- Previous Article
- DCDS Home
- This Issue
-
Next Article
Steady stratified periodic gravity waves with surface tension I: Local bifurcation
Steady stratified periodic gravity waves with surface tension II: Global bifurcation
1. | University of Missouri, Columbia, MO 65201 |
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. |
[1] |
Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109 |
[2] |
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 |
[3] |
Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 |
[4] |
Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465 |
[5] |
Shengfu Deng. Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3419-3435. doi: 10.3934/dcds.2014.34.3419 |
[6] |
Jing Cui, Guangyue Gao, Shu-Ming Sun. Controllability and stabilization of gravity-capillary surface water waves in a basin. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2035-2063. doi: 10.3934/cpaa.2021158 |
[7] |
Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control and Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315 |
[8] |
Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153 |
[9] |
Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108 |
[10] |
Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217 |
[11] |
Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259 |
[12] |
Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023 |
[13] |
Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 |
[14] |
Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103 |
[15] |
Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523 |
[16] |
Walter A. Strauss. Vorticity jumps in steady water waves. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101 |
[17] |
Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1 |
[18] |
Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267 |
[19] |
Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185 |
[20] |
Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]