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

Ruslan Abdulkadirov,Pavel Lyakhov

doi:10.3390/math10050680

Abstract

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in Rn 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 WK˙p,qαMμs and WK˙p,qαN˙μ,rs, 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.

Highlights

Let us consider Rn with n ≥ 2, and a fixed interval with 0 < 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
Mathematics 2022, 10, 680 of our research is to propose new hybrid 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 6= 0

Summary

Introduction

Let us consider Rn with n ≥ 2, and a fixed interval with 0 < T < ∞. Mathematics 2022, 10, 680 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 6= 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 s (Rn ) This allows us to approach establishing the unique existence of local strong. 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 the heat semigroup operator, and the Leray projection.

Preliminaries

Weak Herz-Type Besov–Morrey Space and Its Properties

Proofs of the Theorem 1

Proofs of the Theorem 2

Conclusions