The Stokes conjecture for waves with vorticity
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 6, pp. 861-885.

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far.Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

@article{AIHPC_2012__29_6_861_0,
     author = {Varvaruca, Eugen and Weiss, Georg S.},
     title = {The {Stokes} conjecture for waves with vorticity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {861--885},
     publisher = {Elsevier},
     volume = {29},
     number = {6},
     year = {2012},
     doi = {10.1016/j.anihpc.2012.05.001},
     mrnumber = {2995099},
     zbl = {1317.35209},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.001/}
}
TY  - JOUR
AU  - Varvaruca, Eugen
AU  - Weiss, Georg S.
TI  - The Stokes conjecture for waves with vorticity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 861
EP  - 885
VL  - 29
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.001/
DO  - 10.1016/j.anihpc.2012.05.001
LA  - en
ID  - AIHPC_2012__29_6_861_0
ER  - 
%0 Journal Article
%A Varvaruca, Eugen
%A Weiss, Georg S.
%T The Stokes conjecture for waves with vorticity
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 861-885
%V 29
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.001/
%R 10.1016/j.anihpc.2012.05.001
%G en
%F AIHPC_2012__29_6_861_0
Varvaruca, Eugen; Weiss, Georg S. The Stokes conjecture for waves with vorticity. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 6, pp. 861-885. doi : 10.1016/j.anihpc.2012.05.001. http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.001/

[1] F.J. Almgren, Almgrenʼs Big Regularity Paper: Q-Valued Functions Minimizing Dirichletʼs Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2, World Scientific Monograph Series in Mathematics vol. 1, World Scientific Publishing Co. Inc., River Edge, NJ (2000) | MR

[2] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144 | EuDML | MR | Zbl

[3] C.J. Amick, L.E. Fraenkel, J.F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193-214 | MR | Zbl

[4] Adrian Constantin, Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 no. 4 (2004), 481-527 | MR | Zbl

[5] Daniela De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound. 13 (2011), 223-238 | MR | Zbl

[6] M.-L. Dubreil-Jacotin, Sur la détermination rigoreuse des ondes permanentes périodiques dʼampleur finie, J. Math. Pures Appl. 13 (1934), 217-291 | EuDML | Numdam | MR | Zbl

[7] L.C. Evans, S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. Amer. Math. Soc. 7 no. 1 (1994), 199-219 | MR | Zbl

[8] Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics vol. 74 (1990) | MR | Zbl

[9] Nicola Garofalo, Fang-Hua Lin, Monotonicity properties of variational integrals, A p weights and unique continuation, Indiana Univ. Math. J. 35 no. 2 (1986), 245-268 | MR | Zbl

[10] F. Gerstner, Theorie der Wellen, Abhand. Koen. Boehmischen Gesel. Wiss., Prague (1802)

[11] David Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften vol. 224, Springer-Verlag, Berlin (1983) | MR | Zbl

[12] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics vol. 80, Birkhäuser Verlag, Basel (1984) | MR | Zbl

[13] J.K. Oddson, On the boundary point principle for elliptic equations in the plane, Bull. Amer. Math. Soc. 74 (1968), 666-670 | MR | Zbl

[14] F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math. 79 no. 2 (1993), 161-172 | EuDML | MR | Zbl

[15] P.I. Plotnikov, Proof of the Stokes conjecture in the theory of surface waves, Stud. Appl. Math. 108 no. 2 (2002), 217-244, Dinamika Sploshn. Sredy 57 (1982), 41-76 | MR | Zbl

[16] P. Price, A monotonicity formula for Yang–Mills fields, Manuscripta Math. 43 no. 2–3 (1983), 131-166 | EuDML | MR | Zbl

[17] O. Savin, E. Varvaruca, Existence of steady free-surface water waves with corners of 120° at their crests in the presence of vorticity, submitted for publication.

[18] R.M. Schoen, Analytic aspects of the harmonic map problem, Seminar on Nonlinear Partial Differential Equations, Berkeley, CA, 1983, Math. Sci. Res. Inst. Publ. vol. 2, Springer, New York (1984), 321-358

[19] G.G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Math. and Phys. Papers vol. 1, Cambridge University Press, Cambridge (1880), 225-228

[20] Walter A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.) 47 no. 4 (2010), 671-694 | MR | Zbl

[21] E. Varvaruca, Singularities of Bernoulli free boundaries, Comm. Partial Differential Equations 31 no. 10–12 (2006), 1451-1477 | MR | Zbl

[22] E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations 246 no. 10 (2009), 4043-4076 | MR | Zbl

[23] E. Varvaruca, G.S. Weiss, A geometric approach to generalized Stokes conjectures, Acta Math. 206 (2011), 363-403 | MR | Zbl

[24] G.S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations 23 no. 3–4 (1998), 439-455 | MR | Zbl

[25] G.S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 no. 2 (1999), 317-326 | MR | Zbl

Cited by Sources: