Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev

Abstract

This work guarantees the existence of a positive instant $t = T$ and a unique solution $(u, w) \in {[C ([0, T]; H_{a, σ}^{s} (R^{2}))]}^{3}$ (with $a > 0$ , $σ > 1$ , $s > 0$ and $s \neq 1$ ) for the micropolar equations. Furthermore, we consider the global existence in time of this solution in order to prove the following decay rates: $\begin{matrix} lim_{t \to \infty} t^{\frac{s}{2}} {‖ (u, w) (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = lim_{t \to \infty} t^{\frac{s + 1}{2}} {‖ w (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = lim_{t \to \infty} {‖ (u, w) (t) ‖}_{H_{a, σ}^{λ} (R^{2})} = 0, \forall λ ⩽ s . \end{matrix}$ These limits are established by applying the estimate $\begin{matrix} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{H^{s} (R^{2})} ⩽ {[1 + 2 M^{2}]}^{\frac{1}{2}}, \forall t ⩾ T, \end{matrix}$ where T relies only on $s, μ, ν$ and M (the inequality above is also demonstrated in this paper). Here M is a bound for $‖ (u, w) (t) ‖_{H^{s} (R^{2})}$ (for all $t ⩾ 0$ ) which results from the limits ${lim}_{t \to \infty} t^{\frac{s}{2}} ‖ (u, w) (t) ‖_{{\dot{H}}^{s} (R^{2})} = {lim}_{t \to \infty} ‖ (u, w) (t) ‖_{L^{2} (R^{2})} = 0$ .

Keywords

Micropolar equations decay rates Sobolev–Gevrey spaces

1. Introduction

This paper studies local existence of a unique solution as well as decay rates (if it is assumed that this solution is global in time) for the following 2D Micropolar equations in Sobolev–Gevrey spaces: $\begin{matrix} (1) & \{\begin{matrix} u_{t} + u \cdot \nabla u + \nabla p = (μ + χ) Δ u + χ \nabla \times w, x \in R^{2}, t ⩾ 0, \\ w_{t} + u \cdot \nabla w = γ Δ w + χ \nabla \times u - 2 χ w, x \in R^{2}, t ⩾ 0, \\ div u = 0, x \in R^{2}, t > 0, \\ u (\cdot, 0) = u_{0} (\cdot), w (\cdot, 0) = w_{0} (\cdot), x \in R^{2}, \end{matrix} \end{matrix}$ where $u (x, t) = (u_{1} (x, t), u_{2} (x, t)) \in R^{2}$ denotes the incompressible velocity field, $w (x, t) \in R$ the micro-rotational velocity field and $p (x, t) \in R$ the hydrostatic pressure. The positive constants $μ, χ$ and γ are associated with specific properties of the fluid; more precisely, $μ, γ, χ$ are the kinematic, spin and vortex viscosities, respectively. The initial data for the velocity field, given by $u_{0}$ in (1), is assumed to be divergence free, i.e., $div u_{0} = 0$ . Here $\nabla \times u = D_{1} u_{2} - D_{2} u_{1}$ and $\nabla \times w = (D_{2} w, - D_{1} w)$ .

The local existence, uniqueness and blow-up of solutions for the micropolar system (1) and for its periodic version have been extensively studied in the literature, see for instance [5,7,10–13,17,18,20,22,24–27,29] and references therein.

By considering $w = χ = 0$ and $R^{3}$ instead of $R^{2}$ in (1), W.G. Melo, N.F. Rocha and E. Barbosa [23] proved that there are a positive instant $t = T$ and a unique solution $u \in {[C ([0, T], H_{a, σ}^{s} (R^{3}))]}^{3}$ (with $u_{0} \in H_{a, σ}^{s} (R^{3})$ , $s \in (1 / 2, 3 / 2)$ , $a > 0$ , and $σ ⩾ 1$ ) for the Navier–Stokes system (we also cite [1–3,6,15,16,19] and references therein). Let us inform that the similar 3D MHD case is treated in [14]. Motivated by the paper [23], we are interested in showing which are the assumptions that are necessary in order to guarantee the local existence and uniqueness of solutions for the equations given in (1) in nonhomogeneous Sobolev–Gevrey spaces. More precisely, we present our first result.

Theorem 1.1.
Let $a > 0$ , $σ ⩾ 1$ and $s > 0$ with $s \neq 1$ . Let $(u_{0}, w_{0}) \in H_{a, σ}^{s} (R^{2})$ such that $div u_{0} = 0$ . If $s > 1$ (respectively $s \in (0, 1)$ ), then there exist a time $T = T_{s, μ, χ, γ, u_{0}, w_{0}}$ (respectively $T = T_{s, a, μ, χ, γ, u_{0}, w_{0}}$ ) and a unique solution $(u, w) \in {[C ([0, T]; H_{a, σ}^{ϖ} (R^{2}))]}^{3}$ , for all $ϖ ⩽ s$ , of the Micropolar equations given in ( 1 ).

It is worth to emphasize that the critical case $s = 1$ is not proved in our paper due to a specific technical issue. More precisely, the first integral in (12) below is not finite if $s = 1$ (see also Lemmas 2.5, 2.6, and 2.7 where $s \neq 1$ ). Moreover, local solutions for the Micropolar equations in the specific Sobolev–Gevrey space $H_{a, σ}^{1} (R^{2})$ have not been studied in the literature yet (for the knowledge of the authors). In order to cite some papers related to the (2D or 3D) critical usual Sobolev spaces, see [2,4,6,7,13,17,21,22,26–28] and references therein

By observing Theorem 1.1, we suppose that the solution $(u, w)$ obtained above is global in order to present decay rates related to the spaces $H_{a, σ}^{s} (R^{2})$ and ${\dot{H}}_{a, σ}^{s} (R^{2})$ (where $σ > 1$ , $a > 0$ and $s > 0$ with $s \neq 1$ ). Let us inform that theses rates will be accomplished by applying the following result established by R.H. Guterres, W.G. Melo, J.R. Nunes and C.F. Perusato [13].
Theorem 1.2 (See [13]).

Let $(u_{0}, w_{0}) \in L^{2} (R^{2})$ such that $div u_{0} = 0$ . For a Leray global solution $(u, w)$ of the Micropolar equations ( 1 ), one has

${lim}_{t \to \infty} ‖ (u, w) (t) ‖_{L^{2} (R^{2})} = 0$ ;

${lim}_{t \to \infty} t^{\frac{s}{2}} ‖ (u, w) (t) ‖_{{\dot{H}}^{s} (R^{2})} = 0$ for all $s ⩾ 0$ .

Moreover, if

χ > 0

, one obtains

${lim}_{t \to \infty} t^{\frac{s + 1}{2}} ‖ w (t) ‖_{{\dot{H}}^{s} (R^{2})} = 0$ for all $s ⩾ 0$ .

Our second result establishes the asymptotic behavior of the solution (by assuming its global existence in time) obtained in Theorem 1.1 by extending and improving the steps presented by J. Benameur and L. Jlali [2].

Theorem 1.3.
Let $a > 0$ , $σ > 1$ , and $s > 0$ with $s \neq 1$ . Assume that $(u, w) \in C ([0, \infty); H_{a, σ}^{s} (R^{2}))$ is a global solution for the Micropolar equations ( 1 ). Then,
${lim}_{t \to \infty} ‖ (u, w) (t) ‖_{H_{a, σ}^{s} (R^{2})} = 0$ ;

${lim}_{t \to \infty} t^{\frac{s}{2}} ‖ (u, w) (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ .
Moreover, if $χ > 0$ , one has
${lim}_{t \to \infty} t^{\frac{s + 1}{2}} ‖ w (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ .

Under the same assumptions exposed in Theorems 1.1 and 1.2, it is important to point out the following observations:
It is easy to check that Theorem 1.2(ii) implies the following limit: $\begin{array}{l} (2) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{{\dot{H}}^{s} (R^{2})} = 0, \end{array}$ since $\begin{array}{l} (3) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{{\dot{H}}^{s} (R^{2})} = lim_{t \to \infty} t^{- \frac{s}{2}} [t^{\frac{s}{2}} {‖ (u, w) (t) ‖}_{{\dot{H}}^{s} (R^{2})}] = 0, \forall s ⩾ 0 . \end{array}$

Notice also that the limit $\begin{array}{l} (4) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{H^{s} (R^{2})} = 0, \forall s ⩾ 0, \end{array}$ is a direct consequence of Theorem 1.2(i), (2), and the elementary inequality $\begin{array}{l} ‖ f ‖_{H^{s} (R^{2})}^{2} ⩽ 2^{s} [{(2 π)}^{2} ‖ f ‖_{L^{2} (R^{2})}^{2} + ‖ f ‖_{{\dot{H}}^{s} (R^{2})}^{2}], \forall s ⩾ 0 . \end{array}$ Under the hypotheses presented in Theorems 1.1 and 1.3, we can conclude the following remarks:

By applying Theorem 1.3(ii), the same way as in (3), we obtain $\begin{array}{l} (5) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} = 0 . \end{array}$ As a result, by using Lemma 2.3 below, Theorem 1.2(i) and the limit (5), one deduces $\begin{array}{l} (6) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{H_{a, σ}^{s} (R^{2})} = 0 . \end{array}$ This proves Theorem 1.3(i).

Note that $\begin{array}{l} (7) & lim_{t \to \infty} {‖ (u, w) (t) ‖}_{H_{a, σ}^{λ} (R^{2})} = 0, lim_{t \to \infty} {‖ (u, w) (t) ‖}_{H^{λ} (R^{2})} = 0, \forall λ ⩽ s, \end{array}$ provided that the embeddings $H_{a, σ}^{s} (R^{2}) ↪ H_{a, σ}^{λ} (R^{2})$ and $H_{a, σ}^{λ} (R^{2}) ↪ H^{λ} (R^{2})$ are both valid.

By using the fact that $H_{a, σ}^{λ} (R^{2}) ↪ {\dot{H}}_{a, σ}^{λ} (R^{2})$ (for all $λ ⩾ 0$ ), the first limit in (7) assures that $\begin{matrix} lim_{t \to \infty} {‖ (u, w) (t) ‖}_{{\dot{H}}_{a, σ}^{λ} (R^{2})} = 0, \forall λ \in [0, s] . \end{matrix}$

The most important observation is that Theorem 1.3(iii) assures that the micro-rotational velocity field $w (t)$ decays faster than the velocity field $u (t)$ (see Theorem 1.2), provided that $χ > 0$ .
Notations 1.4.
The main notations and definitions of this paper are listed below:
Denote the standard inner product in $C^{n}$ : $\begin{matrix} x \cdot y : = x_{1} {\overline{y}}_{1} + x_{2} {\overline{y}}_{2} + \dots + x_{n} {\overline{y}}_{n} \end{matrix}$ and let the norm induced by this product be $\begin{matrix} | x | : = \sqrt{| x_{1} |^{2} + | x_{2} |^{2} + \dots + | x_{n} |^{2}}, \end{matrix}$ with $x = (x_{1}, x_{2}, \dots, x_{n})$ , $y = (y_{1}, y_{2}, \dots, y_{n}) \in C^{n}$ ( $n \in N$ ).

Define Fourier transform of f by $\begin{array}{rcl} F (f) (ξ) = \hat{f} (ξ) : = \int_{R^{2}} e^{- i ξ \cdot x} f (x) d x, \forall ξ \in R^{2}, \end{array}$ and its inverse by $\begin{array}{rcl} F^{- 1} (f) (ξ) : = {(2 π)}^{- 2} \int_{R^{2}} e^{i ξ \cdot x} f (x) d x, \forall ξ \in R^{2} . \end{array}$

Let $s \in R$ . ${\dot{H}}^{s} (R^{2})$ denotes the homogeneous Sobolev space $\begin{matrix} {f \in S^{'} (R^{2}) : \int_{R^{2}} | ξ |^{2 s} {| \hat{f} (ξ) |}^{2} d ξ < \infty}, \end{matrix}$ where $S^{'} (R^{2})$ is the set of tempered distributions. It is assumed that the ${\dot{H}}^{s} (R^{2})$ -norm is given by $\begin{array}{rcl} ‖ f ‖_{{\dot{H}}^{s} (R^{2})}^{2} : = \int_{R^{2}} | ξ |^{2 s} {| \hat{f} (ξ) |}^{2} d ξ, \end{array}$ Furthermore, the ${\dot{H}}^{s} (R^{2})$ -inner product is given by $\begin{array}{rcl} {⟨ f, g ⟩}_{{\dot{H}}^{s} (R^{2})} : = \int_{R^{2}} | ξ |^{2 s} \hat{f} (ξ) \cdot \hat{g} (ξ) d ξ . \end{array}$

Assume $s \in R$ . The nonhomogeneous Sobolev space $H^{s} (R^{2})$ is established by $\begin{matrix} {f \in S^{'} (R^{2}) : \int_{R^{2}} {(1 + | ξ |^{2})}^{s} {| \hat{f} (ξ) |}^{2} d ξ < \infty} . \end{matrix}$ This space is assumed to be endowed with the $H^{s} (R^{2})$ -norm $\begin{array}{rcl} ‖ f ‖_{H^{s} (R^{2})}^{2} : = \int_{R^{2}} {(1 + | ξ |^{2})}^{s} {| \hat{f} (ξ) |}^{2} d ξ . \end{array}$ Moreover, the $H^{s} (R^{2})$ -inner product is given by $\begin{array}{rcl} {⟨ f, g ⟩}_{H^{s} (R^{2})} : = \int_{R^{2}} {(1 + | ξ |^{2})}^{s} \hat{f} (ξ) \cdot \hat{g} (ξ) d ξ . \end{array}$

Let $a > 0, σ ⩾ 1$ and $s \in R$ . The Sobolev–Gevrey space $\begin{matrix} {\dot{H}}_{a, σ}^{s} (R^{2}) : = {f \in S^{'} (R^{2}) : \int_{R^{2}} | ξ |^{2 s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ < \infty}, \end{matrix}$ is endowed with the ${\dot{H}}_{a, σ}^{s} (R^{2})$ -norm $\begin{array}{rcl} ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} : = \int_{R^{2}} | ξ |^{2 s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ . \end{array}$

Assume $a > 0, σ ⩾ 1$ and $s \in R$ . The nonhomogeneous Sobolev–Gevrey space is given by $\begin{matrix} H_{a, σ}^{s} (R^{2}) : = {f \in S^{'} (R^{2}) : ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} = \int_{R^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ < \infty} . \end{matrix}$

The constants that appear in this paper may change line to line; however, they have the same notation. In addition, $C_{q, r}$ denotes the constants that depend on q and r, for example.

2. Proof of Theorem 1.1

First of all, we present some results that will play a key role in the proof of the local existence and uniqueness of solutions for the micropolar equations (1) in Sobolev–Gevrey spaces.

Lemma 2.1 (See [1]).

The inequality below is valid: $\begin{array}{l} e^{a | ξ |^{\frac{1}{σ}}} ⩽ e^{a max {| ξ - η |, | η |}^{\frac{1}{σ}}} e^{\frac{a}{σ} min {| ξ - η |, | η |}^{\frac{1}{σ}}}, \forall ξ, η \in R^{2}, a ⩾ 0, σ ⩾ 1 . \end{array}$

Proof.
For more details see [1]. □

It follows directly from Lemma 2.1 the statement below.
Lemma 2.2 (See [1]).

Let $ξ, η \in R^{2}, a ⩾ 0$ , and $σ ⩾ 1$ . Then, it holds $\begin{array}{l} (8) & e^{a | ξ |^{\frac{1}{σ}}} ⩽ e^{a | ξ - η |^{\frac{1}{σ}}} e^{a | η |^{\frac{1}{σ}}} . \end{array}$

Proof.
For more details see [1]. □

The next result establishes standard embeddings involving the spaces $L^{2} (R^{2})$ , ${\dot{H}}_{a, σ}^{s} (R^{2})$ and $H_{a, σ}^{s} (R^{2})$ .
Lemma 2.3.
Let $s ⩾ 0, a ⩾ 0, σ ⩾ 1$ and $f \in H_{a, σ}^{s} (R^{2})$ . Then, the following inequalities are valid: $\begin{array}{l} (9) & ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} & ⩽ 2^{s} [e^{2 a} {(2 π)}^{2} ‖ f ‖_{L^{2} (R^{2})}^{2} + ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2}] ⩽ 2^{s} [e^{2 a} + 1] ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} . \end{array}$
Proof.
In fact, note that, by using Parseval’s identity, one obtains $\begin{array}{l} ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} & = \int_{| ξ | ⩽ 1} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ + \int_{| ξ | > 1} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ \\ ⩽ 2^{s} e^{2 a} {(2 π)}^{2} ‖ f ‖_{L^{2} (R^{2})}^{2} + 2^{s} ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} . \end{array}$ It demonstrates the first inequality in (9).

By applying the last equality above, one infers $\begin{array}{l} 2^{s} e^{2 a} {(2 π)}^{2} ‖ f ‖_{L^{2} (R^{2})}^{2} + 2^{s} ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ 2^{s} [e^{2 a} + 1] \int_{R^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} (ξ) |}^{2} d ξ \\ = 2^{s} [e^{2 a} + 1] ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} . \end{array}$ Therefore, the proof of the second inequality in (9) is given. □

Let us present our interpolation results involving the spaces ${\dot{H}}_{a, σ}^{s} (R^{2})$ and $L^{1} (R^{2})$ (see [1,3] for the case 3D).
Lemma 2.4.
Let $a ⩾ 0$ , $σ ⩾ 1$ , and $s ⩾ 0$ . Then, the following inequality holds: $\begin{matrix} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} ⩽ 2^{2 s - 3} π^{- 4} ({‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{f} ‖}_{L^{1} (R^{2})}^{2} ‖ g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} + {‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{g} ‖}_{L^{1} (R^{2})}^{2} ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2}) . \end{matrix}$
Proof.
This result is a consequence of Lemma 2.1. In fact, notice first that $\begin{array}{l} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ {(2 π)}^{- 4} \int_{R^{2}} (\int_{| η | ⩽ | ξ - η |} | ξ |^{s} e^{a | ξ |^{\frac{1}{σ}}} | \hat{f} (ξ - η) | | \hat{g} (η) | d η \\ + \int_{| η | > | ξ - η |} | ξ |^{s} e^{a | ξ |^{\frac{1}{σ}}} | \hat{f} (ξ - η) | | \hat{g} (η) | d η)^{2} d ξ . \end{array}$ By using basic arguments, it is easy to check that $\begin{array}{l} | ξ |^{s} & ⩽ {[| ξ - η | + | η |]}^{s} ⩽ 2^{s} {[max {| ξ - η |, | η |}]}^{s} . \end{array}$ Now, we are interested in applying Lemma 2.1 in order to obtain $\begin{array}{l} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ {(2 π)}^{- 4} 2^{2 s + 1} [\int_{R^{2}} {(\int_{R^{2}} | ξ - η |^{s} e^{a | ξ - η |^{\frac{1}{σ}}} | \hat{f} (ξ - η) | e^{\frac{a}{σ} | η |^{\frac{1}{σ}}} | \hat{g} (η) | d η)}^{2} d ξ \\ + \int_{R^{2}} {(\int_{R^{2}} e^{\frac{a}{σ} | ξ - η |^{\frac{1}{σ}}} | \hat{f} (ξ - η) | | η |^{s} e^{a | η |^{\frac{1}{σ}}} | \hat{g} (η) | d η)}^{2} d ξ] . \end{array}$ Consequently, it follows from Young’s inequality for convolutions that $\begin{matrix} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} ⩽ 2^{2 s - 3} π^{- 4} ({‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{f} ‖}_{L^{1} (R^{2})}^{2} ‖ g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} + {‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{g} ‖}_{L^{1} (R^{2})}^{2} ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2}) . \end{matrix}$ □

Notice that we can change the proof of Lemma 2.4 to obtain $\begin{array}{l} (10) & ‖ f g ‖_{H_{a, σ}^{s} (R^{2})}^{2} ⩽ 2^{2 s - 3} π^{- 4} ({‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{f} ‖}_{L^{1} (R^{2})}^{2} ‖ g ‖_{H_{a, σ}^{s} (R^{2})}^{2} + {‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{g} ‖}_{L^{1} (R^{2})}^{2} ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2}), \end{array}$ provided that $a ⩾ 0$ , $σ ⩾ 1$ , and $s ⩾ 0$ .

As a consequence of Lemma 2.4, we state the following result.
Lemma 2.5.
Let $a ⩾ 0$ , $σ ⩾ 1$ , and $s > 1$ . Then, the following inequality holds: $\begin{array}{l} (11) & \begin{matrix} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ \frac{2^{3 s} π^{- 1} e^{2 a}}{s - 1} \\ \times (‖ f ‖_{L^{2} (R^{2})}^{2} ‖ g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} + ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} ‖ g ‖_{L^{2} (R^{2})}^{2} + ‖ f ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} ‖ g ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2}) . \end{matrix} \end{array}$
Proof.
By applying Cauchy–Schwarz’s inequality, one infers $\begin{array}{l} (12) & {‖ e^{\frac{a}{σ} | \cdot |^{\frac{1}{σ}}} \hat{g} ‖}_{L^{1} (R^{2})} & ⩽ {(\int_{R^{2}} {(1 + | ξ |^{2})}^{- s} d ξ)}^{\frac{1}{2}} {(\int_{R^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{g} (ξ) |}^{2} d ξ)}^{\frac{1}{2}} \\ (13) & = \sqrt{\frac{π}{s - 1}} ‖ g ‖_{H_{a, σ}^{s} (R^{2})} . \end{array}$ Thus (11) follows directly from Lemma 2.4, (13) and Lemma 2.3. □

By combining (10) and (13), we have $\begin{array}{l} (14) & ‖ f g ‖_{H_{a, σ}^{s} (R^{2})}^{2} & ⩽ \frac{2^{2 s - 2} π^{- 3}}{s - 1} ‖ f ‖_{H_{a, σ}^{s} (R^{2})}^{2} ‖ g ‖_{H_{a, σ}^{s} (R^{2})}^{2}, \end{array}$ provided that $a ⩾ 0$ , $σ ⩾ 1$ , and $s > 1$ .

The lemma below presents an interpolation property involving the space ${\dot{H}}^{s} (R^{2})$ , and it has been proved by J.-Y. Chemin [9].
Lemma 2.6 (See [9]).

Let $(s_{1}, s_{2}) \in R^{2}$ such that $s_{1} < 1$ and $s_{1} + s_{2} > 0$ . Then, there exists a positive constant $C_{s_{1}, s_{2}}$ such that, for all $f, g \in {\dot{H}}^{s_{1}} (R^{2}) \cap {\dot{H}}^{s_{2}} (R^{2})$ , we have $\begin{matrix} ‖ f g ‖_{{\dot{H}}^{s_{1} + s_{2} - 1} (R^{2})} ⩽ C_{s_{1}, s_{2}} (‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})} ‖ g ‖_{{\dot{H}}^{s_{2}} (R^{2})} + ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})} ‖ g ‖_{{\dot{H}}^{s_{1}} (R^{2})}) . \end{matrix}$ If $s_{1} < 1$ , $s_{2} < 1$ and $s_{1} + s_{2} > 0$ , then there is a positive constant $C_{s_{1}, s_{2}}$ such that $\begin{matrix} ‖ f g ‖_{{\dot{H}}^{s_{1} + s_{2} - 1} (R^{2})} ⩽ C_{s_{1}, s_{2}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})} ‖ g ‖_{{\dot{H}}^{s_{2}} (R^{2})} . \end{matrix}$

Proof.
For more details see [9]. □

Lemma 2.6 will be extended for Sobolev–Gevrey spaces as follows.
Lemma 2.7.
Let $a > 0$ , $σ ⩾ 1$ and $(s_{1}, s_{2}) \in R^{2}$ , such that $s_{1} < 1$ and $s_{1} + s_{2} > 0$ . Then, there exists a positive constant $C_{s_{1}, s_{2}}$ such that, for all $f, g \in {\dot{H}}_{a, σ}^{s_{1}} (R^{2}) \cap {\dot{H}}_{a, σ}^{s_{2}} (R^{2})$ , we have $\begin{array}{l} (15) & ‖ f g ‖_{{\dot{H}}_{a, σ}^{s_{1} + s_{2} - 1} (R^{2})} ⩽ C_{s_{1}, s_{2}} [‖ f ‖_{{\dot{H}}_{a, σ}^{s_{1}} (R^{2})} ‖ g ‖_{{\dot{H}}_{a, σ}^{s_{2}} (R^{2})} + ‖ f ‖_{{\dot{H}}_{a, σ}^{s_{2}} (R^{2})} ‖ g ‖_{{\dot{H}}_{a, σ}^{s_{1}} (R^{2})}] . \end{array}$ If $s_{1} < 1$ , $s_{2} < 1$ and $s_{1} + s_{2} > 0$ , then there is a positive constant $C_{s_{1}, s_{2}}$ such that $\begin{array}{l} (16) & ‖ f g ‖_{{\dot{H}}_{a, σ}^{s_{1} + s_{2} - 1} (R^{2})} ⩽ C_{s_{1}, s_{2}} ‖ f ‖_{{\dot{H}}_{a, σ}^{s_{1}} (R^{2})} ‖ g ‖_{{\dot{H}}_{a, σ}^{s_{2}} (R^{2})} . \end{array}$
Proof.
We aim to apply Lemma 2.6. Thus, $\begin{array}{l} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s_{1} + s_{2} - 1} (R^{2})}^{2} & = {(2 π)}^{- 4} \int_{R^{2}} | ξ |^{2 s_{1} + 2 s_{2} - 2} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{f} * \hat{g} (ξ) |}^{2} d ξ \\ ⩽ {(2 π)}^{- 4} \int_{R^{2}} | ξ |^{2 s_{1} + 2 s_{2} - 2} {(\int_{R^{2}} e^{a | ξ |^{\frac{1}{σ}}} | \hat{f} (ξ - η) | | \hat{g} (η) | d η)}^{2} d ξ . \end{array}$ Moreover, the inequality (8) implies the following results: $\begin{array}{l} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s_{1} + s_{2} - 1} (R^{2})}^{2} & ⩽ {(2 π)}^{- 4} \int_{R^{2}} | ξ |^{2 s_{1} + 2 s_{2} - 2} {[(e^{a | \cdot |^{\frac{1}{σ}}} | \hat{f} |) * (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{g} |)] (ξ)}^{2} d ξ \\ = {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{f} |) F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{g} |) ‖}_{{\dot{H}}^{s_{1} + s_{2} - 1} (R^{2})}^{2} . \end{array}$ Now, we are ready to apply Lemma 2.6 and, consequently, deduce (15). On the other hand, if $s_{1}, s_{2} < 1$ and $s_{1} + s_{2} > 0$ , use Lemma 2.6 again in order to obtain $\begin{array}{l} ‖ f g ‖_{{\dot{H}}_{a, σ}^{s_{1} + s_{2} - 1} (R^{2})} & ⩽ C_{s_{1}, s_{2}} {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{f} |) ‖}_{{\dot{H}}^{s_{1}} (R^{2})} {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{g} |) ‖}_{{\dot{H}}^{s_{2}} (R^{2})} \\ = C_{s_{1}, s_{2}} ‖ f ‖_{{\dot{H}}_{a, σ}^{s_{1}} (R^{2})} ‖ g ‖_{{\dot{H}}_{a, σ}^{s_{2}} (R^{2})} . \end{array}$ □

It is well known that the following inequality holds for the micropolar equations (1) (see [10]): $\begin{array}{l} (17) & {‖ (u, w) (t) ‖}_{L^{2} (R^{2})} ⩽ {‖ (u_{0}, w_{0}) ‖}_{L^{2} (R^{2})}, \forall t ⩾ 0 . \end{array}$

We enunciate the next elementary result, which follows from basic tools obtained in Calculus.
Lemma 2.8.
Let $a, b > 0$ . Then, $λ^{a} e^{- b λ} ⩽ a^{a} {(e b)}^{- a}$ for all $λ > 0$ .

Below, we enunciate the fixed point theorem that will be applied in this paper. Lemma 2.9 (See [8]).

Let $(X, ‖ \cdot ‖)$ be a Banach space, $L : X \to X$ continuous linear operator and $B : X \times X \to X$ continuous bilinear operator, i.e., there exist positive constants $C_{1}$ and $C_{2}$ such that $\begin{matrix} ‖ L (x) ‖ ⩽ C_{1} ‖ x ‖, ‖ B (x, y) ‖ ⩽ C_{2} ‖ x ‖ ‖ y ‖, \forall x, y \in X . \end{matrix}$ Then, for each $C_{1} \in (0, 1)$ and $x_{0} \in X$ that satisfy $4 C_{2} ‖ x_{0} ‖ < {(1 - C_{1})}^{2}$ , one has that the equation $\begin{matrix} a = x_{0} + B (a, a) + L (a), a \in X, \end{matrix}$ admits a solution $x \in X$ . Moreover, x obeys the inequality $‖ x ‖ ⩽ \frac{2 ‖ x_{0} ‖}{1 - C_{1}}$ and it is the only one such that $‖ x ‖ ⩽ \frac{1 - C_{1}}{2 C_{2}}$ .

Proof.
For more details see [8] and references therein. □
Proof of Theorem 1.1.
By applying the heat semigroup $e^{(μ + χ) Δ (t - τ)}$ , with $τ \in [0, t]$ , in the first equation of (1), and, subsequently, integrating the result over the interval $[0, t]$ , we obtain $\begin{array}{l} \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} u_{τ} d τ + \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} [u \cdot \nabla u + \nabla p - χ (\nabla \times w)] d τ \\ = (μ + χ) \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} Δ u d τ . \end{array}$ Now, use integration by parts in order to deduce $\begin{array}{l} u (t) = e^{(μ + χ) Δ t} u_{0} - \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} [u \cdot \nabla u + \nabla p] d τ + χ \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} (\nabla \times w) d τ . \end{array}$ Let us recall that Helmontz’s projector $P_{H}$ (see [19] and references therein) is well defined and $\begin{array}{l} (18) & P_{H} (u \cdot \nabla u) = u \cdot \nabla u + \nabla p, F [P_{H} (f)] (ξ) = \hat{f} (ξ) - \frac{\hat{f} (ξ) \cdot ξ}{| ξ |^{2}} ξ . \end{array}$ Consequently, one can write1
¹
The j-th spatial derivative is denoted by $D_{j} = \partial / \partial x_{j}$ ( $j = 1, 2$ ).

$\begin{array}{l} (19) & u (t) = e^{(μ + χ) Δ t} u_{0} - \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (u_{j} u)] d τ + χ \int_{0}^{t} e^{(μ + χ) Δ (t - τ)} (\nabla \times w) d τ, \end{array}$ since u is divergence free.

Analogously, by considering the field w, we deduce the equality below. $\begin{array}{l} (20) & \begin{matrix} w (t) & = e^{γ Δ t} w_{0} - \int_{0}^{t} e^{γ Δ (t - τ)} \sum_{j = 1}^{2} D_{j} (u_{j} w) d τ \\ + χ \int_{0}^{t} e^{γ Δ (t - τ)} (\nabla \times u) d τ - 2 χ \int_{0}^{t} e^{γ Δ (t - τ)} w d τ . \end{matrix} \end{array}$ By (19) and (20), one obtains $\begin{array}{l} (21) & (u, w) (t) & = (e^{(μ + χ) Δ t} u_{0}, e^{γ Δ t} w_{0}) + B ((u, w), (u, w)) (t) + L (u, w) (t), \end{array}$ where $\begin{matrix} (22) & B ((z, v), (φ, ϕ)) (t) = \int_{0}^{t} (- e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)], - e^{γ Δ (t - τ)} [\sum_{j = 1}^{2} D_{j} (z_{j} ϕ)]) d τ, \end{matrix}$ and $\begin{array}{l} (23) & L (z, v) (t) = \int_{0}^{t} (χ e^{(μ + χ) Δ (t - τ)} (\nabla \times v), χ e^{γ Δ (t - τ)} (\nabla \times z) - 2 χ e^{γ Δ (t - τ)} v) d τ . \end{array}$ Notice that L is a linear operator and B is a bilinear operator. Furthermore, $φ, z : R^{2} \to R^{2}$ and $ϕ, v : R^{2} \to R$ belong to appropriate spaces that will be revealed next. In order to examine $(u, w) (t)$ in $H_{a, σ}^{s} (R^{2})$ , let us estimate $B ((w, v), (γ, ϕ)) (t)$ and $L (z, v) (t)$ in this same space.

At first, let us estimate $L (z, v) (t)$ in $H_{a, σ}^{s} (R^{2})$ . Thus, we deduce $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} (\nabla \times v) ‖}_{H_{a, σ}^{s} (R^{2})}^{2} & = \int_{R^{2}} | ξ |^{2} e^{- 2 (μ + χ) (t - τ) | ξ |^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{v} (ξ) |}^{2} d ξ, \end{array}$ since $| \hat{\nabla \times v} | = | ξ | | \hat{v} |$ . As a result, by using Lemma 2.8, it follows that $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} (\nabla \times v) ‖}_{H_{a, σ}^{s} (R^{2})} ⩽ C_{μ, χ} {(t - τ)}^{- \frac{1}{2}} {‖ (z, v) ‖}_{H_{a, σ}^{s} (R^{2})} . \end{array}$ By integrating over $[0, t]$ ( $t \in [0, T]$ ) the above estimate, we conclude $\begin{array}{l} (24) & \int_{0}^{t} {‖ χ e^{(μ + χ) Δ (t - τ)} (\nabla \times v) ‖}_{H_{a, σ}^{s} (R^{2})} d τ & ⩽ C_{μ, χ} T^{\frac{1}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} . \end{array}$ Similarly, one gets $\begin{array}{l} (25) & \int_{0}^{t} {‖ χ e^{γ Δ (t - τ)} (\nabla \times z) ‖}_{H_{a, σ}^{s} (R^{2})} d τ ⩽ C_{χ, γ} T^{\frac{1}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))}, \end{array}$ since $| \hat{\nabla \times z} | ⩽ | ξ | | \hat{z} |$ . On the other hand, it is valid that $\begin{array}{l} {‖ e^{γ Δ (t - τ)} v ‖}_{H_{a, σ}^{s} (R^{2})}^{2} ⩽ \int_{R^{2}} e^{- 2 γ (t - τ) | ξ |^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{v} (ξ) |}^{2} d ξ ⩽ {‖ (z, v) ‖}_{H_{a, σ}^{s} (R^{2})}^{2} . \end{array}$ By integrating over $[0, t]$ , the above estimative, we conclude $\begin{array}{l} (26) & \int_{0}^{t} {‖ 2 χ e^{γ Δ (t - τ)} v ‖}_{H_{a, σ}^{s} (R^{2})} d τ & ⩽ C_{χ} T^{\frac{1}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))}, \forall t \in [0, T], \end{array}$ if it is assumed that $T < 1$ . By (23), we can assure that (24), (25) and (26) imply $\begin{array}{l} (27) & {‖ L (z, v) (t) ‖}_{H_{a, σ}^{s} (R^{2})} ⩽ C_{μ, χ, γ} T^{\frac{1}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T], H_{a, σ}^{s} (R^{2}))}, \forall t \in [0, T] . \end{array}$

Now, let us estimate $B ((z, v), (φ, ϕ)) (t)$ in $H_{a, σ}^{s} (R^{2})$ . To this end, we shall divide the proof into two cases:

1º Case: Assume that $s > 1$ .

Notice that, by applying (18), one can write2
²
The tensor product is given by $f \otimes g : = (g_{1} f, g_{2} f)$ , where $g = (g_{1}, g_{2})$ .

$\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{H_{a, σ}^{s} (R^{2})}^{2} \\ ⩽ \int_{R^{2}} e^{- 2 (μ + χ) (t - τ) | ξ |^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \sum_{j = 1}^{2} F [D_{j} (φ_{j} z)] (ξ) |}^{2} d ξ \\ ⩽ \int_{R^{2}} | ξ |^{2} e^{- 2 (μ + χ) (t - τ) | ξ |^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| F (z \otimes φ) (ξ) |}^{2} d ξ . \end{array}$ As a result, by using Lemma 2.8, it follows $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{μ, χ} {(t - τ)}^{- \frac{1}{2}} ‖ z \otimes φ ‖_{H_{a, σ}^{s} (R^{2})} . \end{array}$ Similarly, we can write $\begin{array}{l} {‖ e^{γ Δ (t - τ)} [\sum_{j = 1}^{2} D_{j} (z_{j} ϕ)] ‖}_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{γ} {(t - τ)}^{- \frac{1}{2}} ‖ ϕ \otimes z ‖_{H_{a, σ}^{s} (R^{2})} . \end{array}$ Consequently, one gets $\begin{array}{l} {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{μ, χ, γ} [\int_{0}^{t} {(t - τ)}^{- \frac{1}{2}} ‖ z \otimes φ ‖_{H_{a, σ}^{s} (R^{2})} d τ \\ (28) & + \int_{0}^{t} {(t - τ)}^{- \frac{1}{2}} ‖ ϕ \otimes z ‖_{H_{a, σ}^{s} (R^{2})} d τ] . \end{array}$ On the other hand, it is true that $\begin{array}{l} ‖ z \otimes φ ‖_{H_{a, σ}^{s} (R^{2})}^{2} & = {(2 π)}^{- 4} \sum_{j, k = 1}^{2} \int_{R^{2}} {(1 + | ξ |^{2})}^{s} e^{2 a | ξ |^{\frac{1}{σ}}} {| \hat{φ_{j}} * \hat{z_{k}} (ξ) |}^{2} d ξ \\ ⩽ C \int_{R^{2}} {(1 + | ξ |^{2})}^{s} {[\int_{R^{2}} e^{a | ξ |^{\frac{1}{σ}}} | \hat{z} (η) | | \hat{φ} (ξ - η) | d η]}^{2} d ξ . \end{array}$ By using (8), we conclude $\begin{array}{l} ‖ z \otimes φ ‖_{H_{a, σ}^{s} (R^{2})}^{2} & ⩽ C \int_{R^{2}} {(1 + | ξ |^{2})}^{s} {[(e^{a | ξ |^{\frac{1}{σ}}} | \hat{z} (ξ) |) * (e^{a | ξ |^{\frac{1}{σ}}} | \hat{φ} (ξ) |)]}^{2} d ξ \\ = C {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{z} |) F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{φ} |) ‖}_{H^{s} (R^{2})}^{2} . \end{array}$ Hence, by the inequality (14) with $a = 0$ , one has $\begin{array}{l} ‖ z \otimes φ ‖_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{s} {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{z} |) ‖}_{H^{s} (R^{2})} {‖ F^{- 1} (e^{a | \cdot |^{\frac{1}{σ}}} | \hat{φ} |) ‖}_{H^{s} (R^{2})} = C_{s} ‖ z ‖_{H_{a, σ}^{s} (R^{2})} ‖ φ ‖_{H_{a, σ}^{s} (R^{2})}, \end{array}$ since $s > 1$ . Replacing this result in (28), one obtains $\begin{array}{l} {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{s, μ, χ, γ} [\int_{0}^{t} {(t - τ)}^{- \frac{1}{2}} ‖ z ‖_{H_{a, σ}^{s} (R^{2})} ‖ φ ‖_{H_{a, σ}^{s} (R^{2})} d τ \\ + \int_{0}^{t} {(t - τ)}^{- \frac{1}{2}} ‖ ϕ ‖_{H_{a, σ}^{s} (R^{2})} ‖ z ‖_{H_{a, σ}^{s} (R^{2})} d τ] . \end{array}$ Therefore, we deduce $\begin{array}{l} (29) & {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{H_{a, σ}^{s} (R^{2})} & ⩽ C_{s, μ, χ, γ} T^{\frac{1}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} \end{array}$ for all $t \in [0, T]$ . By noticing that $L : X \times Y \to X \times Y$ is a continuous linear operator (see (23) and (27)) and $B : {(X \times Y)}^{2} \to X \times Y$ is a continuous bilinear operator (see (22) and (29)), where $Y = C ([0, T]; H_{a, σ}^{s} (R^{2}))$ and $X = Y^{2}$ (with $s > 1$ ), it is enough to apply Lemma 2.9 and consider T small enough in order to obtain a unique solution $(u, w) \in {[C ([0, T]; H_{a, σ}^{s} (R^{2}))]}^{3}$ for the equation (21). More specifically, choose $\begin{array}{l} T < min {{[{(4 C_{s, μ, χ, γ} {‖ (u_{0}, w_{0}) ‖}_{H_{a, σ}^{s} (R^{2})})}^{\frac{1}{2}} + C_{μ, χ, γ}]}^{- 4}, C_{μ, χ, γ}^{- 2}, 1}, \end{array}$ where $C_{μ, χ, γ}$ and $C_{s, μ, χ, γ}$ are given in (27) and (29), respectively; and $\begin{matrix} {‖ (e^{(μ + χ) Δ t} u_{0}, e^{γ Δ t} w_{0}) ‖}_{H_{a, σ}^{s} (R^{2})} ⩽ {‖ (u_{0}, w_{0}) ‖}_{H_{a, σ}^{s} (R^{2})} . \end{matrix}$

2º Case: Consider that $s \in (0, 1)$ .

At first, let us estimate $B ((z, v), (φ, ϕ)) (t)$ in ${\dot{H}}_{a, σ}^{s} (R^{2})$ . By applying (18) and Cauchy–Schwarz’s inequality, one can write $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} \\ ⩽ \int_{R^{2}} | ξ |^{4 - 2 s} e^{- 2 (μ + χ) (t - τ) | ξ |^{2}} | ξ |^{4 s - 2} e^{2 a | ξ |^{\frac{1}{σ}}} {| F (z \otimes φ) (ξ) |}^{2} d ξ . \end{array}$ As a result, by using Lemma 2.8, it follows $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ \frac{C_{s, μ, χ}}{{(t - τ)}^{2 - s}} ‖ z \otimes φ ‖_{{\dot{H}}_{a, σ}^{2 s - 1} (R^{2})}^{2}, \end{array}$ since $0 < s < 1$ . On the other hand, by using Lemma 2.7, one infers $\begin{array}{l} ‖ z \otimes φ ‖_{{\dot{H}}_{a, σ}^{2 s - 1} (R^{2})}^{2} & = \sum_{j, k = 1}^{2} ‖ φ_{j} z_{k} ‖_{{\dot{H}}_{a, σ}^{2 s - 1} (R^{2})}^{2} ⩽ C_{s} ‖ z ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} ‖ φ ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2}, \end{array}$ provided that $0 < s < 1$ . Therefore, one deduces $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} & ⩽ \frac{C_{s, μ, χ}}{{(t - τ)}^{\frac{2 - s}{2}}} {‖ (z, v) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} {‖ (φ, ϕ) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} . \end{array}$ By integrating over $[0, t]$ ( $t \in [0, T]$ ) the above estimate, we conclude $\begin{array}{l} (30) & \begin{array}{l} \int_{0}^{t} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} d τ \\ ⩽ C_{s, μ, χ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} . \end{array} \end{array}$ Similarly, we can write $\begin{array}{l} (31) & \begin{array}{l} \int_{0}^{t} {‖ e^{γ Δ (t - τ)} [\sum_{j = 1}^{2} D_{j} (z_{j} ϕ)] ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} d τ \\ ⩽ C_{s, γ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} . \end{array} \end{array}$ By (22), we can assure that (30) and (31) imply $\begin{array}{l} (32) & {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})} & ⩽ C_{s, μ, χ, γ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; {\dot{H}}_{a, σ}^{s} (R^{2}))} . \end{array}$ Let us estimate $B ((z, v), (φ, ϕ)) (t)$ in $L^{2} (R^{2})$ . Following a similar process to the one presented above, we have, by using Parseval’s identity, that $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{L^{2} (R^{2})}^{2} \\ = {(2 π)}^{- 2} \int_{R^{2}} {| F {e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)]} (ξ) |}^{2} d ξ . \end{array}$ As a result, by using (18), we get $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{L^{2} (R^{2})}^{2} \\ ⩽ {(2 π)}^{- 2} \int_{R^{2}} | ξ |^{4 - 2 s} e^{- 2 (μ + χ) (t - τ) | ξ |^{2}} | ξ |^{2 s - 2} {| F (z \otimes φ) (ξ) |}^{2} d ξ . \end{array}$ As a result, by using Lemma 2.8, it follows $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{L^{2} (R^{2})} & ⩽ \frac{C_{s, μ, χ}}{{(t - τ)}^{\frac{2 - s}{2}}} ‖ z \otimes φ ‖_{{\dot{H}}^{s - 1} (R^{2})}, \end{array}$ On the other hand, by utilizing Lemma 2.6 (provided that $0 < s < 1$ ), one has $\begin{array}{l} ‖ z \otimes φ ‖_{{\dot{H}}^{s - 1} (R^{2})}^{2} & = \sum_{j, k = 1}^{2} ‖ φ_{j} z_{k} ‖_{{\dot{H}}^{s - 1} (R^{2})}^{2} ⩽ C_{s} ‖ z ‖_{{\dot{H}}^{s} (R^{2})}^{2} ‖ φ ‖_{L^{2} (R^{2})}^{2} . \end{array}$ Thereby, as $H_{a, σ}^{s} (R^{2}) ↪ {\dot{H}}^{s} (R^{2})$ ( $s ⩾ 0$ ) and by applying Lemma 2.3, we deduce $\begin{array}{l} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{L^{2} (R^{2})} & ⩽ \frac{C_{s, μ, χ}}{{(t - τ)}^{\frac{2 - s}{2}}} {‖ (z, v) ‖}_{H_{a, σ}^{s} (R^{2})} {‖ (φ, ϕ) ‖}_{H_{a, σ}^{s} (R^{2})} . \end{array}$ By integrating the above estimate over $[0, t]$ ( $t \in [0, T]$ ), we conclude $\begin{array}{l} (33) & \begin{array}{l} \int_{0}^{t} {‖ e^{(μ + χ) Δ (t - τ)} P_{H} [\sum_{j = 1}^{2} D_{j} (φ_{j} z)] ‖}_{L^{2} (R^{2})} d τ \\ ⩽ C_{s, μ, χ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} . \end{array} \end{array}$ Similarly, we can obtain $\begin{array}{l} (34) & \begin{array}{l} \int_{0}^{t} {‖ e^{γ Δ (t - τ)} [\sum_{j = 1}^{2} D_{j} (z_{j} ϕ)] ‖}_{L^{2} (R^{2})} d τ \\ ⩽ C_{s, γ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} . \end{array} \end{array}$ By using the definition (22) and applying (33) and (34), one concludes $\begin{array}{l} (35) & {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{L^{2} (R^{2})} & ⩽ C_{s, μ, χ, γ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} \end{array}$ for all $t \in [0, T]$ . Finally, by using Lemma 2.3, (32), (35) and the fact that $H_{a, σ}^{s} (R^{2}) ↪ {\dot{H}}_{a, σ}^{s} (R^{2})$ ( $s ⩾ 0$ ), it results $\begin{array}{l} (36) & \begin{array}{l} {‖ B ((z, v), (φ, ϕ)) (t) ‖}_{H_{a, σ}^{s} (R^{2})} \\ ⩽ C_{s, a, μ, χ, γ} T^{\frac{s}{2}} {‖ (z, v) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} {‖ (φ, ϕ) ‖}_{L^{\infty} ([0, T]; H_{a, σ}^{s} (R^{2}))} \end{array} \end{array}$ for all $t \in [0, T]$ . Choose $\begin{matrix} T < min {{[{(4 C_{s, a, μ, χ, γ} {‖ (u_{0}, w_{0}) ‖}_{H_{a, σ}^{s} (R^{2})})}^{\frac{1}{2}} + C_{μ, χ, γ}]}^{- \frac{4}{s}}, C_{μ, χ, γ}^{- 2}, 1}, \end{matrix}$ where $C_{μ, χ, γ}$ and $C_{s, a, μ, χ, γ}$ is given in (27) and (36), respectively, and apply Lemma 2.9 to obtain the desired result.

Lastly, by assuming that $ϖ ⩽ s$ , it follows that $(u, w) \in {[C ([0, T]; H_{a, σ}^{ϖ} (R^{2}))]}^{3}$ since $H_{a, σ}^{s} (R^{2}) ↪ H_{a, σ}^{ϖ} (R^{2})$ . □

3. Proof of Theorem 1.3(ii) and (iii)

At first, we present some lemmas that will play a key role in the proof of the decay rates given in Theorem 1.3(ii) and (iii). The first one is based on the paper [2].

Lemma 3.1.

Let $(s_{1}, s_{2}) \in R^{2}$ such that $s_{1} < 1 < s_{2}$ . Then, there is a positive constant $C_{s_{1}, s_{2}}$ such that $\begin{matrix} ‖ \hat{f} ‖_{L^{1} (R^{2})} ⩽ C_{s_{1}, s_{2}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})}^{\frac{s_{2} - 1}{s_{2} - s_{1}}} ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})}^{\frac{1 - s_{1}}{s_{2} - s_{1}}} . \end{matrix}$

Proof.

Let c be an arbitrary positive constant. Note that $\begin{array}{l} ‖ \hat{f} ‖_{L^{1} (R^{2})} = \int_{R^{2}} | \hat{f} (ξ) | d ξ = \int_{| ξ | > c} | \hat{f} (ξ) | d ξ + \int_{| ξ | ⩽ c} | \hat{f} (ξ) | d ξ . \end{array}$ By applying Cauchy–Schwarz’s inequality, we have $\begin{array}{l} \int_{| ξ | ⩽ c} | \hat{f} (ξ) | d ξ ⩽ {(\int_{| ξ | ⩽ c} | ξ |^{- 2 s_{1}} d ξ)}^{\frac{1}{2}} {(\int_{R^{2}} | ξ |^{2 s_{1}} {| \hat{f} (ξ) |}^{2} d ξ)}^{\frac{1}{2}} = \sqrt{\frac{2 π}{2 - 2 s_{1}}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})} c^{1 - s_{1}} . \end{array}$ Similarly, one obtains $\begin{array}{l} \int_{| ξ | > c} | \hat{f} (ξ) | d ξ ⩽ {(\int_{| ξ | > c} | ξ |^{- 2 s_{2}} d ξ)}^{\frac{1}{2}} {(\int_{R^{2}} | ξ |^{2 s_{2}} {| \hat{f} (ξ) |}^{2} d ξ)}^{\frac{1}{2}} = \sqrt{\frac{2 π}{2 s_{2} - 2}} ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})} c^{1 - s_{2}} . \end{array}$ Consequently, we can write $\begin{array}{l} ‖ \hat{f} ‖_{L^{1} (R^{2})} ⩽ \sqrt{\frac{2 π}{2 - 2 s_{1}}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})} c^{1 - s_{1}} + \sqrt{\frac{2 π}{2 s_{2} - 2}} ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})} c^{1 - s_{2}} = : g (c) . \end{array}$ It is easy to see that g attains its minimum at $c_{0} = {[\frac{(s_{2} - 1) \sqrt{\frac{2 π}{2 s_{2} - 2}} ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})}}{(1 - s_{1}) \sqrt{\frac{2 π}{2 - 2 s_{1}}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})}}]}^{\frac{1}{s_{2} - s_{1}}}$ . Thus, $\begin{array}{l} ‖ \hat{f} ‖_{L^{1} (R^{2})} ⩽ g (c_{0}) ⩽ C_{s_{1}, s_{2}} ‖ f ‖_{{\dot{H}}^{s_{1}} (R^{2})}^{\frac{s_{2} - 1}{s_{2} - s_{1}}} ‖ f ‖_{{\dot{H}}^{s_{2}} (R^{2})}^{\frac{1 - s_{1}}{s_{2} - s_{1}}} . \end{array}$ □

It is important to point out that if $0 < s < 1$ , then one can assume $s_{1} = s$ and $s_{2} = s + 1$ in Lemma 3.1 in order to obtain the following interpolation inequality: $\begin{array}{l} (37) & ‖ \hat{f} ‖_{L^{1} (R^{2})} ⩽ C_{s} ‖ f ‖_{{\dot{H}}^{s} (R^{2})}^{s} ‖ f ‖_{{\dot{H}}^{s + 1} (R^{2})}^{1 - s} . \end{array}$ Actually, this is the inequality that will be applied in this paper.

Lemma 3.2.

There is an instant $t = T$ that depends only on $s, μ, γ$ and $‖ (u_{0}, w_{0}) ‖_{H^{s} (R^{2})}$ such that $\begin{matrix} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} ⩽ {[1 + 2 {‖ (u_{0}, w_{0}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}]}^{\frac{1}{2}}, \forall t \in [0, T] . \end{matrix}$

Proof.

By applying the Fourier Transform and taking the scalar product in $C^{2}$ of the first equation of (1) with $\hat{u} (t)$ , one has $\begin{array}{l} (38) & \frac{1}{2} \partial_{t} {| \hat{u} (t) |}^{2} + (μ + χ) | \hat{\nabla u} |^{2} & = - Re [\hat{u} \cdot \hat{u \cdot \nabla u}] + χ Re [\hat{u} \cdot \hat{\nabla \times w}] . \end{array}$ Similarly, considering the second equation of (1), one obtains $\begin{array}{l} (39) & \frac{1}{2} \partial_{t} {| \hat{w} (t) |}^{2} + γ | \hat{\nabla w} |^{2} + 2 χ | \hat{w} |^{2} & = - Re [\hat{w} \cdot \hat{u \cdot \nabla w}] + χ Re [\hat{w} \cdot \hat{\nabla \times u}] . \end{array}$ By using (38) and (39) and the fact that $\hat{u} \cdot \hat{\nabla \times w} = \hat{\nabla \times u} \cdot \hat{w}$ , it follows that $\begin{array}{l} 3 \frac{1}{2} \partial_{t} {| (\hat{u}, \hat{w}) (t) |}^{2} + (μ + χ) | \hat{\nabla u} |^{2} + γ | \hat{\nabla w} |^{2} + 2 χ | \hat{w} |^{2} \\ = - Re [\hat{u} \cdot \hat{u \cdot \nabla u} + \hat{w} \cdot \hat{u \cdot \nabla w} - 2 χ \hat{\nabla \times u} \cdot \hat{w}] . \end{array}$ By applying Cauchy–Schwarz’s inequality, we obtain $\begin{array}{l} 2 Re [\hat{\nabla \times u} \cdot \hat{w}] ⩽ 2 | \hat{\nabla u} | | \hat{w} | ⩽ | \hat{\nabla u} |^{2} + | \hat{w} |^{2} . \end{array}$ Therefore, $\begin{array}{l} \frac{1}{2} \partial_{t} {| (\hat{u}, \hat{w}) (t) |}^{2} + μ | \hat{\nabla u} |^{2} + γ | \hat{\nabla w} |^{2} + χ | \hat{w} |^{2} ⩽ - Re [\hat{u} \cdot \hat{u \cdot \nabla u} + \hat{w} \cdot \hat{u \cdot \nabla w}] . \end{array}$ As a consequence, one has $\begin{array}{l} \partial_{t} {| (\hat{u}, \hat{w}) (t) |}^{2} + 2 θ {| (\hat{\nabla u}, \hat{\nabla w}) |}^{2} ⩽ - 2 Re [\hat{u} \cdot \hat{u \cdot \nabla u} + \hat{w} \cdot \hat{u \cdot \nabla w}], \end{array}$ where $θ = min {μ, γ}$ . Multiplying the inequality above by $| ξ |^{2 s} e^{2 t | ξ |}$ , where $t ⩾ 0$ , and integrating over $ξ \in R^{2}$ , we have $\begin{array}{l} 2 Re {⟨ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)), F^{- 1} (e^{t | \cdot |} ({\hat{u}}_{t}, {\hat{w}}_{t}) (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} + 2 θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ - 2 Re [{⟨ F^{- 1} (e^{t | \cdot |} \hat{u} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \cdot \nabla u} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} \\ (40) & + {⟨ F^{- 1} (e^{t | \cdot |} \hat{w} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \cdot \nabla w} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})}] . \end{array}$ On the other hand, one obtains $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} & = Re {⟨ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)), F^{- 1} (| \cdot | e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} \\ + Re {⟨ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)), F^{- 1} (e^{t | \cdot |} ({\hat{u}}_{t}, {\hat{w}}_{t}) (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} . \end{array}$ Thereby, by applying Cauchy–Schwarz’s inequality, it results that $\begin{array}{l} 2 Re {⟨ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)), F^{- 1} (e^{t | \cdot |} ({\hat{u}}_{t}, {\hat{w}}_{t}) (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} \\ ⩾ \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ - 2 {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}, \end{array}$ since $| (\hat{\nabla u}, \hat{\nabla w}) | = | ξ | | (\hat{u}, \hat{w}) |$ . Once again, by using Cauchy–Schwarz’s inequality, one has $\begin{array}{l} 2 Re {⟨ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)), F^{- 1} (e^{t | \cdot |} ({\hat{u}}_{t}, {\hat{w}}_{t}) (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} \\ ⩾ \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ (41) & - \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} - θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} . \end{array}$ Thus, replace the inequality (41) in (40) in order to reach $\begin{array}{l} \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ - 2 Re [{⟨ F^{- 1} (e^{t | \cdot |} \hat{u} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \cdot \nabla u} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} \\ (42) & + {⟨ F^{- 1} (e^{t | \cdot |} \hat{w} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \cdot \nabla w} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})}] . \end{array}$ On the other hand, notice that $\begin{array}{l} \hat{\nabla w} \cdot \hat{u \otimes w} (ξ) & = - \hat{w} \cdot \hat{u \cdot \nabla w} (ξ), \end{array}$ provided that $div u = 0$ . As a result, we get $\begin{array}{l} {⟨ F^{- 1} (e^{t | \cdot |} \hat{w} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \cdot \nabla w} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} & = - {⟨ F^{- 1} (e^{t | \cdot |} \hat{\nabla w} (t)), F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ⟩}_{{\dot{H}}^{s} (R^{2})} . \end{array}$ Hence, (42) can be rewritten, by using Cauchy–Schwarz’s inequality, as follows: $\begin{array}{l} \underline{} & \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ + 2 {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} \\ (43) & \times [{‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes u} (t)) ‖}_{{\dot{H}}^{s} (R^{2})} + {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}] . \end{array}$ From now on, we shall continue this demonstration by studying two cases.

1º Case: Assume that $s > 1$ :

By using Lemma 2.5, one has $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} & = \sum_{j, k = 1}^{2} {‖ F^{- 1} (e^{t | \cdot |} \hat{w_{j} u_{k}} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ C_{s} e^{2 t} [{‖ F^{- 1} (e^{t | \cdot |} \hat{w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} {‖ u (t) ‖}_{L^{2} (R^{2})}^{2} \\ + {‖ F^{- 1} (e^{t | \cdot |} \hat{u}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} {‖ w (t) ‖}_{L^{2} (R^{2})}^{2} \\ + {‖ F^{- 1} (e^{t | \cdot |} \hat{w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} {‖ F^{- 1} (e^{t | \cdot |} \hat{u} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}], \end{array}$ where $C_{s}$ is a positive constant. From (17), it results $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})} & ⩽ C_{s, u_{0}, w_{0}} e^{t} [{‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} \\ + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}], \end{array}$ where $C_{s, u_{0}, w_{0}}$ is a positive constant that relies only on s and $‖ (u_{0}, w_{0}) ‖_{L^{2} (R^{2})}$ . Consider that $T^{*} > 0$ is fixed and apply Young’s inequality in order to get $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})} & ⩽ C_{s, u_{0}, w_{0}, T^{*}} [1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}], \forall t \in [0, T^{*}] . \end{array}$ Consequently, (43) becomes $\begin{array}{l} \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ + C_{s, u_{0}, w_{0}, T^{*}} {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} [1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}] \end{array}$ for all $t \in [0, T^{*}]$ . By using Cauchy–Schwarz’s inequality, one infers $\begin{array}{l} \frac{d}{d t} [1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}] \\ ⩽ C_{s, θ, u_{0}, w_{0}, T^{*}} {[1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}]}^{2}, \forall t \in [0, T^{*}] . \end{array}$ Thus, by integrating the inequality above over $[0, t]$ ( $t \in [0, T^{*}]$ ), we reach $\begin{array}{l} (44) & φ (t) ⩽ φ (0) + C_{s, θ, u_{0}, w_{0}, T^{*}} \int_{0}^{t} φ {(τ)}^{2} d τ, \forall t \in [0, T^{*}], \end{array}$ where $\begin{array}{l} φ (t) = 1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}, \forall t \in [0, T^{*}] . \end{array}$ Let us denote $T^{'} = {[8 C_{s, θ, u_{0}, w_{0}, T^{*}} φ (0)]}^{- 1}$ (where $C_{s, θ, u_{0}, w_{0}, T^{*}}$ is given in (44)) and $T^{″} = sup {t \in [0, T^{*}) : φ (τ) ⩽ 2 φ (0), \forall τ \in [0, t]}$ . As a consequence, we assure that $\begin{array}{l} φ (t) & ⩽ φ (0) + C_{s, θ, u_{0}, w_{0}, T^{*}} \int_{0}^{t} φ {(τ)}^{2} d τ ⩽ φ (0) + 4 C_{s, θ, u_{0}, w_{0}, T^{*}} φ {(0)}^{2} t \\ ⩽ φ (0) [1 + 4 C_{s, θ, u_{0}, w_{0}, T^{*}} φ (0) T^{'}] ⩽ 2 φ (0) \end{array}$ for all $t \in [0, T]$ , where $T = \frac{1}{2} min {T^{″}, T^{'}}$ . Rewriting the result above obtained, we have $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 {‖ (u_{0}, w_{0}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}, \forall t \in [0, T] . \end{array}$ (Notice that T depends only on $s, θ, ‖ (u_{0}, w_{0}) ‖_{H^{s} (R^{2})}$ .)

2º Case: Assume that $s \in (0, 1)$ :

Note that, by utilizing Lemma 2.4, we get $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} & = \sum_{j, k = 1}^{2} {‖ F^{- 1} (e^{t | \cdot |} \hat{w_{j} u_{k}} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ C_{s} \sum_{j, k = 1}^{2} [{‖ e^{t | \cdot |} {\hat{w}}_{j} (t) ‖}_{L^{1} (R^{2})} {‖ F^{- 1} (e^{t | \cdot |} {\hat{u}}_{k} (t)) ‖}_{{\dot{H}}^{s} (R^{2})} \\ + {‖ e^{t | \cdot |} {\hat{u}}_{k} (t) ‖}_{L^{1} (R^{2})} {‖ F^{- 1} (e^{t | \cdot |} {\hat{w}}_{j} (t)) ‖}_{{\dot{H}}^{s} (R^{2})}]^{2}, \end{array}$ where $C_{s}$ is a positive constant. Now, apply the inequality (37) in order to obtain $\begin{array}{l} (45) & {‖ F^{- 1} (e^{t | \cdot |} \hat{u \otimes w} (t)) ‖}_{{\dot{H}}^{s} (R^{2})} & ⩽ C_{s} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{s + 1} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s + 1} (R^{2})}^{1 - s} . \end{array}$ On the other hand, it is also true that $\begin{array}{l} (46) & {‖ F^{- 1} (e^{t | \cdot |} (\hat{\nabla u}, \hat{\nabla w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} & = {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s + 1} (R^{2})}^{2}, \end{array}$ since $| (\hat{\nabla u}, \hat{\nabla w}) | = | ξ | | (\hat{u}, \hat{w}) |$ . By replacing (45) and (46) in (43), we obtain $\begin{array}{l} \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + θ {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s + 1} (R^{2})}^{2} \\ ⩽ \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ + C_{s} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s + 1} (R^{2})}^{2 - s} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{s + 1} . \end{array}$ By using Young’s inequality, one infers $\begin{array}{l} \frac{d}{d t} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + \frac{θ}{2} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s + 1} (R^{2})}^{2} \\ ⩽ \frac{1}{θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} + C_{s, θ} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{\frac{2 s + 2}{s}} . \end{array}$ Once again, by applying Young’s inequality, we have $\begin{array}{l} \frac{d}{d t} [1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}] & ⩽ C_{s, θ} {[1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}]}^{\frac{s + 1}{s}} . \end{array}$ Thus, by integrating the inequality above over $[0, t]$ ( $t ⩾ 0$ ), we reach $\begin{array}{l} 1 + {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} \\ ⩽ [1 + {‖ (u_{0}, w_{0}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}] + C_{s, θ} \int_{0}^{t} {[1 + {‖ F^{- 1} (e^{τ | \cdot |} (\hat{u}, \hat{w}) (τ)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}]}^{\frac{s + 1}{s}} d τ . \end{array}$ As it was done in the first case, we obtain $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 {‖ (u_{0}, w_{0}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2}, \forall t \in [0, T] . \end{array}$ (Notice that T depends only on $s, θ, ‖ (u_{0}, w_{0}) ‖_{{\dot{H}}^{s} (R^{2})}$ .) □

Now, recall that the limit (4) (since $(u, w) \in C ([0, \infty); H^{s} (R^{2}))$ ) assures that there is a positive constant M such that $\begin{array}{l} (47) & {‖ (u, w) (t) ‖}_{H^{s} (R^{2})} ⩽ M, \forall t ⩾ 0 . \end{array}$ The result below is useful to present the proof of Theorem 1.3(ii) and (iii).

Lemma 3.3.

There is an instant $t = T$ that depends only on $s, μ, γ$ and M such that $\begin{matrix} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} ⩽ {[1 + 2 M^{2}]}^{\frac{1}{2}}, \forall t ⩾ T, \end{matrix}$ where M is given in ( 47 ).

Proof.

By considering the system $\begin{matrix} (48) & \{\begin{matrix} v_{t} + v \cdot \nabla v + \nabla p = (μ + χ) Δ v + χ \nabla \times b, x \in R^{2}, t ⩾ 0, \\ b_{t} + v \cdot \nabla b = γ Δ b + χ \nabla \times v - 2 χ b, x \in R^{2}, t ⩾ 0, \\ div v = 0, x \in R^{2}, t > 0, \\ v (\cdot, 0) = u (\cdot, T_{1}), b (\cdot, 0) = w (\cdot, T_{1}), x \in R^{2}, \end{matrix} \end{matrix}$ where $T_{1} ⩾ 0$ is arbitrary, we obtain, by following the proof of Lemma 3.2, a constant T (which depends only on $s, θ, M$ ) such that $\begin{array}{l} {‖ F^{- 1} (e^{t | \cdot |} (\hat{v}, \hat{b}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 {‖ (v, b) (0) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} = 1 + 2 {‖ (u, w) (T_{1}) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 M^{2} \end{array}$ for all $t \in [0, T]$ . In particular, we infer $\begin{array}{l} {‖ F^{- 1} (e^{T | \cdot |} (\hat{v}, \hat{b}) (T)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 M^{2}, \end{array}$ that is, $\begin{array}{l} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (T + T_{1})) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 M^{2} . \end{array}$ Now, suppose that $t ⩾ T$ in order to obtain (for $T_{1} = t - T ⩾ 0$ ) $\begin{array}{l} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 M^{2}, \forall t ⩾ T . \end{array}$ □

Let us rewrite Theorem 1.3(ii) and (iii) as a proposition.

Proposition 3.4.

The following limits hold:

${lim}_{t \to \infty} t^{\frac{s}{2}} ‖ (u, w) (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ ;

${lim}_{t \to \infty} t^{\frac{s + 1}{2}} ‖ w (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ , if $χ > 0$ .

Proof.

By Lemma 3.3, it results that $\begin{array}{l} (49) & {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})}^{2} ⩽ 1 + 2 M^{2} = : M_{1}^{2}, \forall t ⩾ T . \end{array}$ By applying Young’s inequality, we have $\begin{array}{l} {‖ (u, w) (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & = \int_{R^{2}} | ξ |^{2 s} e^{2 a | ξ |^{\frac{1}{σ}}} {| (\hat{u}, \hat{w}) (t) |}^{2} d ξ ⩽ C_{T, σ, a} \int_{R^{2}} | ξ |^{2 s} e^{T | ξ |} {| (\hat{u}, \hat{w}) (t) |}^{2} d ξ . \end{array}$ Now, use Cauchy–Schwarz’s inequality and (49) in order to conclude that $\begin{array}{l} t^{\frac{s}{2}} {‖ (u, w) (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ C_{T, σ, a} t^{\frac{s}{2}} {(\int_{R^{2}} | ξ |^{2 s} {| (\hat{u}, \hat{w}) (t) |}^{2} d ξ)}^{\frac{1}{2}} {(\int_{R^{2}} | ξ |^{2 s} e^{2 T | ξ |} {| (\hat{u}, \hat{w}) (t) |}^{2} d ξ)}^{\frac{1}{2}} \\ = C_{T, σ, a} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} [t^{\frac{s}{2}} {‖ (u, w) (t) ‖}_{{\dot{H}}^{s} (R^{2})}] \\ ⩽ C_{T, σ, a} M_{1} [t^{\frac{s}{2}} {‖ (u, w) (t) ‖}_{{\dot{H}}^{s} (R^{2})}] \end{array}$ for all $t ⩾ T$ . Lastly, by applying Theorem 1.2(ii), it results that ${lim}_{t \to \infty} t^{\frac{s}{2}} ‖ (u, w) (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ . This proves the item (i).

Analogously, one obtains $\begin{array}{l} t^{\frac{s + 1}{2}} {‖ w (t) ‖}_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} & ⩽ C_{T, σ, a} t^{\frac{s + 1}{2}} {(\int_{R^{2}} | ξ |^{2 s} {| \hat{w} (t) |}^{2} d ξ)}^{\frac{1}{2}} {(\int_{R^{2}} | ξ |^{2 s} e^{2 T | ξ |} {| (\hat{u}, \hat{w}) (t) |}^{2} d ξ)}^{\frac{1}{2}} \\ = C_{T, σ, a} {‖ F^{- 1} (e^{T | \cdot |} (\hat{u}, \hat{w}) (t)) ‖}_{{\dot{H}}^{s} (R^{2})} [t^{\frac{s + 1}{2}} {‖ w (t) ‖}_{{\dot{H}}^{s} (R^{2})}] \\ ⩽ C_{T, σ, a} M_{1} [t^{\frac{s + 1}{2}} {‖ w (t) ‖}_{{\dot{H}}^{s} (R^{2})}] \end{array}$ for all $t ⩾ T$ . Lastly, by applying Theorem 1.2(iii), one has ${lim}_{t \to \infty} t^{\frac{s + 1}{2}} ‖ w (t) ‖_{{\dot{H}}_{a, σ}^{s} (R^{2})}^{2} = 0$ , if $χ > 0$ . This proves the item (ii). □

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Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev–Gevrey spaces

Abstract

Keywords

1. Introduction

Lemma 2.1 (See [1]).

Proof. For more details see [1]. □ It follows directly from Lemma 2.1 the statement below. Lemma 2.2 (See [1]).

References

Proof.
For more details see [1]. □

It follows directly from Lemma 2.1 the statement below.
Lemma 2.2 (See [1]).