Dispersion relations for periodic water waves with surface tension and discontinuous vorticity

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August 2014, 34(8): 3109-3123. doi: 10.3934/dcds.2014.34.3109

Dispersion relations for periodic water waves with surface tension and discontinuous vorticity

1.	Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

Received August 2013 Revised September 2013 Published January 2014

Abstract
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We derive the dispersion relation for water waves with surface tension and having a piecewise constant vorticity distribution. More precisely, we consider here two scenarios; the first one is that of a flow with constant non-zero vorticity adjacent to the flat bed while above this layer of vorticity we assume the flow to be irrotational. The second type of flow has a layer of non-vanishing vorticity adjacent to the free surface and is irrotational below.

Keywords: piece-wise constant vorticity, Capillary-gravity water waves, dispersion relation..

Mathematics Subject Classification: Primary: 35Q31, 35Q35, 76D33, 76D45; Secondary: 12D1.

Citation: Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

References:

[1]	D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, J. Fluid Mech., 714 (2013), 463. doi: 10.1017/jfm.2012.490. Google Scholar
[2]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar
[3]	A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, 81 (2011). doi: 10.1137/1.9781611971873. Google Scholar
[4]	A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity,, Commun. Pure Appl. Anal., 11 (2012), 1397. doi: 10.3934/cpaa.2012.11.1397. Google Scholar
[5]	A. Constantin, Mean velocities in a Stokes wave,, Arch. Ration. Mech. Anal., 207 (2013), 907. doi: 10.1007/s00205-012-0584-6. Google Scholar
[6]	A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1. Google Scholar
[7]	A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773. Google Scholar
[8]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar
[9]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. doi: 10.1002/cpa.3046. Google Scholar
[10]	A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar
[11]	A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133. doi: 10.1007/s00205-011-0412-4. Google Scholar
[12]	A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity,, Comm. Pure Appl. Math., 60 (2007), 911. doi: 10.1002/cpa.20165. Google Scholar
[13]	A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33. doi: 10.1007/s00205-010-0314-x. Google Scholar
[14]	M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436. doi: 10.1137/100792330. Google Scholar
[15]	J. Escher, Regularity of rotational travelling water waves,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602. doi: 10.1098/rsta.2011.0458. Google Scholar
[16]	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. doi: 10.1016/j.jde.2011.03.023. Google Scholar
[17]	D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity,, SIAM J. Math. Anal, 42 (2010), 3103. doi: 10.1137/100801408. Google Scholar
[18]	D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity,, J. Math. Fluid Mech., 14 (2012), 249. doi: 10.1007/s00021-011-0056-z. Google Scholar
[19]	D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface,, Nonlinear Anal. Real World Appl., 14 (2013), 1034. doi: 10.1016/j.nonrwa.2012.08.015. Google Scholar
[20]	D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves,, Commun. Pure Appl. Anal., 11 (2012), 1453. doi: 10.3934/cpaa.2012.11.1453. Google Scholar
[21]	R. S. Johnson, A modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar
[22]	I. G. Jonsson, Wave-current interactions. In: The Sea ,, Wiley, (1990). Google Scholar
[23]	J. Lighthill, Waves in Fluids,, Cambridge Mathematical Library, (2001). doi: 10.1007/s002050100160. Google Scholar
[24]	C. I. Martin, Local bifurcation and regularity for steady periodic capillary-gravity water waves with constant vorticity,, Nonlinear Anal. Real World Appl., 14 (2013), 131. doi: 10.1016/j.nonrwa.2012.05.007. Google Scholar
[25]	C. I. Martin, Local bifurcation for steady periodic capillary water waves with constant vorticity,, J. Math. Fluid Mech., 15 (2013), 155. doi: 10.1007/s00021-012-0096-z. Google Scholar
[26]	C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity,, , (). Google Scholar
[27]	A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129. Google Scholar
[28]	B.-V. Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity,, Int. Math. Res. Not., 17 (2011), 3858. doi: 10.1093/imrn/rnq235. Google Scholar
[29]	G. Thomas and G. Klopman, Wave-Current Interactions in the Nearshore Region,, WIT, (1997). Google Scholar
[30]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[31]	E. Wahlén, Steady periodic capillary-gravity waves with vorticity,, SIAM J. Math. Anal., 38 (2006), 921. doi: 10.1137/050630465. Google Scholar
[32]	E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468. doi: 10.1016/j.jde.2008.10.005. Google Scholar
[33]	S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. doi: 10.1137/080721583. Google Scholar
[34]	L.-J. Wang, Regularity of traveling periodic stratified water waves with vorticity,, Nonlinear Anal., 81 (2013), 247. doi: 10.1016/j.na.2012.11.009. Google Scholar

show all references

References:

[1]	D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, J. Fluid Mech., 714 (2013), 463. doi: 10.1017/jfm.2012.490. Google Scholar
[2]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar
[3]	A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, 81 (2011). doi: 10.1137/1.9781611971873. Google Scholar
[4]	A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity,, Commun. Pure Appl. Anal., 11 (2012), 1397. doi: 10.3934/cpaa.2012.11.1397. Google Scholar
[5]	A. Constantin, Mean velocities in a Stokes wave,, Arch. Ration. Mech. Anal., 207 (2013), 907. doi: 10.1007/s00205-012-0584-6. Google Scholar
[6]	A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1. Google Scholar
[7]	A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773. Google Scholar
[8]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar
[9]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. doi: 10.1002/cpa.3046. Google Scholar
[10]	A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar
[11]	A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133. doi: 10.1007/s00205-011-0412-4. Google Scholar
[12]	A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity,, Comm. Pure Appl. Math., 60 (2007), 911. doi: 10.1002/cpa.20165. Google Scholar
[13]	A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33. doi: 10.1007/s00205-010-0314-x. Google Scholar
[14]	M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436. doi: 10.1137/100792330. Google Scholar
[15]	J. Escher, Regularity of rotational travelling water waves,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602. doi: 10.1098/rsta.2011.0458. Google Scholar
[16]	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. doi: 10.1016/j.jde.2011.03.023. Google Scholar
[17]	D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity,, SIAM J. Math. Anal, 42 (2010), 3103. doi: 10.1137/100801408. Google Scholar
[18]	D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity,, J. Math. Fluid Mech., 14 (2012), 249. doi: 10.1007/s00021-011-0056-z. Google Scholar
[19]	D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface,, Nonlinear Anal. Real World Appl., 14 (2013), 1034. doi: 10.1016/j.nonrwa.2012.08.015. Google Scholar
[20]	D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves,, Commun. Pure Appl. Anal., 11 (2012), 1453. doi: 10.3934/cpaa.2012.11.1453. Google Scholar
[21]	R. S. Johnson, A modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar
[22]	I. G. Jonsson, Wave-current interactions. In: The Sea ,, Wiley, (1990). Google Scholar
[23]	J. Lighthill, Waves in Fluids,, Cambridge Mathematical Library, (2001). doi: 10.1007/s002050100160. Google Scholar
[24]	C. I. Martin, Local bifurcation and regularity for steady periodic capillary-gravity water waves with constant vorticity,, Nonlinear Anal. Real World Appl., 14 (2013), 131. doi: 10.1016/j.nonrwa.2012.05.007. Google Scholar
[25]	C. I. Martin, Local bifurcation for steady periodic capillary water waves with constant vorticity,, J. Math. Fluid Mech., 15 (2013), 155. doi: 10.1007/s00021-012-0096-z. Google Scholar
[26]	C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity,, , (). Google Scholar
[27]	A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129. Google Scholar
[28]	B.-V. Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity,, Int. Math. Res. Not., 17 (2011), 3858. doi: 10.1093/imrn/rnq235. Google Scholar
[29]	G. Thomas and G. Klopman, Wave-Current Interactions in the Nearshore Region,, WIT, (1997). Google Scholar
[30]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[31]	E. Wahlén, Steady periodic capillary-gravity waves with vorticity,, SIAM J. Math. Anal., 38 (2006), 921. doi: 10.1137/050630465. Google Scholar
[32]	E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468. doi: 10.1016/j.jde.2008.10.005. Google Scholar
[33]	S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. doi: 10.1137/080721583. Google Scholar
[34]	L.-J. Wang, Regularity of traveling periodic stratified water waves with vorticity,, Nonlinear Anal., 81 (2013), 247. doi: 10.1016/j.na.2012.11.009. Google Scholar

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