'} | ⩽ 3} \sum_{i = 1}^{n} 2^{j} {(2^{j n} \partial_{i} ψ_{0} (2^{j} \cdot) * (({\dot{S}}_{j^{'} - 2} g_{k}) (Δ_{j^{'}} f_{i}))}, \end{aligned}$ which yields $\begin{matrix} (3.3) & \begin{aligned} [b] ‖ I ‖_{L^{p}} & ⩽ C \sum_{| j - j^{'} | ⩽ 3} \sum_{i = 1}^{n} 2^{j} {‖ ({\dot{S}}_{j^{'} - 2} g_{k}) (Δ_{j^{'}} f_{i}) ‖}_{L^{p}} \\ ⩽ C ‖ g ‖_{L^{p_{2}}} \sum_{| j - j^{'} | ⩽ 3} 2^{j} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}} . \end{aligned} \end{matrix}$ For estimate $II$ , by an argument similar to the one above, we first note that $\begin{matrix} [{\dot{S}}_{j^{'} - 2} f_{i}, Δ_{j}] (\partial_{i} Δ_{j^{'}} g_{k}) = 0, if | j - j^{'} | ⩾ 4 . \end{matrix}$ Then, using $\nabla \cdot {\dot{S}}_{j^{'} - 2} f = 0$ and integration by parts, it holds that $\begin{aligned} II = & \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} {({\dot{S}}_{j^{'} - 2} f_{i}) Δ_{j} (\partial_{i} Δ_{j^{'}} g_{k}) - Δ_{j} ({\dot{S}}_{j^{'} - 2} f_{i}) (\partial_{i} Δ_{j^{'}} g_{k})} \\ = & \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} 2^{j n} \int_{R^{n}} ψ_{0} (2^{j} (x - y)) ({\dot{S}}_{j^{'} - 2} f_{i} (x) - {\dot{S}}_{j^{'} - 2} f_{i} (y)) (\partial_{i} Δ_{j^{'}} g_{k}) (y) d y \\ = & \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} 2^{j (n + 1)} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) ({\dot{S}}_{j^{'} - 2} f_{i} (x) - {\dot{S}}_{j^{'} - 2} f_{i} (y)) (Δ_{j^{'}} g_{k}) (y) d y \\ = & \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} 2^{j (n + 1)} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) \int_{0}^{1} ((x - y) \cdot \nabla) \\ \times ({\dot{S}}_{j^{'} - 2} f_{i} (x + τ (y - x))) d τ (Δ_{j^{'}} g_{k}) (y) d y \\ = & \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} \int_{R^{n}} \partial_{i} ψ_{0} (z) \int_{0}^{1} (z \cdot \nabla) ({\dot{S}}_{j^{'} - 2} f_{i} (x + τ 2^{- j} z)) d τ (Δ_{j^{'}} g_{k}) (x - 2^{- j} z) d z . \end{aligned}$ Therefore, $\begin{aligned} [b] | II | & ⩽ \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 3} \int_{R^{n}} | \partial_{i} ψ_{0} (z) | \int_{0}^{1} | z | | \nabla ({\dot{S}}_{j^{'} - 2} f_{i} (x + τ 2^{- j} z)) | d τ | (Δ_{j^{'}} g_{k}) (x - 2^{- j} z) | d z \\ ⩽ C ‖ \nabla f ‖_{L^{\infty}} \sum_{| j - j^{'} | ⩽ 3} \int_{R^{n}} | z | | \nabla ψ_{0} (z) | | (Δ_{j^{'}} g_{k}) (x - 2^{- j} z) | d z, \end{aligned}$ which leads us to $\begin{matrix} (3.4) & \begin{aligned} ‖ II ‖_{L^{p}} & ⩽ C ‖ \nabla f ‖_{L^{\infty}} \sum_{| j - j^{'} | ⩽ 3} \int_{R^{n}} | z | | \nabla ψ_{0} (z) | {‖ (Δ_{j^{'}} g_{k}) (\cdot - 2^{- j} z) ‖}_{L^{p}} d z \\ ⩽ C ‖ \nabla f ‖_{L^{r_{1}}} \sum_{| j - j^{'} | ⩽ 3} ‖ Δ_{j^{'}} g ‖_{L^{r_{2}}} . \end{aligned} \end{matrix}$ For $III$ , we have that $\begin{aligned} III & = \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 1} {\dot{S}}_{j^{'} - 2} (f_{i} - {\dot{S}}_{j - 2} f_{i}) \partial_{i} Δ_{j^{'}} Δ_{j} g_{k} \\ = \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 1} {\dot{S}}_{j^{'} - 2} (\sum_{m = j - 1}^{j^{'} - 1} Δ_{m} f_{i}) \partial_{i} Δ_{j} Δ_{j^{'}} g_{k} \\ = \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 1} {\dot{S}}_{j^{'} - 2} (Δ_{j - 1} f_{i} + Δ_{j} f_{i}) \partial_{i} Δ_{j} Δ_{j^{'}} g_{k} . \end{aligned}$ Applying the $L^{p}$ -norm, we arrive at $\begin{matrix} (3.5) & \begin{aligned} ‖ III ‖_{L^{p}} & ⩽ \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 1} (‖ Δ_{j - 1} f_{i} ‖_{L^{\infty}} + ‖ Δ_{j} f_{i} ‖_{L^{\infty}}) ‖ \partial_{i} Δ_{j} Δ_{j^{'}} g_{k} ‖_{L^{p}} \\ ⩽ \sum_{i = 1}^{n} \sum_{| j - j^{'} | ⩽ 1} (2^{- j + 1} ‖ Δ_{j - 1} \nabla f_{i} ‖_{L^{\infty}} + 2^{- j} ‖ Δ_{j} \nabla f_{i} ‖_{L^{\infty}}) 2^{j} ‖ Δ_{j} Δ_{j^{'}} g_{k} ‖_{L^{p}} \\ ⩽ C ‖ \nabla f ‖_{L^{r_{1}}} \sum_{| j - j^{'} | ⩽ 1} ‖ Δ_{j^{'}} g ‖_{L^{r_{2}}} . \end{aligned} \end{matrix}$ For the parcel $IV$ , we can decompose $\begin{aligned} IV & = \sum_{i = 1}^{n} Δ_{j} \partial_{i} R (f_{i} - {\dot{S}}_{j - 2} f_{i}, g_{k}) + \sum_{i = 1}^{n} {Δ_{j} R ({\dot{S}}_{j - 2} f_{i}, \partial_{i} g_{k}) - R ({\dot{S}}_{j - 2} f_{i}, Δ_{j} \partial_{i} g_{k})} \\ = {IV}_{1} + {IV}_{2} . \end{aligned}$ Since $\sum_{i = 1}^{n} \partial_{i} Δ_{j^{'}} (f_{i} - {\dot{S}}_{j - 2} f_{i}) = 0$ , it follows that $\begin{aligned} {IV}_{1} & = \sum_{i = 1}^{n} \partial_{i} Δ_{j} {\sum_{| j^{'} - j^{″} | ⩽ 1} Δ_{j^{'}} (f_{i} - {\dot{S}}_{j - 2} f_{i}) Δ_{j^{″}} g_{k}} \\ = \sum_{i = 1}^{n} Δ_{j} {\sum_{| j^{'} - j^{″} | ⩽ 1} \sum_{j^{'} ⩾ j - 3} (Δ_{j^{'}} f_{i} - {\dot{S}}_{j - 2} Δ_{j^{'}} f_{i}) Δ_{j^{″}} \partial_{i} g_{k}} \\ = \sum_{i = 1}^{n} \sum_{| j^{'} - j^{″} | ⩽ 1} \sum_{j^{'} ⩾ j - 3} 2^{j n} \int_{R^{n}} ψ_{0} (2^{j} (x - y)) (Δ_{j^{'}} f_{i} (y) - {\dot{S}}_{j - 2} Δ_{j^{'}} f_{i} (y)) Δ_{j^{″}} \partial_{i} g_{k} (y) d y \\ = \sum_{i = 1}^{n} \sum_{| j^{'} - j^{″} | ⩽ 1} \sum_{j^{'} ⩾ j - 3} 2^{j} 2^{j n} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) (Δ_{j^{'}} f_{i} (y) - {\dot{S}}_{j - 2} Δ_{j^{'}} f_{i} (y)) Δ_{j^{″}} g_{k} (y) d y \\ = \sum_{i = 1}^{n} \sum_{| j^{'} - j^{″} | ⩽ 1} \sum_{j^{'} ⩾ j - 3} 2^{j} {(2^{j n} \partial_{i} ψ_{0} (2^{j} \cdot)) * (Δ_{j^{'}} f_{i} - {\dot{S}}_{j - 2} Δ_{j^{'}} f_{i}) Δ_{j^{″}} g_{k})} \end{aligned}$ Then $\begin{matrix} (3.6) & \begin{aligned} ‖ {IV}_{1} ‖_{L^{p}} & ⩽ C \sum_{i = 1}^{n} \sum_{| j^{'} - j^{″} | ⩽ 1} \sum_{j^{'} ⩾ j - 3} 2^{j} (‖ Δ_{j^{'}} f_{i} ‖_{L^{p}} + ‖ {\dot{S}}_{j - 2} Δ_{j^{'}} f_{i} ‖_{L^{p}}) ‖ Δ_{j^{″}} g_{k} ‖_{L^{\infty}} \\ ⩽ C ‖ g ‖_{L^{p_{2}}} \sum_{j^{'} ⩾ j - 3} 2^{j} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}} . \end{aligned} \end{matrix}$ On the other hand, note that $\begin{aligned} {IV}_{2} & = \sum_{i = 1}^{n} \sum_{| j^{'} - j^{″} | ⩽ 1} [Δ_{j} ((Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i}) Δ_{j^{″}} \partial_{i} g_{k}) - (Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i}) (Δ_{j^{″}} Δ_{j} \partial_{i} g_{k})] \\ = \sum_{i = 1}^{n} \sum_{j - 1 ⩾ j^{'} ⩾ j - 3} \sum_{| j^{'} - j^{″} | ⩽ 1} [Δ_{j}, Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i}] Δ_{j^{″}} \partial_{i} g_{k} . \end{aligned}$ Also, we can write $\begin{aligned} [Δ_{j}, Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i}] Δ_{j^{″}} \partial_{i} g_{k} \\ = 2^{j n} \int_{R^{n}} ψ_{0} (2^{j} (x - y)) (Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (y) - Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (x)) Δ_{j^{″}} \partial_{i} g_{k} (y) d y \\ = 2^{j (n + 1)} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) (Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (y) - Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (x)) Δ_{j^{″}} g_{k} (y) d y \\ = 2^{j (n + 1)} \int_{R^{n}} \partial_{i} ψ_{0} (2^{j} (x - y)) \int_{0}^{1} ((y - x) \cdot \nabla) (Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (x + τ (y - x)) d τ Δ_{j^{″}} g_{k} (y) d y \\ = \int_{R^{n}} \partial_{i} ψ_{0} (z) \int_{0}^{1} (z \cdot \nabla) (Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i} (x - τ 2^{- j} z) d τ Δ_{j^{″}} g_{k} (x - 2^{- j} z) d z . \end{aligned}$ Hence, $\begin{aligned} ‖ [Δ_{j}, Δ_{j^{'}} {\dot{S}}_{j - 2} f_{i}] Δ_{j^{″}} \partial_{i} g_{k} ‖_{L^{p}} & ⩽ C \sum_{m = 1}^{n} ‖ Δ_{j^{'}} {\dot{S}}_{j - 2} \partial_{m} f_{i} ‖_{L^{\infty}} ‖ Δ_{j^{″}} g_{k} ‖_{L^{p}} \\ ⩽ C ‖ \nabla f ‖_{L^{\infty}} ‖ Δ_{j^{″}} g_{k} ‖_{L^{p}}, \end{aligned}$ and $\begin{matrix} (3.7) & ‖ {IV}_{2} ‖_{L^{p}} ⩽ C ‖ \nabla f ‖_{L^{r_{2}}} \sum_{| j - j^{'} | ⩽ 5} ‖ Δ_{j^{'}} g_{k} ‖_{L^{r_{1}}} . \end{matrix}$ Summing up the estimates (3.3), (3.4), (3.5), (3.6) and (3.7), we obtain $\begin{aligned} {‖ ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g ‖}_{L^{p}} \\ ⩽ C ‖ \nabla f ‖_{L^{r_{2}}} \sum_{| j - j^{'} | ⩽ 5} ‖ Δ_{j^{'}} g_{k} ‖_{L^{r_{1}}} + C ‖ g ‖_{L^{p_{2}}} (\sum_{| j - j^{'} | ⩽ 3} 2^{j} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}} + \sum_{j^{'} ⩾ j - 3} 2^{j} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}}) . \end{aligned}$ Multiplying by $2^{j s}$ and computing the $l^{q} (Z)$ -norm, we can estimate $\begin{aligned} {(\sum_{j \in Z} 2^{j q s} {‖ ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g ‖}_{L^{p}}^{q})}^{1 / q} \\ ⩽ C ‖ \nabla f ‖_{L^{r_{2}}} {(\sum_{j \in Z} \sum_{| j - j^{'} | ⩽ 5} 2^{j q s} ‖ Δ_{j^{'}} g_{k} ‖_{L^{r_{1}}}^{q})}^{1 / q} + C ‖ g ‖_{L^{p_{2}}} {(\sum_{j \in Z} \sum_{| j - j^{'} | ⩽ 3} 2^{j (s + 1) q} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}}^{q})}^{1 / q} \\ + C ‖ g ‖_{L^{p_{2}}} {(\sum_{j \in Z} \sum_{j^{'} ⩾ j - 3} 2^{j (s + 1) q} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}}^{q})}^{1 / q} \\ : = K_{1} + K_{2} + K_{3} . \end{aligned}$ Now, observing that $\begin{matrix} \sum_{j \in Z} \sum_{| j - j^{'} | ⩽ 5} 2^{j q s} ‖ Δ_{j^{'}} h ‖_{L^{p}}^{q} = \sum_{k = - 5}^{5} 2^{- k q s} \sum_{j \in Z} 2^{(j + k) q s} ‖ Δ_{j + k} h ‖_{L^{p}}^{q} ⩽ C \sum_{j \in Z} ‖ Δ_{j} h ‖_{L^{p}}^{q}, \end{matrix}$ we have that $\begin{matrix} K_{1} ⩽ C ‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}}, \end{matrix}$ and similarly $K_{2} ⩽ C ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s + 1}}$ .

For $K_{3}$ , note that $\begin{aligned} K_{3} & = C ‖ g ‖_{L^{p_{2}}} {(\sum_{j \in Z} \sum_{j^{'} ⩾ j - 3} 2^{(j - j^{'}) (s + 1) q} {(2^{j^{'} (s + 1)} ‖ Δ_{j^{'}} f ‖_{L^{p_{1}}})}^{q})}^{1 / q} \\ = C ‖ g ‖_{L^{p_{2}}} {(\sum_{k ⩾ - 3} 2^{- k q (s + 1)} \sum_{j \in Z} {(2^{(j + k) (s + 1)} ‖ Δ_{j + k} f ‖_{L^{p_{1}}})}^{q})}^{1 / q} \\ = C ‖ g ‖_{L^{p_{2}}} {(\sum_{k ⩾ - 3} 2^{- k q (s + 1)})}^{1 / q} {(\sum_{j \in Z} 2^{j q (s + 1)} ‖ Δ_{j} f ‖_{L^{p_{1}}}^{q})}^{1 / q} \\ ⩽ C ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s + 1}} . \end{aligned}$ Therefore, $\begin{matrix} {(\sum_{j \in Z} 2^{s j q} {‖ ({\dot{S}}_{j - 2} f \cdot \nabla) Δ_{j} g - Δ_{j} (f \cdot \nabla) g ‖}_{L^{p}}^{q})}^{1 / q} ⩽ C (‖ \nabla f ‖_{L^{r_{2}}} ‖ g ‖_{{\dot{B}}_{r_{1}, q}^{s}} + ‖ g ‖_{L^{p_{2}}} ‖ f ‖_{{\dot{B}}_{p_{1}, q}^{s + 1}}) . \end{matrix}$ This completes the proof of (ii). □

4. An approximate linear iteration problem and local-in-time solvability

In order to prove the local existence to (1.3), we consider the approximate linear iteration problem $\begin{matrix} (4.1) & \{\begin{array}{l} \partial_{t} ω_{n + 1} + (u_{n} \cdot \nabla) ω_{n + 1} = κ \partial_{1} ρ_{n + 1} in R^{2} \times (0, \infty), \\ \partial_{t} ρ_{n + 1} + (u_{n} \cdot \nabla) ρ_{n + 1} = κ u_{2, n + 1} in R^{2} \times (0, \infty), \\ u_{n + 1} = \nabla^{⊥} {(- Δ)}^{- 1} ω_{n + 1} in R^{2} \times (0, \infty), \\ ω_{n + 1} |_{t = 0} = S_{n + 2} ω_{0}, ρ_{n + 1} |_{t = 0} = S_{n + 2} ρ_{0} in R^{2} . \end{array} \end{matrix}$ From (4.1), we provide uniform estimates for the sequence ${(ω_{n}, ρ_{n})}_{n \in N}$ and then obtain a solution for (1.3).

Uniform estimates. Applying $Δ_{j}$ in (4.1), taking the product with $Δ_{j} ω_{n + 1}$ in ${\dot{H}}^{- 1}$ and the product with $Δ_{j} ρ_{n + 1}$ in $L^{2}$ in the first and second equations, respectively, and using the divergence-free condition $\nabla \cdot Δ_{j} u_{n} = 0$ , we obtain that $\begin{aligned} {⟨ Δ_{j} \partial_{t} ω_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} = {⟨ [u_{n} \cdot \nabla, Δ_{j}] ω_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} + κ {⟨ Δ_{j} \partial_{1} ρ_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}}, \\ {⟨ Δ_{j} \partial_{t} ρ_{n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} = {⟨ [u_{n} \cdot \nabla, Δ_{j}] ρ_{n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} + κ {⟨ Δ_{j} u_{2, n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} . \end{aligned}$ Adding the two above inequalities and using the properties $\begin{matrix} (4.2) & \begin{aligned} {⟨ (u_{n} \cdot \nabla) Λ^{- 1} Δ_{j} ω_{n + 1}, Λ^{- 1} Δ_{j} ω_{n + 1} ⟩}_{L^{2}} = {⟨ (u_{n} \cdot \nabla) Δ_{j} ρ_{n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} = 0, \\ {⟨ Δ_{j} \partial_{1} ρ_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} + {⟨ Δ_{j} u_{2, n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} = 0, \end{aligned} \end{matrix}$ we arrive at $\begin{aligned} [b] \frac{1}{2} \frac{d}{d t} (‖ Δ_{j} ω_{n + 1} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ Δ_{j} ρ_{n + 1} ‖_{L^{2}}^{2}) ⩽ & {‖ [u_{n} \cdot \nabla, Λ^{- 1} Δ_{j}] ω_{n + 1} ‖}_{L^{2}} ‖ Δ_{j} ω_{n + 1} ‖_{{\dot{H}}^{- 1}} \\ + {‖ [u_{n} \cdot \nabla, Δ_{j}] ρ_{n + 1} ‖}_{L^{2}} ‖ Δ_{j} ρ_{n + 1} ‖_{L^{2}} . \end{aligned}$ Integrating over $(0, t)$ , it follows that $\begin{aligned} \frac{1}{2} ({‖ Δ_{j} ω_{n + 1} (t) ‖}_{{\dot{H}}^{- 1}}^{2} + {‖ Δ_{j} ρ_{n + 1} (t) ‖}_{L^{2}}^{2}) \\ ⩽ \frac{1}{2} ({‖ Δ_{j} ω_{n + 1} (0) ‖}_{{\dot{H}}^{- 1}}^{2} + {‖ Δ_{j} ρ_{n + 1} (0) ‖}_{L^{2}}^{2}) \\ + \int_{0}^{t} ({‖ [u_{n} (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] ω_{n + 1} (τ) ‖}_{L^{2}} + {‖ [u_{n} (τ) \cdot \nabla, Δ_{j}] ρ_{n + 1} (τ) ‖}_{L^{2}}) ({‖ Δ_{j} ω_{n + 1} (τ) ‖}_{{\dot{H}}^{- 1}}^{2} \\ + {‖ Δ_{j} ρ_{n + 1} (τ) ‖}_{L^{2}}^{2})^{\frac{1}{2}} d τ . \end{aligned}$ Thus, by Grönwall’s inequality (see Proposition 1.2 in [6, page 24]), we have $\begin{aligned} {‖ Δ_{j} ω_{n + 1} (t) ‖}_{{\dot{H}}^{- 1}} + {‖ Δ_{j} ρ_{n + 1} (t) ‖}_{L^{2}} \\ ⩽ {‖ Δ_{j} ω_{n + 1} (0) ‖}_{{\dot{H}}^{- 1}} + {‖ Δ_{j} ρ_{n + 1} (0) ‖}_{L^{2}} \\ + \int_{0}^{t} {‖ [u_{n} (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] ω_{n + 1} (τ) ‖}_{L^{2}} + {‖ [u_{n} (τ) \cdot \nabla, Δ_{j}] ρ_{n + 1} (τ) ‖}_{L^{2}} d τ . \end{aligned}$ Multiplying by $2^{s j}$ , applying the $l^{q} (Z)$ -norm and Lemma 2.1, it follows that $\begin{aligned} {‖ ω_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1}} + {‖ ρ_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s}} \\ ⩽ C {‖ ω_{n + 1} (0) ‖}_{{\dot{B}}_{2, q}^{s - 1}} + C {‖ ρ_{n + 1} (0) ‖}_{{\dot{B}}_{2, q}^{s}} \\ + C \int_{0}^{t} {(\sum_{j \in Z} 2^{s j q} {‖ [u_{n} (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] ω_{n + 1} (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} \\ + {(\sum_{j \in Z} 2^{s j q} {‖ [u_{n} (τ) \cdot \nabla, Δ_{j}] ρ_{n + 1} (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} d τ . \end{aligned}$ From Remark 3.2 and Lemma 3.1, we have $\begin{aligned} [b] & {(\sum_{j \in Z} 2^{s j q} {‖ [u_{n} \cdot \nabla, Λ^{- 1} Δ_{j}] ω_{n + 1} ‖}_{L^{2}}^{q})}^{\frac{1}{q}} \\ ⩽ C (‖ \nabla u_{n} ‖_{L^{\infty}} ‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1}} + ‖ ω_{n + 1} ‖_{L^{2}} ‖ u_{n} ‖_{{\dot{B}}_{\infty, q}^{s}}) \\ ⩽ C ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}, \\ {(\sum_{j \in Z} 2^{s j q} {‖ [u_{n} \cdot \nabla, Δ_{j}] ρ_{n + 1} ‖}_{L^{2}}^{q})}^{\frac{1}{q}} \\ ⩽ C (‖ \nabla u_{n} ‖_{L^{\infty}} ‖ ρ_{n + 1} ‖_{{\dot{B}}_{2, q}^{s}} + ‖ \nabla ρ_{n + 1} ‖_{L^{\infty}} ‖ u_{n} ‖_{{\dot{B}}_{2, q}^{s}}) \\ ⩽ C ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ρ_{n + 1} ‖_{B_{2, q}^{s}} . \end{aligned}$ Then, $\begin{matrix} (4.3) & \begin{aligned} [b] {‖ ω_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1}} + {‖ ρ_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s}} ⩽ & C {‖ ω_{n + 1} (0) ‖}_{{\dot{B}}_{2, q}^{s - 1}} + C {‖ ρ_{n + 1} (0) ‖}_{{\dot{B}}_{2, q}^{s}} \\ + C \int_{0}^{t} ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} (‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ_{n + 1} ‖_{B_{2, q}^{s}}) d τ . \end{aligned} \end{matrix}$ On the other hand, taking the product with $ω_{n + 1}$ in ${\dot{H}}^{- 1}$ and the product with $ρ_{n + 1}$ in $L^{2}$ in the first and second equations of (4.1), respectively, and using the divergence-free condition $\nabla \cdot Δ_{j} u_{n} = 0$ , we obtain that $\begin{aligned} [b] \frac{1}{2} \frac{d}{d t} (‖ ω_{n + 1} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ ρ_{n + 1} ‖_{L^{2}}^{2}) & ⩽ {‖ Λ^{- 1} \nabla \cdot (u_{n} \otimes ω_{n + 1}) ‖}_{L^{2}} ‖ ω_{n + 1} ‖_{{\dot{H}}^{- 1}} \\ ⩽ C ‖ u_{n} ‖_{L^{2}} ‖ ω_{n + 1} ‖_{L^{\infty}} ‖ ω_{n + 1} ‖_{{\dot{H}}^{- 1}} \\ ⩽ C ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ω_{n + 1} ‖_{{\dot{H}}^{- 1}} . \end{aligned}$ Here, we have used the equality $u_{n} = \nabla^{⊥} {(- Δ)}^{- 1} ω_{n}$ , Lemma 2.1, Remark 2.2, the embedding ${\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1} ↪ B_{2, q}^{s - 1}$ and the property (4.2). Now, we integrate over $(0, t)$ and we use a Grönwall-type inequality to get $\begin{matrix} (4.4) & \begin{aligned} {‖ ω_{n + 1} (t) ‖}_{{\dot{H}}^{- 1}} + {‖ ρ_{n + 1} (t) ‖}_{L^{2}} ⩽ & C {‖ ω_{n + 1} (0) ‖}_{{\dot{H}}^{- 1}} + C {‖ ρ_{n + 1} (0) ‖}_{L^{2}} \\ + C \int_{0}^{t} {‖ ω_{n} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} {‖ ω_{n + 1} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} d τ . \end{aligned} \end{matrix}$ Combining (4.3) and (4.4), we obtain $\begin{aligned} {‖ ω_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + {‖ ρ_{n + 1} (t) ‖}_{B_{2, q}^{s}} \\ ⩽ {‖ ω_{n + 1} (0) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + {‖ ρ_{n + 1} (0) ‖}_{B_{2, q}^{s}} \\ + C \int_{0}^{t} {‖ ω_{n} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ({‖ ω_{n + 1} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + {‖ ρ_{n + 1} (τ) ‖}_{B_{2, q}^{s}}) d τ . \end{aligned}$ Doing $A_{n + 1} (t) : = ‖ ω_{n + 1} (t) ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ_{n + 1} (t) ‖_{B_{2, q}^{s}}$ , we have $\begin{matrix} A_{n + 1} (t) ⩽ A_{n + 1} (0) + C \int_{0}^{t} A_{n} (τ) A_{n + 1} (τ) d τ . \end{matrix}$ Since $‖ ω_{n + 1} (0) ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ⩽ C ‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}$ and $‖ ρ_{n + 1} (0) ‖_{B_{2, q}^{s}} ⩽ C ‖ ρ_{0} ‖_{B_{2, q}^{s}}$ , Grönwall’s inequality yields $\begin{matrix} A_{n + 1} (t) ⩽ C_{0} A_{0} exp (C_{1} \int_{0}^{t} A_{n} (τ) d τ), \end{matrix}$ where $A_{0} = ‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ_{0} ‖_{B_{2, q}^{s}}$ and the constants $C_{0}, C_{1} > 0$ are independent of n. We wish to show that there exist $P > 0$ and $T > 0$ satisfying $\begin{matrix} (4.5) & A_{n + 1} (t) ⩽ P A_{0}, for all t \in (0, T) and n \in N . \end{matrix}$ In fact, for $n = 0$ , note that $\begin{matrix} A_{1} (t) ⩽ C_{0} A_{0} exp (C_{1} t A_{0}) ⩽ P A_{0}, for all t \in (0, T_{1}), \end{matrix}$ where $T_{1} : = \frac{1}{C_{1} A_{0}} log (\frac{P}{C_{0}})$ . Similarly, for $n = 1$ we have $\begin{matrix} A_{2} (t) ⩽ C_{0} A_{0} exp (C_{1} \int_{0}^{t} A_{1} (τ) d τ) ⩽ C_{0} A_{0} exp (C_{1} t P A_{0}) ⩽ P A_{0}, \end{matrix}$ for all $t \in (0, T_{2})$ , where $T_{2} : = \frac{1}{C_{1} P A_{0}} log (\frac{P}{C_{0}})$ . Denoting $T_{3} : = min {T_{1}, T_{2}}$ and making the same calculations for $n = 2$ , we arrive at $\begin{matrix} A_{3} (t) ⩽ P A_{0}, for all t \in (0, T_{3}) . \end{matrix}$ Thus, continuing in the same way, we obtain (4.5) by induction.

Continuity of the sequence. Our intent now is to show that the sequences ${ω_{n}}_{n \in N}$ and ${ρ_{n}}_{n \in N}$ belong to $C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $C ([0, T]; B_{2, q}^{s} (R^{2}))$ , respectively. For that, considering the equality $(f \cdot \nabla) g = \nabla \cdot (f \otimes g)$ for all divergence free vector fields f, Remark 2.2, Lemma 2.3, the embedding ${\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1} ↪ B_{2, q}^{s - 1}$ , the following estimates $\begin{aligned} {‖ (u_{n} \cdot \nabla) ω_{n + 1} ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} ⩽ C ‖ u_{n} \otimes ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap L^{2}} ⩽ C ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}, \\ {‖ (u_{n} \cdot \nabla) ρ_{n + 1} ‖}_{B_{2, q}^{s - 1}} ⩽ C ‖ ω_{n} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} ‖ ρ_{n + 1} ‖_{B_{2, q}^{s}}, \end{aligned}$ estimate (4.5), and the two first equations of (4.1), we have that $\partial_{t} ω_{n + 1} \in L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $\partial_{t} ρ_{n + 1} \in L^{\infty} (0, T; B_{2, q}^{s - 1} (R^{2}))$ . Thus, $\begin{aligned} ω_{n + 1} \in W^{1, \infty} ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \subset C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ_{n + 1} \in W^{1, \infty} ([0, T]; B_{2, q}^{s - 1} (R^{2})) \subset C ([0, T]; B_{2, q}^{s - 1} (R^{2})) . \end{aligned}$ For $k \in N$ and n fixed, we denote $y_{k}^{n} : = {\dot{S}}_{k} ω_{n + 1}$ and $z_{k}^{n} : = S_{k} ρ_{n + 1}$ . We claim that $y_{k}^{n} \to ω_{n + 1}$ in $L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $z_{k}^{n} \to ρ_{n + 1}$ in $L^{\infty} (0, T; B_{2, q}^{s} (R^{2}))$ , respectively, as $k \to \infty$ . Using the Littlewood–Paley operators, we can also write $\begin{matrix} \partial_{t} Δ_{j} ω_{n + 1} + ({\dot{S}}_{j - 2} u_{n} \cdot \nabla) Δ_{j} ω_{n + 1} = κ Δ_{j} \partial_{1} ρ_{n + 1} + ({\dot{S}}_{j - 2} u_{n} \cdot \nabla) Δ_{j} ω_{n + 1} - Δ_{j} (u_{n} \cdot \nabla) ω_{n + 1}, \end{matrix}$ for each $j \in N$ . Since $Δ_{j} ω_{n + 1}$ and $Δ_{j} ρ_{n + 1}$ are absolutely continuous functions from $[0, T]$ to $L^{2} (R^{2})$ and $\nabla \cdot S_{j - 2} u_{n} = 0$ , we obtain that $\begin{aligned} {‖ Δ_{j} ω_{n + 1} (t) ‖}_{{\dot{H}}^{- 1}} ⩽ & {‖ Δ_{j} ω_{n + 1} (0) ‖}_{{\dot{H}}^{- 1}} + | κ | \int_{0}^{t} {‖ Δ_{j} \partial_{1} ρ_{n + 1} (τ) ‖}_{{\dot{H}}^{- 1}} d τ \\ + \int_{0}^{t} {‖ ({\dot{S}}_{j - 2} u_{n} (τ) \cdot \nabla) Δ_{j} ω_{n + 1} (τ) - Δ_{j} (u_{n} (τ) \cdot \nabla) ω_{n + 1} (τ) ‖}_{{\dot{H}}^{- 1}} d τ, \\ {‖ Δ_{j} ρ_{n + 1} (t) ‖}_{L^{2}} ⩽ & {‖ Δ_{j} ρ_{n + 1} (0) ‖}_{L^{2}} + | κ | \int_{0}^{t} {‖ Δ_{j} u_{2, n + 1} (τ) ‖}_{L^{2}} d τ \\ + \int_{0}^{t} {‖ Δ_{j} (u_{n} (τ) \cdot \nabla) ρ_{n + 1} ‖}_{L^{2}} d τ . \end{aligned}$ It follows that $\begin{aligned} {‖ ω_{n + 1} (t) - y_{k}^{n} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1}} \\ ⩽ C {(\sum_{j > k} 2^{j (s - 1) q} {‖ Δ_{j} ω_{n + 1} (t) ‖}_{L^{2}}^{q})}^{1 / q} \\ ⩽ C {(\sum_{j > k} 2^{j (s - 1) q} {‖ Δ_{j} ω_{n + 1} (0) ‖}_{L^{2}}^{q})}^{1 / q} + C | κ | \int_{0}^{t} {(\sum_{j > k} 2^{j (s - 1) q} {‖ Δ_{j} \partial_{1} ρ_{n + 1} (τ) ‖}_{L^{2}}^{q})}^{1 / q} d τ \\ + C \int_{0}^{t} {(\sum_{j > k} 2^{j (s - 1) q} {‖ ({\dot{S}}_{j - 2} u_{n} (τ) \cdot \nabla) Δ_{j} ω_{n + 1} (τ) - Δ_{j} (u_{n} (τ) \cdot \nabla) ω_{n + 1} (τ) ‖}_{L^{2}}^{q})}^{1 / q} d τ, \\ {‖ ω_{n + 1} (t) - y_{k}^{n} (t) ‖}_{{\dot{H}}^{- 1}} ⩽ C {‖ Δ_{j} ω_{n + 1} (t) ‖}_{{\dot{H}}^{- 1}} \\ ⩽ C {‖ Δ_{j} ω_{n + 1} (0) ‖}_{{\dot{H}}^{- 1}} + C | κ | \int_{0}^{t} {‖ Δ_{j} \partial_{1} ρ_{n + 1} (τ) ‖}_{{\dot{H}}^{- 1}} d τ \\ + C \int_{0}^{t} {‖ Δ_{j} (u_{n} (τ) \cdot \nabla) ω_{n + 1} (τ) ‖}_{{\dot{H}}^{- 1}} d τ, \\ {‖ ρ_{n + 1} (t) - z_{k}^{n} (t) ‖}_{B_{2, q}^{s}} \\ ⩽ C {(\sum_{j > k} 2^{j s q} {‖ Δ_{j} ρ_{n + 1} (t) ‖}_{L^{2}}^{q})}^{1 / q} \\ ⩽ C {(\sum_{j > k} 2^{j s q} {‖ Δ_{j} ρ_{n + 1} (0) ‖}_{L^{2}}^{q})}^{1 / q} + C | κ | \int_{0}^{t} {(\sum_{j > k} 2^{j s q} {‖ Δ_{j} u_{2, n + 1} (τ) ‖}_{L^{2}}^{q})}^{1 / q} d τ \\ + C \int_{0}^{t} {(\sum_{j > k} 2^{j s q} {‖ Δ_{j} (u_{n} (τ) \cdot \nabla) ρ_{n + 1} (τ) ‖}_{L^{2}}^{q})}^{1 / q} d τ . \end{aligned}$ As $ω_{n + 1} (t) \in {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})$ and $ρ_{n + 1} (t) \in B_{2, q}^{s} (R^{2})$ , by Lemmas 2.3 and 3.3, the R.H.S. of the three above estimates go to zero as $k \to \infty$ , and then the desired claim follows. Moreover, we get $\begin{aligned} (4.6) & \begin{aligned} {‖ y_{k}^{n} (t^{'}) - y_{k}^{n} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} \\ = {‖ {\dot{S}}_{k} (ω_{n + 1} (t^{'}) - ω_{n + 1} (t)) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} \\ ⩽ {(\sum_{j ⩽ k + 1} 2^{(s - 1) j q} {‖ Δ_{j} (ω_{n + 1} (t^{'}) - ω_{n + 1} (t)) ‖}_{L^{2}}^{q})}^{1 / q} \\ + \sum_{j ⩽ k + 1} {‖ Δ_{j} (ω_{n + 1} (t^{'}) - ω_{n + 1} (t)) ‖}_{{\dot{H}}^{- 1}} \\ ⩽ C 2^{k + 1} {‖ ω_{n + 1} (t^{'}) - ω_{n + 1} (t) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}}, \\ {‖ z_{k}^{n} (t^{'}) - z_{k}^{n} (t) ‖}_{B_{2, q}^{s}} \\ = {‖ S_{k} (ρ_{n + 1} (t^{'}) - ρ_{n + 1} (t)) ‖}_{B_{2, q}^{s}} ⩽ {(\sum_{j = - 1}^{k + 1} 2^{s j q} {‖ Δ_{j} (ρ_{n + 1} (t^{'}) - ρ_{n + 1} (t)) ‖}_{L^{2}}^{q})}^{1 / q} \\ ⩽ C 2^{k + 1} {‖ ρ_{n + 1} (t^{'}) - ρ_{n + 1} (t) ‖}_{B_{2, q}^{s - 1}} . \end{aligned} \end{aligned}$ Thus, ${y_{k}^{n}}_{k \in N} \subset C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and ${z_{k}^{n}}_{k \in N} \subset C ([0, T]; B_{2, q}^{s} (R^{2}))$ . Then $\begin{matrix} (4.7) & {ω_{n}}_{n \in N} \subset C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) and {ρ_{n}}_{n \in N} \subset C ([0, T]; B_{2, q}^{s} (R^{2})) . \end{matrix}$

Convergence and local solution. Now, we show that ${ω_{n}}_{n \in N}$ and ${ρ_{n}}_{n \in N}$ converge in $C ([0, T]; {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $C ([0, T]; B_{2, q}^{s - 1} (R^{2}))$ , for some $T > 0$ , respectively. For that, we consider the following system $\begin{matrix} (4.8) & \{\begin{array}{l} \partial_{t} \overline{ω_{n + 1}} + (\overline{u_{n}} \cdot \nabla) ω_{n + 1} + (u_{n - 1} \cdot \nabla) \overline{ω_{n + 1}} = κ \partial_{1} \overline{ρ_{n + 1}} in R^{2} \times (0, \infty), \\ \partial_{t} \overline{ρ_{n + 1}} + (\overline{u_{n}} \cdot \nabla) ρ_{n + 1} + (u_{n - 1} \cdot \nabla) \overline{ρ_{n + 1}} = κ \overline{u_{2, n + 1}} in R^{2} \times (0, \infty), \\ \overline{u_{n + 1}} = \nabla^{⊥} {(- Δ)}^{- 1} \overline{ω_{n + 1}} in R^{2} \times (0, \infty), \\ \overline{ω_{n + 1}} ∣_{t = 0} = Δ_{n + 1} ω_{0}, \overline{ρ_{n + 1}} ∣_{t = 0} = Δ_{n + 1} ρ_{0} in R^{2}, \end{array} \end{matrix}$ where $\overline{ω_{n + 1}} : = ω_{n + 1} - ω_{n}$ and $\overline{ρ_{n + 1}} : = ρ_{n + 1} - ρ_{n}$ .

We take the ${\dot{H}}^{- 1}$ -product with $\overline{ω_{n + 1}}$ and the $L^{2}$ -product with $\overline{ρ_{n + 1}}$ in (4.8) to obtain $\begin{aligned} {⟨ \partial_{t} \overline{ω_{n + 1}}, \overline{ω_{n + 1}} ⟩}_{{\dot{H}}^{- 1}} + {⟨ (\overline{u_{n}} \cdot \nabla) ω_{n + 1}, \overline{ω_{n + 1}} ⟩}_{{\dot{H}}^{- 1}} + {⟨ (u_{n - 1} \cdot \nabla) \overline{ω_{n + 1}}, \overline{ω_{n + 1}} ⟩}_{{\dot{H}}^{- 1}} = {⟨ κ \partial_{1} \overline{ρ_{n + 1}}, \overline{ω_{n + 1}} ⟩}_{{\dot{H}}^{- 1}}, \\ {⟨ \partial_{t} \overline{ρ_{n + 1}}, \overline{ρ_{n + 1}} ⟩}_{L^{2}} + {⟨ (\overline{u_{n}} \cdot \nabla) ρ_{n + 1}, \overline{ρ_{n + 1}} ⟩}_{L^{2}} = {⟨ κ \overline{u_{2, n + 1}}, \overline{ρ_{n + 1}} ⟩}_{L^{2}} . \end{aligned}$ Next, adding the above two equalities, using the property ${⟨ \partial_{1} \overline{ρ_{n + 1}}, \overline{ω_{n + 1}} ⟩}_{{\dot{H}}^{- 1}} + {⟨ \overline{u_{2, n + 1}}, \overline{ρ_{n + 1}} ⟩}_{L^{2}} = 0$ , the equality $(f \cdot \nabla) g = \nabla \cdot (f \otimes g)$ for all divergence free vector fields f, Cauchy–Schwarz and Hölder inequalities, Lemma 2.1, Remark 2.2, the embedding ${\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1} ↪ L^{2}$ and the estimate (4.5) to arrive at $\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ \overline{ρ_{n + 1}} ‖_{L^{2}}^{2}) \\ ⩽ {‖ (\overline{u_{n}} \cdot \nabla) ω_{n + 1} ‖}_{{\dot{H}}^{- 1}} ‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}} + {‖ (u_{n - 1} \cdot \nabla) \overline{ω_{n + 1}} ‖}_{{\dot{H}}^{- 1}} ‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}} \\ + {‖ (\overline{u_{n}} \cdot \nabla) ρ_{n + 1} ‖}_{L^{2}} ‖ \overline{ρ_{n + 1}} ‖_{L^{2}} \\ ⩽ C ‖ \overline{u_{n}} ‖_{L^{2}} ‖ ω_{n + 1} ‖_{L^{\infty}} ‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}} + C ‖ u_{n - 1} ‖_{L^{\infty}} ‖ \overline{ω_{n + 1}} ‖_{L^{2}} ‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}} \\ + C ‖ \overline{u_{n}} ‖_{L^{2}} ‖ ρ_{n + 1} ‖_{L^{\infty}} ‖ \overline{ρ_{n + 1}} ‖_{L^{2}} \\ ⩽ C P A_{0} (‖ \overline{ω_{n}} ‖_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + ‖ \overline{ω_{n + 1}} ‖_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}}) {(‖ \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ \overline{ρ_{n + 1}} ‖_{L^{2}}^{2})}^{\frac{1}{2}} . \end{aligned}$ Integrating over $(0, t)$ and using a Grönwall-type inequality, we have that $\begin{matrix} (4.9) & \begin{aligned} {‖ \overline{ω_{n + 1}} (t) ‖}_{{\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (t) ‖}_{L^{2}} ⩽ & {‖ \overline{ω_{n + 1}} (0) ‖}_{{\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (0) ‖}_{L^{2}} \\ + C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ω_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} d τ . \end{aligned} \end{matrix}$ On the other hand, we applies $Δ_{j}$ to the first and second equations in (4.8), and afterwards take the ${\dot{H}}^{- 1}$ -product with $Δ_{j} \overline{ω_{n + 1}}$ and the $L^{2}$ -product with $Δ_{j} \overline{ρ_{n + 1}}$ in order to get $\begin{aligned} {⟨ \partial_{t} Δ_{j} ω_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} \\ = - {⟨ Δ_{j} (\overline{u_{n}} \cdot \nabla) ω_{n + 1}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} + {⟨ [u_{n - 1} \cdot \nabla, Λ^{- 1} Δ_{j}] \overline{ω_{n + 1}}, Λ^{- 1} Δ_{j} ω_{n + 1} ⟩}_{L^{2}} \\ + κ {⟨ Δ_{j} \partial_{1} \overline{ρ_{n + 1}}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}}, \\ {⟨ \partial_{t} Δ_{j} ρ_{n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} \\ = - {⟨ Δ_{j} (\overline{u_{n}} \cdot \nabla) ρ_{n + 1}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} + {⟨ [u_{n - 1} \cdot \nabla, Δ_{j}] \overline{ρ_{n + 1}}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} \\ + κ {⟨ Δ_{j} \overline{u_{2, n + 1}}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} . \end{aligned}$ Adding the two previous equalities, employing the property $\begin{matrix} {⟨ Δ_{j} \partial_{1} \overline{ρ_{n + 1}}, Δ_{j} ω_{n + 1} ⟩}_{{\dot{H}}^{- 1}} + {⟨ Δ_{j} \overline{u_{2, n + 1}}, Δ_{j} ρ_{n + 1} ⟩}_{L^{2}} = 0, \end{matrix}$ and using Cauchy–Schwarz and Hölder inequalities, we obtain that $\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ Δ_{j} \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ Δ_{j} \overline{ρ_{n + 1}} ‖_{L^{2}}^{2}) \\ ⩽ ({‖ Δ_{j} (\overline{u_{n}} \cdot \nabla) ω_{n + 1} ‖}_{{\dot{H}}^{- 1}} + {‖ [u_{n - 1} \cdot \nabla, Λ^{- 1} Δ_{j}] \overline{ω_{n + 1}} ‖}_{L^{2}} \\ + {‖ Δ_{j} (\overline{u_{n}} \cdot \nabla) ρ_{n + 1} ‖}_{L^{2}} + {‖ [u_{n - 1} \cdot \nabla, Δ_{j}] \overline{ρ_{n + 1}} ‖}_{L^{2}}) {(‖ Δ_{j} \overline{ω_{n + 1}} ‖_{{\dot{H}}^{- 1}}^{2} + ‖ Δ_{j} \overline{ρ_{n + 1}} ‖_{L^{2}}^{2})}^{\frac{1}{2}} . \end{aligned}$ Integrating over $(0, t)$ and using a Grönwall-type inequality, it follows that $\begin{aligned} {‖ Δ_{j} \overline{ω_{n + 1}} (t) ‖}_{{\dot{H}}^{- 1}} + {‖ Δ_{j} \overline{ρ_{n + 1}} (t) ‖}_{L^{2}} \\ ⩽ {‖ Δ_{j} \overline{ω_{n + 1}} (0) ‖}_{{\dot{H}}^{- 1}} + ‖ Δ_{j} \overline{ρ_{n + 1}} (0) ‖_{L^{2}} \\ + \int_{0}^{t} ‖ Δ_{j} (\overline{u_{n}} (τ) \cdot \nabla) ω_{n + 1} (τ) ‖_{{\dot{H}}^{- 1}} d τ + \int_{0}^{t} ‖ [u_{n - 1} (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] \overline{ω_{n + 1}} (τ) ‖_{L^{2}} d τ \\ + \int_{0}^{t} ‖ Δ_{j} (\overline{u_{n}} (τ) \cdot \nabla) ρ_{n + 1} (τ) ‖_{L^{2}} d τ + \int_{0}^{t} ‖ [u_{n - 1} (τ) \cdot \nabla, Δ_{j}] \overline{ρ_{n + 1}} (τ) ‖_{L^{2}} d τ . \end{aligned}$ Taking into account Lemma 2.1, multiplying by $2^{(s - 1) j}$ and taking the $l^{q} (Z)$ -norm, we have that $\begin{matrix} (4.10) & \begin{aligned} {‖ \overline{ω_{n + 1}} (t) ‖}_{{\dot{B}}_{2, q}^{s - 2}} + {‖ \overline{ρ_{n + 1}} (t) ‖}_{{\dot{B}}_{2, q}^{s - 1}} \\ ⩽ C {‖ \overline{ω_{n + 1}} (0) ‖}_{{\dot{B}}_{2, q}^{s - 2}} + C {‖ \overline{ρ_{n + 1}} (0) ‖}_{{\dot{B}}_{2, q}^{s - 1}} \\ + C \int_{0}^{t} {‖ (\overline{u_{n}} (τ) \cdot \nabla) ω_{n + 1} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2}} d τ \\ + C \int_{0}^{t} {(\sum_{j \in Z} 2^{(s - 1) j q} {‖ [u_{n - 1} (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] \overline{ω_{n + 1}} (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} d τ \\ + C \int_{0}^{t} {‖ (\overline{u_{n}} (τ) \cdot \nabla) ρ_{n + 1} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1}} d τ \\ + C \int_{0}^{t} {(\sum_{j \in Z} 2^{(s - 1) j q} {‖ [u_{n - 1} (τ) \cdot \nabla, Δ_{j}] \overline{ρ_{n + 1}} (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} d τ \\ : = I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} . \end{aligned} \end{matrix}$ First, thanks to the embedding ${\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1} ↪ L^{2}$ and the inequality $\begin{matrix} (4.11) & {‖ \overline{ω_{n + 1}} (0) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (0) ‖}_{B_{2, q}^{s - 1}} ⩽ C 2^{- n} (‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ_{0} ‖_{B_{2, q}^{s - 1}}) = C A_{0} 2^{- n}, \end{matrix}$ we can estimate $I_{1}$ and $I_{2}$ . For $I_{3}$ and $I_{5}$ , let us first note that by the equality $(f \cdot \nabla) g = \nabla \cdot (f \otimes g)$ for all divergence free vector fields f, Hölder inequality, Remark 2.2, the equality $u_{n} = \nabla^{⊥} {(- Δ)}^{- 1} ω_{n}$ , the embedding ${\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1} ↪ B_{2, q}^{s - 1}$ and estimate (4.5), we have that $\begin{aligned} {‖ \nabla \cdot (\overline{u_{n}} \otimes ω_{n + 1}) ‖}_{{\dot{B}}_{2, q}^{s - 2}} \\ ⩽ C ‖ \overline{u_{n}} \otimes ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1}} ⩽ C (‖ \overline{u_{n}} ‖_{L^{\infty}} ‖ ω_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1}} + ‖ ω_{n + 1} ‖_{L^{\infty}} ‖ \overline{u_{n}} ‖_{{\dot{B}}_{2, q}^{s - 1}}) \\ ⩽ 2 C P A_{0} ‖ \overline{ω_{n}} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}}, \\ {‖ (\overline{u_{n}} \cdot \nabla) ρ_{n + 1} ‖}_{{\dot{B}}_{2, q}^{s - 1}} \\ ⩽ C (‖ \overline{u_{n}} ‖_{L^{\infty}} ‖ \nabla ρ_{n + 1} ‖_{{\dot{B}}_{2, q}^{s - 1}} + ‖ \nabla ρ_{n + 1} ‖_{L^{\infty}} ‖ \overline{u_{n}} ‖_{{\dot{B}}_{2, q}^{s - 1}}) \\ ⩽ 2 C P A_{0} ‖ \overline{ω_{n}} ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} . \end{aligned}$ Then, $\begin{matrix} (4.12) & I_{3} + I_{5} ⩽ 4 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} d τ . \end{matrix}$ In view of Remark 3.2, Lemma 3.1 and using the same arguments to estimate $I_{3}$ and $I_{5}$ , we see that $\begin{aligned} I_{4} & ⩽ C \int_{0}^{t} {‖ \nabla u_{n - 1} (τ) ‖}_{L^{\infty}} {‖ \overline{ω_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2}} + {‖ \overline{ω_{n + 1}} (τ) ‖}_{L^{2}} {‖ u_{n - 1} (τ) ‖}_{{\dot{B}}_{\infty, q}^{s - 1}} d τ \\ ⩽ 2 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} d τ, \\ I_{6} & ⩽ C \int_{0}^{t} {‖ \nabla u_{n - 1} (τ) ‖}_{L^{\infty}} {‖ \overline{ρ_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1}} + {‖ \overline{ρ_{n + 1}} (τ) ‖}_{L^{\infty}} {‖ u_{n - 1} (τ) ‖}_{{\dot{B}}_{2, q}^{s}} d τ \\ ⩽ 2 C P A_{0} \int_{0}^{t} {‖ \overline{ρ_{n + 1}} (τ) ‖}_{B_{2, q}^{s - 1}} d τ . \end{aligned}$ Thus, $\begin{matrix} (4.13) & I_{4} + I_{6} ⩽ 2 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (τ) ‖}_{B_{2, q}^{s - 1}} d τ . \end{matrix}$ Combining (4.9) and (4.10), and using (4.11), (4.12) and (4.13), it holds that $\begin{matrix} (4.14) & \begin{aligned} [b] & {‖ \overline{ω_{n + 1}} (t) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (t) ‖}_{B_{2, q}^{s - 1}} \\ ⩽ C A_{0} 2^{- n} + 5 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} d τ \\ + 3 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n + 1}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (τ) ‖}_{B_{2, q}^{s - 1}} d τ . \end{aligned} \end{matrix}$ By Grönwall inequality, it follows that $\begin{matrix} {‖ \overline{ω_{n + 1}} (t) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \overline{ρ_{n + 1}} (t) ‖}_{B_{2, q}^{s - 1}} ⩽ (C A_{0} 2^{- n} + 5 C P A_{0} \int_{0}^{t} {‖ \overline{ω_{n}} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} d τ) e^{3 C P A_{0} t} . \end{matrix}$ Denoting $\overline{A_{n + 1}} (t) : = ‖ \overline{ω_{n + 1}} (t) ‖_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + ‖ \overline{ρ_{n + 1}} (t) ‖_{B_{2, q}^{s - 1}}$ , $f (t) : = C A_{0} e^{3 C P A_{0} t}$ and $g (t) : = 5 C P A_{0} e^{3 C P A_{0} t}$ , we observe that $\begin{matrix} \overline{A_{n + 1}} (t) ⩽ 2^{- n} f (t) + g (t) \int_{0}^{t} \overline{A_{n}} (τ) d τ . \end{matrix}$ Following the iterative process, we arrive at $\begin{aligned} \overline{A_{n + 1}} (t) & ⩽ 2^{- n} f (t) + g (t) \int_{0}^{t} \overline{A_{n}} (τ) d τ \\ ⩽ 2^{- n} f (t) + g (t) \int_{0}^{t} (2^{- (n - 1)} f (τ) + g (τ) \int_{0}^{τ} \overline{A_{n - 1}} (τ^{'}) d τ^{'}) d τ \\ ⩽ 2^{- n} f (t) + g (t) t (2^{- (n - 1)} f (t) + g (t) \int_{0}^{t} \overline{A_{n - 1}} (τ) d τ) \\ = 2^{- n} f (t) (1 + 2 g (t) t) + g {(t)}^{2} t \int_{0}^{t} \overline{A_{n - 1}} (τ) d τ \\ ⩽ 2^{- n} f (t) (1 + 2 g (t) t + {(2 g (t) t)}^{2}) + g {(t)}^{3} t^{2} \int_{0}^{t} \overline{A_{n - 2}} (τ) d τ \\ ⋮ \\ ⩽ 2^{- n} f (t) \sum_{i = 0}^{n - 1} {(2 g (t) t)}^{i} + g {(t)}^{n} t^{n - 1} \int_{0}^{t} \overline{A_{1}} (τ) d τ \\ ⩽ 2^{- n} f (t) \sum_{i = 0}^{n - 1} {(2 g (t) t)}^{i} + 2 P A_{0} g {(t)}^{n} t^{n} . \end{aligned}$ Let $T^{'} ⩽ T$ be such that $g (t) t ⩽ \frac{1}{4}$ for all $t \in (0, T^{'})$ . Then, it is fulfilled that $\sum_{i = 0}^{n - 1} {(2 g (t) t)}^{i} ⩽ \sum_{i = 0}^{n - 1} \frac{1}{2^{i}} < \infty$ and ${(g (t) t)}^{n} ⩽ 4^{- n}$ for all $n \in N$ and $t \in (0, T^{'})$ . Moreover, there exists $C_{3} > 0$ such that $f (t) ⩽ C_{3}$ for all $t \in (0, T^{'})$ . Therefore, for some constant $C_{4} > 0$ , we have $\begin{matrix} \overline{A_{n + 1}} (t) ⩽ C_{3} 2^{- n} + 2 P A_{0} 4^{- n} ⩽ C_{4} 2^{- n} . \end{matrix}$ Let $n, m \in N$ be such that $n > m$ , then there exists $C_{5} > 0$ satisfying $\begin{matrix} {‖ (ω_{n} - ω_{m}) (t) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ (ρ_{n} - ρ_{m}) (t) ‖}_{B_{2, q}^{s - 1}} ⩽ \sum_{i = m}^{n - 1} \overline{A_{i + 1}} (t) ⩽ C_{5} \sum_{i = m}^{n - 1} 2^{- i} . \end{matrix}$ This implies that ${ω_{n}}_{n \in N}$ and ${ρ_{n}}_{n \in N}$ are Cauchy in the spaces $L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1})$ and $L^{\infty} (0, T^{'}; B_{2, q}^{s - 1})$ , respectively. Therefore, there are $ω \in L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1})$ and $ρ \in L^{\infty} (0, T^{'}; B_{2, q}^{s - 1})$ such that if $n \to \infty$ , then $\begin{matrix} ω_{n} ⟶ ω in L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}) and ρ_{n} ⟶ ρ in L^{\infty} (0, T^{'}; B_{2, q}^{s - 1}) . \end{matrix}$ Furthermore, since the sequences ${ω_{n}}_{n \in N}$ and ${ρ_{n}}_{n \in N}$ belong to $C ([0, T^{'}]; {\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1})$ and $C ([0, T^{'}]; B_{2, q}^{s})$ , respectively, we have that $ω \in C ([0, T^{'}]; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1})$ and $ρ \in C ([0, T^{'}]; B_{2, q}^{s - 1})$ .

On the other hand, as ${ω_{n}}_{n \in N}$ and ${ρ_{n}}_{n \in N}$ are bounded in $L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1})$ and $L^{\infty} (0, T^{'}; B_{2, q}^{s})$ , respectively, we can extract subsequences ${ω_{n_{j}}}_{j \in N}$ and ${ρ_{n_{j}}}_{j \in N}$ such that $ω_{n_{j}} \overset{*}{⇀} ω$ and $ρ_{n_{j}} \overset{*}{⇀} ρ$ in $L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1})$ and $L^{\infty} (0, T^{'}; B_{2, q}^{s})$ , respectively. Thus, $\begin{matrix} (4.15) & \begin{aligned} ω \in C ([0, T^{'}]; {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ \in C ([0, T^{'}]; B_{2, q}^{s - 1} (R^{2})) \cap L^{\infty} (0, T^{'}; B_{2, q}^{s} (R^{2})), \end{aligned} \end{matrix}$ with $‖ ω ‖_{L^{\infty} (0, T^{'}; {\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1})}, ‖ ρ ‖_{L^{\infty} (0, T^{'}; B_{2, q}^{s})} ⩽ P A_{0}$ , where P and $A_{0}$ are as in (4.5).

Now, with the above convergence in hand, we sketch the convergence of the nonlinearity of the second equation and the coupling term in (4.1). The others follow similarly and are left to the reader. By Hölder’s inequality, Remark 2.2, the identity $u_{n} - u = \nabla^{⊥} {(- Δ)}^{- 1} (ω_{n} - ω)$ , the embedding $B_{2, q}^{s - 1} (R^{2}) ↪ B_{2, q}^{m + 1} (R^{2})$ and $B_{2, q}^{s - 1} (R^{2}) ↪ B_{\infty, q}^{m} (R^{2})$ with $0 ⩽ m ⩽ s - 2$ , Lemma 2.1 and (4.5), it follows that $\begin{aligned} \int_{0}^{t} {‖ (u_{n} (τ) \cdot \nabla) ρ_{n + 1} (τ) - (u (τ) \cdot \nabla) ρ (τ) ‖}_{B_{2, q}^{m + 1}} d τ \\ ⩽ \int_{0}^{t} {‖ ((u_{n} - u) (τ) \cdot \nabla) ρ_{n + 1} (τ) ‖}_{B_{2, q}^{m + 1}} + {‖ (u (τ) \cdot \nabla) (ρ_{n + 1} - ρ) (τ) ‖}_{B_{2, q}^{m + 1}} d τ \\ ⩽ C P A_{0} \int_{0}^{t} ‖ ω_{n} - ω ‖_{B_{2, q}^{s - 2}} + ‖ ρ_{n + 1} - ρ ‖_{B_{2, q}^{m + 2}} + ‖ ρ_{n + 1} - ρ ‖_{B_{2, 2}^{1}} d τ \\ ⩽ 2 C P A_{0} T (‖ ω_{n} - ω ‖_{L^{\infty} (0, T; B_{2, q}^{s - 2})} + ‖ ρ_{n + 1} - ρ ‖_{L^{\infty} (0, T; B_{2, q}^{m + 2})} + ‖ ρ_{n + 1} - ρ ‖_{L^{\infty} (0, T; B_{2, 2}^{1})}) \\ \to 0 as n \to \infty, \end{aligned}$ which implies $\begin{matrix} \int_{0}^{t} (u_{n} (τ) \cdot \nabla) ρ_{n + 1} (τ) d τ ⟶ \int_{0}^{t} (u (τ) \cdot \nabla) ρ (τ) d τ in B_{2, q}^{m + 1} (R^{2}), as n \to \infty . \end{matrix}$ Moreover, since $ρ_{n + 1} ⟶ ρ$ in $L^{\infty} (0, T; B_{2, q}^{m + 1} (R^{2}))$ , we have $\begin{aligned} {‖ \int_{0}^{t} κ (\partial_{1} ρ_{n + 1} (τ) - \partial_{1} ρ (τ)) d τ ‖}_{L^{\infty} (0, T; {\dot{B}}_{2, q}^{m} \cap {\dot{H}}^{- 1})} ⩽ C | κ | T ‖ ρ_{n + 1} - ρ ‖_{L^{\infty} (0, T; B_{2, q}^{m + 1})} ⟶ 0, \\ as n \to \infty, \end{aligned}$ and then $\begin{matrix} \int_{0}^{t} κ \partial_{1} ρ_{n + 1} (τ) d τ ⟶ \int_{0}^{t} κ \partial_{1} ρ (τ) d τ in {\dot{B}}_{2, q}^{m} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}) . \end{matrix}$ So, we can pass the limit in the integral formulation of approximate system (4.1) and obtain that $(ω, ρ)$ is an integral solution for (1.3) in $L^{\infty} (0, T; {\dot{B}}_{2, q}^{m} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \times L^{\infty} (0, T; B_{2, q}^{m + 1} (R^{2}))$ , namely $\begin{matrix} (4.16) & \begin{aligned} ω (t) - ω_{0} = \int_{0}^{t} (u (τ) \cdot \nabla) ω (τ) + κ \partial_{1} ρ (τ) d τ, \\ ρ (t) - ρ_{0} = \int_{0}^{t} (u (τ) \cdot \nabla) ρ (τ) + κ u_{2} (τ) d τ . \end{aligned} \end{matrix}$ Considering $y_{k} = {\dot{S}}_{k} ω$ , $z_{k} = S_{k} ρ$ , using (4.15), and proceeding as in the proof of (4.6) and (4.7), it follows that ${y_{k}}_{k \in N} \subset C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ , ${z_{k}}_{k \in N} \subset C ([0, T]; B_{2, q}^{s} (R^{2}))$ , $y_{k} \to ω$ in $L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $z_{k} \to ρ$ in $L^{\infty} (0, T; B_{2, q}^{s} (R^{2}))$ , as $k \to \infty$ . Then, $ω \in$ $C ([0, T]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ , $ρ \in C ([0, T]; B_{2, q}^{s} (R^{2}))$ , and the integral system (4.16) is indeed verified in $({\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \times B_{2, q}^{s - 1} (R^{2})$ , and consequently $ω \in C^{1} ([0, T]; {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2}))$ and $ρ \in C^{1} ([0, T]; B_{2, q}^{s - 1} (R^{2}))$ , as desired.

Uniqueness. In this part, we suppose that system (1.3) possesses two solutions $(ω^{1}, ρ^{1})$ and $(ω^{2}, ρ^{2})$ with the same initial data $(ω_{0}, ρ_{0})$ and we show that $ω^{1} = ω^{2}$ and $ρ^{1} = ρ^{2}$ . For that, we set $\tilde{ω} : = ω^{2} - ω^{1}$ and $\tilde{ρ} : = ρ^{2} - ρ^{1}$ , respectively. Then, $(\tilde{ω}, \tilde{ρ})$ satisfy the following system $\begin{matrix} (4.17) & \{\begin{array}{l} \partial_{t} \tilde{ω} + (\tilde{u} \cdot \nabla) ω^{2} + (u^{1} \cdot \nabla) \tilde{ω} = κ \partial_{1} \tilde{ρ}, \\ \partial_{t} \tilde{ρ} + (\tilde{u} \cdot \nabla) ρ^{2} + (u^{1} \cdot \nabla) \tilde{ρ} = κ \tilde{u_{2}}, \\ \tilde{u} = \nabla^{⊥} {(- Δ)}^{- 1} \tilde{ω}, \\ \tilde{ω} ∣_{t = 0} = 0, \tilde{ρ} ∣_{t = 0} = 0 . \end{array} \end{matrix}$ Considering the spaces $L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1})$ and $L^{\infty} (0, T; B_{2, q}^{s - 1})$ in (4.17) and employing the argument used in (4.8) to estimate $\tilde{ω}$ and $\tilde{ρ}$ , we obtain an inequality similar to inequality (4.14) as follows $\begin{matrix} {‖ \tilde{ω} (t) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \tilde{ρ} (t) ‖}_{B_{2, q}^{s - 1}} ⩽ C P A_{0} \int_{0}^{t} {‖ \tilde{ω} (τ) ‖}_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + {‖ \tilde{ρ} (τ) ‖}_{B_{2, q}^{s - 1}} d τ . \end{matrix}$ Therefore, by Grönwall’s inequality we obtain $‖ \tilde{ω} (t) ‖_{{\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1}} + ‖ \tilde{ρ} (t) ‖_{B_{2, q}^{s - 1}} ⩽ 0$ for all $t \in [0, T]$ . Thus, $‖ \tilde{ω} ‖_{L^{\infty} (0, T; {\dot{B}}_{2, q}^{s - 2} \cap {\dot{H}}^{- 1})} = ‖ \tilde{ρ} ‖_{L^{\infty} (0, T; B_{2, q}^{s - 1})} = 0$ , implying that $ω^{1} = ω^{2}$ and $ρ^{1} = ρ^{2}$ , which shows the uniqueness of solution to (1.3).

5. Long-time solvability

In this section we prove the long-time solvability of (1.3) for large values of $| κ |$ . We start with a proposition containing a blow-up criterion.

Proposition 5.1.

Let s and q be such that $s > 2$ with $1 ⩽ q ⩽ \infty$ or $s = 2$ with $q = 1$ . For $ω_{0} \in {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})$ and $ρ_{0} \in B_{2, q}^{s} (R^{2})$ , consider $(ω, ρ)$ the corresponding solution of ( 1.3 ) satisfying $\begin{aligned} ω \in C ([0, T); {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T); {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ \in C ([0, T); B_{2, q}^{s} (R^{2})) \cap C^{1} ([0, T); B_{2, q}^{s - 1} (R^{2})), \end{aligned}$ where $T > 0$ is an existence time. If $\int_{0}^{T} ‖ \nabla ρ (t) ‖_{L^{\infty}} + ‖ \nabla u (t) ‖_{L^{\infty}} d t < \infty$ , then there exists $T^{'} > T$ such that $(ω, ρ)$ can be extended to $[0, T^{'})$ with $\begin{aligned} ω \in C ([0, T^{'}); {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T^{'}); {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ \in C ([0, T^{'}); B_{2, q}^{s} (R^{2})) \cap C^{1} ([0, T^{'}); B_{2, q}^{s - 1} (R^{2})) . \end{aligned}$

Proof.

By standard procedures used to estimate ω and ρ in Besov norms, we obtain the following estimates: $\begin{matrix} (5.1) & \begin{aligned} {‖ ω (t) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + {‖ ρ (t) ‖}_{B_{2, q}^{s}} \\ ⩽ C {‖ ω (0) ‖}_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + C {‖ ρ (0) ‖}_{B_{2, q}^{s}} \\ + \int_{0}^{t} {(\sum_{j \in Z} 2^{s j q} {‖ [u (τ) \cdot \nabla, Λ^{- 1} Δ_{j}] ω (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} d τ \\ + \int_{0}^{t} {(\sum_{j \in Z} 2^{s j q} {‖ [u (τ) \cdot \nabla, Δ_{j}] ρ (τ) ‖}_{L^{2}}^{q})}^{\frac{1}{q}} d τ . \end{aligned} \end{matrix}$ If we denote by I and J the penultimate and last term of the previous inequality, respectively, by Lemma 3.1, we have that $\begin{aligned} I ⩽ C (‖ \nabla u ‖_{L^{\infty}} ‖ ω ‖_{{\dot{B}}_{2, q}^{s - 1}} + ‖ ω ‖_{L^{\infty}} ‖ u ‖_{{\dot{B}}_{2, q}^{s}} ⩽ C ‖ ω ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} (‖ \nabla u ‖_{L^{\infty}} + ‖ ω ‖_{L^{\infty}}), \\ J ⩽ C (‖ \nabla u ‖_{L^{\infty}} ‖ ρ ‖_{{\dot{B}}_{2, q}^{s}} + ‖ \nabla ρ ‖_{L^{\infty}} ‖ u ‖_{{\dot{B}}_{2, q}^{s}} ⩽ C (‖ ω ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ ‖_{B_{2, q}^{s}}) (‖ \nabla u ‖_{L^{\infty}} + ‖ ω ‖_{L^{\infty}}) . \end{aligned}$ Denoting $z_{s, q} (t) : = ‖ ω (t) ‖_{{\dot{B}}_{2, q}^{s - 1} \cap {\dot{H}}^{- 1}} + ‖ ρ (t) ‖_{B_{2, q}^{s}}$ and employing the above inequalities in (5.1), it holds $\begin{matrix} z_{s, q} (t) ⩽ C z_{s, q} (0) + C \int_{0}^{t} z_{s, q} (τ) ({‖ ω (τ) ‖}_{L^{\infty}} + {‖ \nabla ρ (τ) ‖}_{L^{\infty}} + {‖ \nabla u (τ) ‖}_{L^{\infty}}) d τ . \end{matrix}$ Then, by using the inequality $‖ ω ‖_{L^{\infty}} ⩽ 2 ‖ \nabla u ‖_{L^{\infty}}$ and the Grönwall-type inequality, it follows that there exists a constant $C_{6} > 0$ such that $\begin{matrix} (5.2) & z_{s, q} (t) ⩽ z_{s, q} (0) exp (C_{6} \int_{0}^{t} {‖ \nabla ρ (τ) ‖}_{L^{\infty}} + {‖ \nabla u (τ) ‖}_{L^{\infty}} d τ), \end{matrix}$ for all $t \in [0, T)$ . Thus, by standard arguments, $(ω, ρ)$ can be continued to $[0, T^{'}]$ for some $T^{'} > T$ , whenever $\begin{matrix} \int_{0}^{T} {‖ \nabla ρ (t) ‖}_{L^{\infty}} + {‖ \nabla u (t) ‖}_{L^{\infty}} d t < \infty . \end{matrix}$ □

Long-time solvability. For $ω_{0} \in {\dot{B}}_{2, q}^{s} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})$ , $ρ_{0} \in B_{2, q}^{s + 1} (R^{2})$ , let $(ω, ρ)$ be the solution of system (1.3) satisfying $\begin{aligned} ω \in C ([0, T^{*}); {\dot{B}}_{2, q}^{s} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T^{*}); {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ \in C ([0, T^{*}); B_{2, q}^{s + 1} (R^{2})) \cap C^{1} ([0, T^{*}); B_{2, q}^{s} (R^{2})), \end{aligned}$ with maximal existence time $T^{*} > 0$ . If $T^{*} = \infty$ , we are done. Assume that $T^{*} < \infty$ . Denoting $V^{\pm} : = ω \pm Λ ρ$ , we can use Duhamel’s principle to get $\begin{matrix} V^{\pm} (t) = e^{\pm κ R_{1} t} V_{0} - \int_{0}^{t} e^{\pm κ R_{1} (τ - t)} (f \pm Λ g) (τ) d τ, \end{matrix}$ where $f = (u \cdot \nabla) ω$ and $g = (u \cdot \nabla) ρ$ . For $0 ⩽ t ⩽ T^{*}$ , we define $\begin{matrix} M (t) : = \int_{0}^{t} {‖ V^{\pm} (τ) ‖}_{{\dot{B}}_{\infty, 1}^{0}} d τ . \end{matrix}$ In what follows, we continue to use the notation $z_{s + 1, q} (0) = ‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}} + ‖ ρ_{0} ‖_{B_{2, q}^{s + 1}}$ . We first consider the case $s = 2$ with $q = 1$ . We can estimate $\begin{aligned} M (t) & ⩽ C \int_{0}^{t} {‖ e^{\pm κ R_{1} τ} V_{0} ‖}_{{\dot{B}}_{\infty, 1}^{0}} d τ + C \int_{0}^{t} {‖ \int_{0}^{τ} e^{\pm κ R_{1} (τ^{'} - τ)} (f \pm Λ g) (τ^{'}) d τ^{'} ‖}_{{\dot{B}}_{\infty, 1}^{0}} d τ \\ : = K_{1} + K_{2} . \end{aligned}$ For $K_{1}$ , we use Hölder’s inequality and Lemma 2.5 with $r = \infty$ to get $\begin{aligned} K_{1} & ⩽ C t^{1 - \frac{1}{γ}} {‖ e^{\pm κ R_{1} \cdot} V_{0} ‖}_{L^{γ} (0, \infty; {\dot{B}}_{\infty, 1}^{0})} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} ‖ V_{0} ‖_{{\dot{B}}_{2, 1}^{1}} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{2, 1} (0) . \end{aligned}$ For $K_{2}$ , we employ the Minkowski and Hölder inequalities, Lemma 2.5 and Remark 2.2 to obtain $\begin{aligned} K_{2} ⩽ & C \int_{0}^{t} \int_{0}^{τ} {‖ e^{\pm κ R_{1} (τ^{'} - τ)} (f \pm Λ g) (τ^{'}) ‖}_{{\dot{B}}_{\infty, 1}^{0}} d τ^{'} d τ \\ = & C \int_{0}^{t} \int_{τ^{'}}^{t} {‖ e^{\pm κ R_{1} (τ^{'} - τ)} (f \pm Λ g) (τ^{'}) ‖}_{{\dot{B}}_{\infty, 1}^{0}} d τ d τ^{'} \\ ⩽ & C t^{1 - \frac{1}{γ}} \int_{0}^{t} {‖ e^{\pm κ R_{1} (τ^{'} - \cdot)} (f \pm Λ g) (τ^{'}) ‖}_{L^{γ} (τ^{'}, t; {\dot{B}}_{\infty, 1}^{0})} d τ^{'} \\ ⩽ & C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} {‖ (f \pm Λ g) (τ^{'}) ‖}_{{\dot{B}}_{2, 1}^{1}} d τ^{'} \\ ⩽ & C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} ({‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{0}} {‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{2}} + {‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{0}} {‖ ρ (τ) ‖}_{{\dot{B}}_{2, 1}^{3}} \\ + {‖ ρ (τ) ‖}_{{\dot{B}}_{2, 1}^{2}} {‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{1}}) d τ \\ ⩽ & C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} ({‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{2} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, 1}^{3}}^{2}) d τ . \end{aligned}$ Thus, for each $0 < t < T^{*}$ , we use (5.2), the embedding ${\dot{B}}_{\infty, 1}^{0} ↪ L^{\infty}$ and the equality $u = \nabla^{⊥} {(- Δ)}^{- 1} ω$ to get $\begin{aligned} M (t) & ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} (z_{2, 1} (0) + \int_{0}^{t} {‖ ω (τ) ‖}_{{\dot{B}}_{2, 1}^{2} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, 1}^{3}}^{2} d τ) \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} (z_{3, 1} (0) + z_{3, 1} {(0)}^{2} \int_{0}^{t} e^{C_{6} M (τ)} d τ) \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{3, 1} (0) (1 + z_{3, 1} (0) t e^{C_{6} M (t)}) . \end{aligned}$ Now, we deal with the case $s > 2$ with $1 ⩽ q ⩽ \infty$ . For each $1 ⩽ q ⩽ \infty$ , we take $2 < r ⩽ \infty$ such that $q ⩽ r$ . Note that since $s - 1 > 1$ we have the nonhomogeneous embedding $B_{\infty, \infty}^{s - 1} ↪ W^{1, \infty}$ , so that we can estimate $‖ V ‖_{{\dot{B}}_{\infty, 1}^{0}}$ by the $B_{\infty, \infty}^{s - 2}$ -norm of V. Thus, we have that $\begin{aligned} M (t) & ⩽ C \int_{0}^{t} {‖ e^{κ R_{1} τ} V_{0} ‖}_{B_{\infty, \infty}^{s - 2}} d τ + C \int_{0}^{t} {‖ \int_{0}^{τ} e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) d τ^{'} ‖}_{B_{\infty, \infty}^{s - 2}} d τ \\ : = K_{3} + K_{4} . \end{aligned}$ We now estimate $K_{3}$ . Using the embedding ${\dot{B}}_{\infty, q}^{s - 2} ↪ {\dot{B}}_{\infty, \infty}^{s - 2}$ , Hölder’s inequality and Lemma 2.5, we can estimate $\begin{aligned} \int_{0}^{t} {‖ e^{κ R_{1} τ} V_{0} ‖}_{{\dot{B}}_{\infty, \infty}^{s - 2}} d τ & ⩽ \int_{0}^{t} {‖ e^{κ R_{1} τ} V_{0} ‖}_{{\dot{B}}_{\infty, q}^{s - 2} (R^{2})} d τ \\ ⩽ t^{1 - \frac{1}{γ}} {‖ e^{κ R_{1} τ} V_{0} ‖}_{L^{γ} (0, \infty; {\dot{B}}_{\infty, q}^{s - 2})} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} ‖ V_{0} ‖_{{\dot{B}}_{2, q}^{s - 1}} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{s, q} (0) . \end{aligned}$ Also, by Lemma 2.4, we have that $\begin{aligned} \int_{0}^{t} {‖ e^{κ R_{1} τ} V_{0} ‖}_{L^{\infty}} d τ & ⩽ t^{1 - \frac{1}{γ}} {‖ e^{κ R_{1} τ} V_{0} ‖}_{L^{γ} (0, \infty; L^{\infty})} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} ‖ V_{0} ‖_{L^{2}} . \end{aligned}$ Therefore, $\begin{matrix} K_{3} ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{s, q} (0) . \end{matrix}$ We proceed to estimate $K_{4}$ . Hölder’s inequality, the embedding ${\dot{B}}_{\infty, q}^{s - 2} ↪ {\dot{B}}_{\infty, \infty}^{s - 2}$ and Lemma 2.5 yield $\begin{aligned} \int_{0}^{t} {‖ \int_{0}^{τ} e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) d τ^{'} ‖}_{{\dot{B}}_{\infty, \infty}^{s - 2}} d τ \\ ⩽ C \int_{0}^{t} \int_{0}^{τ} {‖ e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) ‖}_{{\dot{B}}_{\infty, q}^{s - 2}} d τ^{'} d τ \\ = C \int_{0}^{t} \int_{τ^{'}}^{t} {‖ e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) ‖}_{{\dot{B}}_{\infty, q}^{s - 2}} d τ d τ^{'} \\ ⩽ C t^{1 - \frac{1}{γ}} \int_{0}^{t} {‖ e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) ‖}_{L^{γ} (τ^{'}, t; {\dot{B}}_{\infty, q}^{s - 2})} d τ^{'} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} {‖ (f + Λ g) (τ) ‖}_{{\dot{B}}_{2, q}^{s - 1}} d τ \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} ({‖ ω (τ) ‖}_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, q}^{s + 1}}^{2}) d τ . \end{aligned}$ Here we have proceeded as for the term $K_{2}$ in the case $s = 2$ , $q = 1$ . Also, by Lemma 2.4, the continuity of $R_{l}$ in $L^{2}$ and the embedding $B_{2, q}^{s + 1} ↪ B_{2, q}^{s} ↪ L^{2}$ , we arrive at $\begin{aligned} \int_{0}^{t} {‖ \int_{0}^{τ} e^{κ R_{1} (τ^{'} - τ)} (f + Λ g) (τ^{'}) d τ^{'} ‖}_{L^{\infty}} d τ^{'} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} {‖ (f + Λ g) (τ^{'}) ‖}_{L^{2}} d τ^{'} \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} ({‖ ω (τ) ‖}_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, q}^{s + 1}}^{2}) d τ . \end{aligned}$ It follows that $\begin{matrix} K_{4} ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} \int_{0}^{t} {‖ ω (τ) ‖}_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, q}^{s + 1}}^{2} d τ^{'} . \end{matrix}$ Thus, for each $0 < t < T^{*}$ , we have $\begin{aligned} M (t) & ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} (z_{s, q} (0) + \int_{0}^{t} {‖ ω (τ) ‖}_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}}^{2} + {‖ ρ (τ) ‖}_{B_{2, q}^{s + 1}}^{2} d τ) \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} (z_{s + 1, q} (0) + z_{s + 1, q} {(0)}^{2} \int_{0}^{t} e^{C_{6} M (τ)} d τ) \\ ⩽ C t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{s + 1, q} (0) (1 + z_{s + 1, q} (0) t e^{C_{6} M (t)}) . \end{aligned}$ Therefore, for both cases of s and q such that $s = 2$ with $q = 1$ or $s > 2$ with $1 ⩽ q ⩽ \infty$ , we have that there exists $C_{7} > 0$ such that $\begin{matrix} (5.3) & M (t) ⩽ C_{7} t^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{s + 1, q} (0) (1 + z_{s + 1, q} (0) t e^{C_{6} M (t)}) . \end{matrix}$ Next, for each $0 < T < \infty$ we define $\tilde{T} = sup D_{T}$ , where $\begin{matrix} D_{T} = {t \in [0, T] \cap [0, T^{*}) | M (t) ⩽ C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0)} . \end{matrix}$ We first show that $\tilde{T} = min {T, T^{*}}$ . We proceed by contradiction. So, assume on the contrary that $\tilde{T} < min {T, T^{*}}$ . We have that there exists $T_{1}$ such that $\tilde{T} < T_{1} < min {T, T^{*}}$ . It follows that $\begin{aligned} ω \in C ([0, T_{1}]; {\dot{B}}_{2, q}^{s - 1} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})) \cap C^{1} ([0, T_{1}]; {\dot{B}}_{2, q}^{s - 2} (R^{2}) \cap {\dot{H}}^{- 1} (R^{2})), \\ ρ \in C ([0, T_{1}]; B_{2, q}^{s + 1} (R^{2})) \cap C^{1} ([0, T_{1}]; B_{2, q}^{s} (R^{2})), \end{aligned}$ $M (t)$ is uniformly continuous on $[0, T_{1}]$ , and $\begin{matrix} (5.4) & M (\tilde{T}) ⩽ C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0) . \end{matrix}$ We now take a $| κ |$ large enough so that $\begin{matrix} (5.5) & | κ |^{\frac{1}{γ}} ⩾ 2 (1 + (‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}} + ‖ ρ_{0} ‖_{B_{2, q}^{s + 1}}) T exp (C_{6} C_{7} T^{1 - \frac{1}{γ}} (‖ ω_{0} ‖_{{\dot{B}}_{2, q}^{s} \cap {\dot{H}}^{- 1}} + ‖ ρ_{0} ‖_{B_{2, q}^{s + 1}}))) . \end{matrix}$ Using (5.3), (5.4) and (5.5), we obtain that $\begin{aligned} M (\tilde{T}) & ⩽ C_{7} {(\tilde{T})}^{1 - \frac{1}{γ}} | κ |^{- \frac{1}{γ}} z_{s + 1, q} (0) (1 + z_{s + 1, q} (0) \tilde{T} exp (C_{6} M (\tilde{T}))) \\ ⩽ C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0) | κ |^{- \frac{1}{γ}} (1 + z_{s + 1, q} (0) T exp (C_{6} C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0))) \\ ⩽ \frac{1}{2} C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0) . \end{aligned}$ Thus, we can choose $T_{2}$ such that $\tilde{T} < T_{2} < T_{1}$ with $M (T_{2}) ⩽ C_{7} T^{1 - \frac{1}{γ}} z_{s, q} (0)$ . This contradicts the definition of $\tilde{T}$ . It follows that $\tilde{T} = min {T, T^{*}}$ when κ verifies (5.5). If $T^{*} < T$ , then $T^{*} = \tilde{T} = sup D_{T}$ and $\begin{matrix} M (t) ⩽ C_{7} T^{1 - \frac{1}{γ}} z_{s + 1, q} (0) < \infty, for all 0 ⩽ t < T^{*} . \end{matrix}$ It follows that $\int_{0}^{T^{*}} ‖ \nabla ρ (t) ‖_{L^{\infty}} + ‖ \nabla u (t) ‖_{L^{\infty}} d t ⩽ C M (T^{*}) < \infty$ and, in view of the blow-up criterion, we obtain a contradiction with the maximality of $T^{*}$ . This concludes the proof.

Footnotes

Acknowledgements

L.C.F. Ferreira was supported by FAPESP (grant: 2020/05618-6) and CNPq (grant: 308799/2019-4), Brazil. L. Kosloff was supported by FAPESP (grant: 2016/15985-0), Brazil.

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