Skip to main content

On the global existence and analyticity of the mild solution for the fractional Porous medium equation

Abstract

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

1 Introduction

This paper considers the following 3D fractional porous medium (FPM) equation:

$$ \textstyle\begin{cases} w_{t}+\alpha \Lambda ^{s}w= \nabla \cdot (w\nabla Qw) \in \mathbb{R}^{3} \times \mathbb{R}^{+} , \\ w(z,0)=w_{0} \in \mathbb{R}^{3}, \end{cases} $$
(FPM)

where \(w=w(z,t)\) represents the density or concentration. The positive coefficient of dissipation \(\alpha > 0\) denotes the viscous property, while \(\alpha = 0 \) is for the corresponding inviscid property. The symbol Q denotes an abstract operator. The term \(\Lambda ^{s}\) is a fractional Laplacian that is defined by the Fourier transform \(\widehat{\Lambda ^{s} w}=|\varphi |^{s} \widehat{w}\). For the sake of simplicity, α is equal to one.

According to [1], Zhou, Xiao, and Zheng were the first to introduce equation (FPM). In their work, they applied the fractional dissipation term \(\alpha \Lambda ^{s}\) to the equation of continuity \(w t+\nabla \cdot (wW)=0\), then Caffarelli and Vázquez [2] established equation (FPM). The potential \(W=-\nabla p\) gives the velocity W, and the pressure or velocity potential is related to w via an abstract operator \(q=Qw\) [3]. There are plenty of real-world applications for porous media problems. The main purpose of this equation is to provide a mathematical description of fluid’s behavior as it permeates porous materials such as biological tissues, rocks, or soil [46]. The fractional porous medium equation is a fundamental equation to model groundwater flow in deep aquifer, which provides the dynamics, transportation of substances, and the effects of injection or pumping activities of groundwater [7, 8]. In addition to fluid flow, the equation has been adapted to facilitate the modeling of heat and mass transfer phenomena occurring within porous media [9, 10]. We refer the focus of the reader to [1116] and the references listed within for more detailed research referring to the physical importance of equation (FPM).

In numerous cases, the abstract pressure Qw is a suitable choice. The simplest example is derived from a groundwater filtration model [17, 18]. Zhou, Xiao, and Chen [19] investigated the strong solution for the more general case \(\alpha =0\) and \(Qw=(-\Delta )^{-m}w=\Lambda ^{-ms}\), \(0< m<1\), of equation (FPM) in the Besov spaces \(B_{p,\infty}^{s}\). Further, they constructed the existence of local solution for the initial data in the space \(B_{1,\infty}^{s}\). Lin and Zhang [20] obtained the mean field equation by considering the critical case, i.e., \(s=1\). Regarding the existence of solutions and the uniqueness of these equations, the authors suggest the reader refers to the work of Zhou and Biler [1, 21] and their references.

The aggregation equations, which explain aggregation behaviors and collective dynamics in the structure of continuous media [22, 23], is another model that is similar to the one described above. Applications of this equation can be found in domains such as biology, physics, chemistry, and dynamics of populations. The operator Q in the aggregation model may also be expressed in the form of convolution operator with kernel \(\mathcal{J}\) as follows: \(Qw=\mathcal{J}*w\). The Newton potential \(|z|^{\gamma}\) [24] and the exponential potential \(-e^{-|z|}\) [25] are two examples of typical kernels. Further research in this regard can be found in [26, 27] and the related references cited there.

Further, by considering the corresponding initial data, equation (FPM) can be modified to the following form [28]:

$$ \begin{gathered} w_{t}+\alpha \Lambda ^{s}w+ u\cdot \nabla w= -w(\nabla \cdot u); \\ u=-\nabla Qw. \end{gathered} $$
(1)

The turbulent velocity associated with the fractional porous medium equations is not strictly divergence free, and this allows that (1) can be compared to the geostrophic model. Moreover, as the divergence free vector u follows the condition \((\nabla \cdot u =0)\), equation (1) thus contains the quasi-geostrophic equation [29, 30].

The singularity of an abstract pressure component Qw that ensures the well-posedness or produces the blow-up solutions is one of the most significant challenges related to equation (FPM). The existence of local solution with large initial data belonging to Besov spaces was established by Zhou, Xiao, and Zheng [1], while the global existence of solution was established for small initial data. In addition to that, a blow-up criterion was provided for the solution. The existence of solution and the blow-up condition for equation (FPM) related to its pressure \(Qw=\mathcal{J}*w\), where \(\mathcal{J}(x)=e^{-|z|}\), in the Sobolev space were obtained by Li and Rodrigo [31]. In addition, Wu and Zhang’s work [32] advanced their previous investigation to the case in which \(\nabla \mathcal{J} \in W^{1,1}\) and \(\mathcal{J}(x)=e^{-|z|}\). For \(\nabla \mathcal{J} \in L^{1}\), Xiao and Zhou [3] showed the existence of local solutions in Fourier–Besov spaces with large initial data, and they obtained the existence of global solutions with small initial data. These results come from the fact that convolution \(\mathcal{J}*w\) and its gradient \(\nabla \mathcal{J}*w\) can be controlled in Besov spaces. More recently Zhou, Xiao, and Zheng [28] established the existence of local solutions with large initial data and the global existence of solutions when the initial data is small in homogenous Besov spaces \(\dot{B}_{p, q}^{s}\) under the following general condition:

$$ \Vert \nabla P w \Vert _{\dot{B}_{p, q}^{s}} \leq C \Vert w \Vert _{\dot{B}_{p, q}^{s+ \sigma}}. $$

Motivated by the previously mentioned studies and considering the above condition, we achieved the existence of global solution and analyticity to the solution for the three-dimensional fractional porous medium equation in the following critical space:

$$ {\mathcal{X}^{a}}:= \biggl\{ g \in \mathcal{Z}^{\prime} \bigl( \mathbb{R}^{3} \bigr)\biggm| \int _{\mathbb{R}^{3}} \vert \varphi \vert ^{a} \bigl\vert \widehat{g}(\varphi ) \bigr\vert \,d \varphi < \infty , a \in \mathbb{R} \biggr\} . $$

Regarding the function space mentioned before, Lei and Lin [33] established the global existence of mild solution to the classical Navier–Stokes (NS) equation in the critical space \(C (\mathbb{R}_{+}, \mathcal{X}^{-1} ) \cap L^{1} ( \mathbb{R}_{+}, \mathcal{X}^{1} )\). Bae [34] recently showed the results of Lei and Lin [33] in a little modified way, which is given as follows:

$$ \mathcal{L}_{t}^{\infty} \mathcal{X}^{s}:= \biggl\{ g \in \mathcal{Z}^{ \prime} \bigl(\mathbb{R}^{3} \times \mathbb{R}_{+} \bigr): \int _{ \mathbb{R}^{3}} \Bigl[\sup_{0 \leq t< +\infty} \vert \varphi \vert ^{s} \bigl\vert \widehat{g}(\varphi , t) \bigr\vert \Bigr]\,d \varphi < +\infty \biggr\} $$

with

$$ \Vert g \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{s}}= \int _{\mathbb{R}^{3}} \Bigl[\sup_{0 \leq t< +\infty} \vert \varphi \vert ^{s} \bigl\vert \widehat{g}(\varphi , t) \bigr\vert \Bigr]\,d \varphi $$

and

$$ L_{t}^{1} \mathcal{X}^{1}:= \biggl\{ g \in \mathcal{Z}^{\prime} \bigl( \mathbb{R}^{3} \times \mathbb{R}_{+} \bigr): \int _{\mathbb{R}^{3}} \int _{0}^{+\infty} \vert \varphi \vert \bigl\vert \widehat{g}(\varphi , \eta ) \bigr\vert \,d \eta\,d \varphi < +\infty \biggr\} $$

with

$$ \Vert g \Vert _{L_{t}^{1} \mathcal{X}^{1}}= \int _{\mathbb{R}^{3}} \int _{0}^{t} \vert \varphi \vert \bigl\vert \widehat{g}(\varphi , \eta ) \bigr\vert \,d \eta\,d \varphi . $$

In the present investigation, we use the approaches proposed by Lei and Bae [33, 34] to obtain the existence of global solution and analyticity to the solution of equation (FPM). In this paper, \(f \lesssim g \) is used to denote \(f \leq Cg\), where C represents positive constants (different values may be taken in different places). The symbol denotes the Fourier transform of f. The following is the arrangement of this research article: In Sect. 2, we give the statements of the two main theorems. In Sect. 3, we show the proof of Theorem 2.1, and the proof of Theorem 2.2 is presented in Sect. 4.

2 Main results

The primary purpose of this research is to determine the existence of global solution to equation (FPM). The following is the related result.

Theorem 2.1

Let \(\frac{1}{2} \leq s \leq 1\) and there exists a constant \(\varepsilon _{0}>0\) depending on the value of s such that for all initial data \(w_{0}\) belongs to \(\mathcal{X}^{1-2 s}\) satisfies the condition

$$ \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}}< \varepsilon _{0}, $$

then equation (FPM) has a unique global in time solution

$$ w \in \mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s} \cap L_{t}^{1} \mathcal{X}^{1} $$

such that

$$ \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}+ \Vert u \Vert _{L_{t}^{1} \mathcal{X}^{1}} \lesssim \Vert u_{0} \Vert _{\mathcal{X}^{1-2 s}}. $$

The next objective of this paper is to work on the Gevrey class regularity of solution to equation (FPM). The Gevrey class regularity to the solution for the classical NS equations has been the focus of significant research; for instance, see [35, 36] and the references therein. The specific result is the following.

Theorem 2.2

Let \(\frac{1}{2} \leq s \leq 1\) and then there exists a constant \(\kappa _{0}>0\) depending on the value of s such that for all initial data \(u_{0}\) belongs to \(\mathcal{X}^{1-2 s}\) satisfying the condition

$$ \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}} \leq \kappa _{0}. $$

The global solution established in Theorem 2.1can be analytic in a way that

$$ \bigl\Vert e^{\sqrt{t} \vert \delta \vert ^{s}} w \bigr\Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}+ \bigl\Vert e^{\sqrt{t} \vert \delta \vert ^{s}} w \bigr\Vert _{L_{t}^{1} \mathcal{X}^{1}} \lesssim \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}}, $$

where \(e^{\sqrt{t}|\delta |^{s}}\) is defined as a Fourier multiplier with symbol \(e^{\sqrt{t}|\varphi |^{s}}\).

Throughout this paper, \(A \lesssim B\) represents \(A \leq C B\) depending on some constant \(C>0\).

3 Proof of Theorem 2.1

The proof of Theorem 2.1 is presented in this section. To prove our key results, we first state the following lemma.

Lemma 3.1

([37])

Suppose \(\frac{1}{2}\le s \le 1\), then we have the following inequality:

$$ \begin{aligned} \vert \varphi \vert ^{2(1-s )}\le 2^{1-2s} \bigl( \vert \zeta \vert \vert \varphi -\zeta \vert ^{1-2s }+ \vert \zeta \vert ^{1-2s } \vert \varphi -\zeta \vert \bigr) \end{aligned} $$

for any \(\varphi ,\,\zeta \in \mathbb{R}^{3}\).

Proof of Theorem 2.1

To get the solution of equation (FPM), we can transform equation (FPM) into the following integral form:

$$ w = G_{s} (t)w_{0} + \int _{0}^{t} G_{s} (t-\eta )\nabla \cdot (w \nabla Qw)\,d\eta , $$
(2)

where \(G_{s}(t):=e^{-t\Lambda ^{s}}\).

Applying Fourier transform to the above integral form, we get

$$ \widehat{w}(\varphi ,t)=e^{-t|\cdot |^{s}}\widehat{w}_{0}( \varphi )+ \int _{0}^{t}e^{-(t-\eta )|\cdot |^{s}}\iota \varphi \cdot \int _{ \mathbb{R}^{3}} \widehat{w}( \varphi -\zeta , \eta ) \widehat{ \nabla Qw}(\zeta , \eta )\,d \zeta\,d\eta . $$
(3)

The multiplication of \(|\varphi |^{1-2s} \) to both sides gives us the following:

$$ \begin{aligned}[b] \vert \varphi \vert ^{1-2s}\widehat{w}(\varphi ,t)\lesssim{}& e^{-t \vert \cdot \vert ^{s}} \vert \varphi \vert ^{1-2s} \bigl\vert \widehat{w}_{0}( \varphi ) \bigr\vert \\ & {}+ \int _{0}^{t}e^{-(t-\eta ) \vert \cdot \vert ^{s}} \int _{\mathbb{R}^{3}} \vert \varphi \vert ^{2-2 s} \bigl\vert \widehat{w}(\varphi -\zeta , \eta ) \bigr\vert \bigl\vert \widehat{w}( \zeta , \eta ) \bigr\vert \,d\zeta\,d\eta . \end{aligned} $$
(4)

By using Lemma 3.1, the nonlinear term can be estimated as follows:

$$ \begin{aligned}[b] & \int _{0}^{t} \biggl[ \int _{\mathbb{R}^{3}} \vert \varphi \vert ^{2-2 s} \bigl\vert \widehat{u}(\varphi -\zeta , \eta )\bigr|\bigl| \widehat{w}(\zeta , \eta ) \bigr\vert \,d \zeta \biggr]\,d \eta \\ &\quad \lesssim 2^{2(1-s)} \biggl[ \int _{0}^{\infty} \bigl\vert \cdot \| \widehat{w}( \cdot , \eta ) \bigr\vert \,d \eta \biggr] * \Bigl[\sup _{0 \leq t< +\infty} \vert \cdot \vert ^{1-2 s} \bigl\vert \widehat{w}(\cdot , t) \bigr\vert \Bigr]. \end{aligned} $$
(5)

Considering equations (4) and (5) and applying Young’s inequality, we have

$$ \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}} \leq \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}}+2^{2(1-2 s)} \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}} \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}. $$
(6)

Equation (3) shows that w is estimated in \(\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}\).

Next we need to estimate w in \(L_{t}^{1} \mathcal{X}^{1}\). Multiplying by \(|\varphi |\) both sides of equation (3), we have

$$ \begin{aligned}[b] \vert \varphi \vert \bigl\vert \widehat{w}(\varphi , t) \bigr\vert \lesssim{}& \vert \varphi \vert ^{2 s} e^{-t \vert \varphi \vert ^{ s}} \vert \xi \vert ^{1-2 s} \bigl\vert \widehat{u}_{0}(\varphi ) \bigr\vert \\ & {}+ \int _{0}^{t} \vert \varphi \vert ^{2 s} e^{-(t-\eta ) \vert \varphi \vert ^{ s}} \int _{ \mathbb{R}^{3}} \vert \varphi \vert ^{2-2 s} \bigl\vert \widehat{w}(\varphi -\zeta , \eta ) \bigr\vert \bigl\vert \widehat{w}( \zeta , \eta ) \bigr\vert \,d \zeta\,d \eta . \end{aligned} $$
(7)

Applying \(L^{1}_{t}\) to inequality (7), utilizing \(\int _{0}^{+\infty}|\varphi |^{2 s} e^{-t|\xi |^{ s}}\,d t<\infty \) and Young’s inequality, we have

$$ \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}} \lesssim \Vert w_{0} \Vert _{ \mathcal{X}^{1-2 s}}+2^{2(1-2 s)} \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}} \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}. $$
(8)

Denote

$$ S := \mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s} \cap L_{t}^{1} \mathcal{X}^{1} $$

and

$$ H:= \Vert u \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}+ \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}}, $$

then from equations (3) and (8), we have

$$ \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}+ \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}} \lesssim \Vert u_{0} \Vert _{\mathcal{X}^{1-2 s}}+2^{2(1-2 s)} \bigl( \Vert w \Vert _{\mathcal{L}_{t}^{\infty} \mathcal{X}^{1-2 s}}+ \Vert w \Vert _{L_{t}^{1} \mathcal{X}^{1}} \bigr)^{2}, $$

that is,

$$ H \lesssim 2^{2(1-2 s)} H^{2}+ \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}}{.} $$

By choosing

$$ \psi =1-4 \cdot 2^{2(1-2 s)} \Vert w_{0} \Vert _{\mathcal{X}^{1-2 s}} $$

and applying the Banach fixed point principle, it is easy to achieve the existence of global solution in S for small initial data belonging to \(\mathcal{X}^{1-2 s}\). □

4 Proof of Theorem 2.2

The lemma stated below is helpful in constructing the proof of Theorem 2.2.

Lemma 4.1

[38] Let \(0<\pi \leq t<\infty \) and \(0 \leq s \leq 1\), then there holds the following inequality:

$$ t \vert m \vert ^{s}-\frac{1}{2} \bigl(t^{2}- \pi ^{2} \bigr) \vert m \vert ^{2 s}-\pi \vert m-n \vert ^{s}- \pi \vert n \vert ^{s} \leq \frac{1}{2} $$

for any \(m, n \in \mathbb{R}^{3}\).

The construction of the proof of Theorem 2.2 follows the idea of Lemarié-Rieusset [36], where he showed the analyticity for the solution, i.e.,

$$ \sup_{0< t< \infty} \sup_{\varphi \in \mathbb{R}^{3}} e^{\sqrt{t} \vert \varphi \vert ^{s}} \vert \varphi \vert ^{2 s} \bigl\vert \widehat{w}(\varphi , t) \bigr\vert < \infty{.} $$

Suppose \(f(z, t)=e^{\sqrt{t}|\delta |^{s}} w(z, t)\), considering the integral form (2), we can write

$$ f(z, t)=e^{\sqrt{t}|\delta |^{s}} G_{ s}(t) w_{0}- \int _{0}^{t} e^{ \sqrt{t}|\delta |^{s}} G_{s} (t-\eta )\nabla \cdot (w\nabla Qw)\,d\eta . $$

We can easily get that

$$ \begin{aligned} \bigl\vert \widehat{f}(\varphi , t) \bigr\vert \lesssim{}& e^{\sqrt{t} \vert \varphi \vert ^{s}-t \vert \varphi \vert ^{2 s}} \bigl\vert \widehat{w}_{0}(\varphi ) \bigr\vert \\ & {}+ \int _{0}^{t} e^{\sqrt{t} \vert \varphi \vert ^{s}-(t-\eta ) \vert \varphi \vert ^{2 s}- \sqrt{\eta} ( \vert \varphi -\zeta \vert ^{s}+ \vert \zeta \vert ^{s} )} \vert \varphi \vert \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \widehat{w}(\varphi -\zeta , \eta ) \bigr\vert \bigl\vert \widehat{w}(\zeta , \eta ) \bigr\vert \,d \zeta \biggr)\,d \eta . \end{aligned} $$

Noticing that

$$ e^{\sqrt{t}|\varphi |^{s}-\frac{1}{2} t|\varphi |^{2 s}}=e^{- \frac{1}{2} (\sqrt{t}|\varphi |^{s}-1 )^{2}+\frac{1}{2}} \leq e^{\frac{1}{2}}{} $$
(9)

and Lemma 4.1, we have

$$ e^{\sqrt{t}|\varphi |^{s}-\frac{1}{2}(t-\eta )|\varphi |^{2 s}-\sqrt{ \eta} (|\varphi -\zeta |^{s}+|\zeta |^{s} )} \leq e^{ \frac{1}{2}}{.} $$
(10)

Utilizing inequalities (9), (10), we have

$$ \begin{aligned}[b] \bigl\vert \widehat{f}(\varphi , t) \bigr\vert \lesssim{}& e^{-\frac{1}{2} t \vert \varphi \vert ^{s}} \bigl\vert \widehat{w}_{0}(\varphi ) \bigr\vert \\ & {}+ \int _{0}^{t} e^{-\frac{1}{2}(t-\eta ) \vert \varphi \vert ^{2 s}} \vert \varphi \vert \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \widehat{w}(\varphi -\zeta , \eta ) \bigr\vert \bigl\vert \widehat{w}(\zeta , \eta ) \bigr\vert \,d \zeta \biggr)\,d \eta. \end{aligned} $$
(11)

The rest of the theorem follows similar steps as in the proof of Theorem 2.1. That’s why the remaining details are skipped here.

5 Conclusion

In this paper we considered the fractional porous medium equation and established the existence of global solution in the corresponding critical function spaces for small initial data belonging to these spaces. These spaces were previously considered related to the existence of solution for the classical case of Navier–Stokes equations [33]. The existence of solution for equation (FPM) was previously studied in various function spaces, for instance, [19, 28]. This paper extended the study of equation (FPM) to Lei and Lin spaces and achieved the existence of global solution for small initial data. Moreover, this paper also provided the analyticity of the solution of equation (FPM).

Availability of data and materials

Not applicable

References

  1. Zhou, X., Xiao, W., Zheng, T.: Well-posedness and blowup criterion of generalized porous medium equation in Besov spaces. Electron. J. Differ. Equ. 2015(261), 1 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Caffarelli, L.A., Vazquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Xiao, W., Zhou, X.: On the generalized porous medium equation in Fourier-Besov spaces. J. Math. Study 53(3), 316–328 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1988)

    MATH  Google Scholar 

  5. Bhatti, M.M., Michaelides, E.E.: Oldroyd 6-constant electro-magneto-hydrodynamic fluid flow through parallel micro-plates with heat transfer using Darcy-Brinkman-Forchheimer model: A parametric investigation (2023)

  6. Othman, M.I., Fekry, M., Marin, M.: Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating. Struct. Eng. Mech. 73(6), 621–629 (2020)

    Google Scholar 

  7. Fetter, C.W.: Applied Hydrogeology, 4th edn. Prentice Hall, New York (2001)

    Google Scholar 

  8. Ghiţă, C., Pop, N., Cioban, H.: Quasi-static behavior as a limit process of a dynamical one for an anisotropic hardening material. Comput. Mater. Sci. 52(1), 217–225 (2012)

    Article  Google Scholar 

  9. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.S.: In: Fundamentals of Heat and Mass Transfer. 2007, pp. 939–940. John Wiley, Hoboken (1985)

    Google Scholar 

  10. Abo-Dahab, S.M., Abouelregal, A.E., Marin, M.: Generalized thermoelastic functionally graded on a thin slim strip non-Gaussian laser beam. Symmetry 12(7), 1094 (2020)

    Article  Google Scholar 

  11. Vázquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, USA (2007)

    MATH  Google Scholar 

  12. Abidin, M.Z., Chen, J.: Global well-posedness and analyticity of generalized porous medium equation in Fourier-Besov-Morrey spaces with variable exponent. Mathematics 9(5), 498 (2021)

    Article  Google Scholar 

  13. Zhang, L., Li, S.: Regularity of weak solutions of the Cauchy problem to a fractional porous medium equation. Bound. Value Probl. 2015, 1 (2015)

    Article  MathSciNet  Google Scholar 

  14. Marin, M.: Generalized solutions in elasticity of micropolar bodies with voids. Rev. Acad. Canar. Cienc. 8(1), 101–106 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Shannon, A.G., Özkan, E.: Some aspects of interchanging difference equation orders. Notes Number Theory Discrete Math. 28(3), 507–516 (2022)

    Article  Google Scholar 

  16. Luo, L., Zhou, J.: Global existence and blow-up to the solutions of a singular porous medium equation with critical initial energy. Bound. Value Probl. 2016(1), 1 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, Chelmsford (2013)

    MATH  Google Scholar 

  18. Aronson, D.G.: The porous medium equation. In: Nonlinear Diffusion Problems, pp. 1–46. Springer, Germany (1986)

    Google Scholar 

  19. Zhou, X., Xiao, W., Chen, J.: Fractional porous medium and mean field equations in Besov spaces. Electron. J. Differ. Equ. 2014, 199 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Lin, F., Zhang, P.: On the hydrodynamic limit of Ginzburg-Landau wave vortices. Commun. Pure Appl. Math. 55(7), 831–856 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Biler, P., Imbert, C., Karch, G.: Barenblatt profiles for a nonlocal porous medium equation. C. R. Math. 349(11–12), 641–645 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Blanchet, A., Carrillo, J.A., Masmoudi, N.: Infinite time aggregation for the critical Patlak-Keller-Segel model in \(\mathbb{R}^{2}\). Commun. Pure Appl. Math. 61(10), 1449–1481 (2008)

    Article  MATH  Google Scholar 

  23. Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68(7), 1601 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, D., Zhang, X.: Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels. Commun. Pure Appl. Anal. 9(6), 1591 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, D., Rodrigo, J.L.: Wellposedness and regularity of solutions of an aggregation equation. Rev. Mat. Iberoam. 26(1), 261–294 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Karch, G., Suzuki, K.: Blow-up versus global existence of solutions to aggregation equations. Appl. Math. 38, 243–258 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Laurent, T.: Local and global existence for an aggregation equation. Commun. Partial Differ. Equ. 32(12), 1941–1964 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, X., Xiao, W., Zheng, T.: Well-posedness and blowup criterion of generalized porous medium equation in Besov spaces. Electron. J. Differ. Equ. 2015(261), 1 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Chen, Q., Zhang, Z.: Global well-posedness of the 2D critical dissipative quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinear Anal., Theory Methods Appl. 67(6), 1715–1725 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, H., Zhang, Z.: A frequency localized maximum principle applied to the 2D quasi-geostrophic equation. Commun. Math. Phys. 301(1), 105–129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, D., Rodrigo, J.L.: Wellposedness and regularity of solutions of an aggregation equation. Rev. Mat. Iberoam. 26(1), 261–294 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu, G., Zhang, Q.: Global well-posedness of the aggregation equation with supercritical dissipation in Besov spaces. J. Appl. Math. Mech. 93(12), 882–894 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Lei, Z., Lin, F.H.: Global mild solutions of Navier-Stokes equations. Commun. Pure Appl. Math. 64(9), 1297–1304 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bae, H.: Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations. Proc. Am. Math. Soc. 143(7), 2887–2892 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton (2002)

    MATH  Google Scholar 

  37. Ye, Z.: Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations. Ann. Mat. Pura Appl. 195, 1111–1121 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ferreira, L.C., Villamizar-Roa, E.J.: Exponentially-stable steady flow and asymptotic behavior for the magnetohydrodynamic equations. Commun. Math. Sci. 9(2), 499–516 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research has no external funding.

Author information

Authors and Affiliations

Authors

Contributions

MZA worked on the problem and wrote the original draft. MM revised the mathematical calculations, made corrections and did several improvements. All authors have read and approved the final version of the manuscript.

Corresponding author

Correspondence to Muhammad Marwan.

Ethics declarations

Ethics approval and consent to participate

Not applicable

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abidin, M.Z., Marwan, M. On the global existence and analyticity of the mild solution for the fractional Porous medium equation. Bound Value Probl 2023, 107 (2023). https://doi.org/10.1186/s13661-023-01794-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-023-01794-3

Keywords