Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces

We study the well-posedness and regularity of the generalized Navier–Stokes equations with initial data in a new critical space View the MathML sourceQα;∞β,−1(Rn)=∇⋅(Qαβ(Rn))n, View the MathML sourceβ∈(12,1), which is larger than some known critical homogeneous Besov spaces. Here View the MathML sourceQαβ(Rn) is a space defined as the set of all measurable functions with View the MathML sourcesup(l(I))2(α+β−1)−n∫I∫I|f(x)−f(y)|2|x−y|n+2(α−β+1)dxdy<∞ Turn MathJax on where the supremum is taken over all cubes I with edge length l(I)l(I) and edges parallel to the coordinate axes in RnRn. In order to study the well-posedness and regularity, we give a Carleson measure characterization of View the MathML sourceQαβ(Rn) by investigating a new type of tent spaces and an atomic decomposition of the predual for View the MathML sourceQαβ(Rn). In addition, our regularity results apply to the incompressible Navier–Stokes equations with initial data in View the MathML sourceQα;∞1,−1(Rn).