Solitary gravity-capillary water waves with point vortices

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July 2016, 36(7): 3927-3959. doi: 10.3934/dcds.2016.36.3927

Solitary gravity-capillary water waves with point vortices

Kristoffer Varholm ^1,

1.	Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received May 2015 Revised November 2015 Published March 2016

Abstract
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We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.

Keywords: vorticity., Steady water waves, solitary waves, Euler equations, bifurcation theory.

Mathematics Subject Classification: Primary: 35Q31; Secondary: 35C07, 76B2.

Citation: Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927

References:

[1]	K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude, Dokl. Akad. Nauk SSSR, 294 (1987), 1033-1037.
[2]	G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007. doi: 10.1002/cpa.20365.
[3]	A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, ().
[4]	A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. 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), Philadelphia, PA, 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), 591-603. 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), 481-527. 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), 133-175. 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), 33-67. doi: 10.1007/s00205-010-0314-x.
[10]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. 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), 497-522. 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), 321-340. 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., ().
[15]	M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419. 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), 1436-1456. doi: 10.1137/100792330.
[17]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not. IMRN, (2009), 4578-4596. 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), 1888-1909. doi: 10.1016/j.jde.2008.01.012.
[19]	M. Ehrnström and E. Wahlén, Trimodal steady water waves, Arch. Ration. Mech. Anal., 216 (2015), 449-471. 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), 2932-2949. doi: 10.1016/j.jde.2011.03.023.
[21]	T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-0-387-21607-2.
[22]	F. Gerstner, Theorie der wellen, Annalen Der Physik, 32 (1809), 412-445. doi: 10.1002/andp.18090320808.
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 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), 1530-1538. 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), 775-786. 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), 4279-4283. doi: 10.1002/grl.50838.
[27]	V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521 (electronic). doi: 10.1137/040621168.
[28]	V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491-509. 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, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.
[30]	D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.
[31]	J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978.
[32]	C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 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), 523-541.
[34]	A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.
[35]	A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.
[36]	B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity, Monatsh. Math., 174 (2014), 459-475. doi: 10.1007/s00605-013-0583-1.
[37]	C. C. Mei, The applied dynamics of ocean surface waves, Ocean Engineering, 11 (1984), p321. doi: 10.1016/0029-8018(84)90033-7.
[38]	A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 ().
[39]	P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. 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., Berlin, 1996. doi: 10.1515/9783110812411.
[41]	J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity, Nonlinearity, 26 (2013), 1529-1564. 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), 698-744. doi: 10.1002/cpa.20213.
[43]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
[44]	J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math., 56 (1996), 207-217.
[45]	K. Varholm, Water Waves with Compactly Supported Vorticity, Master's thesis, Norwegian University of Science and Technology, 2014.
[46]	E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.
[47]	S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.
[48]	S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. 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), 190-194. 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), 1033-1037.
[2]	G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., 64 (2011), 975-1007. doi: 10.1002/cpa.20365.
[3]	A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers,, preprint, ().
[4]	A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. 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), Philadelphia, PA, 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), 591-603. 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), 481-527. 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), 133-175. 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), 33-67. doi: 10.1007/s00205-010-0314-x.
[10]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. 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), 497-522. 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), 321-340. 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., ().
[15]	M. Ehrnström, J. Escher and G. Villari, Steady water waves with multiple critical layers: Interior dynamics, J. Math. Fluid Mech., 14 (2012), 407-419. 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), 1436-1456. doi: 10.1137/100792330.
[17]	M. Ehrnström, H. Holden and X. Raynaud, Symmetric waves are traveling waves, Int. Math. Res. Not. IMRN, (2009), 4578-4596. 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), 1888-1909. doi: 10.1016/j.jde.2008.01.012.
[19]	M. Ehrnström and E. Wahlén, Trimodal steady water waves, Arch. Ration. Mech. Anal., 216 (2015), 449-471. 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), 2932-2949. doi: 10.1016/j.jde.2011.03.023.
[21]	T. W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-0-387-21607-2.
[22]	F. Gerstner, Theorie der wellen, Annalen Der Physik, 32 (1809), 412-445. doi: 10.1002/andp.18090320808.
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 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), 1530-1538. 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), 775-786. 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), 4279-4283. doi: 10.1002/grl.50838.
[27]	V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521 (electronic). doi: 10.1137/040621168.
[28]	V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), 491-509. 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, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.
[30]	D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.
[31]	J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978.
[32]	C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Applied Mathematical Sciences, Springer-Verlag, New York, 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), 523-541.
[34]	A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. I, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.
[35]	A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. II, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.
[36]	B.-V. Matioc, Global bifurcation for water waves with capillary effects and constant vorticity, Monatsh. Math., 174 (2014), 459-475. doi: 10.1007/s00605-013-0583-1.
[37]	C. C. Mei, The applied dynamics of ocean surface waves, Ocean Engineering, 11 (1984), p321. doi: 10.1016/0029-8018(84)90033-7.
[38]	A. Nekrasov, On steady waves,, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 ().
[39]	P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. 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., Berlin, 1996. doi: 10.1515/9783110812411.
[41]	J. Shatah, S. Walsh and C. Zeng, Travelling water waves with compactly supported vorticity, Nonlinearity, 26 (2013), 1529-1564. 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), 698-744. doi: 10.1002/cpa.20213.
[43]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
[44]	J.-M. Vanden-Broeck, Periodic waves with constant vorticity in water of infinite depth, IMA J. Appl. Math., 56 (1996), 207-217.
[45]	K. Varholm, Water Waves with Compactly Supported Vorticity, Master's thesis, Norwegian University of Science and Technology, 2014.
[46]	E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.
[47]	S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.
[48]	S. Walsh, Steady stratified periodic gravity waves with surface tension {II}: Global bifurcation, Discrete Contin. Dyn. Syst., 34 (2014), 3287-3315. 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), 190-194. doi: 10.1007/BF00913182.

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