Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces

: The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in R n for n ≥ 2 on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov–Morrey and Herz spaces considered in known works on Navier–Stokes equations. Morrey–Sobolev and Besov–Morrey spaces based on weak-Herz space denoted as W ˙ K α p , q M s µ and W ˙ K α p , q ˙ N s µ , r , respectively, represent new properties and interpolations. This class of spaces and its developed properties could also be employed to study elliptic, parabolic, and conservation-law type PDEs.


Introduction
Let us consider R n with n ≥ 2, and a fixed interval with 0 < T < ∞. The incompressible Navier-Stokes equations system in R n × (0, T) is written in the form where vector values u and f denote the velocity of the fluid and external forces acting on the fluid, respectively. The scalar value p represents the pressure. The nonstationary Navier-Stokes equations are invariant under the following change of scaling: u λ (x, t) = λu(λx, λ 2 t), p λ (x, t) = λ 2 p(λx, λ 2 t), ∀t > 0.
There are several works [7,10,11], where properties of the Besov-Morrey space were provided, and they also included related K θ,r -method real interpolations. The properties of the Besov-weak Herz space BWK α,s p,q,r were explored in [12]. Herz-type BesovK α,p q B s β anḋ K α,p q F s β Triebel-Lizorkin spaces were considered in [6,7]. These spaces were introduced to explore global solutions of NSE in the case that f = 0 and to prove the Jawerth-Franke P = (P ij ) 1≤i,j≤n , P ij = δ ij + R i R j , i, j = 1, ..., n, where (δ ij ) 1≤i,j≤n is the Kronecker symbol and R j , j = 1, ..., n are the Riesz transforms that can be represented by using Fourier transform: where f ∈ S(R n ). From the Calderón-Zygmund operator theory, for 1 < p < ∞, 0 ≤ µ < n, the boundedness of Riesz transform R j on the Morrey space M p,µ implies that P is bounded on M p,µ , as it was remarked in [10].
The Navier-Stokes equations can be transformed into an integral formula where Functions that satisfy (2) are called mild solutions of the NSE. Applying P to (1), we have where A = −P∆ is the Stokes operator.
In [15], mild solutions were constructed for L α,∞ (0, ∞; L p,∞ (R n )), where 2/α + n/p = 3 and max{1, n/3} < p < ∞. In [10], these properties were extended to homogeneous Besov-Morrey spaceṄ s q,µ,r (R n ), and especially estimates of heat semigroup operator e t∆ . According to interpolations and Lemma 2.3 from [12] for Besov-weak Herz space, we prove the interpolation of the proposed weak Herz-type Besov-Morrey spaces. The motivation of our research is to propose new hybrid spaces (weak Herz-type Besov-Morrey spaces), which contain the properties of several global spaces (Herz, Besov-Morrey spaces), and explore mild solutions of the incompressible Navier-Stokes equations with f = 0. Mild solutions were researched on Besov-Morrey and Herz spaces with proper interpolations, embeddings, and estimates in [9,10], respectively. Herz-type BesovK α,p q B s β and Triebel-Lizorkin spacesK α,p q F s β were engaged in [6], and BWK α,s p,q,r Besov-weak Herz spaces in [12]. In our manuscript, we explore weak Herz-type Besov-Morrey spaces WK α p,qṄ s µ,r (R n ), which were not met in other publications. Therefore, it would be reasonable to provide their properties and study mild solutions of NSE on such spaces.
Our main results are Suppose that a measurable function u on R n × (0, T) is a mild solution of (4) and satisfies where C = C(n, µ, q, α, γ, δ, s, p, r) is independent of u, u 0 and T.
Extension of (weak) Herz and Besov-Morrey spaces to (weak) Herz-type Besov-Morrey spaces allows enlarging their properties, especially embedding, interpolations and wavelet characterizations. Moreover, atomic partition and oscillations in [1,2] make it possible to receive useful estimates and properties of solutions of nonlinear PDEs and investigate the similar extension on Triebel-Lizorkin-Morrey spaces researching mild solutions of NSE with f = 0.
Our main contribution to the theory of Navier-Stokes equations is providing estimates in Theorems 1 and 2, which can state the maximal Lorentz regularity of a function u in WK α p,qṄ s µ,r (R n ). This allows us to approach establishing the unique existence of local strong or weak solutions to (1) for arbitrary large initial data u 0 and large external force f . The maximal Lorentz regularity is exploited for Besov-Morrey space in [10].
The current problems of nonlinear PDEs need new tools, such as embedding, wavelet characterization, real (K-and J-types), and complex interpolations. In our manuscript, we provide and prove K-real interpolations for Herz-type Besov-Morrey spaces, which allow us to imply useful estimates in Lemma 1 that engage not only the heat semigroup operator, but also the Leray projection. In [10] in Lemma 2.2 for Besov-Morrey spaces, Leray projection was not considered, while for weak Herz space, it was shown in [14] (Corollary 1). Combining Besov-Morrey and weak Herz spaces into Herz-type Besov-Morrey spaces allows us to imply new estimates in Lemma 1, by real interpolation of new proposed spaces.
The remaining of the paper is organized as follows. Section 2 is devoted to function spaces and some necessary statements from references. Section 3 defines weak Herz-type Besov-Morrey space and proves the interpolation of Theorem 3 and Lemma 1, providing essential properties and inequalities. Section 4 and 5 are devoted to proofs of Theorems 1 and 2, respectively.

Definition 2.
With the same conditions as in Definition 1, one defines the homogeneous weak Herz space WK α p,q (R n ) as the space of measurable functions such that The definition and the basic properties of Morrey and Besov-Morrey spaces were reviewed in [6,10,11].
From [13] we recall the definition of the Lorentz space that is applied in the proofs of the theorems.
In particular, L p,∞ agrees with the weak- Let us provide Proposition 2.2 and Corollary 2.1 in [14].
Then we have the following estimate: provided that one of the following cases holds: Some properties of the operators e t∆ and e t∆ P are investigated and proved in [14], which gives us a necessary estimate.
Let B r (x 0 ) be the open ball in R n centered at x 0 and radius r > 0. The definition of the Morrey-type (weak) Herz space is provided in [14].
For weak Morrey-type Herz space, we substitute the norm of the WK α p,q instead the norm of theK α p,q .
As in the [10,11] for s ∈ R and 1 ≤ p < ∞, the homogeneous weak Sobolev-Morrey- Additionally, the Herz-type Sobolev space can be defined by means of the Riesz potential I s = (−∆) s/2 , as in [16], defined as

Weak Herz-Type Besov-Morrey Space and Its Properties
Let S(R n ) and S (R n ) be the Schwarz space and the tempered distributions space, respectively. Let φ ∈ S(R n ) be a non-negative radial function such that Let us define the homogeneous weak Herz-type Besov-Morrey space.
Definition 5. For 1 ≤ p < ∞, 0 < q ≤ ∞, 0 ≤ µ < n and s, α ∈ R, the homogeneous weak Herz-type Besov-Morrey space WK α p,qṄ s µ,r with r ∈ [1, ∞] is the set of f ∈ S /P, where P is the set of polynomials, such that F −1 φ k * f ∈ WK α p,q M µ and We denote the localization operators of the Littlewood-Paley decomposition as The space WK α p,qṄ s µ,r is Banach and in particular, WK 0 p,pṄ s 0,r corresponds to the homogeneous Besov space with weak-Lebesgue space, which implies the K θ,r -method realinterpolation properties.
Proof. Let f = f 0 + f 1 with f i ∈ WK α p,q M s i µ , i = 0, 1. By using Lemma 2.3 from [12] for weak Herz-type Sobolev space, we note that WK α,s p,q is an Herz-Sobolev space and it holds that where C is a constant. Therefore, It follows that Multiplying the previous inequality by 2 js and s − s 0 = −θ(s 0 − s 1 ), we obtain and then (see Lemma 3.1.3 from [17]) we can conclude that To prove the reverse inequality of (14), note that by using Lemma 2.3 from [12], again we have
for every t > 0, r ∈ [1, ∞] and f ∈ S /P. For all inequalities α − β ≤ s 0 − s and C is a constant. [14] for Herz-type Sobolev-Morrey spaces. Now we use the Lemma 2.2 (i) from [10] and to get

Proof. (1) We use inequality
(2) As in first part of this proof we can use Corollary 2.1 (iv) in [14] with respect to weak Herz space Particularly we estimate the norm of the weak Herz-type Morrey space Then it follows that Now by applying the Lemma 2.2 (ii) from [10] and properties of first part of this proof to (18), it implies that Additionally, if α − β ≤ s 0 − s, then we receive (16).
Examples 1 and 2 demonstrate functions belonging to weak Herz-type Besov-Morrey spaces that satisfy inequalities in Lemma 1.
In function space theory [18,19], it could be useful to provide a norm of WK α p,qṄ s µ,r , defined by derivatives and differences, equivalent to the norm in Definition 5. In the case of Besov spaces, such an approach was used in [20,21], where the authors established the equivalence between the norms defined by Fourier analytic tools and by derivatives and differences, respectively. Theorems 1 and 2 allow to provide the maximal Lorentz regularity theorem of Stokes and Navier-Stokes equations. They can help in establishing the unique existence of local strong solutions to Navier-Stokes equation on proposed weak Herz-type Besov-Morrey spaces, as it is made in [10] for homogeneous Besov-Morrey spaces and in [15] in Lorentz spaces.
The properties of Herz-type Besov-Morrey spaces, such as the interpolations in Theorem 3 and the inequalities in Lemma 1, can be also used to study other nonlinear PDEs. For example, a mathematical model of waves on shallow water surfaces described by Korteweg-de Vries equation [22]; the Keller-Segel system [23] presents a cellular chemotaxis model; and Fokker-Planck equations [24] demonstrate models of anomalous diffusion processes. Developing atomic decomposition, oscillations, real and complex interpolations can advance the study of the WK α p,qṄ s µ,r spaces, especially observing them not only with the Fourier approach ( [25]), but by the finite difference approach, in the same fashion of Besov spaces in [26,27].

Conclusions
This article focused on mild solutions of the incompressible Navier-Stokes equations with external forces on R n for n ≥ 2 on Herz-type Besov-Morrey spaces. We introduced real interpolations on WK α p,q M s µ and WK α p,qṄ s µ,r and discussed some useful properties, which were proved in Theorem 3. The inequalities in Lemma 1 were extended from WK α p,q (R n ), M s µ (R n ), andṄ s p,q,r (R n ) into WK α p,q M s µ (R n ) and WK α p,qṄ s µ,r (R n ). Applying such properties, we achieved some estimates for mild solutions of Navier-Stokes equations, described in Theorems 1 and 2.
The function spaces theory propagates not only for nonlinear PDEs and abstract harmonic analysis, but for global and geometric analysis. For example, Besov and Triebel-Lizorkin spaces are defined on the Riemannian manifold, Lie groups, and fractals. Weak Herz-type Besov-Morrey spaces can be applied, for instance, in Riemannian geometry, global and geometric analysis, pseudo-differential operator theory, and approximation theory.
The provided estimates can be helpful to explore mild solutions of Navier-Stokes equations and imply the existence and uniqueness of weak and strong solutions. Theorems 1.2-1.4 from [10] show the uniqueness of strong solutions for Navier-Stokes equations, from properties of mild solutions on Besov-Morrey spaces. Future works could focus on obtaining some features of weak Herz-type Besov-Morrey spaces, such as their interpolations, atomic decompositions, and representation via finite differences. Combining (weak) Herz and Triebel-Lizorkin-Morrey spaces may be useful for further studying nonlinear PDEs.