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 Qα;∞β,−1(Rn)=∇⋅(Qαβ(Rn))n, β∈(12,1), which is larger than some known critical homogeneous Besov spaces. Here Qαβ(Rn) is a space defined as the set of all measurable functions withsup(l(I))2(α+β−1)−n∫I∫I|f(x)−f(y)|2|x−y|n+2(α−β+1)dxdy<∞ 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 Qαβ(Rn) by investigating a new type of tent spaces and an atomic decomposition of the predual for Qαβ(Rn). In addition, our regularity results apply to the incompressible Navier–Stokes equations with initial data in Qα;∞1,−1(Rn).