On the regularity of deep-water waves with general vorticity distributions
Author:
Bogdan-Vasile Matioc
Journal:
Quart. Appl. Math. 70 (2012), 393-405
MSC (2010):
Primary 35J25, 76B03, 76B47
DOI:
https://doi.org/10.1090/S0033-569X-2012-01261-1
Published electronically:
March 1, 2012
MathSciNet review:
2953110
Full-text PDF Free Access
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Abstract: We prove that the streamlines and the profile of traveling deep-water waves with Hölder continuous vorticity function are smooth, provided there are no stagnation points in the flow. In addition, if the vorticity function is real analytic, then so is the profile of both solitary and periodic traveling deep-water waves. Finally, by choosing appropriate weighted Sobolev spaces, we show that the streamlines beneath the surface of a periodic traveling water wave are in fact real analytic, provided the vorticity function is merely integrable against a cubic weight.
References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI https://doi.org/10.1002/cpa.3160120405
- Cherif Amrouche and Šárka Nečasová, Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), 2001, pp. 265–274. MR 1844267
- Sigurd Angenent, Parabolic equations for curves on surfaces. I. Curves with $p$-integrable curvature, Ann. of Math. (2) 132 (1990), no. 3, 451–483. MR 1078266, DOI https://doi.org/10.2307/1971426
- Walter Craig and Ana-Maria Matei, On the regularity of the Neumann problem for free surfaces with surface tension, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2497–2504. MR 2302570, DOI https://doi.org/10.1090/S0002-9939-07-08776-X
- Adrian Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417. MR 1819940, DOI https://doi.org/10.1088/0305-4470/34/7/313
- Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI https://doi.org/10.1007/s00222-006-0002-5
- Adrian Constantin, Mats Ehrnström, and Gabriele Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl. 9 (2008), no. 4, 1336–1344. MR 2422547, DOI https://doi.org/10.1016/j.nonrwa.2007.03.003
- Adrian Constantin and Joachim Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math. 15 (2004), no. 6, 755–768. MR 2144685, DOI https://doi.org/10.1017/S0956792504005777
- Adrian Constantin and Joachim Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2) 173 (2011), no. 1, 559–568. MR 2753609, DOI https://doi.org/10.4007/annals.2011.173.1.12
- Adrian Constantin and Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. MR 2027299, DOI https://doi.org/10.1002/cpa.3046
- Adrian Constantin and Walter Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. MR 2604871, DOI https://doi.org/10.1002/cpa.20299
- Constantin, A. and W. Strauss. “Periodic traveling gravity water waves with discontinuous vorticity.”, preprint.
- Dubreil-Jacotin, M.-L. “Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie.” J. Math. Pures Appl. 13, (1934): 217–291.
- Joachim Escher and Bogdan-Vasile Matioc, A moving boundary problem for periodic Stokesian Hele-Shaw flows, Interfaces Free Bound. 11 (2009), no. 1, 119–137. MR 2487025, DOI https://doi.org/10.4171/IFB/205
- Joachim Escher and Gieri Simonett, Analyticity of the interface in a free boundary problem, Math. Ann. 305 (1996), no. 3, 439–459. MR 1397432, DOI https://doi.org/10.1007/BF01444233
- Gerstner, F. “Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile.” Ann. Phys. 2, (1809): 412–445.
- Gilbarg, D. and T. S. Trudinger. Elliptic Partial Differential Equations of Second Order, New York: Springer–Verlag, 1998.
- Groves, M. D. and E. Wahlén. “On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water.” preprint.
- David Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. , posted on (2006), Art. ID 23405, 13. MR 2272104, DOI https://doi.org/10.1155/IMRN/2006/23405
- David Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys. 14 (2007), no. 1, 1–7. MR 2287829, DOI https://doi.org/10.2991/jnmp.2007.14.1.1
- David Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 87–95. MR 2434727, DOI https://doi.org/10.2991/jnmp.2008.15.s2.7
- David Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal. 42 (2010), no. 6, 3103–3111. MR 2763714, DOI https://doi.org/10.1137/100801408
- Henry, D. “Regularity for steady periodic capillary water waves with vorticity.”, preprint.
- Henry, D. “Analyticity of free surface for periodic traveling capillary-gravity water waves with vorticity.”, preprint.
- D. Kinderlehrer, L. Nirenberg, and J. Spruck, Regularity in elliptic free boundary problems, J. Analyse Math. 34 (1978), 86–119 (1979). MR 531272, DOI https://doi.org/10.1007/BF02790009
- Hans Lewy, A note on harmonic functions and a hydrodynamical application, Proc. Amer. Math. Soc. 3 (1952), 111–113. MR 49399, DOI https://doi.org/10.1090/S0002-9939-1952-0049399-9
- James Lighthill, Waves in fluids, Cambridge University Press, Cambridge-New York, 1978. MR 642980
- Robert B. Lockhart, Fredholm properties of a class of elliptic operators on noncompact manifolds, Duke Math. J. 48 (1981), no. 1, 289–312. MR 610188
- Matioc, A.–V. “On particle trajectories in linear deep-water waves,” to appear in Commun. Pure Appl. Anal.
- Matioc, B.–V. “Analyticity of the streamlines for periodic traveling water waves with bounded vorticity.” Int. Math. Res. Not. 17, (2011): 3858–3871.
- Vera Mikyoung Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal. 37 (2006), no. 5, 1482–1521. MR 2215274, DOI https://doi.org/10.1137/040621168
- Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645
References
- Agmon, S., Douglis, A., and L. Nirenberg. “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions”. I. Comm. Pure Appl. Math. 12, (1959): 623–727. MR 0125307 (23:A2610)
- Amrouche, C. and S. Necǎsová. “Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition.” Math. Bohem. 126, no. 2 (2001): 265–274. MR 1844267 (2002e:35057)
- Angenent, S. “Parabolic equations for curves on surfaces.” Ann. of Math. (2) 132, no. 2 (1990): 451–483. MR 1078266 (91k:35102)
- Craig, W. and A.–M. Matei. “On the regularity of the Neumann problem for the free surfaces with surface tension.” Proc. Amer. Math. Soc. 135 (2007): 2497–2504. MR 2302570 (2008a:35297)
- Constantin, A. “On the deep water wave motion.” J. Phys. A 34, no. 7 (2001): 1405–1417. MR 1819940 (2002b:76010)
- Constantin, A. “The trajectories of particles in Stokes waves.” Invent. Math. 166, (2006): 523–535. MR 2257390 (2007j:35240)
- Constantin, A., Ehrnström, M., and G. Villari. “Particle trajectories in linear deep-water waves.” Nonlinear Anal. Real World Appl. 9, (2008): 1336–1344. MR 2422547 (2009f:35265)
- Constantin, A. and J. Escher. “Symmetry of steady deep-water waves with vorticity.” European J. Appl. Math. 15, (2004): 755–768. MR 2144685 (2006b:76013)
- Constantin, A. and J. Escher. “Analyticity of periodic traveling free surface water waves with vorticity.” Ann. of Math. (2) 173 (2011): 559–568. MR 2753609.
- Constantin, A. and W. Strauss. “Exact steady periodic water waves with vorticity.” Comm. Pure Appl. Math. 57, no. 4 (2004): 481–527. MR 2027299 (2004i:76018)
- Constantin, A. and W. Strauss. “ Pressure beneath a Stokes wave.” Comm. Pure Appl. Math. 63, no. 4 (2010): 533–557. MR 2604871
- Constantin, A. and W. Strauss. “Periodic traveling gravity water waves with discontinuous vorticity.”, preprint.
- Dubreil-Jacotin, M.-L. “Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie.” J. Math. Pures Appl. 13, (1934): 217–291.
- Escher, J. and B.–V. Matioc. “A moving boundary problem for periodic Stokesian Hele-Shaw flows.” Interfaces Free Bound. 11, (2009): 119–137. MR 2487025 (2010f:35448)
- Escher, J. and G. Simonett. “Analyticity of the interface in a free boundary problem.” Math. Ann. 305, (1996): 439–459. MR 1397432 (98d:35242)
- Gerstner, F. “Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile.” Ann. Phys. 2, (1809): 412–445.
- Gilbarg, D. and T. S. Trudinger. Elliptic Partial Differential Equations of Second Order, New York: Springer–Verlag, 1998.
- Groves, M. D. and E. Wahlén. “On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water.” preprint.
- Henry, D. “The trajectories of particles in deep-water Stokes waves.” Int. Math. Res. Not. Art. ID 23405 (2006): 1–13. MR 2272104 (2007k:76017)
- Henry, D. “Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves.” J. Nonlinear Math. Phys. 14, no. 1 (2007): 1–7. MR 2287829 (2007k:76018)
- Henry, D. “On Gerstner’s water wave.” J. Nonlinear Math. Phys. 15, (2008): 87–95. MR 2434727 (2009k:76020)
- Henry, D. “Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity.” SIAM J. Math. Anal. 42 (2010): 3103–3111. MR 2763714
- Henry, D. “Regularity for steady periodic capillary water waves with vorticity.”, preprint.
- Henry, D. “Analyticity of free surface for periodic traveling capillary-gravity water waves with vorticity.”, preprint.
- Kinderlehrer, D., Nirenberg, L., and J. Spruck. “Regularity in elliptic free boundary value problems I.” J. Anal. Math. 34, (1978): 86–119. MR 531272 (83d:35060)
- Lewy, A. “A note on harmonic functions and a hydrodynamical application.” Proc. Amer. Math. Soc. 3, (1952): 111–113. MR 0049399 (14:168c)
- Lighthill, J. Waves in fluids, Cambridge: Cambridge University Press, 1978. MR 642980 (84g:76001a)
- Lockhart, R. B. “Fredholm properties of a class of elliptic operators on non-compact manifolds.” Duke Math. J. 48, (1981): 289–312. MR 610188 (82j:35050)
- Matioc, A.–V. “On particle trajectories in linear deep-water waves,” to appear in Commun. Pure Appl. Anal.
- Matioc, B.–V. “Analyticity of the streamlines for periodic traveling water waves with bounded vorticity.” Int. Math. Res. Not. 17, (2011): 3858–3871.
- Hur, V. M. “Global bifurcation theory of deep-water waves with vorticity.” SIAM J. Math. Anal. 37, no. 5 (2005): 1482–1521. MR 2215274 (2007e:76025)
- Triebel, H. Interpolation Theory. Function Spaces. Differential Operators, Heidelberg: Johann Ambrosius Barth Verlag, 1995. MR 1328645 (96f:46001)
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Additional Information
Bogdan-Vasile Matioc
Affiliation:
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Email:
matioc@ifam.uni-hannover.de
Keywords:
Deep-water waves,
streamlines,
vorticity
Received by editor(s):
November 17, 2010
Published electronically:
March 1, 2012
Article copyright:
© Copyright 2012
Brown University