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 - A, 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. 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. 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, (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. 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. 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. 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.

[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.

[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. 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. 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. doi: 10.1007/BF00251412.

[15]

B. Kinsman, Wind Waves,, Prentice Hall, (1965). doi: 10.1029/JZ066i008p02411.

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie,, Ann. Math., 93 (1925), 264. 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. 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.

[19]

C. Mei, The applied dynamics of ocean surface waves,, World Scientific Pub. Co. Inc., 11 (1984). 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, (1951).

[21]

H. Okamoto, On the problem of water waves of permanent configuration,, Nonlinear Anal., 14 (1990), 469. 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. 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. 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.

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. Roy. Soc. London Ser. A, 399 (1985), 391. 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), 115.

[27]

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

[28]

E. Wahlén, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367. 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. doi: 10.1098/rsta.2007.2003.

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion,, PhD thesis, (2008).

[31]

S. Walsh, Stratified and steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. 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. doi: 10.1080/14786440508635350.

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids,, The Macmillan Co., (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. 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. 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, (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. 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. 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. 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.

[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.

[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. 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. 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. doi: 10.1007/BF00251412.

[15]

B. Kinsman, Wind Waves,, Prentice Hall, (1965). doi: 10.1029/JZ066i008p02411.

[16]

T. Levi-Civita, Détermination rigoureuse de ondes permanentes d'ampleur finie,, Ann. Math., 93 (1925), 264. 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. 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.

[19]

C. Mei, The applied dynamics of ocean surface waves,, World Scientific Pub. Co. Inc., 11 (1984). 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, (1951).

[21]

H. Okamoto, On the problem of water waves of permanent configuration,, Nonlinear Anal., 14 (1990), 469. 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. 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. 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.

[25]

J. Toland and M. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. Roy. Soc. London Ser. A, 399 (1985), 391. 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), 115.

[27]

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

[28]

E. Wahlén, Steady periodic capillary waves with vorticity,, Ark. Mat., 44 (2006), 367. 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. doi: 10.1098/rsta.2007.2003.

[30]

E. Wahlén, On Some Nonlinear Aspects of Wave Motion,, PhD thesis, (2008).

[31]

S. Walsh, Stratified and steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. 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. doi: 10.1080/14786440508635350.

[34]

C.-S. Yih, Dynamics of Nonhomogeneous Fluids,, The Macmillan Co., (1965).

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