Abstract
We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.
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Almgren, F. J. Jr., Almgren’s Big Regularity Paper. World Scientific Monograph Series in Mathematics, 1. World Scientific, River Edge, NJ, 2000.
2 Alt, H. W. & Caffarelli, L. A., Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325 (1981), 105–144.
3 Amick, C. J., Fraenkel, L. E. & Toland, J. F., On the Stokes conjecture for the wave of extreme form. Acta Math., 148 (1982), 193–214.
4 Amick, C. J. & Toland, J. F., On solitary water-waves of finite amplitude. Arch. Ration. Mech. Anal., 76 (1981), 9–95.
Andersson, J. & Weiss, G. S., A parabolic free boundary problem with Bernoulli type condition on the free boundary. J. Reine Angew. Math., 627 (2009), 213–235.
Caffarelli, L. A. & Vázquez, J. L., A free-boundary problem for the heat equation arising in flame propagation. Trans. Amer. Math. Soc., 347 (1995), 411–441.
Chen, B. & Saffman, P. G., Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Math., 62 (1980), 1–21.
Constantin, A. & Strauss, W., Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math., 57 (2004), 481–527.
— Rotational steady water waves near stagnation. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2227–2239.
Constantin, A. & Varvaruca, E., Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Ration. Mech. Anal., 199 (2011), 33–67.
Evans, L. C., Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Regional Conference Series in Mathematics, 74. Amer. Math. Soc., Providence, RI, 1990.
12 Evans, L. C. & Müller, S., Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity. J. Amer. Math. Soc., 7 (1994), 199–219.
Garabedian, P. R., A remark about pointed bubbles. Comm. Pure Appl. Math., 38 (1985), 609–612.
Garofalo, N. & Petrosyan, A., Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math., 177 (2009), 415–461.
GiustiE., Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, 80 Birkhäuser, Basel, 1984.
Keady, G. & Norbury, J., On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc., 83 (1978), 137–157.
18 Krasovskiĭ, Yu. P., On the theory of steady-state waves of finite amplitude. Zh. Vychisl. Mat. i Mat. Fiz., 1 (1961), 836–855 (Russian). English translation in U.S.S.R. Comput. Math. and Math. Phys., 1 (1961), 996–1018.
McLeod, J. B., The Stokes and Krasovskii conjectures for the wave of greatest height. Stud. Appl. Math., 98 (1997), 311–333.
Pacard, F., Partial regularity for weak solutions of a nonlinear elliptic equation. Manuscripta Math., 79 (1993), 161–172.
Plotnikov, P. I., Justification of the Stokes conjecture in the theory of surface waves. Dinamika Sploshn. Sredy, 57 (1982), 41–76 (Russian). English traslation in Stud. Appl. Math., 108 (2002), 217–244.
Plotnikov, P. I. & Toland, J. F., Convexity of Stokes waves of extreme form. Arch. Ration. Mech. Anal., 171 (2004), 349–416.
Price, P., A monotonicity formula for Yang–Mills fields. Manuscripta Math., 43 (1983), 131–166.
Savin, O. & Varvaruca, E., Existence of steady free-surface waves with corners of 120° at their crests in the presence of vorticity. In preparation.
Schoen, R. M., Analytic aspects of the harmonic map problem, in Seminar on Nonlinear Partial Differential Equations (Berkeley, CA, 1983), Math. Sci. Res. Inst. Publ., 2, pp. 321–358. Springer, New York, 1984.
Shargorodsky, E. & Toland, J. F., Bernoulli free-boundary problems. Mem. Amer. Math. Soc., 196:914 (2008).
Spielvogel, E. R., A variational principle for waves of infinite depth. Arch. Ration. Mech. Anal., 39 (1970), 189–205.
Stokes, G. G., Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, in Mathematical and Physical Papers, Vol. I, pp. 225–228. Cambridge University Press, Cambridge, 1880.
Toland, J. F., On the existence of a wave of greatest height and Stokes’s conjecture. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363 (1978), 469–485.
Vanden-Broeck, J.-M., Some new gravity waves in water of finite depth. Phys. Fluids, 26 (1983), 2385–2387.
Varvaruca, E., Singularities of Bernoulli free boundaries. Comm. Partial Differential Equations, 31 (2006), 1451–1477.
— Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth. Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1345–1362.
— On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations, 246 (2009), 4043–4076.
Varvaruca, E. & Weiss, G. S., The Stokes conjecture for waves with vorticity. In preparation.
Wahlén, E., Steady water waves with a critical layer. J. Differential Equations, 246 (2009), 2468–2483.
Weiss, G. S., Partial regularity for weak solutions of an elliptic free boundary problem. Comm. Partial Differential Equations, 23 (1998), 439–455.
— Partial regularity for a minimum problem with free boundary. J. Geom. Anal., 9 (1999), 317–326.
— A singular limit arising in combustion theory: fine properties of the free boundary. Calc. Var. Partial Differential Equations, 17 (2003), 311–340.
— Some new nonlinear frequency formulas and applications. In preparation.
Weiss, G. S. & Zhang, G., A free boundary approach to two-dimensional steady capillary gravity water waves. Submitted.
Wu, S., Almost global wellposedness of the 2-D full water wave problem. Invent. Math., 177 (2009), 45–135.
Zufiria, J. A., Nonsymmetric gravity waves on water of infinite depth. J. Fluid Mech., 181 (1987), 17–39.
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Dedicated to John Toland on the occasion of his 60th birthday.
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Varvaruca, E., Weiss, G.S. A geometric approach to generalized Stokes conjectures. Acta Math 206, 363–403 (2011). https://doi.org/10.1007/s11511-011-0066-y
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DOI: https://doi.org/10.1007/s11511-011-0066-y