Well-posedness and regularity of generalized Navier-Stokes equations in some Critical $Q-$spaces

Abstract

We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space $Q_{\alpha;\infty}^{\beta,-1}(\mathbb{R}^{n})=\nabla\cdot(Q_{\alpha}^{\beta}(\mathbb{R}^{n}))^{n}, \beta\in({1/2},1)$ which is larger than some known critical homogeneous Besov spaces. Here $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ is a space defined as the set of all measurable functions with $$\sup(l(I))^{2(\alpha+\beta-1)-n}\int_{I}\int_{I}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2(\alpha-\beta+1)}}dxdy<\infty$$ where the supremum is taken over all cubes $I$ with the edge length $l(I)$ and the edges parallel to the coordinate axes in $\mathbb{R}^{n}.$ In order to study the well-posedness and regularity, we give a Carleson measure characterization of $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ by investigating a new type of tent spaces and an atomic decomposition of the predual for $Q_{\alpha}^{\beta}(\mathbb{R}^{n}).$ In addition, our regularity results apply to the incompressible Navier-Stokes equations with initial data in $Q_{\alpha;\infty}^{1,-1}(\mathbb{R}^{n}).$