The vorticity equations in a half plane with measures as initial data

Abe, Ken

doi:10.1016/j.anihpc.2020.10.002

Abe, Ken ¹

¹ Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku Osaka, 558-8585, Japan

Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1055-1094.

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Résumé

We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an $L^{1}$ -estimate of a solution operator for the vorticity equations associated with the Stokes equations.

Reçu le : 2019-10-12
Révisé le : 2020-09-24
Accepté le : 2020-10-20

DOI : 10.1016/j.anihpc.2020.10.002

Classification : 35Q35, 35K90
Mots-clés : Vorticity equations, Half plane, Finite measures

Affiliations des auteurs :

Abe, Ken ¹

¹ Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku Osaka, 558-8585, Japan

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Abe, Ken. The vorticity equations in a half plane with measures as initial data. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1055-1094. doi : 10.1016/j.anihpc.2020.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.002/

Bibliographie
Cité par

[1] Abe, K. Global well-posedness of the two-dimensional exterior Navier-Stokes equations for non-decaying data, Arch. Ration. Mech. Anal., Volume 227 (2018), pp. 69-104 | DOI | MR

[2] Abe, K.; Giga, Y. Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Math., Volume 211 (2013), pp. 1-46 | DOI | MR | Zbl

[3] Adams, R.A.; Fournier, J.J.F. Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003 | MR | Zbl

[4] Ben-Artzi, M. Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Ration. Mech. Anal., Volume 128 (1994), pp. 329-358 | DOI | MR | Zbl

[5] Bergh, J.; Löfström, J. Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976 | MR | Zbl

[6] Borchers, W.; Miyakawa, T. $L^{2}$ decay for the Navier-Stokes flow in halfspaces, Math. Ann., Volume 282 (1988), pp. 139-155 | DOI | MR | Zbl

[7] Brezis, H. Remarks on the preceding paper by M. Ben-Artzi “Global solutions of two-dimensional Navier-Stokes and Euler equations”, Arch. Ration. Mech. Anal., Volume 128 (1994), pp. 359-360 | DOI | MR | Zbl

[8] Carpio, A. Asymptotic behavior for the vorticity equations in dimensions two and three, Commun. Partial Differ. Equ., Volume 19 (1994), pp. 827-872 | DOI | MR | Zbl

[9] Cottet, G.-H. Équations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci., Sér. 1 Math., Volume 303 (1986), pp. 105-108 | MR | Zbl

[10] Desch, W.; Hieber, M.; Prüss, J. $L^{p}$ -theory of the Stokes equation in a half space, J. Evol. Equ., Volume 1 (2001), pp. 115-142 | DOI | MR | Zbl

[11] Folland, G.B. Real Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999 | MR | Zbl

[12] Galdi, G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics, Springer, New York, 2011 | MR | Zbl

[13] Gallagher, I.; Gallay, T. Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., Volume 332 (2005), pp. 287-327 | DOI | MR | Zbl

[14] Gallagher, I.; Gallay, T.; Lions, P.-L. On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity, Math. Nachr., Volume 278 (2005), pp. 1665-1672 | DOI | MR | Zbl

[15] Gallay, T.; Wayne, C.E. Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $R^{2}$ , Arch. Ration. Mech. Anal., Volume 163 (2002), pp. 209-258 | DOI | MR | Zbl

[16] Gallay, T.; Wayne, C.E. Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Commun. Math. Phys., Volume 255 (2005), pp. 97-129 | DOI | MR | Zbl

[17] Giga, M.-H.; Giga, Y.; Saal, J. Nonlinear Partial Differential Equations, Birkhäuser Boston, Inc., Boston, MA, 2010 | MR | Zbl

[18] Giga, Y.; Kambe, T. Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Commun. Math. Phys., Volume 117 (1988), pp. 549-568 | DOI | MR | Zbl

[19] Giga, Y.; Matsui, S.; Shimizu, Y. On estimates in Hardy spaces for the Stokes flow in a half space, Math. Z., Volume 231 (1999), pp. 383-396 | DOI | MR | Zbl

[20] Giga, Y.; Miyakawa, T. Navier-Stokes flow in $R^{3}$ with measures as initial vorticity and Morrey spaces, Commun. Partial Differ. Equ., Volume 14 (1989), pp. 577-618 | DOI | MR | Zbl

[21] Giga, Y.; Miyakawa, T.; Osada, H. Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal., Volume 104 (1988), pp. 223-250 | DOI | MR | Zbl

[22] Johnston, H.; Liu, J.-G. Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., Volume 199 (2004), pp. 221-259 | DOI | MR | Zbl

[23] Kato, T. The Navier-Stokes equation for an incompressible fluid in $R^{2}$ with a measure as the initial vorticity, Differ. Integral Equ., Volume 7 (1994), pp. 949-966 | MR | Zbl

[24] Kozono, H.; Yamazaki, M. Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space $L^{n, \infty}$ , Houst. J. Math., Volume 21 (1995), pp. 755-799 | MR | Zbl

[25] Liu, J.-G.; Liu, J.; Pego, R.L. Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1443-1487 | DOI | MR | Zbl

[26] Maekawa, Y. Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit, Adv. Differ. Equ., Volume 18 (2013), pp. 101-146 | MR | Zbl

[27] Miyakawa, T. Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations, Kyushu J. Math., Volume 50 (1996), pp. 1-64 | DOI | MR | Zbl

[28] Miyakawa, T.; Yamada, M. Planar Navier-Stokes flows in a bounded domain with measures as initial vorticities, Hiroshima Math. J., Volume 22 (1992), pp. 401-420 | DOI | MR | Zbl

[29] Rudin, W. Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987 | MR | Zbl

[30] Saal, J. The Stokes operator with Robin boundary conditions in solenoidal subspaces of $L^{1} (R_{+}^{n})$ and $L^{\infty} (R_{+}^{n})$ , Commun. Partial Differ. Equ., Volume 32 (2007), pp. 343-373 | DOI | MR | Zbl

[31] Seregin, G. Liouville theorem for 2D Navier-Stokes equations in a half space, J. Math. Sci. (N.Y.), Volume 210 (2015), pp. 849-856 | DOI | MR

[32] Sohr, H. The Navier-Stokes equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[33] Solonnikov, V.A. Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator, Usp. Mat. Nauk, Volume 58 (2003), pp. 123-156 | MR | Zbl

[34] Solonnikov, V.A. On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci. (N.Y.), Volume 114 (2003), pp. 1726-1740 | DOI | MR | Zbl

[35] Stein, E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl

[36] Stein, E.M.; Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971 | MR

[37] Yamazaki, M. The Navier-Stokes equations in the weak- $L^{n}$ space with time-dependent external force, Math. Ann., Volume 317 (2000), pp. 635-675 | DOI | MR | Zbl

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