In a domain ω ⊂ Rn, consider a weak solution u of the Navier-Stokes equations in the class u ε L∞(0, T;Ln(ω)). If lim supt→t*-0 ||u(t)||nn-||u(t*)||nn is small at each point of t* ε (0, T), then u is regular on ω̄ × (0, T). As an application, we give a precise characterization of the singular time; i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T* < T, then either lim supt→T*-0 ||u(t)||Ln(ω) = +∞, or u(t) oscillates in Ln(ω) around the weak limit wlimt→ T*-0 u(t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in Ln(ω) becomes regular.
|ジャーナル||Advances in Differential Equations|
|出版物ステータス||Published - 1997 12 1|
ASJC Scopus subject areas
- Applied Mathematics