Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces - Wang Yuzhao; Xiao Jie (Advances in Nonlinear Analysis, 2017)
As an essential extension of the well known case β∈(12,1]{β\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case β∈(1,∞){β\kern-1.0pt\in\kern-1.0pt(1,∞)}, this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation (-Δ)12<β<∞{(-Δ)^{{\frac{1}{2}}<β<∞}} through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space (Q-s=-α)n{(Q_{-s=-α})^{n}}, the BMO-Sobolev space ((-Δ)-s2⁢BMO)n{((-Δ)^{-{\frac{s}{2}}}BMO)^{n}}, the Lip-Sobolev space ((-Δ)-s2⁢Lip⁢α)n{((-Δ)^{-{\frac{s}{2}}}Lipα)^{n}}, and the Besov space (B˙∞,∞s)n{(\dot{B}^{s}_{∞,∞})^{n}}.