On the existence of steady periodic capillary-gravity stratified water waves

Henry, David; Matioc, Bogdan-Vasile

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 955-974.

We prove the existence of small steady periodic capillary-gravity water waves for stratified flows, where we allow for stagnation points in the flow. We establish the existence of both laminar and non-laminar flow solutions for the governing equations. This is achieved using bifurcation theory and estimates based on the ellipticity of the system, where we regard, in turn, the mass-flux and surface tension as bifurcation parameters.

Classification : 35Q35, 76B70, 76B47

[1] A. Constantin, On the deep water wave motion, J. Phys. A 34 (2001), 1405–1417. | MR | Zbl

[2] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523–535. | MR | Zbl

[3] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), 591–603. | MR | Zbl

[4] A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech. 498 (2004), 171–181. | MR | Zbl

[5] A. Constantin and J. Escher, Analyticity of periodic travelling free surface water waves with vorticity, Ann. of Math. (2) 173 (2011), 559–568. | MR | Zbl

[6] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (4) (2004), 481–527. | MR | Zbl

[7] A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), 911–950. | MR | Zbl

[8] A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), 2227–2239. | MR | Zbl

[9] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Ration. Mech. Anal. 199 (2011), 33–67. | MR | Zbl

[10] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. | MR | Zbl

[11] M.-L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides hétérogènes, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1932), 814–819. | JFM | Zbl

[12] M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie, J. Math. Pures Appl. (9) 13 (1934), 217–291. | EuDML | Numdam | MR | Zbl

[13] M. L. Dubreil-Jacotin, Sur les théorèmes d’existence relatifs aux ondes permanentes périodiques a deux dimensions dans les liquides hétérogènes, J. Math. Pures Appl. (9) 9 (1937), 43–67. | Numdam | Zbl

[14] J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations 251 (2011), 2932–2949. | MR | Zbl

[15] J. Escher, A.-V. Matioc and B.-V. Matioc, Classical solutions and stability results for Stokesian Hele-Shaw flows, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 9 (2010), 325–349. | Numdam | MR | Zbl

[16] D. Gilbarg and T. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, New York, 1998. | MR | Zbl

[17] D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), 87–95. | MR

[18] D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech. 14 (2012), 249–254. | MR | Zbl

[19] D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal. 42 (2010), 3103–3111. | MR | Zbl

[20] D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Comm. Pure Appl. Anal. 11 (2012), 1453–1464. | MR | Zbl

[21] R. S. Johnson, “A Modern Introduction to the Mathematical Theory of Water Waves”, Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl

[22] T. Kato, “Perturbation Theory for Linear Operators”, Springer-Verlag, Berlin Heidelberg, 1995. | MR | Zbl

[23] Lord Kelvin, Vibrations of a columnar vortex, Phil. Mag. 10 (1880), 155–168. | JFM

[24] J. Lighthill, “Waves in Fluids”, Cambridge University Press, Cambridge, 1978. | MR | Zbl

[25] R. R. Long, Some aspects of the flow of stratified fluids. Part I : A theoretical investigation, Tellus 5 (1953), 42–57. | MR

[26] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birkhäuser, Basel, 1995. | MR | Zbl

[27] B.-V. Matioc, Analyticity of the streamlines for periodic travelling water waves with bounded vorticity, Int. Math. Res. Not. IMRN 17 (2011), 3858–3871. | MR | Zbl

[28] B.-V. Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math. 70 (2) (2012), 393–405. | MR | Zbl

[29] J. F. Toland, Errata to: Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), 413-414. | MR | Zbl

[30] R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981), 513–573. | EuDML | Numdam | MR | Zbl

[31] E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal. 39 (2008), 1686–1692. | MR | Zbl

[32] E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal. 38 (2006), 921–943. | MR | Zbl

[33] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations 246 (2009), 2468–2483. | MR | Zbl

[34] S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal. 41 (2009), 1054–1105. | MR | Zbl

[35] S. Walsh, Some criteria for the symmetry of stratified water waves, Wave Motion 46 (2009), 350–362. | MR | Zbl

[36] S. Walsh, Steady periodic gravity waves with surface tension, preprint.

[37] C.-S. Yih, Exact solutions for steady two-dimensional flow of a stratified fluid, J. Fluid Mech. 9 (1960), 161–174. | MR | Zbl