Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces

Abstract

As an essential extension of the well known case β∈(12,1]{\beta\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case β∈(1,∞){\beta\kern-1.0pt\in\kern-1.0pt(1,\infty)}, 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<∞{(-\Delta)^{{\frac{1}{2}}<\beta<\infty}} through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space (Q-s=-α)n{(Q_{-s=-\alpha})^{n}}, the BMO-Sobolev space ((-Δ)-s2⁢BMO)n{((-\Delta)^{-{\frac{s}{2}}}\mathrm{BMO})^{n}}, the Lip-Sobolev space ((-Δ)-s2⁢Lip⁢α)n{((-\Delta)^{-{\frac{s}{2}}}\mathrm{Lip}\alpha)^{n}}, and the Besov space (B˙∞,∞s)n{(\dot{B}^{s}_{\infty,\infty})^{n}}.

Keywords

• incompressible navier–stokes system with dissipation
• well/ill-posedness for mild solutions
• generalized carleson measure spaces
• 30h25
• 31c15
• 35q30
• 42b37
• 46e35