Well-posedness of Naiver–Stokes equations in logarithmic Q spaces covering $BMO^{-1}$ and its fractional counterpart

Source abstract

Abstract In this article, we introduce a new logarithmic Q -type space $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to study the well-posedness of the classical/fractional Naiver–Stokes equations. We show that $\nabla \cdot (Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n}))^{n}$ covers the well-known critical spaces $BMO^{-1}(\mathbb R^{n}), Q_{\alpha }^{-1}(\mathbb R^{n})$ and $\mathcal {Q}_{0}^{-1}(\mathbb R^{n})$ for the classical Naiver–Stokes equations. Moreover, it covers the fractional counterparts $BMO^{-(2\beta -1)}(\mathbb R^{n}), Q_{\alpha }^{\beta ,-1}(\mathbb R^{n})$ and even the largest critical space $\dot {B}^{-(2\beta -1)}_{\infty ,\infty }(\mathbb R^{n}).$ In doing so, we first establish some basic properties of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb {R}^{n}).$ Then, via the fractional heat semigroups, we characterize the extension of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to $\mathscr H_{K_{\ln }^{(l,k)}}^{p,\lambda }(\mathbb R_+^{n+1})$ which is a function space related to the weight function $K_{\ln }^{(l,k)}(\cdot )$ . This extension provides a semigroup characterization of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ . With this in hand, we establish the well-posedness of mild solutions to fractional Naiver–Stokes equations and fractional magneto-hydrodynamic equations, respectively, with small data in $\nabla \cdot \left (Q_{\ln ,\frac {4(1-\beta )}{n}}^{2,k,l+2(1-\beta )}(\mathbb {R}^{n})\right )^{n}$ for $k\in \mathbb {N}$ and $l>n+2\beta -4.$