Analyticity of periodic traveling free surface water waves with vorticity. (English) Zbl 1228.35076
Summary: We prove that the profile of a periodic traveling wave propagating at the surface of water above a flat bed in a flow with a real analytic vorticity must be real analytic, provided the wave speed exceeds the horizontal fluid velocity throughout the flow. The real analyticity of each streamline beneath the free surface holds even if the vorticity is only Hölder continuously differentiable.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35R35 | Free boundary problems for PDEs |
References:
| [1] | C. J. Amick, L. E. Fraenkel, and J. F. Toland, ”On the Stokes conjecture for the wave of extreme form,” Acta Math., vol. 148, pp. 193-214, 1982. · Zbl 0495.76021 · doi:10.1007/BF02392728 |
| [2] | S. Angenent, ”Parabolic equations for curves on surfaces. I. Curves with \(p\)-integrable curvature,” Ann. of Math., vol. 132, iss. 3, pp. 451-483, 1990. · Zbl 0789.58070 · doi:10.2307/1971426 |
| [3] | B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton, NJ: Princeton Univ. Press, 2003. · Zbl 1021.47044 |
| [4] | A. Constantin, ”The trajectories of particles in Stokes waves,” Invent. Math., vol. 166, iss. 3, pp. 523-535, 2006. · Zbl 1108.76013 · doi:10.1007/s00222-006-0002-5 |
| [5] | A. Constantin, M. Ehrnström, and E. Wahlén, ”Symmetry of steady periodic gravity water waves with vorticity,” Duke Math. J., vol. 140, iss. 3, pp. 591-603, 2007. · Zbl 1151.35076 · doi:10.1215/S0012-7094-07-14034-1 |
| [6] | A. Constantin and W. Strauss, ”Exact steady periodic water waves with vorticity,” Comm. Pure Appl. Math., vol. 57, iss. 4, pp. 481-527, 2004. · Zbl 1038.76011 · doi:10.1002/cpa.3046 |
| [7] | A. Constantin and W. Strauss, ”Rotational steady water waves near stagnation,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., vol. 365, iss. 1858, pp. 2227-2239, 2007. · Zbl 1152.76328 · doi:10.1098/rsta.2007.2004 |
| [8] | A. Constantin and W. Strauss, ”Pressure beneath a Stokes wave,” Comm. Pure Appl. Math., vol. 63, iss. 4, pp. 533-557, 2010. · Zbl 1423.76061 |
| [9] | A. D. D. Craik, ”The origins of water wave theory,” Ann. Rev. Fluid Mech., vol. 36, pp. 1-28, 2004. · Zbl 1076.76011 · doi:10.1146/annurev.fluid.36.050802.122118 |
| [10] | A. F. Teles da Silva and D. H. Peregrine, ”Steep, steady surface waves on water of finite depth with constant vorticity,” J. Fluid Mech., vol. 195, pp. 281-302, 1988. · doi:10.1017/S0022112088002423 |
| [11] | J. Escher and G. Simonett, ”Analyticity of the interface in a free boundary problem,” Math. Ann., vol. 305, iss. 3, pp. 439-459, 1996. · Zbl 0857.76086 · doi:10.1007/BF01444233 |
| [12] | L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge: Cambridge Univ. Press, 2000, vol. 128. · Zbl 0947.35002 · doi:10.1017/CBO9780511569203 |
| [13] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001. · Zbl 1042.35002 |
| [14] | R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge: Cambridge Univ. Press, 1997. · Zbl 0892.76001 · doi:10.1017/CBO9780511624056 |
| [15] | D. Kinderlehrer, L. Nirenberg, and J. Spruck, ”Regularity in elliptic free boundary problems,” J. Analyse Math., vol. 34, pp. 86-119 (1979), 1978. · Zbl 0402.35045 · doi:10.1007/BF02790009 |
| [16] | J. Ko and W. Strauss, ”Effect of vorticity on steady water waves,” J. Fluid Mech., vol. 608, pp. 197-215, 2008. · Zbl 1145.76330 · doi:10.1017/S0022112008002371 |
| [17] | J. Ko and W. Strauss, ”Large-amplitude steady rotational water waves,” Eur. J. Mech. B Fluids, vol. 27, iss. 2, pp. 96-109, 2008. · Zbl 1149.76011 · doi:10.1016/j.euromechflu.2007.04.004 |
| [18] | H. Lewy, ”A note on harmonic functions and a hydrodynamical application,” Proc. Amer. Math. Soc., vol. 3, pp. 111-113, 1952. · Zbl 0046.41706 · doi:10.2307/2032464 |
| [19] | J. Lighthill, Waves in Fluids, Cambridge: Cambridge Univ. Press, 1978. · Zbl 0375.76001 |
| [20] | J. F. Toland, ”Stokes waves,” Topol. Methods Nonlinear Anal., vol. 7, iss. 1, pp. 1-48, 1996. · Zbl 0897.35067 |
| [21] | E. Varvaruca, ”Singularities of Bernoulli free boundaries,” Comm. Partial Differential Equations, vol. 31, iss. 10-12, pp. 1451-1477, 2006. · Zbl 1111.35137 · doi:10.1080/03605300600635012 |
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