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July  2012, 11(4): 1453-1464. doi: 10.3934/cpaa.2012.11.1453

On the regularity of steady periodic stratified water waves

David Henry 1, and  Bogdan--Vasile Matioc 2,

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

Received  June 2011 Revised  September 2011 Published  January 2012

In this paper we prove regularity results for steady periodic stratified water waves, where we allow for the effects of surface tension. Our results concern stratified water waves, without stagnation points, which exist in three distinct physical regimes, namely: capillary, capillary-gravity, and gravity water waves. We prove, for all three types of waves, that, when the Bernoulli function is Hölder continuous and the variable density function has a first derivative which is Hölder continuous, then the free-surface profile is the graph of a smooth function. Furthermore, we show that the streamlines are analytic a priori for capillary stratified waves, whereas for gravity and capillary-gravity stratified waves the streamlines are smooth in general, and analytic in an unstable regime. Moreover, if the Bernoulli function and the streamline density function are both real analytic functions then all of the streamlines, including the wave profile, are real analytic for all gravity, capillary, and capillary-gravity stratified waves.
Keywords: surface tension, Stratified water waves, analyticity, streamlines, free surface, implicit function theorem..
Mathematics Subject Classification: 35Q35, 76B03, 76B45, 76B47, 76B7.
Citation: David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1453-1464. doi: 10.3934/cpaa.2012.11.1453
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.   Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature,, Ann. Math., 132 (1990), 451.  doi: 10.2307/1971426.  Google Scholar

[3]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[6]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,'', CBMS-NSF Conference Series in AppliedMathematics, (2011).   Google Scholar

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

A. Constantin and J. Escher, Analyticity of periodic travelling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

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

[10]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 53 (2010), 533.  doi: 10.1002/cpa.20165.  Google Scholar

[11]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Rational Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[12]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[13]

G. D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude,, J. Fluid Mech., 2 (1957), 532.  doi: 10.1017/S0022112057000348.  Google Scholar

[14]

Yu V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations,'', Springer-Verlag, (1998).   Google Scholar

[15]

M. Ehrnstrom, On the streamlines and particle paths of gravitational water waves,, Nonlinearity, 21 (2008), 1141.  doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[16]

J. Escher, Regularity of rotational travelling water waves,, Philos. Trans. R. Soc. Lond. Ser. A, ().   Google Scholar

[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.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[18]

J. Escher and G. Simonett, Analyticity of the interface in a free boundary problem,, Math. Ann., 305 (1996), 439.  doi: 10.1007/BF01444233.  Google Scholar

[19]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., (1809), 412.   Google Scholar

[20]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1.  doi: 10.1155/IMRN/2006/21630.  Google Scholar

[21]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., (2008), 87.  doi: 10.2991/jnmp.2008.15.s2.7.  Google Scholar

[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.  doi: 10.1137/100801408.  Google Scholar

[23]

D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity,, J. Math. Fluid Mech., (): 00021.  doi: 10.1007/s00021-011-0056-z.  Google Scholar

[24]

D. Henry, Regularity for steady periodic capillary water waves with vorticity,, Philos. Trans. R. Soc. Lond. Ser. A, ().   Google Scholar

[25]

D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Scuola Norm. Sup. Pisa, ().   Google Scholar

[26]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'', Cambridge Univ. Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[27]

W. Kinnersley, Exact large amplitude capillary waves on sheets of fluid,, J. Fluid Mech., 77 (1976), 229.  doi: 10.1017/S0022112076002085.  Google Scholar

[28]

B. Kinsman, "Wind Waves,'', Prentice Hall, (1965).   Google Scholar

[29]

J. Lighthill, "Waves in Fluids,", Cambridge Univ. Press, (1978).   Google Scholar

[30]

R. R. Long, Some aspects of the flow of stratified fluids. Part I : A theoretical investigation,, Tellus, 5 (1953), 42.  doi: 10.1111/j.2153-3490.1953.tb01035.x.  Google Scholar

[31]

B. V. Matioc, Analyticity of the streamlines for periodic travelling water waves with bounded vorticity,, Int. Math. Res. Not., 17 (2011), 3858.   Google Scholar

[32]

B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, Quart. Appl. Math., ().   Google Scholar

[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 \textbf{771}, 771 (1980), 360.   Google Scholar

[34]

C. B. Morrey, "Multiple Integrals in the Calculus of Variations,'', Springer, (1966).   Google Scholar

[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.   Google Scholar

[36]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[37]

R. E. L. Turner, Internal waves in fluids with rapidly varying density,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (1981), 513.   Google Scholar

[38]

S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.  doi: 10.1137/080721583.  Google Scholar

[39]

S. Walsh, Steady periodic gravity waves with surface tension,, preprint., ().   Google Scholar

[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.  doi: 10.1017/S0022112096003539.  Google Scholar

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.   Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces, Part I. Curves with p-integrable curvature,, Ann. Math., 132 (1990), 451.  doi: 10.2307/1971426.  Google Scholar

[3]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[6]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,'', CBMS-NSF Conference Series in AppliedMathematics, (2011).   Google Scholar

[7]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[8]

A. Constantin and J. Escher, Analyticity of periodic travelling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

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

[10]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 53 (2010), 533.  doi: 10.1002/cpa.20165.  Google Scholar

[11]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Rational Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[12]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[13]

G. D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude,, J. Fluid Mech., 2 (1957), 532.  doi: 10.1017/S0022112057000348.  Google Scholar

[14]

Yu V. Egorov and M. A. Shubin, "Foundations of the Classical Theory of Partial Differential Equations,'', Springer-Verlag, (1998).   Google Scholar

[15]

M. Ehrnstrom, On the streamlines and particle paths of gravitational water waves,, Nonlinearity, 21 (2008), 1141.  doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[16]

J. Escher, Regularity of rotational travelling water waves,, Philos. Trans. R. Soc. Lond. Ser. A, ().   Google Scholar

[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.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[18]

J. Escher and G. Simonett, Analyticity of the interface in a free boundary problem,, Math. Ann., 305 (1996), 439.  doi: 10.1007/BF01444233.  Google Scholar

[19]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., (1809), 412.   Google Scholar

[20]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1.  doi: 10.1155/IMRN/2006/21630.  Google Scholar

[21]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., (2008), 87.  doi: 10.2991/jnmp.2008.15.s2.7.  Google Scholar

[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.  doi: 10.1137/100801408.  Google Scholar

[23]

D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity,, J. Math. Fluid Mech., (): 00021.  doi: 10.1007/s00021-011-0056-z.  Google Scholar

[24]

D. Henry, Regularity for steady periodic capillary water waves with vorticity,, Philos. Trans. R. Soc. Lond. Ser. A, ().   Google Scholar

[25]

D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Scuola Norm. Sup. Pisa, ().   Google Scholar

[26]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'', Cambridge Univ. Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[27]

W. Kinnersley, Exact large amplitude capillary waves on sheets of fluid,, J. Fluid Mech., 77 (1976), 229.  doi: 10.1017/S0022112076002085.  Google Scholar

[28]

B. Kinsman, "Wind Waves,'', Prentice Hall, (1965).   Google Scholar

[29]

J. Lighthill, "Waves in Fluids,", Cambridge Univ. Press, (1978).   Google Scholar

[30]

R. R. Long, Some aspects of the flow of stratified fluids. Part I : A theoretical investigation,, Tellus, 5 (1953), 42.  doi: 10.1111/j.2153-3490.1953.tb01035.x.  Google Scholar

[31]

B. V. Matioc, Analyticity of the streamlines for periodic travelling water waves with bounded vorticity,, Int. Math. Res. Not., 17 (2011), 3858.   Google Scholar

[32]

B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, Quart. Appl. Math., ().   Google Scholar

[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 \textbf{771}, 771 (1980), 360.   Google Scholar

[34]

C. B. Morrey, "Multiple Integrals in the Calculus of Variations,'', Springer, (1966).   Google Scholar

[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.   Google Scholar

[36]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[37]

R. E. L. Turner, Internal waves in fluids with rapidly varying density,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 8 (1981), 513.   Google Scholar

[38]

S. Walsh, Stratified steady periodic water waves,, SIAM J. Math. Anal., 41 (2009), 1054.  doi: 10.1137/080721583.  Google Scholar

[39]

S. Walsh, Steady periodic gravity waves with surface tension,, preprint., ().   Google Scholar

[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.  doi: 10.1017/S0022112096003539.  Google Scholar

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