On the radius of analyticity of solutions to the three-dimensional Euler equations
HTML articles powered by AMS MathViewer
- by Igor Kukavica and Vlad Vicol PDF
- Proc. Amer. Math. Soc. 137 (2009), 669-677 Request permission
Abstract:
We address the problem of analyticity of smooth solutions $u$ of the incompressible Euler equations. If the initial datum is real–analytic, the solution remains real–analytic as long as $\int _{0}^{t} \left \Vert {\nabla u(\cdot ,s)}\right \Vert _{L^\infty } ds< \infty$. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on $\exp {\int _{0}^{t} \left \Vert {\nabla u(\cdot ,s)}\right \Vert _{L^\infty }}ds$. In particular, we answer in the positive a question posed by Levermore and Oliver.References
- S. Alinhac and G. Métivier, Propagation of local analyticity for the Euler equation, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 1–4. MR 812279, DOI 10.1090/pspum/043/812279
- Claude Bardos, Analyticité de la solution de l’équation d’Euler dans un ouvert de $R^{n}$, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 5, Aii, A255–A258 (French, with English summary). MR 425393
- C. Bardos and S. Benachour, Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de $R^{n}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 647–687 (French). MR 454413
- Claude Bardos, Said Benachour, and Martin Zerner, Analyticité des solutions périodiques de l’équation d’Euler en deux dimensions, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 17, Aiii, A995–A998. MR 410094
- Saïd Benachour, Analyticité des solutions périodiques de l’équation d’Euler en trois dimensions, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Aii, A107–A110 (French, with English summary). MR 425323
- M. S. Baouendi and C. Goulaouic, Problèmes de Cauchy pseudo-différentiels analytiques, Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque, No. 34-35, Soc. Math. France, Paris, 1976, pp. 27–41 (French). MR 0481385
- Jerry L. Bona, Zoran Grujić, and Henrik Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 6, 783–797 (English, with English and French summaries). MR 2172859, DOI 10.1016/j.anihpc.2004.12.004
- Jerry L. Bona, Zoran Grujić, and Henrik Kalisch, Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip, J. Differential Equations 229 (2006), no. 1, 186–203. MR 2265624, DOI 10.1016/j.jde.2006.04.013
- J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762
- Peter Constantin, Edriss S. Titi, and Jesenko Vukadinovic, Dissipativity and Gevrey regularity of a Smoluchowski equation, Indiana Univ. Math. J. 54 (2005), no. 4, 949–969. MR 2164412, DOI 10.1512/iumj.2005.54.2653
- David G. Ebin and Jerrold E. Marsden, Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid, Bull. Amer. Math. Soc. 75 (1969), 962–967. MR 246328, DOI 10.1090/S0002-9904-1969-12315-3
- C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989), no. 2, 359–369. MR 1026858, DOI 10.1016/0022-1236(89)90015-3
- Andrew B. Ferrari and Edriss S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations 23 (1998), no. 1-2, 1–16. MR 1608488, DOI 10.1080/03605309808821336
- Zoran Grujić and Igor Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differential Equations 154 (1999), no. 1, 42–54. MR 1685650, DOI 10.1006/jdeq.1998.3562
- Zoran Grujić and Igor Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal. 152 (1998), no. 2, 447–466. MR 1607936, DOI 10.1006/jfan.1997.3167
- V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189
- Igor Kukavica, Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity 8 (1995), no. 2, 113–129. MR 1328590
- Igor Kukavica, On the dissipative scale for the Navier-Stokes equation, Indiana Univ. Math. J. 48 (1999), no. 3, 1057–1081. MR 1736969, DOI 10.1512/iumj.1999.48.1748
- Tosio Kato, Nonstationary flows of viscous and ideal fluids in $\textbf {R}^{3}$, J. Functional Analysis 9 (1972), 296–305. MR 0481652, DOI 10.1016/0022-1236(72)90003-1
- Pierre Gilles Lemarié-Rieusset, Une remarque sur l’analyticité des solutions milds des équations de Navier-Stokes dans $\textbf {R}^3$, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 3, 183–186 (French, with English and French summaries). MR 1748305, DOI 10.1016/S0764-4442(00)00103-8
- Pierre-Gilles Lemarié-Rieusset, Nouvelles remarques sur l’analyticité des solutions milds des équations de Navier-Stokes dans $\Bbb R^3$, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 443–446 (French, with English and French summaries). MR 2057722, DOI 10.1016/j.crma.2004.01.015
- Daniel Le Bail, Analyticité locale pour les solutions de l’équation d’Euler, Arch. Rational Mech. Anal. 95 (1986), no. 2, 117–136 (French). MR 850093, DOI 10.1007/BF00281084
- C. David Levermore and Marcel Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations 133 (1997), no. 2, 321–339. MR 1427856, DOI 10.1006/jdeq.1996.3200
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- Marcel Oliver and Edriss S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations 174 (2001), no. 1, 55–74. MR 1844523, DOI 10.1006/jdeq.2000.3927
- Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32–43. MR 0430568, DOI 10.1016/0022-1236(75)90052-x
Additional Information
- Igor Kukavica
- Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
- MR Author ID: 314775
- Email: kukavica@usc.edu
- Vlad Vicol
- Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
- MR Author ID: 846012
- ORCID: setImmediate$0.00243841196800898$2
- Email: vicol@usc.edu
- Received by editor(s): November 13, 2007
- Published electronically: September 16, 2008
- Additional Notes: Both authors were supported in part by the NSF grant DMS-0604886.
- Communicated by: Matthew J. Gursky
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 669-677
- MSC (2000): Primary 76B03, 35L60
- DOI: https://doi.org/10.1090/S0002-9939-08-09693-7
- MathSciNet review: 2448589