Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces

The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [ J. Funct. Anal. , 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant \begin{document}$\mathbb{C}$\end{document} such that if the initial data \begin{document}$(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$\end{document} satisfies \begin{document}$\begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}$ \end{document} for \begin{document}$p, q, α$\end{document} with \begin{document}$1 \frac{1}{3}, 1 , then global existence of solutions with large initial vertical velocity component is established.

The above system (1) has been studied by several authors. Schmuck [36] and Ryham [35] obtained the global existence of weak solutions with Neumann and Dirichlet boundary conditions respectively. Li [32] studied the quasineutral limit in periodic domain. When Ω = R 3 , Joseph [31] established the existence of a unique smooth local solution for smooth initial data by making use of Kato's semigroup ideas. Zhao et al. [12,13,40,41,42] studied the local and global well-posedness in the critical Lebesgue spaces, modulation spaces, Triebel-Lizorkin spaces and Besov spaces.
In particular, if n = c = 0, then (1) becomes the problem related to the classical Navier-Stokes equations which has been widely studied during the past eighty more years. Leray [22] proved the global existence of weak solutions of (2), but the uniqueness and regularity of weak solutions are remaining the big open problems. It has been proved that the Cauchy problem (2) is globally well-posed for small initial data in a series of function spaces including particularly the following critical spaces:Ḣ (3 < p < ∞), BMO −1 , see Fujita and Kato [14], Kato [19], Kozono and Yamazaki [8], Koch and Tataru [20]; Xiao [29,38] proved this property in the space Q −1 α (R 3 ). Biswas [4] introduced V θ,p and homogeneous Besov type spaces B −δ p,∞ and then established Gevrey regularity of a class of dissipative equations in B −δ p,∞ and V θ,p . Biswas and Swanson [5] studied Gevrey regularity of Navier-Stokes equations with space-periodic boundary conditions in FḂ −1+N −N/p p,p . Bae on [2] studied the Gevrey regularity of Lei-Lin solutions [21] of Navier-Stokes equations in FḂ −1 1,1 . Recently, Bae, Biswas and Tadmor [3] obtained analyticity of Navier-Stokes equations in critical Besov spacesḂ −1+N/p p,q . Let S be the Schwartz class of rapidly decreasing functions, S be the space of tempered distributions. F and F −1 denote Fourier and inverse Fourier transforms of L 1 functions, respectively, which are defined by More generally, Fourier transform of any f ∈ S , given by (Ff, g) = (f, Fg) for any g ∈ S. Let C be the annulus {ξ ∈ R 3 : 3 4 ≤ |ξ| ≤ 8 3 } and D(Ω) be a space of smooth compactly supported functions on the domain Ω. There exist radial functions χ and ϕ, valued in the interval [0, 1], belonging respectively to D(B(0, 4 3 )) and D(C), and such that Define the setC = B(0, 2 3 ) + C. Then we have From now on, we write h = F −1 ϕ andh = F −1 χ. The homogeneous dyadic blocks ∆ j and S j are defined by It is well-known that if either s < 3/p or s = 3/p and r = 1, thenḂ s p,r (R 3 ) is a Banach space.
Let us now recall the definition of the Chemin-Lerner space L ρ (0, T ;Ḃ s p,r (R 3 )): for 0 < T ≤ ∞, s ∈ R and 1 ≤ p, r, ρ ≤ ∞ (with the usual convention if r = ∞ or ρ = ∞). The Chemin-Lerner space is defined by We define the usual space L ρ (0, T ;Ḃ s p,r (R 3 )) associated with the norm By Minkowski's inequality, it is readily to verify that A constantC exists which satisfies the following properties: if s 1 and s 2 are real numbers such that s 1 < s 2 and θ ∈ (0, 1), then we have, for any (p, r, ρ, ρ 1 , ρ 2 ) ∈ [1, ∞] 5 and any 1/ρ = θ/ρ 1 + (1 − θ)/ρ 2 , BecauseḂ 3 p p,1 is embedded in L ∞ , we deduce that whenever 1 ≤ p ≤ ∞, the product of two functions inḂ 3 p p,1 is also inḂ 3 p p,1 and such that for some constant The homogeneous paraproduct of v and u is defined by The homogeneous remainder of v and u is defined by We have the following Bony's decomposition For any operator T :Ḃ s p,r →Ḃ s p,r , we set u TḂ s p,r := T u Ḃs p,r . Let Λ be the Fourier multiplier whose symbol is given by |ξ| 1 = 3 i=1 |ξ i | and λ = 0, 1. We emphasize that here Λ ≡ Λ 1 is quantified by the 1 norm rather than the usual 2 norm associated with Λ 2 := (−∆) 1/2 . e λ √ tΛ is a Fourier multiplier operator whose symbol is given by e λ √ t|ξ|1 . A function f is said to be Gevrey regular if e √ tΛ f Ḃs p,r < +∞, for some s ∈ R and 1 ≤ p, r ≤ ∞. We mention that the finiteness of the corresponding Gevrey norm implies that the functions are analytic.
As a consequence of working with Gevrey norms, we obtain higher-order derivatives of solutions in Besov and Lebesgue spaces.
The solution (u, n, c) in Theorem 1.1 enables us to establish the following estimates on the high-order derivatives in Besov and Lebesgue spaces, that is, for all 0 < t < T , there exist positive constants C 1 , C 2 , C 3 andC such that (i) If m > 0, then (iv) If k 2 > −2 + 3/q and 1 < q ≤ 3 2 , then It is worth mentioning that for any t ∈ (0, T ) in Theorem 1.1, we obtain the solution (u(t), n(t), . Motivated by [39] concerning the global well-posedness of 3D incompressible Navier-Stokes equations with the third component of the initial velocity field being large, applying the local existence mentioned in Theorem 1.1 (in this case λ = 0) as well as the standard continuity argument, the system (1) has a unique global solution. Our main result reads as follows: There exists a positive constant C such that if the initial data (u 0 , n 0 , c 0 ) satisfies Then the system (1) admits a unique global-in-time solution such that (u, n, c) ∈ Θ ∩ Θ C .
We mention that our results do not impose any smallness conditions on the third component u 3 0 of the initial velocity field. It improves the recent result of [42], where the exponent form of the initial smallness condition is replaced by a polynomial form.
Notations. Throughout the paper,c andC stand for harmless constants. Let A and B be two operators, we denote [A; B] := AB − BA. For X a Banach space and I an interval of R, we denote by C(I; X) in the set of continuous functions on I with value in X. (d j ) j∈Z will be a generic element of 1 (Z) such that d j ≥ 0 and j∈Z d j = 1.

NPPNS 1627
For the estimate of the second term I 2 , using an argument similar to the one used in I 1 , we obtain immediately Now we tackle with the most difficult term I 3 . We can first split I 3 into the following three terms for m = 1, 2, , 3: where Moreover, since I 32 can be treated similarly to I 33 , we treat I 31 and I 33 only. We first consider I 31 , T (e λ √ tΛḂ −2+3/q+2/r 2 q,1 ) (29) By Lemmas 2.1 and 2.2, we have T (e λ √ tΛḂ −2+3/q+2/r 2 q,1 ) (30) Summing over m = 1, 2, 3, we obtain the estimate of I 3 .
Proof. The proof of Proposition 2 is easy, so we skip it here.
thus, from above let η > 0 be a sufficiently small constant and we can define We can choose T = min{T 1 , T 2 , T 3 } and we can take η sufficiently small and T ≤ T. Let ).

4K
for i = 1 or i = 2, applying Proposition 1, we thus get that there exists a positive time T > 0 such that the system (33) has a unique solution (ū,n,c) on [0, T ]. Therefore, applying Proposition 2 yields that the local Gevrey regularity of solution to the system (1).

4.
Proof of Corollary 1: Decay of Besov norms and Lebesgue norms. Theorem 1.1 tells us that the solution is locally in the Gevrey regular, that is, the energy bound (u, n, c) Θ < ∞ for λ = 1. Specifically, we can show that a solution (u, n, c) satisfies As we have showed the solution (u, n, c) of system (1) in Gevrey classes for which the estimate (40) holds, thanks to (3), (40) and using the same argument as Corollary 3.3 in [1], for m > 0, there exists positive constan C such that By using the relation between homogeneous Besov spaces and homogeneous Triebel-Lizorkin spacesḞ s p0,p0 (the definition of Triebel-Lizorkin spaces, see [9]), note that p0 → 2 for p 0 ≤ 2 andḞ s p0,2 =Ẇ s,p0 := (−∆) −s/2 L p0 , one thus haṡ Applying (42) and (40), along the same line with (41), for k 1 > −1 + 3/p and 1 < p ≤ 2, there exists a positive constant C 1 so that There exist positive constants C 2 and C 3 , for k 2 > −2 + 3/q, 1 < q ≤ 3 2 , whence a similar argument as that in (43) gives rise to We complete the proof of Corollary 1.

5.
Proof of Theorem 1.2: Global existence with large vertical velocity component. The goal of this section is to present the proof of the global existence of system (1) with large vertical velocity component. Let u = (u 1 , u 2 , u 3 ) = (u h , u 3 ) and div h u h = ∂ 1 u 1 + ∂ 2 u 2 . ).
• The estimate of u h and u 3 . The first equation of (1) implies that and and u 3 Proof. We first prove (54), applying the operator ∆ j to (52). Denote T ε (x) := √ x 2 + ε 2 , taking L 2 inner product of the resulting equation with as that in [11], then let ε → 0, an integration by parts yields Thanks to [11,27], there exists a positive constantc p fulfilling From (56) and (57), we thus infer the inequality from which, we obtain the desired estimate (54).
Finally, as the proof of (59), let us apply ∆ j to (53), we need to make some modifications as that [11], by virtue of Lemmas 2.6,5.1 and Proposition 3, we have Multiplying (60) by 2 j(−1+3/p) and summing up over j yield (78). Therefore, we complete the proof of Proposition 4.
• The estimate of n. Let consider the second equation of (1), we have ∂ t n − ∆n = −u · ∇n − ∇ · (n∇φ). (61) Proof. Applying the operator ∆ j to (61), as the same estimates (56), we obtain Thanks to [11,27], there exists a positive constantc so that It is clear that, formally, we have the following homogeneous Bony's paraproduct decomposition u∇n = T u ∇n + R(u, ∇n) + T ∇n u.
Proof of Theorem 1.2. Let T * be a maximal time of existence introducted in Theorem 1.1. Hence, to prove Theorem 1.2, we only need to prove that T * = ∞ with (u, n, c) ∈ Θ ∩ Θ C provided that there holds (7). Let η, ϑ be positive constants, which will be determined later on, we define Applying Propositions 5 and 6, we obtain Using the Cauchy-Schwartz inequality ab ≤ ε 2 a 2 + 1 2ε b 2 for ε = 2 Thanks to (76) and Proposition 4, we have We deduce from (76) and Proposition 4 that Thanks to (74), taking C > 0 large enough in (7) follows that which show that We thus obtain Thus (81) contradicts with the definition (74), in view of the standard continuity argument, we can conclude that T := T * . Therefore, we complete the proof of Theorem 1.2.