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Dispersion relations for periodic water waves with surface tension and discontinuous vorticity
A characterization of the symmetric steady water waves in terms of the underlying flow
1. | University of Vienna, Nordbergstraße 15, 1090, Vienna, Austria |
References:
[1] |
H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[2] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[3] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[4] |
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. |
[5] |
A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181.
doi: 10.1017/S0022112003006773. |
[6] |
A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768.
doi: 10.1017/S0956792504005777. |
[7] |
A. Constantin and J. Escher, Analyticity of periodic travelling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[8] |
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. |
[9] |
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. |
[10] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[11] |
W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations, 13 (1988), 603-633.
doi: 10.1080/03605308808820554. |
[12] |
J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, to appear. |
[13] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 2001. |
[15] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[16] |
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. |
[17] |
V. M. Hur, Symmetry of steady periodic water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2203-2214.
doi: 10.1098/rsta.2007.2002. |
[18] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[19] |
B. Kinsman, Wind Waves, Prentice Hall, New Jersey, 1965.
doi: 10.1029/JZ066i008p02411. |
[20] |
A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140. |
[21] |
A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves, J. Evol. Equ., 12 (2012), 481-494.
doi: 10.1007/s00028-012-0141-7. |
[22] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
[23] |
P. R. Garabedian, Surface waves of finite depth, J. Anal. Math., 14 (1965), 161-169.
doi: 10.1007/BF02806385. |
[24] |
J. F. Toland, On the symmetry theory for Stokes waves of finite and infinite depth, in Trends in applications of mathematics to mechanics (Nice, 1998), Chapman & Hall/CRC, Boca Raton, FL, 2000, 207-217. |
[25] |
G. Tulzer, On the symmetry of steady periodic water waves with stagnation points, Commun. Pure Appl. Anal., 11 (2012), 1577-1586.
doi: 10.3934/cpaa.2012.11.1577. |
[26] |
H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, Adv. Ser. Nonlinear Dynam., 20, World Scientific Pub. Co. Inc., River Edge, NJ, 2001. |
[27] |
E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
[28] |
S. Walsh, Some criteria for the symmetry of stratified water waves, Wave Motion, 46 (2009), 350-362.
doi: 10.1016/j.wavemoti.2009.06.008. |
show all references
References:
[1] |
H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[2] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[3] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971873. |
[4] |
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. |
[5] |
A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181.
doi: 10.1017/S0022112003006773. |
[6] |
A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768.
doi: 10.1017/S0956792504005777. |
[7] |
A. Constantin and J. Escher, Analyticity of periodic travelling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[8] |
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. |
[9] |
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. |
[10] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[11] |
W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations, 13 (1988), 603-633.
doi: 10.1080/03605308808820554. |
[12] |
J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, to appear. |
[13] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 2001. |
[15] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[16] |
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. |
[17] |
V. M. Hur, Symmetry of steady periodic water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2203-2214.
doi: 10.1098/rsta.2007.2002. |
[18] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[19] |
B. Kinsman, Wind Waves, Prentice Hall, New Jersey, 1965.
doi: 10.1029/JZ066i008p02411. |
[20] |
A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140. |
[21] |
A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves, J. Evol. Equ., 12 (2012), 481-494.
doi: 10.1007/s00028-012-0141-7. |
[22] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
[23] |
P. R. Garabedian, Surface waves of finite depth, J. Anal. Math., 14 (1965), 161-169.
doi: 10.1007/BF02806385. |
[24] |
J. F. Toland, On the symmetry theory for Stokes waves of finite and infinite depth, in Trends in applications of mathematics to mechanics (Nice, 1998), Chapman & Hall/CRC, Boca Raton, FL, 2000, 207-217. |
[25] |
G. Tulzer, On the symmetry of steady periodic water waves with stagnation points, Commun. Pure Appl. Anal., 11 (2012), 1577-1586.
doi: 10.3934/cpaa.2012.11.1577. |
[26] |
H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, Adv. Ser. Nonlinear Dynam., 20, World Scientific Pub. Co. Inc., River Edge, NJ, 2001. |
[27] |
E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
[28] |
S. Walsh, Some criteria for the symmetry of stratified water waves, Wave Motion, 46 (2009), 350-362.
doi: 10.1016/j.wavemoti.2009.06.008. |
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