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Variants of Arnold's Stability Results for 2D Euler Equations

Published online by Cambridge University Press:  20 November 2018

Michael Taylor
Michael Taylor*
Affiliation:
Mathematics Department, University of North Carolina, Chapel Hill, NC 27599, USA e-mail: met@email.unc.edu
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Abstract

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We establish variants of stability estimates in norms somewhat stronger than the ${{H}^{1}}$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains.

Keywords

35Q35
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 1 , 01 March 2010 , pp. 163 - 170
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Arnold, V. I. and Khesin, B. A., Topological methods in hydrodynamics. Applied Mathematical Sciences, Springer-Verlag, New York, 1998.Google Scholar
[2] Beale, T., Kato, T., and Majda, A., Remarks on the breakdown of smooth solutions of the 3-D Euler equations. Comm. Math. Phys. 94(1984), no. 1, 61–66. doi:10.1007/BF01212349Google Scholar
[3] Brézis, H. and Gallouet, T., Nonlinear Schrödinger evolutions. Nonlinear Anal. 4(1980), no. 4, 677–681. doi:10.1016/0362-546X(80)90068-1Google Scholar
[4] Brézis, H. and Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differential Equations 5(1980), no. 7, 773–789. doi:10.1080/03605308008820154Google Scholar
[5] Chang, D.-C., Dafni, G., and Stein, E., Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in ℝ n . Trans. Amer. Math. Soc. 351(1999), no. 4, 1605–1661. doi:10.1090/S0002-9947-99-02111-XGoogle Scholar
[6] Grenier, E., Jones, C., Rousset, F., and Sandstede, B., Viscous perturbations of marginally stable Euler flow and finite-time Melnikov theory. Nonlinearity 18(2005), no. 2, 465–483. doi:10.1088/0951-7715/18/2/001Google Scholar
[7] Kato, T., On classical solutions of the two dimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25(1967), 188–200.Google Scholar
[8] Judovič, V. I., Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz 3(1963), 1032–1066.Google Scholar
[9] Marchioro, C. and Pulvirenti, M., Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, New York, 1994.Google Scholar
[10] Taylor, M., Partial differential equations, Vols. 1–3. Applied Mathematical Sciences 115–116. Springer-Verlag, New York, 1996.Google Scholar