# American Institute of Mathematical Sciences

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

## 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

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.
Citation: David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure and 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), 623-727. [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, (). [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. [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., (): 00021.  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, (). [25] D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Scuola Norm. Sup. Pisa, (). [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] B. Kinsman, "Wind Waves,'' Prentice Hall, New Jersey, 1965. [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., (). [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, (). [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. [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., (): 00021.  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, (). [25] D. Henry and B. Matioc, On the existence of steady periodic capillary-gravity stratified water waves,, Ann. Scuola Norm. Sup. Pisa, (). [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] B. Kinsman, "Wind Waves,'' Prentice Hall, New Jersey, 1965. [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., (). [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.
 [1] Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287 [2] Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241 [3] Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109 [4] Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465 [5] Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 [6] Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022053 [7] Jing Cui, Guangyue Gao, Shu-Ming Sun. Controllability and stabilization of gravity-capillary surface water waves in a basin. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2035-2063. doi: 10.3934/cpaa.2021158 [8] Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025 [9] Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767 [10] Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure and Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379 [11] Miles H. Wheeler. On stratified water waves with critical layers and Coriolis forces. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4747-4770. doi: 10.3934/dcds.2019193 [12] Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153 [13] Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108 [14] Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217 [15] Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259 [16] Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023 [17] Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523 [18] Delia Ionescu-Kruse. Short-wavelength instabilities of edge waves in stratified water. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2053-2066. doi: 10.3934/dcds.2015.35.2053 [19] Jeongwhan Choi, Tao Lin, Shu-Ming Sun, Sungim Whang. Supercritical surface waves generated by negative or oscillatory forcing. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1313-1335. doi: 10.3934/dcdsb.2010.14.1313 [20] Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185

2020 Impact Factor: 1.916