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The Navier-Stokes equations in nonendpoint borderline Lorentz spaces. (English) Zbl 1326.35249

Summary: 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 L. Escauriaza et al. [Russ. Math. Surv. 58, No. 2, 211–250 (2003); translation from Usp. Mat. Nauk 58, No. 2, 3–44 (2003; Zbl 1064.35134)], 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.

MSC:

35Q30 Navier-Stokes equations

Citations:

Zbl 1064.35134
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References:

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