On the global solvability of the generalised Navier-Stokes system in critical Besov spaces

TL;DR

The paper studies global solvability of a generalized Navier–Stokes system with fractional diffusion $(-\Delta)^{\alpha}$ and a nonlinear convective term $J_m(u)\cdot\nabla u$ in $\mathbb{R}^n$ for $n\ge2$, focusing on critical Besov spaces. It develops a Lorentz–Besov framework and proves maximal regularity for the linear fractional heat semigroup, then establishes nonlinear estimates for $J_m$ and a fixed-point argument to obtain global existence and uniqueness of strong solutions under small data for $\alpha>1/2$ and all $m\ge1$, treating $1<m<2$ and $m\ge2$ separately. The results include local well-posedness without the smallness condition and provide a priori estimates for solutions in the critical spaces with precise time–integrability properties. This work extends the well-posedness theory of the GNS to Besov spaces with a generalized convective power, leveraging Lorentz maximal regularity to handle the nonlinear structure.

Abstract

This paper is devoted to the global solvability of the Navier-Stokes system with fractional Laplacian $(-Δ)^α$ in $\mathbb{R}^{n}$ for $n\geq2$, where the convective term has the form $(|u|^{m-1}u)\cdot\nabla u$ for $m\geq1$. By establishing the estimates for the difference $|u_{1}|^{m-1}u_{1}-|u_{2}|^{m-1}u_{2}$ in homogeneous Besov spaces, and employing the maximal regularity property of $(-Δ)^α$ in Lorentz spaces, we prove global existence and uniqueness of the strong solution of the Navier-Stokes in critical Besov spaces for both $m=1$ and $m>1$

Theorems & Definitions (16)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• Proposition 2.2
• Lemma 2.3: bcd2011, § 2.6 or mwz2012, § 3.1
• Proposition 2.4
• Proposition 2.5
• Remark 2.6
• Proposition 2.7
• Proposition 2.8