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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[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., (). 

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[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., (). 

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

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