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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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., ().   Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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., ().   Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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