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

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449. doi: 10.1007/s00205-014-0812-3. Google Scholar
[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. Google Scholar
[21]	T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001). doi: 10.1007/978-0-387-21607-2. Google Scholar
[22]	F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412. doi: 10.1002/andp.18090320808. Google Scholar
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482. doi: 10.1137/040621168. Google Scholar
[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. Google Scholar
[29]	R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar
[30]	D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013). doi: 10.1090/surv/188. Google Scholar
[31]	J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978). Google Scholar
[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. Google Scholar
[33]	S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523. Google Scholar
[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. Google Scholar
[37]	C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984). doi: 10.1016/0029-8018(84)90033-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[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. Google Scholar
[47]	S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. doi: 10.1137/080721583. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	W. Craig and C. Sulem, Numerical simulation of gravity waves,, J. Comput. Phys., 108 (1993), 73. doi: 10.1006/jcph.1993.1164. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. Deny and J. L. Lions, Les espaces du type de Beppo Levi,, Ann. Inst. Fourier, 5 (): 305. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	M. Ehrnström and E. Wahlén, Trimodal steady water waves,, Arch. Ration. Mech. Anal., 216 (2015), 449. doi: 10.1007/s00205-014-0812-3. Google Scholar
[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. Google Scholar
[21]	T. W. Gamelin, Complex Analysis,, Undergraduate Texts in Mathematics, (2001). doi: 10.1007/978-0-387-21607-2. Google Scholar
[22]	F. Gerstner, Theorie der wellen,, Annalen Der Physik, 32 (1809), 412. doi: 10.1002/andp.18090320808. Google Scholar
[23]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	V. M. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482. doi: 10.1137/040621168. Google Scholar
[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. Google Scholar
[29]	R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997). doi: 10.1017/CBO9780511624056. Google Scholar
[30]	D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs,, American Mathematical Society, (2013). doi: 10.1090/surv/188. Google Scholar
[31]	J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978). Google Scholar
[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. Google Scholar
[33]	S. Mardare, On Poincaré and de Rham's theorems,, Rev. Roumaine Math. Pures Appl., 53 (2008), 523. Google Scholar
[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. Google Scholar
[37]	C. C. Mei, The applied dynamics of ocean surface waves,, Ocean Engineering, 11 (1984). doi: 10.1016/0029-8018(84)90033-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1. Google Scholar
[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. Google Scholar
[47]	S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054. doi: 10.1137/080721583. Google Scholar
[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. Google Scholar
[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. Google Scholar

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