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

TL;DR

This paper proves the global existence and uniqueness of strong solutions to a generalized Navier-Stokes system with fractional Laplacian in critical Besov spaces, extending results to nonlinear convective terms and fractional diffusion.

Contribution

It introduces new estimates in Besov spaces for the nonlinear term and employs maximal regularity in Lorentz spaces to establish global solvability for both linear and nonlinear cases.

Findings

Established global existence and uniqueness of solutions

Extended analysis to nonlinear convective terms

Utilized maximal regularity in Lorentz spaces

Abstract

This paper is devoted to the global solvability of the Navier-Stokes system with fractional Laplacian $(-\Delta)^{\alpha}$ 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 $(-\Delta)^{\alpha}$ 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 $m>1$