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May  2017, 37(5): 2669-2680. doi: 10.3934/dcds.2017114

Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity

Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Aníbal Fernandes, Recife, Pernambuco, Brazil

*The author is supported by Science Foundation Ireland (SFI) under the grant 13/CDA/2117

Received  June 2016 Revised  December 2016 Published  February 2017

Fund Project: The author is supported by Science Foundation Ireland (SFI) under the grant 13/CDA/2117

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Hölder spaces.

Citation: Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114
References:
[1]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 81 (2011), xii+315. doi: 10.1137/1.9781611971873. Google Scholar

[2]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397. Google Scholar

[3]

A. Constantin, The flow beneath a periodic travelling surface water wave, J. Phys. A: Math. Theor., 48 (2015), 1397-143001(25pp). doi: 10.1088/1751-8113/48/14/143001. Google Scholar

[4]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777. Google Scholar

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773. Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. Google Scholar

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4. Google Scholar

[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. doi: 10.1007/s00205-010-0314-x. Google Scholar

[10]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, Journal de Mathématiques Pures et Appliquées, 13 (1934), 217-291. Google Scholar

[11]

J. Escher and B. V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232. Google Scholar

[12]

J. EscherA. 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. doi: 10.1016/j.jde.2011.03.023. Google Scholar

[13]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007(14 pp). doi: 10.1142/S1402925112400074. Google Scholar

[14]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. A., 370 (2012), 1616-1628. doi: 10.1098/rsta.2011.0449. Google Scholar

[15]

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), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015. Google Scholar

[16]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889. Google Scholar

[17]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8. Google Scholar

[18]

D. Henry and A. V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786. doi: 10.1017/S0308210512001990. Google Scholar

[19]

D. Henry and B. V. Matioc, On the regularity of steady periodic stratified water waves, Comm. Pure Appl. Anal., 11 (2012), 1453-1464. Google Scholar

[20]

D. Henry and B. V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974. Google Scholar

[21]

D. Henry and B. V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and parabolic equations, Springer Proc. Math. Stat., Springer, Cham, 119 (2015), 159-177. doi: 10.1007/978-3-319-12547-3_7. Google Scholar

[22]

D. Henry and S. Sastre-Gomez, Steady Periodic Waver Waves Bifurcating for Fixed-Depth Rotational Flows with Discontinuous Vorticity, Differential and Integral Equations, 2017.Google Scholar

[23] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. Google Scholar
[24]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371. Google Scholar

[25]

M. J. Lighthill, Waves in fluids, View issue TOC, 20 (1967), 267-293. doi: 10.1002/cpa.3160200204. Google Scholar

[26]

C. I. Martin and B. V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036. Google Scholar

[27]

C. I. Martin and B. V. Matioc, Steady periodic water waves with unbounded vorticity: Equivalent formulations and existence results, J. Nonlinear Sci., 24 (2014), 633-659. doi: 10.1007/s00332-014-9201-1. Google Scholar

[28]

A. V. Matioc and B. V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z. Google Scholar

[29]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, Journal of Fluid Mechanics, 66 (1974), 625-640. doi: 10.1017/S0022112074000413. Google Scholar

[30]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, gravity waves in water of finite depth, Advances in Fluid Mechanics, 10 (1997), 215-319. Google Scholar

[31]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[32]

E. Varvaruca, On the existence of extreme waves and the {S}tokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076. doi: 10.1016/j.jde.2008.12.018. Google Scholar

[33]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized {S}tokes conjectures, Acta Math., 206 (2011), 363-403. doi: 10.1007/s11511-011-0066-y. Google Scholar

[34]

E. Varvaruca and A. Zarnescu, Equivalence of weak formulations of the steady water waves equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1703-1719. doi: 10.1098/rsta.2011.0455. Google Scholar

[35]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943. Google Scholar

[36]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. Google Scholar

[37]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. Google Scholar

show all references

References:
[1]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 81 (2011), xii+315. doi: 10.1137/1.9781611971873. Google Scholar

[2]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397. Google Scholar

[3]

A. Constantin, The flow beneath a periodic travelling surface water wave, J. Phys. A: Math. Theor., 48 (2015), 1397-143001(25pp). doi: 10.1088/1751-8113/48/14/143001. Google Scholar

[4]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777. Google Scholar

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773. Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. Google Scholar

[8]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4. Google Scholar

[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. doi: 10.1007/s00205-010-0314-x. Google Scholar

[10]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, Journal de Mathématiques Pures et Appliquées, 13 (1934), 217-291. Google Scholar

[11]

J. Escher and B. V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232. Google Scholar

[12]

J. EscherA. 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. doi: 10.1016/j.jde.2011.03.023. Google Scholar

[13]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007(14 pp). doi: 10.1142/S1402925112400074. Google Scholar

[14]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. A., 370 (2012), 1616-1628. doi: 10.1098/rsta.2011.0449. Google Scholar

[15]

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), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015. Google Scholar

[16]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889. Google Scholar

[17]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8. Google Scholar

[18]

D. Henry and A. V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786. doi: 10.1017/S0308210512001990. Google Scholar

[19]

D. Henry and B. V. Matioc, On the regularity of steady periodic stratified water waves, Comm. Pure Appl. Anal., 11 (2012), 1453-1464. Google Scholar

[20]

D. Henry and B. V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974. Google Scholar

[21]

D. Henry and B. V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and parabolic equations, Springer Proc. Math. Stat., Springer, Cham, 119 (2015), 159-177. doi: 10.1007/978-3-319-12547-3_7. Google Scholar

[22]

D. Henry and S. Sastre-Gomez, Steady Periodic Waver Waves Bifurcating for Fixed-Depth Rotational Flows with Discontinuous Vorticity, Differential and Integral Equations, 2017.Google Scholar

[23] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. Google Scholar
[24]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371. Google Scholar

[25]

M. J. Lighthill, Waves in fluids, View issue TOC, 20 (1967), 267-293. doi: 10.1002/cpa.3160200204. Google Scholar

[26]

C. I. Martin and B. V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114. doi: 10.1016/j.jde.2014.01.036. Google Scholar

[27]

C. I. Martin and B. V. Matioc, Steady periodic water waves with unbounded vorticity: Equivalent formulations and existence results, J. Nonlinear Sci., 24 (2014), 633-659. doi: 10.1007/s00332-014-9201-1. Google Scholar

[28]

A. V. Matioc and B. V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886. doi: 10.1007/s00220-014-1918-z. Google Scholar

[29]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, Journal of Fluid Mechanics, 66 (1974), 625-640. doi: 10.1017/S0022112074000413. Google Scholar

[30]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, gravity waves in water of finite depth, Advances in Fluid Mechanics, 10 (1997), 215-319. Google Scholar

[31]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[32]

E. Varvaruca, On the existence of extreme waves and the {S}tokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076. doi: 10.1016/j.jde.2008.12.018. Google Scholar

[33]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized {S}tokes conjectures, Acta Math., 206 (2011), 363-403. doi: 10.1007/s11511-011-0066-y. Google Scholar

[34]

E. Varvaruca and A. Zarnescu, Equivalence of weak formulations of the steady water waves equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1703-1719. doi: 10.1098/rsta.2011.0455. Google Scholar

[35]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943. Google Scholar

[36]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. Google Scholar

[37]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. Google Scholar

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