Local well-posedness for the incompressible Euler equations in the critical Besov spaces

Zhou, Yong

Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 773-786.

Dans cet article on établit l’existence et l’unicité de la solution locale de l’équation d’Euler incompressible dans $ℝ$${}^N$, $N\ge 3$, avec des données initiales quelconques appartenant aux espaces de Besov critique $B_{p,1}^{N/p+1}$. De plus, un critère d’explosion est donné en terme du champ de vorticités.

In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in $ℝ$${}^N$, $N\ge 3$, with any given initial data belonging to the critical Besov spaces $B_{p,1}^{N/p+1}$. Moreover, a blowup criterion is given in terms of the vorticity field.

Classification : 76D03, 35Q35, 46E35.
Keywords: well-posedness, Euler equations, Besov spaces
Mot clés : bien-posé, equations d'Euler, espaces de Besov

[1] J.-T. Beale; T. Kato; A. Majda Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys, Volume 94 (1984) no. 1, pp. 61-66 | MR | Zbl

[2] J.-M. Bony Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ecole Norm. Sup. (4), Volume 14 (1981) no. 2, pp. 209-246 | Numdam | MR | Zbl

[3] D. Chae On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces., Comm. Pure Appl. Math, Volume 55 (2002) no. 5, pp. 654-678 | MR | Zbl

[4] J.-Y. Chemin Régularité de la trajectoire des particules d'un fluide parfait incompressible remplissant l'espace, J. Math. Pures Appl. (9), Volume 71 (1992) no. 5, pp. 407-417 | MR | Zbl

[5] J.-Y. Chemin Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998 | MR | Zbl

[6] R. Danchin Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math, Volume 141 (2000) no. 3, pp. 579-614 | MR | Zbl

[7] R. Danchin Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, Volume 26 (2001) no. 7-8, pp. 1183-1233 | MR | Zbl

[8] M. Frazier; B. Jawerth; G. Weiss Littlewood-Paley theory and the study of function spaces (CBMS Regional Conference Series in Mathematics), Volume 79 (1991) | Zbl

[9] T. Kato Nonstationary flows of viscous and ideal fluids in $R^3$, J. Functional Analysis, Volume 9 (1972), pp. 296-305 | MR | Zbl

[10] T. Kato; G. Ponce Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math, Volume 41 (1988) no. 7, pp. 891-907 | MR | Zbl

[11] H. Kozono; Y. Taniuchi Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys., Volume 214 (2000) no. 1, pp. 191-200 | MR | Zbl

[12] H. Kozono; T. Ogawa; Y. Taniuchi The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z, Volume 242 (2002), pp. 251-278 | MR | Zbl

[13] A. Majda Vorticity and the mathematical theory of incompressible fluid flow, Frontiers of the mathematical sciences (New York, 1985) (Comm. Pure Appl. Math.), Volume 39, suppl. (1986), p. S186-S220 | Zbl

[14] T. Runst; W. Sickel Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations., de Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter \& Co., Berlin, 1996 | MR | Zbl

[15] E.M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Monographs in Harmonic Analysis, III (Princeton Mathematical Series), Volume 43 (1993) | Zbl

[16] H. Triebel Theory of function spaces. II., Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992 | Zbl

[17] M. Vishik Hydrodynamics in Besov spaces., Arch. Ration. Mech. Anal, Volume 145 (1998) no. 3, pp. 197-214 | MR | Zbl

[18] M. Vishik Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Ecole Norm. Sup. (4), Volume 32 (1999) no. 6, pp. 769-812 | Numdam | MR | Zbl