August  2014, 34(8): 3241-3285. doi: 10.3934/dcds.2014.34.3241

Steady stratified periodic gravity waves with surface tension I: Local bifurcation

1. 

University of Missouri, Columbia, MO 65201, United States

Received  July 2013 Revised  September 2013 Published  January 2014

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from $\mathcal{T}$ at an eigenvalue of the linearized problem.
Citation: Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241
References:
[1]

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

[2]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.  Google Scholar

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

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A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  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,, submitted., ().  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[8]

M. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291. Google Scholar

[9]

M. Dubreil-Jacotin, Sur les theoremes d'existence relatifs aux ondes permanentes periodiques a deux dimensions dans les liquides heterogenes, J. Math. Pures Appl., 16 (1937), 43-67. Google Scholar

[10]

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

[11]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, ().   Google Scholar

[12]

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

[13]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453.  Google Scholar

[14]

M. Jones and J. Toland, Symmetry and the bifurcation of capillary-gravity waves, Arch. Rational Mech. Anal., 96 (1986), 29-53. doi: 10.1007/BF00251412.  Google Scholar

[15]

B. Kinsman, Wind Waves, Prentice Hall, New Jersey, 1965. doi: 10.1029/JZ066i008p02411.  Google Scholar

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie, Ann. Math., 93 (1925), 264-314. doi: 10.1007/BF01449965.  Google Scholar

[17]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves, J. Evol. Equ., 12 (2012), 481-494. doi: 10.1007/s00028-012-0141-7.  Google Scholar

[18]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140.  Google Scholar

[19]

C. Mei, The applied dynamics of ocean surface waves, World Scientific Pub. Co. Inc., 11 (1984), 321pp. doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[20]

A. I. Nekrasov, The exact theory of steady waves on the surface of a heavy fluid, Izdat. Akad. Nauk SSSR, Moscow, (1951).  Google Scholar

[21]

H. Okamoto, On the problem of water waves of permanent configuration, Nonlinear Anal., 14 (1990), 469-481. doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[22]

H. Okamoto and M. Shōji, The resonance of modes in the problem of two-dimensional capillary-gravity waves, Physica D: Nonlinear Phenomena, 95 (1996), 336-350. doi: 10.1016/0167-2789(96)00071-1.  Google Scholar

[23]

L. Schwartz and L. Vanden-Broeck, Numerical solution of the exact equations for capillary gravity waves, J. Fluid Mech., 95 (1979), 119-139. doi: 10.1017/S0022112079001373.  Google Scholar

[24]

M. Shōji, New bifurcation diagrams in the problem of permanent progressive waves, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 571-613.  Google Scholar

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. Roy. Soc. London Ser. A, 399 (1985), 391-417. doi: 10.1098/rspa.1985.0063.  Google Scholar

[26]

R. E. L. Turner, Traveling waves in natural systems, in Variational and topological methods in the study of nonlinear phenomena (Pisa, 2000), vol. 49 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2002, 115-131.  Google Scholar

[27]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943 (electronic). doi: 10.1137/050630465.  Google Scholar

[28]

E. Wahlén, Steady periodic capillary waves with vorticity, Ark. Mat., 44 (2006), 367-387. doi: 10.1007/s11512-006-0024-7.  Google Scholar

[29]

E. Wahlén, On rotational water waves with surface tension, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2215-2225. doi: 10.1098/rsta.2007.2003.  Google Scholar

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion, PhD thesis, Lund University, 2008. Google Scholar

[31]

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

[32]

S. Walsh, Steady stratified periodic gravity waves with surface tension ii: Global bifurcation,, Preprint., ().   Google Scholar

[33]

J. Wilton, On ripples, Phil. Mag., 29 (1915), 688-700. doi: 10.1080/14786440508635350.  Google Scholar

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids, The Macmillan Co., New York, 1965.  Google Scholar

show all references

References:
[1]

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

[2]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.  Google Scholar

[3]

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

[4]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  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,, submitted., ().  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[8]

M. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291. Google Scholar

[9]

M. Dubreil-Jacotin, Sur les theoremes d'existence relatifs aux ondes permanentes periodiques a deux dimensions dans les liquides heterogenes, J. Math. Pures Appl., 16 (1937), 43-67. Google Scholar

[10]

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

[11]

D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, ().   Google Scholar

[12]

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

[13]

D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453.  Google Scholar

[14]

M. Jones and J. Toland, Symmetry and the bifurcation of capillary-gravity waves, Arch. Rational Mech. Anal., 96 (1986), 29-53. doi: 10.1007/BF00251412.  Google Scholar

[15]

B. Kinsman, Wind Waves, Prentice Hall, New Jersey, 1965. doi: 10.1029/JZ066i008p02411.  Google Scholar

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie, Ann. Math., 93 (1925), 264-314. doi: 10.1007/BF01449965.  Google Scholar

[17]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves, J. Evol. Equ., 12 (2012), 481-494. doi: 10.1007/s00028-012-0141-7.  Google Scholar

[18]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140.  Google Scholar

[19]

C. Mei, The applied dynamics of ocean surface waves, World Scientific Pub. Co. Inc., 11 (1984), 321pp. doi: 10.1016/0029-8018(84)90033-7.  Google Scholar

[20]

A. I. Nekrasov, The exact theory of steady waves on the surface of a heavy fluid, Izdat. Akad. Nauk SSSR, Moscow, (1951).  Google Scholar

[21]

H. Okamoto, On the problem of water waves of permanent configuration, Nonlinear Anal., 14 (1990), 469-481. doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[22]

H. Okamoto and M. Shōji, The resonance of modes in the problem of two-dimensional capillary-gravity waves, Physica D: Nonlinear Phenomena, 95 (1996), 336-350. doi: 10.1016/0167-2789(96)00071-1.  Google Scholar

[23]

L. Schwartz and L. Vanden-Broeck, Numerical solution of the exact equations for capillary gravity waves, J. Fluid Mech., 95 (1979), 119-139. doi: 10.1017/S0022112079001373.  Google Scholar

[24]

M. Shōji, New bifurcation diagrams in the problem of permanent progressive waves, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 571-613.  Google Scholar

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. Roy. Soc. London Ser. A, 399 (1985), 391-417. doi: 10.1098/rspa.1985.0063.  Google Scholar

[26]

R. E. L. Turner, Traveling waves in natural systems, in Variational and topological methods in the study of nonlinear phenomena (Pisa, 2000), vol. 49 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2002, 115-131.  Google Scholar

[27]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943 (electronic). doi: 10.1137/050630465.  Google Scholar

[28]

E. Wahlén, Steady periodic capillary waves with vorticity, Ark. Mat., 44 (2006), 367-387. doi: 10.1007/s11512-006-0024-7.  Google Scholar

[29]

E. Wahlén, On rotational water waves with surface tension, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2215-2225. doi: 10.1098/rsta.2007.2003.  Google Scholar

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion, PhD thesis, Lund University, 2008. Google Scholar

[31]

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

[32]

S. Walsh, Steady stratified periodic gravity waves with surface tension ii: Global bifurcation,, Preprint., ().   Google Scholar

[33]

J. Wilton, On ripples, Phil. Mag., 29 (1915), 688-700. doi: 10.1080/14786440508635350.  Google Scholar

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids, The Macmillan Co., New York, 1965.  Google Scholar

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