In a recent paper [6], the global well-posedness of the two-dimensional Euler equation with vorticity in was proved, where LBMO is a Banach space which is strictly imbricated between and BMO. In the present paper we prove a global result on the inviscid limit of the Navier–Stokes system with data in this space and other spaces with the same BMO flavor. Some results of local uniform estimates on solutions of the Navier–Stokes equations, independent of the viscosity, are also obtained.
Keywords: 2D incompressible Navier–Stokes equations, Inviscid limit, Global well-posedness, BMO-type space
@article{AIHPC_2016__33_2_597_0, author = {Bernicot, Fr\'ed\'eric and Elgindi, Tarek and Keraani, Sahbi}, title = {On the inviscid limit of the {2D} {Navier{\textendash}Stokes} equations with vorticity belonging to {BMO-type} spaces}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {597--619}, publisher = {Elsevier}, volume = {33}, number = {2}, year = {2016}, doi = {10.1016/j.anihpc.2014.12.001}, zbl = {1332.35280}, mrnumber = {3465387}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.12.001/} }
TY - JOUR AU - Bernicot, Frédéric AU - Elgindi, Tarek AU - Keraani, Sahbi TI - On the inviscid limit of the 2D Navier–Stokes equations with vorticity belonging to BMO-type spaces JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 597 EP - 619 VL - 33 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.12.001/ DO - 10.1016/j.anihpc.2014.12.001 LA - en ID - AIHPC_2016__33_2_597_0 ER -
%0 Journal Article %A Bernicot, Frédéric %A Elgindi, Tarek %A Keraani, Sahbi %T On the inviscid limit of the 2D Navier–Stokes equations with vorticity belonging to BMO-type spaces %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 597-619 %V 33 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2014.12.001/ %R 10.1016/j.anihpc.2014.12.001 %G en %F AIHPC_2016__33_2_597_0
Bernicot, Frédéric; Elgindi, Tarek; Keraani, Sahbi. On the inviscid limit of the 2D Navier–Stokes equations with vorticity belonging to BMO-type spaces. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 2, pp. 597-619. doi : 10.1016/j.anihpc.2014.12.001. http://www.numdam.org/articles/10.1016/j.anihpc.2014.12.001/
[1] Some theorems on stable processes, Trans. Amer. Math. Soc., Volume 95 (1960), pp. 263–273 | DOI | MR | Zbl
[2] Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343, 2011 | DOI | MR | Zbl
[3] Self-improving properties for abstract Poincaré type inequalities, Trans. Amer. Math. Soc. (2015) (in press) | DOI | MR | Zbl
[4] On the global well-posedness for Euler equations with unbounded vorticity | arXiv | DOI | Zbl
[5] Sharp constants for composition with a measure-preserving map, Math. Res. Lett., Volume 21 (2014), pp. 937–952 | DOI | MR | Zbl
[6] On the global well-posedness of the 2D Euler equations for a large class of Yudovich type data, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 3, pp. 559–576 | MR | Zbl
[7] Vorticity and Incompressible Flow, Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, Cambridge, 2002 | MR | Zbl
[8] Estimates in the generalized Campanato–John–Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., Volume 118 (2003) no. 1, pp. 1–17 | DOI | MR | Zbl
[9] Weak solutions of 2-D Euler equations with initial vorticity in , J. Differ. Equ., Volume 103 (1993), pp. 323–337 | DOI | MR | Zbl
[10] Perfect Incompressible Fluids, The Clarendon Press, Oxford University Press, New York, 1998 | MR | Zbl
[11] A remark on the inviscid limit for two-dimensional incompressible fluids, Commun. Partial Differ. Equ., Volume 21 (1996) no. 11–12, pp. 1771–1779 (in English, French summary) | MR | Zbl
[12] The bi-dimensional Euler equations in bmo-type space | arXiv
[13] Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type, J. Differ. Equ., Volume 235 (2007) no. 2, pp. 647–657 | DOI | MR | Zbl
[14] Existence de nappes de tourbillon en dimension deux, J. Am. Math. Soc., Volume 4 (1991), pp. 553–586 | MR | Zbl
[15] Concentrations in regularization for 2D incompressible flow, Commun. Pure Appl. Math., Volume 40 (1987), pp. 301–345 | DOI | MR | Zbl
[16] Heat kernels on metric measure spaces, Proceedings of AFRT2012, Geometry and Analysis of Fractals, 2014, pp. 147–208 | DOI | MR | Zbl
[17] Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal., Volume 158 (2001) no. 3, pp. 235–257 | MR | Zbl
[18] Self-improving properties of John–Nirenberg and Poincaré inequalities on space of homogeneous type, J. Funct. Anal., Volume 153 (1998) no. 1, pp. 108–146 | DOI | MR | Zbl
[19] On the nonstationary Navier–Stokes system, Rend. Semin. Mat. Univ. Padova, Volume 32 (1962), pp. 243–260 | Numdam | MR | Zbl
[20] Résultats récents sur les fluides parfaits incompressibles bidimensionnels, Astérisque, Volume 206 (1992), pp. 411–444 (d'après J.-Y. Chemin et J.-M. Delort Séminire Bourbaki, 1991/1992, No. 757) | Numdam | MR | Zbl
[21] 2D Navier–Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal., Volume 104 (1988), pp. 223–250 | DOI | MR | Zbl
[22] Classical and Modern Fourier Analysis, Prentice Hall, New York, 2006
[23] The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity, Math. Res. Lett., Volume 11 (2004) no. 4, pp. 519–528 | DOI | MR | Zbl
[24] Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta Math., Volume 63 (1934), pp. 193–248 | DOI | JFM | MR
[25] Mathematical Topics in Fluid Mechanics, vol. 1, The Clarendon Press, Oxford University Press, New York, 1996
[26] Remarks about the inviscid limit of the Navier–Stokes system, Commun. Math. Phys., Volume 270 (2007) no. 3, pp. 777–788 | DOI | MR | Zbl
[27] On convolution operators leaving spaces invariant, Ann. Mat. Pura Appl., Volume 72 (1966), pp. 295–304 | MR | Zbl
[28] Structures holomorphes à faible régularité spatiale en mécanique des fluides, J. Math. Pures Appl., Volume 74 (1995), pp. 95–104 | MR | Zbl
[29] Some function spaces defined using the mean oscillation over cubes, Ann. Sc. Norm. Super. Pisa, Volume 19 (1965), pp. 593–608 | Numdam | MR | Zbl
[30] Uniformly local estimate for 2D vorticity equation and its application to Euler equations with initial vorticity in BMO, Commun. Math. Phys., Volume 248 (2004), pp. 169–186 | DOI | MR | Zbl
[31] Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Éc. Norm. Supér. (4), Volume 32 (1999) no. 6, pp. 769–812 (in English, French summary) | Numdam | MR | Zbl
[32] Nonstationary flow of an ideal incompressible liquid, Zh. Vychisl. Mat., Volume 3 (1963), pp. 1032–1066 | MR | Zbl
[33] Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., Volume 2 (1995), pp. 27–38 | DOI | MR | Zbl
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