In this paper, we study the generalized porous medium equations with Laplacian and abstract pressure term. By using the Fourier localization argument and the Littlewood-Paley theory, we get global well-posedness results of this equation for small initial data \(u_{0}\) belonging to the critical Fourier-Besov-Morrey spaces. In addition, we also give the Gevrey class regularity of the solution.
We investigate the generalized porous medium equation in the whole space \(\mathbb{R}^{3}\),
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
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 \(1\leq p\leq\infty\).
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
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 \(0\leq\lambda< n,\)
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
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 \(1\leq q\leq p< \infty,\, 0\leq\lambda_{1},\lambda_{2}0\) independent of \(f\) and \(j\) such that
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 \(1\leq p< \infty,\,1\leq \rho,\,q\leq \infty,\,1+\sigma< \alpha< \frac{2+\frac{3}{p'}+\frac{\lambda}{p}+\sigma}{2-\frac{1}{\rho}},\, 0\leq\lambda0\) 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
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. 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
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 \(1\leq p< \infty,\,1\leq q \leq \infty,\,0\leq\lambda< 3,\,1 +\sigma< \alpha0\) 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.