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

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 u0 belonging to the critical Fourier-Besov-Morrey spaces. In addition, we also give the Gevrey class regularity of the solution.


Introduction
W e investigate the generalized porous medium equation in the whole space R 3 ,      u t + µΛ α u = ∇ · (u∇Pu); (t, x) ∈ R + × R 3 , where u = u(t, x) is a real-valued function, which denotes a density or concentration. The dissipative coefficient µ > 0 corresponds to the viscous case, while µ = 0 corresponds to the inviscid case. The fractional Laplacian operator Λ α is defined by Fourier transform as Λ α u = |ξ| αû , 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 µΛ α u to the continuity equation (PME) u t + ∇ · (uV) = 0 given by Caffarelli and Vázquez [2], where the velocity V derives from a potential, V = −∇p and the velocity potential or pressure p is related to u by an abstract operator p = Pu [3].
For µ = 0 and Pu = (−∆) −s u = Λ −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 α p,∞ and they obtained the local solution for any initial data in B α 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 (∇ · 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 −µ √ t|D| α 2 and Clearly, for the fractional porous medium equation, i.e. Pu = Λ −2s u, we get σ = 1 − 2s. If Pu = K * u in the aggregation equation, Wu and Zhang [18] proved a similar result under the condition ∇K ∈ W 1,1 , α ∈ (0, 1). Corresponding to their case we give a same result for σ = 0 when ∇K ∈ L 1 , and also a similar result for σ = 1 when K ∈ L 1 .
Throughout this paper, we use FṄ s p,λ,q to denote the homogenous Fourier Besov-Morrey spaces, C will denote constants which can be different at different places, U V means that there exists a constant C > 0 such that U ≤ CV, and p is the conjugate of p satisfying 1 p + 1 p = 1 for 1 ≤ p ≤ ∞.

Preliminaries and main results
We start with a dyadic decomposition of R n . Suppose and denote ϕ j (ξ) = ϕ(2 −j ξ) and P the set of all polynomials. First, we recall the definition of Morrey spaces which are a complement of L p spaces.
where B(x 0 , r) denotes the ball in R n with center x 0 and radius r.
Our first main result is the following theorem.
Theorem 4. Assume that the abstract operator P satisfies the condition (3). If 0 ≤ λ < 3, 1 ≤ q ≤ ∞, 1 ≤ p < ∞ and max{1 + σ, 0} < α < 2 + 3 p + λ p + σ then there exists a constant C 0 such that for any u 0 ∈ FṄ 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.

then the Cauchy problem
(1) admits a unique analytic solution u, in the sense that We finish this section with a Bernstein type lemma in Fourier variables in Morrey spaces.

The well-posedness
First, we consider the linear nonhomogeneous dissipative equation for which we recall the following result.
If in addition q is finite, then u belongs to C(I; FṄ s p,λ,q ).
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 Using the decomposition of Bony's paraproducts for the fixed j, we havė 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 |γ| = 0, we get Multiplying by 2 j(−2(α−1)+ 3 p + α ρ + λ p +σ) , and taking l q −norm of both sides in the above estimate, we obtain Likewise, we prove that ∑ j∈Z 2 j(−2(α−1)+ 3 p + α ρ + λ p +σ)q I I j q L ρ (I,M λ p ) To evaluate I I I j , we apply the Young inequality (5) in Morrey spaces and Lemma 6 with |γ| = 0, we obtain 2 j(−2(α−1) Taking the l q −norm on both sides in the above estimate and using Hölder's inequalities for series with −2(α − 1) Estimates (10), (11), (12) and (13) yield (9) .
Lemma 9. Let X be a Banach space with norm . X and B : X × X −→ X be a bounded bilinear operator satisfying for all u, v ∈ X and a constant η > 0. Then, if 0 < ε < 1 4η and if y ∈ X such that y X ≤ ε, the equation x := y + B(x, x) has a solution x in X such that x X ≤ 2ε. This solution is the only one in the ball B(0, 2ε). Moreover, the solution depends continuously on y in the sense: if y X ≤ ε, x = y + B(x , x ), and x X ≤ 2ε, then

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 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 = ∇ · (u(τ)∇Pv(τ)). According to Lemma 7 with s = 1 − α + 3 p + λ p + σ and Proposition 8 with ρ = 1, we obtain then the equation (14) has a unique solution in B(0, 2R) := {x ∈ X : x X ≤ 2R}. To prove e −µtΛ α u 0 X < R, notice that e −µtΛ α u 0 is the solution to the dissipative equation with u 0 = u 0 and f = 0. So, Lemma 7 yields If u 0 , then (14) has a unique global solution u ∈ X satisfying

Proof of theorem 5
Proof. To prove Theorem 5, we note a(t, x) := e µ √ t|D| α 2 u(t, x) . Using the integral Equation (14), we obtain In order to obtain the Gevrey class regularity of the solution, we use Lemma 9. Firstly, we start by estimating the term Lu 0 = e − 1 2 µ( √ t|D| α 2 −1) 2 + µ 2 e − 1 2 µtΛ α u 0 . Using the Fourier transform, multiplying by ϕ j and taking the M λ p -norm we obtain p +σ) and taking l q −norm we get We conclude by taking l q −norm that On the other hand, we notice that is uniformly bounded on t ∈ (0, ∞) and τ ∈ [0, t], it sufficient to consider the estimate of e µ √ τ|D| α 2 u∂ i (Pv) for which we prove the flowing lemma. Lemma 10. Let 1 ≤ p < ∞, 1 ≤ q ≤ ∞, 0 ≤ λ < 3, 1 + σ < α < min{2, 2 + 3 p + λ p + σ}, I = [0, T), T ∈ (0, ∞], and set X = L ∞ I; FṄ 1−α+ 3 p + λ p +σ p,λ,q ∩ L 1 I; FṄ There exists a constant C = C(p, q) > 0 depending on p, q such that Proof. Based on the same procedure in the proof of Proposition 8, we evaluate the estimate of is uniformly bounded on τ when α ∈ [0, 2], we obtain The same calculus as in Proposition 8 gives 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.
Conflicts of Interest: "The author declare no conflict of interest."