Abstract
We establish two new estimates for a transport-diffusion equation. As an application we treat the problem of global persistence of the Besov regularity \(B_{p,1}^{\frac{2}{p}+1},\) with \(p \in ]2,+\infty]\) , for the two-dimensional Navier–Stokes equations with uniform bounds on the viscosity. We provide also an inviscid global result.
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References
Bony J.-M. (1981) Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. École Norm. Sup. 14, 209–246
Beale J.T., Kato T., Majda A. (1984) Remarks on the breakdown of smooth solutions for 3-D Euler equations. Commun. Math. Phys. 94, 61–66
Chae D. (2004) Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymptot. Anal. 38(3–4): 339–358
Chemin J.-Y. (1998) Perfect incompressible Fluids. Oxford University Press, New York
Chemin J.-Y. (1999) Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel. J. Anal. Math. 77, 27–50
Danchin R. (1997) Poches de tourbillon visqueuses. J. Math. Pures Appl. (9) 76(7): 609–647
Fujita H., Kato T. (1962) On the nonstationnary Navier–Stokes system. Rend. Sem. Mat. Univ. Padova, 32, 243–260
Hmidi T. (2005) Régularité höldérienne des poches de tourbillon visqueuses. J. Math. Pures Appl. (9) 84(11): 1455–1495
Hmidi T., Keraani S. (2007) Inviscid limit for the two-dimensional Navier–Stokes system in a critical Besov space. Asymptotic Anal. 53(3): 125–138
Hmidi T., Keraani S. (2005) Existence globale pour le système d’Euler incompressible 2-D dans \(B_{\infty,1}^1\) . C. R. Math. Acad. Sci. Paris 341(11): 655–658
Kato T., Ponce G. (1988) Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907
Leray J. (1934) Sur le mouvement d’un liquide visqueux remplissant l’espace. Acta Math. 63, 193–248
Majda A. (1986) Vorticity and the mathematical theory of an incompressible fluid flow. Commun. Pure Appl. Math. 38, 187–220
Serfati P. (1995) Solutions C ∞ en temps, n-log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris 320(5): 555–558
Vishik M. (1998) Hydrodynamics in Besov Spaces. Arch. Rational Mech. Anal 145, 197–214
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Communicated by Y. Brenier
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Hmidi, T., Keraani, S. Incompressible Viscous Flows in Borderline Besov Spaces. Arch Rational Mech Anal 189, 283–300 (2008). https://doi.org/10.1007/s00205-008-0115-7
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DOI: https://doi.org/10.1007/s00205-008-0115-7