The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces
Tóm tắt
It is shown both locally and globally that $${L_t^{\infty}(L_x^{3,q})}$$ solutions to the three-dimensional Navier–Stokes equations are regular provided $${q\neq\infty}$$ . Here $${L_x^{3,q}}$$ , $${0 < q \leq\infty}$$ , is an increasing scale of Lorentz spaces containing $${L^3_x}$$ . Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), which treated the case q = 3. A new local energy bound and a new $${\epsilon}$$ -regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray–Hopf weak solutions in $${L_t^{\infty}(L_x^{3,q})}$$ , $${q\neq\infty}$$ , is also obtained as a consequence.
Tài liệu tham khảo
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