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On the regularity of steady periodic stratified water waves
1. | Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria |
2. | Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany |
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. |
[2] |
S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature, Ann. Math., 132 (1990), 451-483.
doi: 10.2307/1971426. |
[3] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[4] |
A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[5] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[6] |
A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,'' CBMS-NSF Conference Series in AppliedMathematics, Vol. 81, SIAM, Philadelphia, 2011. |
[7] |
A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[8] |
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. |
[9] |
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. |
[10] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 53 (2010), 533-557.
doi: 10.1002/cpa.20165. |
[11] |
A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175.
doi: 10.1007/s00205-011-0412-4. |
[12] |
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. |
[13] |
G. D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech., 2 (1957), 532-540.
doi: 10.1017/S0022112057000348. |
[14] |
Yu V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations,'' Springer-Verlag, Berlin Heidelberg, 1998. |
[15] |
M. Ehrnstrom, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154.
doi: 10.1088/0951-7715/21/5/012. |
[16] |
J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A, to appear. |
[17] |
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, 2932-2949.
doi: 10.1016/j.jde.2011.03.023. |
[18] |
J. Escher and G. Simonett, Analyticity of the interface in a free boundary problem, Math. Ann., 305 (1996), 439-459.
doi: 10.1007/BF01444233. |
[19] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[20] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art. ID 23405 (2006), 1-13.
doi: 10.1155/IMRN/2006/21630. |
[21] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[22] |
D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal., 42 (2010), 3103-3111.
doi: 10.1137/100801408. |
[23] |
D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech., to appear.
doi: 10.1007/s00021-011-0056-z. |
[24] |
D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A, to appear. |
[25] |
D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Scuola Norm. Sup. Pisa, to appear. |
[26] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[27] |
W. Kinnersley, Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech., 77 (1976), 229-241.
doi: 10.1017/S0022112076002085. |
[28] | |
[29] |
J. Lighthill, "Waves in Fluids," Cambridge Univ. Press, Cambridge, 1978. |
[30] |
R. R. Long, Some aspects of the flow of stratified fluids. Part I : A theoretical investigation, Tellus, 5 (1953), 42-57.
doi: 10.1111/j.2153-3490.1953.tb01035.x. |
[31] |
B. V. Matioc, Analyticity of the streamlines for periodic travelling water waves with bounded vorticity, Int. Math. Res. Not., 17 (2011), 3858-3871. |
[32] |
B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math., to appear. |
[33] |
K. Masuda, On the regularity of solutions of the nonstationary Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Equations,'' Lecture Notes in Mathematics 771, 360-370, Springer-Verlag, Berlin, 1980. |
[34] |
C. B. Morrey, "Multiple Integrals in the Calculus of Variations,'' Springer, Berlin, 1966. |
[35] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138. [10.2307/1971426] |
[36] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[37] |
R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (1981), 513-573. |
[38] |
S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.
doi: 10.1137/080721583. |
[39] |
S. Walsh, Steady periodic gravity waves with surface tension, preprint. |
[40] |
C. S. Yih, The role of drift mass in the kinetic energy and momentum of periodic water waves and sound waves, J. Fluid Mech, 331 (1997), 429-438.
doi: 10.1017/S0022112096003539. |
show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. |
[2] |
S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature, Ann. Math., 132 (1990), 451-483.
doi: 10.2307/1971426. |
[3] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[4] |
A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[5] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[6] |
A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,'' CBMS-NSF Conference Series in AppliedMathematics, Vol. 81, SIAM, Philadelphia, 2011. |
[7] |
A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[8] |
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. |
[9] |
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. |
[10] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 53 (2010), 533-557.
doi: 10.1002/cpa.20165. |
[11] |
A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175.
doi: 10.1007/s00205-011-0412-4. |
[12] |
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. |
[13] |
G. D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech., 2 (1957), 532-540.
doi: 10.1017/S0022112057000348. |
[14] |
Yu V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations,'' Springer-Verlag, Berlin Heidelberg, 1998. |
[15] |
M. Ehrnstrom, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154.
doi: 10.1088/0951-7715/21/5/012. |
[16] |
J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A, to appear. |
[17] |
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, 2932-2949.
doi: 10.1016/j.jde.2011.03.023. |
[18] |
J. Escher and G. Simonett, Analyticity of the interface in a free boundary problem, Math. Ann., 305 (1996), 439-459.
doi: 10.1007/BF01444233. |
[19] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. |
[20] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art. ID 23405 (2006), 1-13.
doi: 10.1155/IMRN/2006/21630. |
[21] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[22] |
D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal., 42 (2010), 3103-3111.
doi: 10.1137/100801408. |
[23] |
D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech., to appear.
doi: 10.1007/s00021-011-0056-z. |
[24] |
D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. Ser. A, to appear. |
[25] |
D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Scuola Norm. Sup. Pisa, to appear. |
[26] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge Univ. Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[27] |
W. Kinnersley, Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech., 77 (1976), 229-241.
doi: 10.1017/S0022112076002085. |
[28] | |
[29] |
J. Lighthill, "Waves in Fluids," Cambridge Univ. Press, Cambridge, 1978. |
[30] |
R. R. Long, Some aspects of the flow of stratified fluids. Part I : A theoretical investigation, Tellus, 5 (1953), 42-57.
doi: 10.1111/j.2153-3490.1953.tb01035.x. |
[31] |
B. V. Matioc, Analyticity of the streamlines for periodic travelling water waves with bounded vorticity, Int. Math. Res. Not., 17 (2011), 3858-3871. |
[32] |
B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math., to appear. |
[33] |
K. Masuda, On the regularity of solutions of the nonstationary Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Equations,'' Lecture Notes in Mathematics 771, 360-370, Springer-Verlag, Berlin, 1980. |
[34] |
C. B. Morrey, "Multiple Integrals in the Calculus of Variations,'' Springer, Berlin, 1966. |
[35] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138. [10.2307/1971426] |
[36] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[37] |
R. E. L. Turner, Internal waves in fluids with rapidly varying density, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (1981), 513-573. |
[38] |
S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.
doi: 10.1137/080721583. |
[39] |
S. Walsh, Steady periodic gravity waves with surface tension, preprint. |
[40] |
C. S. Yih, The role of drift mass in the kinetic energy and momentum of periodic water waves and sound waves, J. Fluid Mech, 331 (1997), 429-438.
doi: 10.1017/S0022112096003539. |
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