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Hyperbolic periodic points for chain hyperbolic homoclinic classes
Solitary gravity-capillary water waves with point vortices
1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway |
References:
[1] |
K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude,, Dokl. Akad. Nauk SSSR, 294 (1987), 1033.
|
[2] |
G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.
doi: 10.1002/cpa.20365. |
[3] |
A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, (). Google Scholar |
[4] |
A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.
doi: 10.1088/0305-4470/34/7/313. |
[5] |
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), (2011).
doi: 10.1137/1.9781611971873. |
[6] |
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. |
[7] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.
doi: 10.1002/cpa.3046. |
[8] |
A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.
doi: 10.1007/s00205-011-0412-4. |
[9] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.
doi: 10.1007/s00205-010-0314-x. |
[10] |
W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73.
doi: 10.1006/jcph.1993.1164. |
[11] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach,, Nonlinearity, 5 (1992), 497.
doi: 10.1088/0951-7715/5/2/009. |
[12] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.
doi: 10.1016/0022-1236(71)90015-2. |
[13] |
J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305.
|
[14] |
M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur, finie., (). Google Scholar |
[15] |
M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics,, J. Math. Fluid Mech., 14 (2012), 407.
doi: 10.1007/s00021-011-0068-8. |
[16] |
M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436.
doi: 10.1137/100792330. |
[17] |
M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves,, Int. Math. Res. Not. IMRN, (2009), 4578.
doi: 10.1093/imrn/rnp100. |
[18] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.
doi: 10.1016/j.jde.2008.01.012. |
[19] |
M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449.
doi: 10.1007/s00205-014-0812-3. |
[20] |
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. |
[21] |
T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001).
doi: 10.1007/978-0-387-21607-2. |
[22] |
F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412.
doi: 10.1002/andp.18090320808. |
[23] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).
|
[24] |
M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity,, Phys. D, 237 (2008), 1530.
doi: 10.1016/j.physd.2008.03.015. |
[25] |
D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775.
doi: 10.1017/S0308210512001990. |
[26] |
C. Hirt, S. Claessens, T. Fecher, M. Kuhn, R. Pail and M. Rexer, New ultrahigh-resolution picture of earth's gravity field,, Geophysical Research Letters, 40 (2013), 4279.
doi: 10.1002/grl.50838. |
[27] |
V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482.
doi: 10.1137/040621168. |
[28] |
V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.
doi: 10.4310/MRL.2008.v15.n3.a9. |
[29] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).
doi: 10.1017/CBO9780511624056. |
[30] |
D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013).
doi: 10.1090/surv/188. |
[31] |
J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).
|
[32] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences,, Springer-Verlag, (1994).
doi: 10.1007/978-1-4612-4284-0. |
[33] |
S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523.
|
[34] |
A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I,, Prentice-Hall, (1965). Google Scholar |
[35] |
A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II,, Prentice-Hall, (1965). Google Scholar |
[36] |
B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity,, Monatsh. Math., 174 (2014), 459.
doi: 10.1007/s00605-013-0583-1. |
[37] |
C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984).
doi: 10.1016/0029-8018(84)90033-7. |
[38] |
A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 (). Google Scholar |
[39] |
P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.
doi: 10.1007/s00205-003-0292-3. |
[40] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications,, Walter de Gruyter & Co., (1996).
doi: 10.1515/9783110812411. |
[41] |
J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity,, Nonlinearity, 26 (2013), 1529.
doi: 10.1088/0951-7715/26/6/1529. |
[42] |
J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation,, Comm. Pure Appl. Math., 61 (2008), 698.
doi: 10.1002/cpa.20213. |
[43] |
J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.
|
[44] |
J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth,, IMA J. Appl. Math., 56 (1996), 207. Google Scholar |
[45] |
K. Varholm, Water Waves with Compactly Supported Vorticity,, Master's thesis, (2014). Google Scholar |
[46] |
E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.
doi: 10.1016/j.jde.2008.10.005. |
[47] |
S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.
doi: 10.1137/080721583. |
[48] |
S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation,, Discrete Contin. Dyn. Syst., 34 (2014), 3287.
doi: 10.3934/dcds.2014.34.3287. |
[49] |
V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190.
doi: 10.1007/BF00913182. |
show all references
References:
[1] |
K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude,, Dokl. Akad. Nauk SSSR, 294 (1987), 1033.
|
[2] |
G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., 64 (2011), 975.
doi: 10.1002/cpa.20365. |
[3] |
A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, (). Google Scholar |
[4] |
A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.
doi: 10.1088/0305-4470/34/7/313. |
[5] |
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), (2011).
doi: 10.1137/1.9781611971873. |
[6] |
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. |
[7] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481.
doi: 10.1002/cpa.3046. |
[8] |
A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.
doi: 10.1007/s00205-011-0412-4. |
[9] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.
doi: 10.1007/s00205-010-0314-x. |
[10] |
W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73.
doi: 10.1006/jcph.1993.1164. |
[11] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach,, Nonlinearity, 5 (1992), 497.
doi: 10.1088/0951-7715/5/2/009. |
[12] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.
doi: 10.1016/0022-1236(71)90015-2. |
[13] |
J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305.
|
[14] |
M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur, finie., (). Google Scholar |
[15] |
M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics,, J. Math. Fluid Mech., 14 (2012), 407.
doi: 10.1007/s00021-011-0068-8. |
[16] |
M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers,, SIAM J. Math. Anal., 43 (2011), 1436.
doi: 10.1137/100792330. |
[17] |
M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves,, Int. Math. Res. Not. IMRN, (2009), 4578.
doi: 10.1093/imrn/rnp100. |
[18] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths,, J. Differential Equations, 244 (2008), 1888.
doi: 10.1016/j.jde.2008.01.012. |
[19] |
M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449.
doi: 10.1007/s00205-014-0812-3. |
[20] |
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. |
[21] |
T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001).
doi: 10.1007/978-0-387-21607-2. |
[22] |
F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412.
doi: 10.1002/andp.18090320808. |
[23] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).
|
[24] |
M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity,, Phys. D, 237 (2008), 1530.
doi: 10.1016/j.physd.2008.03.015. |
[25] |
D. Henry and A.-V. Matioc, Global bifurcation of capillary-gravity-stratified water waves,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775.
doi: 10.1017/S0308210512001990. |
[26] |
C. Hirt, S. Claessens, T. Fecher, M. Kuhn, R. Pail and M. Rexer, New ultrahigh-resolution picture of earth's gravity field,, Geophysical Research Letters, 40 (2013), 4279.
doi: 10.1002/grl.50838. |
[27] |
V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482.
doi: 10.1137/040621168. |
[28] |
V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.
doi: 10.4310/MRL.2008.v15.n3.a9. |
[29] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).
doi: 10.1017/CBO9780511624056. |
[30] |
D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013).
doi: 10.1090/surv/188. |
[31] |
J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978).
|
[32] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences,, Springer-Verlag, (1994).
doi: 10.1007/978-1-4612-4284-0. |
[33] |
S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523.
|
[34] |
A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I,, Prentice-Hall, (1965). Google Scholar |
[35] |
A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II,, Prentice-Hall, (1965). Google Scholar |
[36] |
B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity,, Monatsh. Math., 174 (2014), 459.
doi: 10.1007/s00605-013-0583-1. |
[37] |
C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984).
doi: 10.1016/0029-8018(84)90033-7. |
[38] |
A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 (). Google Scholar |
[39] |
P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.
doi: 10.1007/s00205-003-0292-3. |
[40] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications,, Walter de Gruyter & Co., (1996).
doi: 10.1515/9783110812411. |
[41] |
J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity,, Nonlinearity, 26 (2013), 1529.
doi: 10.1088/0951-7715/26/6/1529. |
[42] |
J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation,, Comm. Pure Appl. Math., 61 (2008), 698.
doi: 10.1002/cpa.20213. |
[43] |
J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.
|
[44] |
J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth,, IMA J. Appl. Math., 56 (1996), 207. Google Scholar |
[45] |
K. Varholm, Water Waves with Compactly Supported Vorticity,, Master's thesis, (2014). Google Scholar |
[46] |
E. Wahlén, Steady water waves with a critical layer,, J. Differential Equations, 246 (2009), 2468.
doi: 10.1016/j.jde.2008.10.005. |
[47] |
S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.
doi: 10.1137/080721583. |
[48] |
S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation,, Discrete Contin. Dyn. Syst., 34 (2014), 3287.
doi: 10.3934/dcds.2014.34.3287. |
[49] |
V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190.
doi: 10.1007/BF00913182. |
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