Analyticity estimates for the 3D magnetohydrodynamic equations

1.   Introduction

The system of 3D incompressible MHD equations is given by

where $u = u(t, x)$, $b = b(t, x)$, $\Pi = \Pi(t, x)$ denote the velocity of the fluid, the magnetic field and the modified pressure, respectively. The system (1.1) can be widely applied in many fields, including geophysics, astrophysics, and engineering.

In the case of $b = 0$, the system (1.1) reduces to the classical Navier-Stokes system:

It is well-known that one of the most challenging problems in applied analysis is establishing the global well-posedness of the system (1.2) when the initial data is large. Thus, many papers have tried to obtain the global existence of solutions with small initial data. Giga [1] proved the local well-posedness of the Navier-Stokes equations with the initial data $u_{0}\in L^{p}(\mathbb{R}^{3})$ for some $p\in]3, \infty [$, whereas Kato and Masuda [2] established the endpoint case when $p = 3$. In 1981, Foias and Temam [3] proposed a technique which can be used to derive the analyticity radius of solutions of the system (1.2) and obtained the analyticity of periodic solutions. This result was then generalized by many authors; see [4,5,6,7] and its cited references. It should be emphasise that the analyticity radius estimates given above are lower estimates; in certain cases, the true analyticity radius may be significantly higher. Consequently, Biswas and Foias [8] by solving a new auxiliary ODE for the evolution of the analyticity radius involving the Gevrey class norms to obtain a more intimate connection between the radius of analyticity and the dynamics of the Navier-Stokes equations. Bae et al. [9] investigated the analyticity of the system (1.2) with the initial data in the critical Besov spaces. On this basis, Zhang [10] generalized the result to the initial data belonging to general Besov space.

For the MHD system (1.1), Duvaut and Lions [11], Miao and Yuan [12], and Wang and Wang [13] independently obtained the existence, uniqueness, and regularities of generalized solutions. Yu and Li [14] established the time analyticity radius of the $2$D MHD system with periodic boundary conditon, and they proved that if the initial data is close enough to a stationary solution, then the radius of the solution at $t = 0$ can be arbitrarily large. Wang et al. [15] utilized the Gevrey class method in [3] to prove the analyticity of the solutions to the system (1.1) with the initial data in the Lei-Lin space $\chi^{-1}(\mathbb{R}^{3})$. In [16], the analyticity of periodic solutions to the system (1.1) with initial data in Sobolev spaces $H^{s}(\mathbb{T}^{3})$ with $s > \frac{1}{2}$ had been established. By using the semigroup technique, Xiao and Yuan [17] derived the analytic estimates for small perturbation near the solutions to the generalized MHD system in the critical space $\chi^{1-\alpha}$ with $\frac{1}{2}\leq \alpha\leq 1$. For more results about the Hall-MHD system, the generalized MHD system and magnetohydrodynamics-$\alpha$ system, we may refer to [18,19,20,21] and the references cited therein.

Motivated by [10], this paper focuses on the analyticity of solutions to the system (1.1) when the initial data in $\dot{B}^{s}_{p, q}(\mathbb{R}^{3})$ for some $s\in[-1+\frac{3}{p}, \frac{3}{p}[$, $p\in]1, \infty[$, and $q\in[1, \infty]$. By utilizing the new fixed point theorem, the Mihlin-Hormander multiplier lemma, and the properties of the semigroup operator, lower bounds on the analytic radius of the MHD equations are proven in $\dot{B}^{s}_{p, q}(\mathbb{R}^{3})$ for some $s\in]-1+\frac{3}{p}, \frac{3}{p}[$, $p\in]1, \infty[$, and $q\in[1, \infty]$; in $\dot{B}^{-1+\frac{3}{p}}_{p, q}(\mathbb{R}^{3})$, $p\in]1, \infty[$, and $q\in[1, \infty]$; and $L^{p}(\mathbb{R}^{3})$ with $p\in]3, 6[$, respectively. In particular, it should be noted that by using the classical Bony decomposition, we estimate the nonlinear term in proving Theorems 1.2 and 1.3. Furthermore, our results correspond to Zhang's findings [10] when $b = 0$. Our conclusion is articulated as follows.

Theorem 1.1. Suppose $(u_{0}, b_{0})\in\dot{B}^{s}_{p, q}(\mathbb{R}^{3})$ for some $s\in]-1+\frac{3}{p}, \frac{3}{p}[$, $p\in]1, \infty[$, $q\in[1, \infty]$, and $\Lambda(D) = \sqrt{-\Delta}$. Then, there exist positive constants $t_{0} = t_{0}(\varepsilon, \|(u_{0}, b_{0})\|_{\dot{B}^{s}_{p, q}})$ and $c_{0} = c_{0}(\varepsilon)$ such that for any $\varepsilon < \varepsilon_{0}$, the MHD Eq (1.1) admits a unique solution $(u, b)$ on $[0, t_{0}]$ satisfying

and

Inspired by [10], we consider endpoint cases, i.e., $s = -1+\frac{3}{p}$.

Theorem 1.2. Let $(u_{0}, b_{0})\in \dot{B}^{-1+\frac{3}{p}}_{p, q}(\mathbb{R}^{3})$, $p\in]1, \infty[$, and $q\in[1, \infty[$. Then, there exist positive constants $t_{1} = t_{1}\big(\varepsilon, \|(u_{0}, b_{0})\|_{\dot{B}^{-1+\frac{3}{p}}_{p, q}}\big)$ and $c_{1} = c_{1}(\varepsilon)$ such that for any $0 < \eta < \varepsilon\leq\varepsilon_{1}$, the MHD Eq (1.1) admits a unique solution $(u, b) = (u_{L}, b_{L})+(v, w)$ with $(v, w)\in \mathcal{B}_{r}^{\varepsilon}(T)$ on $[0, t_{1}]$ which satisfies for any $r\in ]1, \infty[$ with $\frac{1}{r}+\frac{3}{p} > 1$ and $(u_{\eta, L}, b_{\eta, L})\overset{def}{ = }(e^{\eta t \Delta}(u_0, b_0))$,

and

where $\mathcal{A} = ({-\frac{1}{3}\ln\|(u_{\eta, L}, b_{\eta, L})\|_{\widetilde{L}_{T}^{2r}(\dot{B}_{p, q}^{-1+\frac{1}{r}+\frac{3}{p}})})^{\frac{1}{2}}}$ and the norm of $\widetilde{L}_{T}^{2r}(\dot{B}_{p, q}^{-1+\frac{1}{r}+\frac{3}{p}})$ is defined by Definition A3 in the Appendix.

Remark 1.1. When $(u_{0}, b_{0})\in \dot{B}^{-1+\frac{3}{p}}_{p, q}(\mathbb{R}^{3})$ with $q\in[1, \infty[$ and $0 < r < \infty$, we can naturally conclude that

which combining with (1.6), we can infer that

Now, we will consider the Gevrey regularity of solutions to the system (1.1) with initial data in $L^{p}(\mathbb{R}^{3})$. To do this, we introduce the following function space:

Definition 1.1. Let $0\leq t\leq T$, then for $p\in]3, 6[$, the definition of the norm of the function space $E_{p}(T)$ is as follows:

where $f$ is a homogeneous tempered distribution, and $\widetilde{L}^{r}_{T}(\dot{B}_{p, q}^{s})$ is the Chemin-Lerner type space where its norm is defined by Definition A3.

Theorem 1.3. Assume that $(u_{0}, b_{0})\in L^{p}(\mathbb{R}^{3})$ with $p\in]3, 6[$, then there exists a sufficiently small constant $\xi$, such that for any $T$ satisfying

the MHD Eq (1.1) admits a unique solution $(u, b) = (u_{L}, b_{L})+(v, w)$ with $(v, w)\in E_{p}(T)$ on $[0, t_{2}]$ which satisfies

and

The remainder of the paper is structured as follows: In Section 2, we show the proof of Theorem 1.1. Sections 3 and 4 are devoted to prove Theorems 1.2 and 1.3, respectively. In Section 5, we shall present some basic facts on Littlewood-Paley theory and functional spaces.

Notation: In this paper, the letter $C$ represents different positive and finite constants. The precise values of these constants are not significant and may differ from one line to another. For $A\lesssim B$, what we mean is a universal constant $C$, such that $A\leq CB$.

2.   The analyticity of solutions to system (1.1) with initial data in Besov space

Definition 2.1. Let $\Lambda(\xi) = \sum_{j = 1}^{3}|\xi_{j}|$, $\lambda > 0$, and $0\leq t\leq T$. For $p, q\in[1, \infty[$, $\varepsilon\in]0, 1[$, and $s\in\mathbb{R}$, the definition of the norm of the function space $B^{\varepsilon}_{s}(T)$ is as follows:

We recall the following Mihlin-Hörmander multiplier lemma on $\mathbb{R}^{n}$ from [22].

Lemma 2.1. ([22]) Let $m(\xi)$ be a complex-valued bounded function on $\mathbb{R}^{n}\setminus{0}$ that obeys for some $0 < A < \infty$

for all multi-indices $|\alpha|\leq [\frac{n}{2}]+1$ and $R > 0$. Then, for all $1 < p < \infty$, $m$ lies in $\mathcal{M}_{p}(\mathbb{R}^{3})$, and the following estimate is valid:

In order to make it easier for the reader, we present the following lemma that outlines the properties of the Heat semigroup operator acting on the initial data. The proof of this lemma can be accessed in [10], and we exclude the details here.

Lemma 2.2. Let $0 < \eta < \varepsilon$ and $m(t, \xi) = e^{-\frac{3\lambda^{2}}{4(1-\varepsilon)}\frac{t}{T}}e^{\lambda\frac{t}{\sqrt{T}}\Lambda(\xi)}e^{-(1-\eta)t|\xi|^{2}}$, then for any $p\in ]1, \infty[$, there exists a constant $C_{\varepsilon, \eta}$ so that

The following lemmas are also necessary for estimating the nonlinear terms.

Lemma 2.3. ([23]) Let $B_{t}(\cdot, \cdot)$ be the bilinear operator defined by

and it has the following expansion

Here, $\vec{\phi} = (\phi_{1}, \phi_{2}, \phi_{3})$ and $\vec{\chi} = (\chi_{1}, \chi_{2}, \chi_{3})$ belong to $\{-1, 1\}^{3}$, and the operators $K_{\phi} = K_{\phi_{1}}\otimes K_{\phi_{2}}\otimes K_{\phi_{3}}$ and $Z_{t, \vec{\phi}, \vec{\chi}} = \sum_{i = 1}^{3}K_{\chi_{i}}L_{t, \phi_{i}, \chi_{i}}$ can be defined by

and

Then, for any $p\in]1, \infty[$, we have

Lemma 2.4. ([10]) Let $u, v\in \mathcal{B}_{s}^{\varepsilon}(T)$ and $B(\cdot, \cdot)$ be the bilinear operator, then for $p\in]1, \infty[$ and $s\in]-1+\frac{3}{p}, \frac{3}{p}[$, there holds

The subsequent lemma is crucial in substantiating our findings.

Lemma 2.5. ([24]) Let $X$ be a Banach space, $L$ be a continuous linear map from $X$ to $X$, and $B$ be a bilinear map from $X\times X$ to $X$. Let us define

If $\|L\|_{\mathcal{L}(X)} < 1$, then for any $x_{0}$ in $X$ such that

the equation

admits a unique solution in the ball of center $0$ and radius $\frac{1-\|L\|_{\mathcal{L}(X)}}{2\|B\|_{\mathcal{B}(x)}}$.

Now, we give the complete proof of Theorem 1.1. Before proceeding, we denote $u_{L}\overset{def}{ = }e^{t\Delta}u_{0}$ and $b_{L}\overset{def}{ = }e^{t\Delta}b_{0}$. Let $\mathbb{P} = Id-\nabla \Delta^{-1} \mathrm{div}$ be the orthogonal projection of $L^2$ over divergence-free vector fields. By applying $\mathbb{P}$ to (1.1), we attain

then the system (1.1) can be equivalently reformulated as

with $B(f, g)\overset{def}{ = }\int_{0}^{t}e^{(t-s)\Delta}\mathbb{P}\nabla\cdot(f\otimes g)\mathrm{d}s$. We deduce from Lemmas 2.2 and A2 that

and

Combining (2.4) and (2.5), we have

Definition 2.1 implies that

Subsequently, to deduce Theorem 1.1 from Lemma 2.5, let $X$ be $\mathcal{B}^{\varepsilon}_{s}(T)$ as in Definition 2.1, $L$ be $0$, and $x_{0}$ be set to $(u_{L}, b_{L})$, then owing to Lemma 2.4 and (2.6), we get

To ensure contraction condition (2.2) holds ture, we take

so that there exist $\delta$ and $T < 1$ sufficiently small to have

Then, the system (2.3) admits a unique solution $(u, b)\in\mathcal{B}^{\varepsilon}_{s}$ in the ball of center $0$ and radius $\frac{1}{2\|B\|_{\mathcal{B}(X)}}$. Moreover, for any $\varepsilon\leq \varepsilon_{0}$ and $T\leq t_{0}(\varepsilon, \|(u_{0}, b_{0})\|_{\dot{B}^{s}_{p, q}})$, we have

3.   The analyticity of solutions to system (1.1) with initial data in critical Besov space

Motivated by [10], this section is to give the proof of Theorem 1.2. In order to do so, we first introduce a functional space.

Definition 3.1. Under the conditions of Definition 2.1, let $s = -1+\frac{3}{p}$, thus the definition of the norm of the function space $\mathcal{B}_{r}^{\varepsilon}(T)$ is as follows:

where $r\in]0, \infty[$ and $\frac{3}{p}+\frac{1}{r} > 1$.

Lemma 3.1. Let $b_{0}\in\dot{B}_{p, q}^{-1+\frac{3}{p}}(\mathbb{R}^{3})$ and $b_{\eta, L} = e^{\eta t\Delta}b_{0}$ for some $\eta\in]0, \varepsilon[$ and $\varepsilon\in]0, 1[$. Then, we have

Proof. By utilizing Bony's decomposition Definition A2 to $b_{L}\otimes b_{L}$, we have

where

For $0 < t\leq T$, denote $b_{L}^{\lambda}\overset{def}{ = }e^{\lambda\frac{t}{\sqrt{T}}\Lambda(D)}b_{L}$. Then, we get

where the operators $Z_{t, \vec{\phi}, \vec{\chi}}$ and $Z_{t, \vec{\phi}, \vec{\psi}}$ are defined by Lemma 2.3 and $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. By using Lemmas 2.2 and 2.3, we deduce

For the case $r > 1$, we can conclude from Lemmas 2.3 and A1 that

By substituting estimates (3.3) and (3.4) into (3.2), we get

Along the same lines of the proof of (3.5), for $p\in[2, \infty[$, we get

where we used the fact that $\frac{3}{p}+\frac{1}{r} > 1$ in the last step. For $p\in]1, 2[$, taking advantage of Lemmas 2.2, 2.3 and A1, we get

By summing up the estimates (3.5)–(3.7), for $p\in]1, \infty[$, we obtain

Meanwhile, for $t\leq T$ and $\varepsilon\in]0, 1[$, by using Lemmas 2.2 and A1, we get that

Combining (3.9) with (3.8), we have

Along the same lines, we obtain

We deduce from the time-interpolation formula that

which together with Definition 3.1 and (3.10) ensures Lemma 3.1. This proof is complete. □

Lemma 3.2. Let $u, b\in \mathcal{B}^{\varepsilon}_{r}(T)$, then we have

Proof. Similar to the proof of Lemma 3.1, let $u^{\lambda}\overset{def}{ = }e^{\lambda\frac{t}{\sqrt{T}}\Lambda(D)}u(t)$, and we obtain for $0\leq t\leq T$

where $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. Then, by using Lemmas 2.3 and A1, we infer that

Thus, we have

Along the same lines of the proof of (3.14), we can show that

By using a method similar to get (3.7) and Lemma A1, for $p\in[2, \infty[$, we obtain

where we used the fact that $\frac{3}{p}+\frac{1}{r} > 1$ in the last step. The case for $p\in]1, 2[$ is similar to that of (3.7). Hence, for $p\in]1, \infty[$, we have

Using a similar approach to (3.10), we derive

and

Combined with Definition 3.1, this proof is complete. □

Lemma 3.3. Let $u\in \mathcal{B}^{\varepsilon}_{r}(T)$ and $b_{0}\in \dot{B}_{p, q}^{-1+\frac{3}{p}}$, then the following inequality holds

Proof. According to the Bony decomposition of Definition A2, we have

By applying an argument similar to the one used to prove (3.2) shows that

where $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. Then, we can conclude from (3.3) and (3.13) that

Along the same lines of the proof of (3.19), it is easy to verify that

By using a method similar to get (3.7) and Lemma A1, for $p\in[2, \infty[$, we obtain

The case for $p\in]1, 2[$ is similar to that of (3.7). Hence, (3.19)–(3.21) implies that for $p\in]1, \infty[$,

The remainder of the argument is analogous to that in Lemma 3.2 and is left to the reader. □

We are now in a position to prove Theorem 1.2. We shall use Lemma 2.5 and denote $(v, w)\overset{def}{ = }(u-u_{L}, b-b_{L})$; thus, from (2.3), we can express

Let $X\overset{def}{ = }\mathcal{B}_{r}^{\varepsilon}(T)$ be defined by Definition 3.1,

Based on Lemmas 3.1–3.3, it turns out that

Let us now examine the conditions (2.2) of Lemma 2.5. Since $(u_{0}, b_{0})\in\dot{B}_{p, q}^{-1+\frac{3}{p}}$, we can choose $\delta$ and $t_{1}(\varepsilon, \eta, \|(u_{0}, b_{0})\|_{\dot{B}_{p, q}^{-1+\frac{3}{p}}})$ sufficiently small, such that

For $T\leq t_{1}$, we set

From (3.25) and (3.26), it suffices to have

Using (3.27), we can show that (2.2) holds, and, therefore, the system (3.23) admits a unique solution $(v, w)\in \mathcal{B}_{r}^{\varepsilon}(T)$ satisfying

Consequently, from (3.26) and (3.28), we have

4.   The Gevrey regularity of solutions to system (1.1) with initial data in $L^{p}$

Following [10], we shall study the Gevrey regularity of solutions to system (1.1) and the instantaneous radius of space analyticity of $(v, w) = (u-u_{L}, b-b_{L})$ with the initial data in $L^{p}$ for $p\in]3, 6[$.

4.1.   The Gevrey regularity of solutions to system (1.1)

Lemma 4.1. Let $b_{0}\in L^{p}(\mathbb{R}^{3})$ and $b_{L} = e^{t\Delta}b_{0}$. Then, for $p\in[2, \infty[$, we have

Proof. By definition of the para-product of $b_{L}$ and $b_{L}$ in Definition A2, we have

where we use the fact that $L^{p}\hookrightarrow\dot{B}^{0}_{p, p}$, and by virtue of Lemma A2, we infer that

By using Lemma A1 and (4.2), for $p\in[2, \infty[$, we obtain

Using (4.1) and (4.3), we can show that

We infer an estimate similar to (3.17) and (3.18) such that

which combined with Definition 1.1 completes the proof of Lemma 4.1. □

Lemma 4.2. Let $u, b\in E_{p}(T)$ with $p\in]3, 6[$ and $\frac{1}{\gamma} = 1-\frac{3}{p}$, then we have

Proof. In a similar way as in the proof of Lemma 4.1, we obtain

where we use the fact that

and

The term $\|\Delta_{l}(T_{b}u)\|_{L_{T}^{\frac{p}{3}}(L^{p})}$ has the same bound as (4.5). By using Lemma A1 and (4.6), for $p\in[2, \infty[$, we obtain

By a suitable modification of the proof of Lemma 4.1, we can show that

This combined with Definition 1.1 ensures Lemma 4.2. □

Lemma 4.3. Let $u\in E_{p}(T)$ and $b_{0}\in L^{p}$ with $p\in]3, 6[$, then we get

Proof. By applying an argument similar to the one used to prove (4.5), as well as Hölder's inequality and (4.2), one shows that

The term with $T_{b_{L}}u$ has the same bound as (4.8). Owing to Lemma A1, we have

From estimates (4.8) and (4.9), it is evident that

which together with Definition 1.1 ensures Lemma 4.3. □

Now, let us present the existence part of Theorem 1.3 by using Lemma 2.5. We take $X\overset{def}{ = }E_{p}(T)$ according to Definition 1.1, and use (3.24) to choose $L$ and $x_{0}$. Then, we deduce from Lemmas 4.1 to 4.3 that

To ensure contraction condition (2.2) holds true, we take $T^{\frac{1}{2\gamma}} \|(u_{0}, b_{0})\|_{L^{p}}\leq \xi$ for a sufficiently small $\xi$, then by using Lemma 2.5, we get that the system (3.23) admits a unique solution $(v, w)\in E_{p}(T)$. We thus obtain the unique solution $(u, b) = (u_{L}+v, b_{L}+w)$ of the system (1.1) on $[0, T]$.

The aim of the next subsection is to study the instantaneous radius of analyticity of $(v, w)$ under the assumptions that the initial data $(u_{0}, b_{0})\in L^{p}(\mathbb{R}^{3})$ for $p\in]3, 6[$.

4.2.   The radius of analyticity of $(v, w)$

Definition 4.1. Under the conditions of Definition 1.1, let $p\in[2, \infty[$, then the definition of the norm of the function space $\mathcal{B}_{p}^{\varepsilon}(T)$ is as follows:

Lemma 4.4. Let $b_{0}\in L^{p}(\mathbb{R}^{3})$ with $2\leq p\leq \infty$ and $b_{L} = e^{t\Delta}b_{0}$ for some $\eta\in]0, \varepsilon[$ and $\varepsilon\in]0, 1[$. Then, we have

Proof. An argument similar to Lemma 3.1 shows that

for the operators $Z_{t, \vec{\phi}, \vec{\chi}}$ and $Z_{t, \vec{\phi}, \vec{\psi}}$ defined by Lemma 2.3 and $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. By virtue of $L^{p}\hookrightarrow\dot{B}^{0}_{p, p}(\mathbb{R}^{3})$ in Theorem 2.40 of [25] and (2.5), we have

By Lemmas 2.3 and A1, for $r > 1$, the following inequality holds true:

By substituting estimates (4.12) and (4.13) into (4.11), we get

Along the same lines as in the proof of (4.14), we get

By summing up the estimates (4.14) and (4.15), for $p\in[2, \infty[$, we obtain

We deduce from Lemmas 2.2 and A1 that for any $t\leq T$ and $\varepsilon\in]0, 1[$,

Combining (4.16) with (4.17), we have

which together with Definition A1, we get the desired result. □

Lemma 4.5. Let $v, w\in \mathcal{B}^{\varepsilon}_{p}(T)$ with $p\in ]3, 6[$, then for $\frac{1}{\gamma} = 1-\frac{3}{p}$, we have

Proof. We recall $v^{\lambda}\overset{def}{ = }e^{\lambda\frac{t}{\sqrt{T}}\Lambda(D)}v(t)$, and follow the same technique as in the proof of Lemma 4.4 to obtain for $0\leq t\leq T$,

where $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. By Lemmas 2.3 and A1, we deduce

We infer from the interpolation inequality that

Combining (4.18) and (4.19), we get

Along the same lines of the proof of (4.20), the term $\|e^{-\frac{3\lambda^{2}}{2(1-\varepsilon)}\frac{t}{T}}e^{\lambda\frac{t}{\sqrt{T}}\Lambda(D)}\Delta_{l}(T_{v}w)\|_{L_{T}^{\frac{p}{3}}(L^{p})}$ admits the same estimate. Moreover, from the same lines to prove (4.15) and Lemma A1, we get that

As a consequence, we obtain

Then, we get

which combines with (4.22) to ensure that

which together with Definition A1 ensures Lemma 4.5. □

Lemma 4.6. Let $w\in \mathcal{B}^{\varepsilon}_{p}(T)$ and $b_{0}\in L^{p}(\mathbb{R}^{3})$ with $p\in]3, 6[$, then the following inequality holds

Proof. According to the Bony decomposition of Definition A2, we have

By an argument similar to the one used to prove Lemma 4.5, we get that

where $(\vec{\phi}, \vec{\chi}, \vec{\psi})\in\{1, -1\}^{3}$. Then, by using (3.3) and (4.18), we infer that

Along the same lines as in the proof of (4.23), it is easy to verify that

Taking advantage of Lemma A1 and a similar way to get (4.15), we have

The remainder of the argument is analogous to that in Lemma 3.2 and is left to the reader. □

We now apply Lemma 2.5 to prove Theorem 1.3. Let $X\overset{def}{ = }\mathcal{B}_{p}^{\varepsilon}(T)$ according to Definition A1, and use (3.24) to choose $L$ and $x_{0}$. Based on Lemmas 4.4–4.6, it turns out that

We set

and we deduce from (4.25) such that

We take $t_{2}$ so small such that

where $\delta$ being sufficiently small, then we have

from which we can show that (2.2) is true. Thus, Lemma 2.5 leads us to conclude that the system (3.23) admits a unique solution $(v, w)\in \mathcal{B}_{p}^{\varepsilon}(T)$ satisfying

which implies

This is (1.9) of Theorem 1.3. (1.10) follows from (1.9) and, thus, we complete the proof of Theorem 1.3.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (No.11801443), Shaanxi Fundamental Science Research Project for Mathematics and Physics (No.22JSQ031 and 23JSQ046) and Young Talent Fund of Association for Science and Technology in Shaanxi, China (No.20230525).

Conflict of interest

The authors declare that there are no conflicts of interest.

Appendix: Littlewood-Paley decomposition and functional spaces

We start with the classical dyadic decomposition in $\mathbb{R}^{3}$; see [25]. Let $\chi, \varphi\in [0, 1]$ be two nonnegative radial functions which are supported, respectively, in the ball $\mathcal{B} = \{\xi\in \mathbb{R}^{3}, |\xi|\leq\frac{4}{3}\}$ and the annulus $\mathcal{C} = \{\xi\in \mathbb{R}^{3}, \frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$ such that

The homogeneous dyadic blocks $\dot{\Delta}_{j}$ and the homogeneous low-frequency cut-off operator $\dot{S}_{j}$ are defined for all $j\in\mathbb{Z}$ by

where $h\overset{def}{ = }\mathcal{F}^{-1}\varphi$ and $\widetilde{h}\overset{def}{ = }\mathcal{F}^{-1}\chi$. Formally, $\dot{\Delta}_{j} = \dot{S}_{j+1}-\dot{S}_{j}$ is a frequency projection to annulus $|\xi|\sim 2^{j}$, and $\dot{S}_{j}$ is a frequency projection to the ball $\{|\xi|\leq C 2^{j}\}$. The homogeneous dyadic blocks $\Delta_{j}$ are defined by

The homogeneous low-frequency cut-off operator $S_{j}$ is defined by

First, we recall the definition of the homogeneous Besov space.

Definition A1. (Besov Spaces) Let $\mathcal{S}'$ be the space of all tempered distributions. For $s$ is a real number and $(p, q)$ is in $[1, \infty]^{2}$, the homogeneous Besov space $\dot{B}^{s}_{p, q}$ is defined by

where

and

Now, we give the definition of the following so-called Bony decomposition.

Definition A2. (Bony's para-product) The para-product of $f$ and $g$ is defined by

The remainder of $f$ and $g$ is defined by

Then, Bony's decomposition reads

Lemma A1. (Bernstein's inequalities; see [25]) Let $\mathcal{B}$ be a ball and $\mathcal{C}$ be an annulus of $\mathbb{R}^3$. A constant $C$ exists such that for positive real number $\lambda$ and nonnegative integer $k$, any smooth homogeneous function $\sigma$ of degree $m$, and any couple of real numbers $(p, q)$ in $[1, \infty]^{2}$ with $q\geq p\geq 1$, we have

with $\sigma(D)u\stackrel{def}{ = }\mathcal{F}^{-1}(\sigma \hat{u})$.

Definition A3. (Chemin-Lerner type space; see [25]) For $T > 0$, $s\in\mathbb{R}$, and $0\leq r, q\leq\infty$, we set

We also need the following lemma to describe the action of the semigroup of the heat equation on distributions with Fourier transforms supported in an annulus.

Lemma A2 ([25]). Let $\mathcal{C}$ be an annulus. Then, there exist positive constants $c$ and $C$ such that for any $p$ in $[1, \infty]$ and any couple $(t, \lambda)$ of positive real numbers, we have