Global well-posedness and analyticity for generalized porous medium equation in critical Fourier-Besov-Morrey spaces

1. Introduction

We investigate the generalized porous medium equation in the whole space \(\mathbb{R}^{3}\),

where \(u=u(t,x)\) is a real-valued function, which denotes a density or concentration. The dissipative coefficient \(\mu>0\) corresponds to the viscous case, while \(\mu=0\) corresponds to the inviscid case. The fractional Laplacian operator \(\Lambda^{\alpha}\) is defined by Fourier transform as \(\widehat{\Lambda^{\alpha}u}=|\xi|^{\alpha}\hat{u}\), and \(P\) is an abstract operator.

The equation (1) was introduced in the first by Zhou et al. [1]. In fact, Equation (1) is obtained by adding the fractional dissipative term \(\mu \Lambda^{\alpha}u\) to the continuity equation (PME) \(u_{t}+\nabla\cdot(u V)=0\) given by Caffarelli and Vázquez [2], where the velocity \(V\) derives from a potential, \(V=-\nabla p\) and the velocity potential or pressure \(p\) is related to \(u\) by an abstract operator \(p=Pu\) [3].

For \(\mu=0\) and \( Pu=(-\Delta)^{-s}u=\Lambda^{-2s}u,\, 0< s< 1\); X. Zhou et al. [4] were interested in finding the strong solutions of the equation (1) which becomes the fractional porous medium equation in the Besov spaces \(B_{p,\infty}^{\alpha}\) and they obtained the local solution for any initial data in \(B_{1,\infty}^{\alpha}\). Moreover, in the critical case \(s=1\), the Equation (1) leads to a mean field equation [4, 5]. Let's take this opportunity to briefly quote some works on the well-posedness and regularity on those equations such as [4, 6] and the references therein.

On the other hand, an another similar model occurs in the aggregation equation, and plays a fundamental role in applied sciences such as physics, biology, chemistry, population dynamics. It describes a collective motion and aggregation phenomena in biology and in mechanics of continuous media [7,8]. In the aggregation equation, the abstract form pressure term \(Pu\) can also be represented by convolution with a kernel \(K\) as \(Pu=K*u\). The typical kernels are the Newton potential \(|x|^{\gamma}\) [9, 10], and the exponent potential \(-e^{-|x|}\) [11, 12]. For more results on this equation, we refer to [13, 14] and the references therein.

Recently, Zhou et al. [1] obtained the local well-posedness in Besov spaces for large initial data, and proved that the solution becomes global if the initial data is small, also, they studied a blowup criterion for the solution.

In addition, we can represent the Equation (1) with the same initial data by

As a consequence, this equation must be compared to the geostrophic model. So, the convective velocity is not absolutely divergence-free for the generalized porous medium equation. Additionally, if we assume that \(v\) is divergence-free vector function (\(\nabla\cdot v=0\)), the form (2) can contain the quasi-geostrophic (Q-G) equation [15, 16].

Inspired by the works [1, 17]; the aim of this paper is to prove the well-posedness results of Equation (1) and to give the Gevrey class regularity of the solution in homogeneous Fourier Besov-Morrey spaces under the condition that the abstract operator \(P\) is commutative with the operator \(e^{-\mu\sqrt{t}|D|^{\frac{\alpha}{2}}}\) and

Clearly, for the fractional porous medium equation, i.e. \(Pu=\Lambda^{-2s}u\), we get \(\sigma=1-2s\). If \(Pu=K*u\) in the aggregation equation, Wu and Zhang [18] proved a similar result under the condition \(\nabla K\in W^{1,1}\), \(\alpha\in (0,1)\). Corresponding to their case we give a same result for \(\sigma=0\) when \(\nabla K\in L^1\), and also a similar result for \(\sigma=1\) when \(K\in L^1\).

Throughout this paper, we use \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) to denote the homogenous Fourier Besov-Morrey spaces, \(C\) will denote constants which can be different at different places, \({\mathsf U}\lesssim{\mathsf V}\) means that there exists a constant \(C>0\) such that \({\mathsf U}\leq C{\mathsf V}\), and \(p'\) is the conjugate of \(p\) satisfying \(\frac{1}{p}+\frac{1}{p'}= 1\) for \(1\leq p\leq\infty\).

2. Preliminaries and main results

We start with a dyadic decomposition of \(\mathcal {\mathbb{R}}^n\). Suppose \(\chi \in C_0^\infty(\mathcal {\mathbb{R}}^n),\;\varphi\in C_0^\infty(\mathcal {\mathbb{R}}^n\setminus \{0\})\) satisfying \begin{gather*} \operatorname{supp}\chi \subset \left\{\xi\in {\mathbb{R}}^n:|\xi|\leq \frac 43\right\},\\ \operatorname{supp}\varphi \subset \left\{\xi\in {\mathbb{R}}^n:\frac 34\leq|\xi|\leq \frac 83\right\},\\ \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n,\\ \sum_{j\in \mathbb{Z}}\varphi(2^{-j}\xi)=1,\quad \xi \in \mathcal {\mathbb{R}}^n\backslash\{0\}, \end{gather*} and denote \(\varphi_{j}(\xi)=\varphi(2^{-j}\xi)\) and \(\mathcal{P}\) the set of all polynomials.

First, we recall the definition of Morrey spaces which are a complement of \(L^{p}\) spaces.

Definition 1. [19] For \(1 \leq p < \infty\), \(0\leq\lambda < n\), the Morrey spaces \(\mathrm{M}_{p}^{\lambda}=\mathrm{M}_{p}^{\lambda}(\mathbb{R}^{n})\) is defined as the set of functions \(f\in L_{loc}^{p}(\mathbb{R}^{n})\) such that

where \(B(x_{0},r)\) denotes the ball in \(\mathbb{R}^{n}\) with center \(x_{0}\) and radius \(r\).

It is easy to see that the injection \(\mathrm{M}_{p_{1}}^{\lambda}\hookrightarrow \mathrm{M}_{p_{2}}^{\mu}\) provided \(\frac{n-\mu}{p_{2}}\geq\frac{n-\lambda}{p_{1}}\) and \( p_{2}\leq p_{1}\), and \(\mathrm{M}_{p}^{0}=L^{p}\).

If \(1\leq p_{1},p_{2},p_{3}< \infty\) and \( 0\leq\lambda_{1},\;\lambda_{2},\,\lambda_{3}< n\) with \( \frac{1}{p_{3}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\) and \( \frac{\lambda_{3}}{p_{3}}=\frac{\lambda_{1}}{p_{1}}+\frac{\lambda_{2}}{p_{2}}\), then we have the H\"{o}lder type inequality \begin{equation*} \|fg\|_{\mathrm{M}_{p_{3}}^{\lambda_{3}}}\leq\|f\|_{\mathrm{M}_{p_{1}}^{\lambda_{1}}} \|g\|_{\mathrm{M}_{p_{2}}^{\lambda_{2}}}\,. \end{equation*}

Also, for \(1\leq p< \infty\) and \(0\leq\lambda< n,\)

for all \(\varphi\in L^{1}\) and \(g\in\mathrm{M}_{p}^{\lambda}\).

Definition 2.(homogeneous Fourier-Besov-Morrey spaces ) Let \(s\in\mathbb{R}, \;0\leq\lambda< n\), \(1\leq p< +\infty \) and \(1\leq q\leq+\infty\). The space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) denotes the set of all \(u\in \mathcal{S'}(\mathbb{R}^{n})/\mathcal{P}\) such that

with suitable modification made when \(q = \infty\).

Note that the space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}(\mathbb{R}^{n})\) equipped with the norm (6) is a Banach space. Since \(\mathrm{M}_{p}^{0}=L^{p}\), we have \(\mathcal{F} \dot{\mathcal{N}}_{p, 0, q}^{s}=F \dot{B}_{p, q}^{s}, \, \mathcal{F} \dot{\mathcal{N}}_{1, 0, q}^{s}=F \dot{B}_{1, q}^{s}=\dot{\mathcal{B}}_{q}^{s}\) and \(\mathcal{F} \dot{\mathcal{N}}_{1, 0, 1}^{-1}=\chi^{-1}\) where \(\dot{\mathcal{B}}_{q}^{s}\) is the Fourier-Herz space and \(\chi^{-1}\) is the Lei-Lin space [20].

Now, we recall the definition of the mixed space-time spaces.

Definition 3. Let \(s\in\mathbb{ R},\;1\leq p< \infty,\; 1\leq q,\rho\leq\infty, \;0\leq\lambda< n\), and \(I=[0,T),\;T\in(0,\infty]\). The space-time norm is defined on \(u(t,x)\) by \begin{eqnarray*} \|u(t,x)\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}= \Big\{\sum_{j\in \mathbb{Z}}2^{jqs}\| \varphi_{j}\widehat{u}\| _{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} ^q \Big\}^{1/q}, \end{eqnarray*} and denote by \(\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\) the set of distributions in \(S'(\mathbb{R}\times\mathbb{R}^{n})/\mathcal{P}\) with finite \(\|.\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})}\) norm. According to Minkowski inequality, we have \begin{equation*} \begin{gathered} L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\hookrightarrow \mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}),\quad \text{if } \rho\leq q, \\ \mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}) \hookrightarrow L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}),\quad \text{if } \rho\geq q\,, \end{gathered} \end{equation*} where \(\|u(t,x)\|_{L^\rho(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})} :=\Big(\int_I\|u(\tau,\cdot)\|^\rho_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}}d\tau\Big)^{1/\rho}\,.\)

Our first main result is the following theorem.

Theorem 4. Assume that the abstract operator \(P\) satisfies the condition (3). If \(0\leq\lambda< 3,\, 1\leq q\leq \infty,\, 1\leq p< \infty\) and \(\max\{1+\sigma,0\}< \alpha< 2+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma\) then there exists a constant \(C_{0}\) such that for any \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\) satisfies \( \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}< C_{0}\mu\), the equation(1) admits a unique global solution \(u\), \begin{equation*} \|u\|_{\mathcal{L}^{\infty}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|u\|_{\mathcal{L}^{1}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq2C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma}} \end{equation*} where \(C\) is a positive constant.

Now, we give some remarks about this result.

Remark 1. The result stated in Theorem 4 is based on the works [3]. In particular, this result remains true if we replace the Fourier-Besov-Morrey space \(\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) by other functional spaces such as Fourier-Herz space \(\mathcal{\dot{B}}_{q}^{ s}\), Fourier-Besov space \(\mathrm{F\dot{B}}_{p,q}^{s}\) and Lei-Lin space \(\chi^{-1}\).

The analyticity of the solution is also an important subject developed by several researchers, particularly with regard to the Navier-Stokes equations, see [17] and its references. In this paper, we will prove the Gevrey class regularity for (1) in the Fourier-Besov-Morrey space. Inspired by this, we have obtained the following specific results.

Theorem 5. Let \(0\leq\lambda< 3,\,1\leq q\leq \infty,\,1\leq p< \infty\) and \(\max\{1+\sigma,0\}< \alpha< \min \{2,2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\}\). There exists a constant \(C_{0}\) such that, if \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\) satisfies \(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}< C_{0}\mu\), then the Cauchy problem (1) admits a unique analytic solution \(u\), in the sense that $$ \|e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u\|_{\mathcal{L}^{\infty}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u\|_{\mathcal{L}^{1}\left([0,\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq2C\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,.$$

We finish this section with a Bernstein type lemma in Fourier variables in Morrey spaces.

Lemma 6.[21] Let \(1\leq q\leq p< \infty,\, 0\leq\lambda_{1},\lambda_{2}< n,\;\frac{n-\lambda_{1}}{p}\leq\frac{n-\lambda_{2}}{q}\), and let \(\gamma\) be a multiindex. If \(supp(\widehat{f})\subset\{|\xi|\leq A2^{j}\}\) then there is a constant \(C>0\) independent of \(f\) and \(j\) such that

3. The well-posedness

First, we consider the linear nonhomogeneous dissipative equation for which we recall the following result.

Lemma 7. [22] Let \(I=[0,T),\;0< T\leq \infty,\,s\in\mathbb{R},\,0\leq\lambda< 3, 1\leq p< \infty\), and \(1\leq q,\rho\leq \infty.\) Assume that \(u_{0}\in \mathcal{F\dot{N}}_{p,\lambda,q}^{s}\) and \(f\in \mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s-\alpha+\frac{\alpha}{\rho}}\right)\). Then the Cauchy problem (8) has a unique solution \(u(t,x)\) such that for all \(\rho_{1}\in[\rho,+\infty]\) \begin{eqnarray*} \mu^{\frac{1}{\rho_{1}}}\|u\|_{\mathcal{L}^{\rho_{1}}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\frac{\alpha}{\rho_{1}} }\right)}\leq \Big(\frac{4}{3}\Big)^{\alpha}\Big(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s}} +\mu^{\frac{1}{\rho}-1}\|f\|_{\mathcal{L}^{\rho}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\frac{\alpha}{\rho}-\alpha})}\Big) \end{eqnarray*} and \begin{eqnarray*} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s}\right)} +\mu\|u\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s+\alpha }\right)} \leq(1+\left(\frac{4}{3}\right)^{\alpha})\left(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{s }}+\|f\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s }\right)}\right). \end{eqnarray*} If in addition \(q\) is finite, then u belongs to \(\mathcal{C}(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{s})\).

Proposition 8. Let \(1\leq p< \infty,\,1\leq \rho,\,q\leq \infty,\,1+\sigma< \alpha< \frac{2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}{2-\frac{1}{\rho}},\, 0\leq\lambda< 3,\,I=[0,T),\,T\in(0,\infty]\), and set \begin{equation*} X=\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\frac{\alpha}{\rho}+\sigma}\right), \end{equation*} with the norm \begin{equation*} \|u\|_{X}=\|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} +\mu\|u\|_{\mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\frac{\alpha}{\rho}+\sigma}\right) }\,. \end{equation*} There exists a constant \(C=C(p,q)>0\) depending on \(p,q\) such that

Proof. Let us introduce some notations about the standard localization operators. We set \begin{align*} u_{j}=\dot{\Delta}_{j}u=\left(\mathscr{F}^{-1} \varphi_{j}\right)* u,\;\;\;\dot{S}_{j}u=\sum_{k\leq j-1}\dot{\Delta}_{k}u,\;\;\; \widetilde{\dot{\Delta}}_{j}u=\sum_{|k-j|\leq 1}\dot{\Delta}_{k}u,\;\;\; \forall j\in \mathbb{Z}\,. \end{align*} Using the decomposition of Bony's paraproducts for the fixed \(j\), we have \begin{align*} \dot{\Delta}_{j}(u\partial_{i}Pv) &=\sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}u \dot{\Delta}_{k}(\partial_{i}Pv))+ \sum_{|k-j|\leq 4}\dot{\Delta}_{j}(\dot{S}_{k-1}(\partial_{i}Pv) \dot{\Delta}_{k}u)+\sum_{k\geq j-3}\dot{\Delta}_{j}(\dot{\Delta}_{k}u \widetilde{\dot{\Delta}}_{k}(\partial_{i}Pv))\\ &=I_{j}+II_{j}+III_{j}\,. \end{align*} To prove this proposition, we can write

We treat the above three terms differently. First, using Young's inequality (5) in Morrey spaces, and Lemma 6 with \(|\gamma|=0\), we get \begin{align*} \|\widehat{I_{j}}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}&\leq \sum_{|k-j|\leq 4}\|\widehat{\dot{S}_{k-1}u \dot{\Delta}_{k}(\partial_{i}P v)}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}\\ &\leq \sum_{|k-j|\leq 4}\|\varphi_{k}\mathcal{F}(\partial_{i}P v)\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}\| \varphi_{l}\hat{u}\|_{L^{\infty}\left(I ,L^{1}\right)}\\ &\leq\sum_{|k-j|\leq 4}\|\varphi_{k}\mathcal{F}(\partial_{i}P v)\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\sum_{l\leq k-2}2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\| \widehat{u}_{l}\|_{L^{\infty}\left(I ,\mathrm{M}_{p}^{\lambda}\right)}\\ &\lesssim\sum_{|k-j|\leq 4}2^{k\sigma}\|\widehat{v}_{k}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda} )}\Big(\sum_{l\leq k-2}2^{l(\alpha-1-\sigma)q'}\Big)^{\frac{1}{q'}} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\\ &\lesssim\sum_{|k-j|\leq 4}2^{k(\alpha-1)}\|\widehat{v}_{k}\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda} )} \|u\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Multiplying by \(2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\), and taking \(l^{q}-\)norm of both sides in the above estimate, we obtain Likewise, we prove that To evaluate \(III_{j}\), we apply the Young inequality (5) in Morrey spaces and Lemma 6 with \(|\gamma|=0\), we obtain \begin{eqnarray*} {2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\|\widehat{III_{j}}\|_{L^{\rho}(I ,\mathrm{M}_{p}^{\lambda})}}\\ &\leq& 2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} \big\|\mathcal{F}(\dot{\Delta}_{k}u\dot{\Delta}_{l}(\partial_{i}P v))\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\nonumber\\ &\leq&2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} \big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\big\|\varphi_{l}\mathcal{F}(\partial_{i}P v)\big\|_{L^{\infty}\left(I,L^{1}\right)}\nonumber\\ &\leq& 2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)}\sum_{k\geq j-3}\sum_{|l-k|\leq 1} 2^{l(\frac{3}{p'}+\frac{\lambda}{p})}\big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} 2^{l\sigma}\big\|\widehat{v}_{l}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\nonumber\\ &\leq& \sum_{k\geq j-3}\sum_{l=-1}^{1}2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+ \frac{\lambda}{p}+\sigma)(j-k)}2^{(\alpha-1)l} \big(2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)k} \big\|\widehat{u}_{k}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}\big)\\ \quad&\times&\big(2^{(l+k)(-(\alpha-1)+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)} \big\|\widehat{v}_{l+k}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\big)\,. \end{eqnarray*} Taking the \(l^{q}-\)norm on both sides in the above estimate and using H\"{o}lder's inequalities for series with \(-2(\alpha-1)+\frac{\alpha}{\rho}+\frac{3}{p'}+\frac{\lambda}{p}+\sigma>0\), we get \begin{eqnarray*} {\Big(\sum_{j\in\mathbb{Z}}2^{j(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)q} \|\widehat{III_{j}}\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})}^{q}\Big)^{\frac{1}{q}}}\\ &\leq&\Big(\sum_{j\in\mathbb{Z}}\Big(\sum_{m\leq 3}\sum_{l=-1}^{1} 2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)m}2^{(\alpha-1)l} 2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)(j-m)}\\ &&\times\big\|\widehat{u}_{j-m}\big\|_{L^{\rho}(I,\mathrm{M}_{p}^{\lambda})} 2^{(-(\alpha-1)+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)(j-m+l)} \big\|\widehat{v}_{j-m+l}\big\|_{L^{\infty}\left(I,\mathrm{M}_{p}^{\lambda}\right)}\Big)^{q}\Big)^{\frac{1}{q}}\\ &\leq&\sum_{l=-1}^{1}\sum_{m\leq 3} 2^{(-2(\alpha-1)+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma)m}2^{(\alpha-1)l} \|u\|_{\mathcal{L}^{\rho}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\alpha}{\rho}+\frac{\lambda}{p}+\sigma}\right)}\\ &&\times\|v\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,\infty}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{eqnarray*} Since \(l^{q} \hookrightarrow l^{\infty}\), we obtain Estimates (10), (11), (12) and (13) yield (9).

Lemma 9. Let \(X\) be a Banach space with norm \(\|.\|_{X}\) and \(B:X\times X\longmapsto X\) be a bounded bilinear operator satisfying \begin{equation*} \|B(u,v)\|_{X}\leq \eta \|u\|_{X}\|v\|_{X} \end{equation*} for all \(u,v\in X \) and a constant \(\eta >0\). Then, if \(0< \varepsilon< \frac{1}{4\eta}\) and if \(y\in X\) such that \(\|y\|_{X}\leq\varepsilon\), the equation \(x:=y+B(x,x)\) has a solution \(\overline{x}\) in \(X\) such that \(\|\overline{x}\|_{X}\leq 2 \varepsilon\). This solution is the only one in the ball \(\overline{B}(0,2\varepsilon)\). Moreover, the solution depends continuously on \(y\) in the sense: if \(\|y'\|_{X}\leq \varepsilon ,\;x'=y'+B(x',x')\), and \(\|x'\|_{X}\leq2\varepsilon\), then \begin{equation*} \|\overline{x}-x'\|_{X}\leq \frac{1}{1-4\varepsilon \eta}\|y-y'\|_{X}\,. \end{equation*}

Proof of theorem 4

Proof. To ensure the existence of global solutions with small initial data, we will use Lemma 9. In the following, we consider the Banach space \begin{equation*} X=\mathcal{L}^{\infty}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{1}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\,. \end{equation*} First, we start with the integral equation

We notice that \(B(u,v)\) can be thought as the solution to the heat Equation (8) with \(u_{0}=0\) and force \(f=\nabla\cdot(u(\tau)\nabla Pv(\tau))\). According to Lemma 7 with \(s=1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\) and Proposition 8 with \(\rho=1 \), we obtain \begin{align*} \|B(u,v)\|_{X}&\leq \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big) \|\nabla\cdot(u\nabla Pv)\|_{\mathcal{L}^{1}\left([0,+\infty);\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+ \frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\\ &\leq \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big)C \mu^{-1} \|u\|_{X}\|v\|_{X}\,. \end{align*} By Lemma 9, we know that if \(\|e^{-\mu t\Lambda^{\alpha}}u_{0}\|_{X}< R\) with \(R=\frac{\mu}{4(1+(\frac{4}{3})^{\alpha})C}\)\\ then the Equation (14) has a unique solution in \(B(0,2R):=\{x\in X:\|x\|_{X}\leq 2R\}\). To prove \(\|e^{-\mu t\Lambda^{\alpha}}u_{0}\|_{X}< R\), notice that \(e^{-\mu t\Lambda^{\alpha}}u_{0}\) is the solution to the dissipative equation with \(u_{0}=u_{0}\) and \(f=0\). So, Lemma 7 yields If \(\|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\leq C_{0}\mu\) with \(C_{0}=\frac{1}{4(1+(\frac{4}{3})^{\alpha})^{2}C}\), then (14) has a unique global solution \(u\in X\) satisfying \begin{eqnarray*} \|u\|_{X} \leq 2 \Big(1+\Big(\frac{4}{3}\Big)^{\alpha}\Big) \|u_{0}\|_{\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{eqnarray*}

Proof of theorem 5

Proof. To prove Theorem 5, we note \(a(t, x) :=e^{\mu \sqrt{t}|D|^{\frac{\alpha}{2}}} u(t, x)\,.\) Using the integral Equation (14), we obtain \begin{align*} a(t, x)&=e^{\mu(\sqrt{t}|D|^{\frac{\alpha}{2}}-\frac{1}{2}t\Lambda^{\alpha})} e^{-\frac{1}{2} \mu t \Lambda^{\alpha}} u_{0}\\ &\quad+\int_{0}^{t}e^{\mu[(\sqrt{t}-\sqrt{\tau})|D|^{\frac{\alpha}{2}}-\frac{1}{2}(t-\tau)\Lambda^{\alpha}]} e^{-\frac{1}{2}\mu(t-\tau)\Lambda^{\alpha}} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} \nabla \cdot(u \nabla(Pu))d \tau\\ &:=L u_{0}+\widetilde{B}(u,u)\,. \end{align*} In order to obtain the Gevrey class regularity of the solution, we use Lemma 9. Firstly, we start by estimating the term \(L u_{0}=e^{-\frac{1}{2}\mu (\sqrt{t}|D|^{\frac{\alpha}{2}}-1)^{2}+\frac{\mu}{2}}e^{-\frac{1}{2} \mu t \Lambda^{\alpha}} u_{0}\,.\) Using the Fourier transform, multiplying by \(\varphi_{j}\) and taking the \(\mathrm{M}_{p}^{\lambda}\)-norm we obtain \begin{equation*} \|\varphi_{j}\widehat{L u_{0}}\|_{\mathrm{M}_{p}^{\lambda}} \leq C e^{-\frac{1}{2}\mu t 2^{j \alpha}(3 / 4)^{\alpha}}\left\|\varphi_{j}\widehat{u_{0 }}\right\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{equation*} Multiplying by \(2^{j(1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\) and taking \(l^{q}-\)norm we get \begin{equation*} \left\|L u_{0}\right\|_{\mathcal{L}^{\infty}\left([0,+\infty) ; \mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} Similarly \begin{equation*} 2^{j(1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\left\|\varphi_{j}\widehat{L u_{0}}\right\|_{L^{1}\left([0,+\infty) ; \mathrm{M}_{p}^{\lambda}\right)} \leq \left(\int_{0}^{\infty} e^{-\frac{1}{2}\mu t 2^{j\alpha}(3 / 4)^{\alpha}} 2^{j \alpha} d t \right)2^{j (1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)}\left\|\varphi_{j}\widehat{u_{0}}\right\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{equation*} We conclude by taking \(l^{q}-\)norm that \begin{equation*} \mu\left\|L u_{0}\right\|_{\mathcal{L}^{1}\left([0,+\infty) ; \mathcal{F\dot{N}}_{p,\lambda, q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} Finally, \begin{equation*} \left\|L u_{0}\right\|_{X} \leq C \left\|u_{0}\right\|_{\mathcal{F\dot{N}}_{p,\lambda, q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}}\,. \end{equation*} On the other hand, we notice that \(\widetilde{B}(u, v)\) as \(\widetilde{B}\left(e^{-\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} a, e^{-\mu\sqrt{\tau} |D|^{\frac{\alpha}{2}}} b\right)\) with \(b :=e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}} v\). Since \(e^{\mu[(\sqrt{t}-\sqrt{\tau})|\xi|^{\frac{\alpha}{2}}-\frac{1}{2}(t-\tau)|\xi|^{\alpha}]}\) is uniformly bounded on \(t \in(0, \infty)\) and \(\tau \in[0, t]\), it sufficient to consider the estimate of \(\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\) for which we prove the flowing lemma.

Lemma 10. Let \(1\leq p< \infty,\,1\leq q \leq \infty,\,0\leq\lambda< 3,\,1 +\sigma< \alpha< \min \{2,2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma\},\, I=[0,T),\,T\in(0,\infty]\), and set \begin{equation*} X=\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\cap \mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)\,. \end{equation*} There exists a constant \(C=C(p,q)>0\) depending on \(p,q\) such that \begin{equation*} \|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\leq C \mu^{-1}\|a\|_{X}\|b\|_{X}\,. \end{equation*}

Proof. Based on the same procedure in the proof of Proposition 8, we evaluate the estimate of \(\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\), in fact, we have for fixed \(j\) \begin{align*} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}(u \partial_{i}(Pv)) &=\sum_{|k-j| \leq 4} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{S}_{k-1} u \dot{\Delta}_{k} \partial_{i}(Pv)\right)\\ &\quad+\sum_{|k-j| \leq 4} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{S}_{k-1} \partial_{i}(Pv) \dot{\Delta}_{k} u\right)\\ &\quad+\sum_{k \geq j-3} \dot{\Delta}_{j} e^{\mu\sqrt{\tau}|D|^{\frac{\alpha}{2}}}\left(\dot{\Delta}_{k} u \widetilde{\Delta}_{k} \partial_{i}(Pv)\right)\\ &:=S_{1,j}+S_{2,j}+S_{3,j}\,. \end{align*} Since \(e^{\mu\sqrt{\tau}\left(|\xi|^{\frac{\alpha}{2}}-|\xi-\eta|^{\frac{\alpha}{2}}-|\eta|^{\frac{\alpha}{2}}\right)}\) is uniformly bounded on \(\tau\) when \(\alpha\in[0,2]\), we obtain \begin{align*} \|\widehat{S_{1,j}}\|_{\mathrm{M}_{p}^{\lambda}} &=\|\sum_{|k-j| \leq 4}\varphi_{j} e^{\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}}\mathscr{F}\big(\dot{S}_{k-1} u \dot{\Delta}_{k} \partial_{i}(Pv)\big)\|_{\mathrm{M}_{p}^{\lambda}}\\ &=\|\sum_{|k-j| \leq 4}\varphi_{j} e^{\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}}\big[(\sum_{l \leq k-2} e^{-\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}} \widehat{a}_{l})* \big(e^{-\mu\sqrt{\tau}|\xi|^{\frac{\alpha}{2}}} \mathscr{F}(\dot{\Delta}_{k} \partial_{i}(Pb))\big) \big]\|_{\mathrm{M}_{p}^{\lambda}}\\ &=\|\sum_{|k-j| \leq 4} \varphi_{j}\int_{\mathbb{R}^{3}} e^{\mu\sqrt{\tau}\big(|\xi|^{\frac{\alpha}{2}}-|\xi-\eta|^{\frac{\alpha}{2}}-|\eta|^{\frac{\alpha}{2}}\big)}\big(\sum_{l \leq k-2} \widehat{a_{l}}\big)(\xi-\eta)\mathscr{F}(\dot{\Delta}_{k} \partial_{i}(Pb))(\eta) d \eta\|_{\mathrm{M}_{p}^{\lambda}}\\ &\leq C\|\sum_{|k-j| \leq 4}\mathscr{F}\big(\dot{S}_{k-1}a \dot{\Delta}_{k} \partial_{i}(Pb)\big)\|_{\mathrm{M}_{p}^{\lambda}}\,. \end{align*} The same calculus as in Proposition 8 gives \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}&2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\| \widehat{S_{1,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}&\lesssim \|a\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|b\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Similarly, we show that \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\|\widehat{S_{2,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}\lesssim\|b\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|a\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Similarly, \begin{equation*} \left\|\widehat{S_{3,j}}\right\|_{\mathrm{M}_{p}^{\lambda}}\leq \sum_{k \geq j-3} \sum_{|l-k| \leq 1}\left\|\mathcal{F}\left(\dot{\Delta}_{k} a \dot{\Delta}_{l}\left(\partial_{i}(Pb)\right)\right)\right\|_{M_{p}^{\lambda}}\,. \end{equation*} Using again the same procedure described in the proof of Proposition 8 we obtain \begin{align*} \Big\{ \sum_{j\in\mathbb{Z}}2^{j(2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma)q}\|\widehat{S_{3,j}}\|_{L^{1}(I ,\mathrm{M}_{p}^{\lambda} )}^q\Big\}^{1/q}\lesssim\|a\|_{\mathcal{L}^{\infty}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \|b\|_{\mathcal{L}^{1}\left(I;\mathcal{F\dot{N}}_{p,\lambda,q}^{ 1+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)}\,. \end{align*} Finally, \begin{equation*} \left\|e^{\mu \sqrt{\tau}|D|^{\frac{\alpha}{2}}}u\partial_{i}(Pv)\right\|_{\mathcal{L}^{1}\left(I; \mathcal{F} \dot{\mathcal{N}}_{p, \lambda, q}^{2-\alpha+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}\right)} \leq C \mu^{-1}\|a\|_{X}\|b\|_{X}\,. \end{equation*}

To finish the proof of Theorem 5, it is easy to obtain the requested result by repeating the same step in the proof of Theorem 4 and Proposition 8.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.