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August  2014, 34(8): 3287-3315. doi: 10.3934/dcds.2014.34.3287

Steady stratified periodic gravity waves with surface tension II: Global bifurcation

Samuel Walsh 1,

1. 

University of Missouri, Columbia, MO 65201

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 global continua of classical solutions that are periodic and traveling. This is accomplished by globally continuing the curves of small-amplitude solutions obtained by the author in [25]. We do this in two ways: first, by means of a degree theoretic theorem in the spirit of Rabinowitz, and second via the analytic continuation method of Dancer.
Keywords: Water waves, surface tension, stratification, bifurcation theory..
Mathematics Subject Classification: Primary: 35C07, 35B32; Secondary: 35A16, 35R35, 76B45, 76B7.
Citation: Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287
References:
[1]

S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure and Appl. Math., 15 (1962), 119-147. doi: 10.1002/cpa.3160150203.

[2]

B. Buffoni, E. Dancer and J. Toland, Sur les ondes de Stokes et une conjecture de levi-civita, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1265-1268. doi: 10.1016/S0764-4442(98)80176-6.

[3]

B. Buffoni, E. Dancer and J. Toland, The regularity and local bifurcation of Stokes waves, Arch. Rational Mech. Anal., 152 (2000), 207-240. doi: 10.1007/s002050000086.

[4]

B. Buffoni, E. Dancer and J. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Rational Mech. Anal., 152 (2000), 241-271. doi: 10.1007/s002050000087.

[5]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation: An Introduction, Princeton University Press, 2003.

[6]

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.

[7]

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.

[8]

E. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., 26 (1973), 359-384.

[9]

E. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192.

[10]

E. Dancer, Global structure of the solutions of nonlinear real analytic eigenvalue problems, Proc. London Math. Soc, 27 (1973), 747-765.

[11]

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.

[12]

T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Rational Mech. Anal., 143 (1998), 1-28. doi: 10.1007/s002050050098.

[13]

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

[14]

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

[15]

H. Kielhöfer, Multiple eigenvalue bifurcation for Fredholm operators, J. Reine Angew. Math., 358 (1985), 104-124. doi: 10.1515/crll.1985.358.104.

[16]

H. Kielhöfer, Bifurcation Theory, 156 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004, An introduction with applications to PDEs.

[17]

Y. Luo and N. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, in Proc. R. Soc. Edinb., Sect. A, 118 (1991), 193-207. doi: 10.1017/S0308210500029048.

[18]

Y. Luo and N. Trudinger, Quasilinear second order elliptic equations with Venttsel boundary conditions, Potential Anal., 3 (1994), 219-243. doi: 10.1007/BF01053434.

[19]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[20]

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

[21]

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

[22]

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.

[23]

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

[24]

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

[25]

S. Walsh, Steady stratified periodic gravity waves with surface tension i: Local bifurcation, Preprint.

[26]

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

show all references

References:
[1]

S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure and Appl. Math., 15 (1962), 119-147. doi: 10.1002/cpa.3160150203.

[2]

B. Buffoni, E. Dancer and J. Toland, Sur les ondes de Stokes et une conjecture de levi-civita, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1265-1268. doi: 10.1016/S0764-4442(98)80176-6.

[3]

B. Buffoni, E. Dancer and J. Toland, The regularity and local bifurcation of Stokes waves, Arch. Rational Mech. Anal., 152 (2000), 207-240. doi: 10.1007/s002050000086.

[4]

B. Buffoni, E. Dancer and J. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Rational Mech. Anal., 152 (2000), 241-271. doi: 10.1007/s002050000087.

[5]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation: An Introduction, Princeton University Press, 2003.

[6]

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.

[7]

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.

[8]

E. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., 26 (1973), 359-384.

[9]

E. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192.

[10]

E. Dancer, Global structure of the solutions of nonlinear real analytic eigenvalue problems, Proc. London Math. Soc, 27 (1973), 747-765.

[11]

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.

[12]

T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Rational Mech. Anal., 143 (1998), 1-28. doi: 10.1007/s002050050098.

[13]

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

[14]

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

[15]

H. Kielhöfer, Multiple eigenvalue bifurcation for Fredholm operators, J. Reine Angew. Math., 358 (1985), 104-124. doi: 10.1515/crll.1985.358.104.

[16]

H. Kielhöfer, Bifurcation Theory, 156 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004, An introduction with applications to PDEs.

[17]

Y. Luo and N. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, in Proc. R. Soc. Edinb., Sect. A, 118 (1991), 193-207. doi: 10.1017/S0308210500029048.

[18]

Y. Luo and N. Trudinger, Quasilinear second order elliptic equations with Venttsel boundary conditions, Potential Anal., 3 (1994), 219-243. doi: 10.1007/BF01053434.

[19]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[20]

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

[21]

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

[22]

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.

[23]

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

[24]

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

[25]

S. Walsh, Steady stratified periodic gravity waves with surface tension i: Local bifurcation, Preprint.

[26]

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

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