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We study the well-posedness and regularity of the generalized Navier-Stokesequations 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 Besovspaces. Here $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ is a space defined as the setof 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 $l(I)$ andthe edges parallel to the coordinate axes in $\mathbb{R}^{n}.$ In order tostudy the well-posedness and regularity, we give a Carleson measurecharacterization of $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ by investigating a newtype of tent spaces and an atomic decomposition of the predual for$Q_{\alpha}^{\beta}(\mathbb{R}^{n}).$ In addition, our regularity results applyto the incompressible Navier-Stokes equations with initial data in$Q_{\alpha;\infty}^{1,-1}(\mathbb{R}^{n}).$

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