Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions <i>d</i> ≥ 4

Li, Kuijie; Wang, Baoxiang

doi:10.1016/j.anihpc.2019.02.003

Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4

Li, Kuijie ; Wang, Baoxiang

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1679-1707.

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Résumé

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier–Stokes equation in higher spatial dimensions $d \geq 4$ . By establishing an ϵ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in ${\dot{B}}_{p, q}^{- 1 + d / p} (R^{d})$ , $d < p, q < \infty$ becomes singular at a finite time $T_{⁎}$ , then

\underset{t \to T_{⁎}}{\lim \sup} {‖ u (t) ‖}_{{\dot{B}}_{p, q}^{- 1 + d / p} (R^{d})} = \infty .

The corresponding result in 3D case has been obtained in [24]. As a by-product, we also prove a regularity criterion for the Leray–Hopf solution in the critical Besov space, which generalizes the results in [17], where blowup criterion in critical Lebesgue space

L^{d} (R^{d})

is addressed.

DOI : 10.1016/j.anihpc.2019.02.003

Mots-clés : Navier–Stokes equation, Blowup criterion, Critical Besov spaces, Higher spatial dimensions

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     author = {Li, Kuijie and Wang, Baoxiang},
     title = {Blowup criterion for {Navier{\textendash}Stokes} equation in critical {Besov} space with spatial dimensions \protect\emph{d} \ensuremath{\geq} 4},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1679--1707},
     publisher = {Elsevier},
     volume = {36},
     number = {6},
     year = {2019},
     doi = {10.1016/j.anihpc.2019.02.003},
     mrnumber = {4002170},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.003/}
}

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Li, Kuijie; Wang, Baoxiang. Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1679-1707. doi : 10.1016/j.anihpc.2019.02.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.02.003/

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