• Previous Article
    Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations
  • DCDS Home
  • This Issue
  • Next Article
    Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian
May  2017, 37(5): 2669-2680. doi: 10.3934/dcds.2017114

Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity

Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Aníbal Fernandes, Recife, Pernambuco, Brazil

*The author is supported by Science Foundation Ireland (SFI) under the grant 13/CDA/2117

Received  June 2016 Revised  December 2016 Published  February 2017

Fund Project: The author is supported by Science Foundation Ireland (SFI) under the grant 13/CDA/2117.

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Hölder spaces.

Citation: Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114
References:
[1]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 81 (2011), xii+315.  doi: 10.1137/1.9781611971873.  Google Scholar

[2]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406.  doi: 10.3934/cpaa.2012.11.1397.  Google Scholar

[3]

A. Constantin, The flow beneath a periodic travelling surface water wave, J. Phys. A: Math. Theor., 48 (2015), 1397-143001(25pp).  doi: 10.1088/1751-8113/48/14/143001.  Google Scholar

[4]

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.  Google Scholar

[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.  Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  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), 33-67.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, Journal de Mathématiques Pures et Appliquées, 13 (1934), 217-291.   Google Scholar

[11]

J. Escher and B. V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232.   Google Scholar

[12]

J. EscherA. V. Matioc and B. V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations, 251 (2011), 2932-2949.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[13]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007(14 pp).  doi: 10.1142/S1402925112400074.  Google Scholar

[14]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. A., 370 (2012), 1616-1628.  doi: 10.1098/rsta.2011.0449.  Google Scholar

[15]

D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043.  doi: 10.1016/j.nonrwa.2012.08.015.  Google Scholar

[16]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037.  doi: 10.1080/03605302.2012.734889.  Google Scholar

[17]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487.  doi: 10.1090/S0033-569X-2013-01293-8.  Google Scholar

[18]

D. Henry and A. V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786.  doi: 10.1017/S0308210512001990.  Google Scholar

[19]

D. Henry and B. V. Matioc, On the regularity of steady periodic stratified water waves, Comm. Pure Appl. Anal., 11 (2012), 1453-1464.   Google Scholar

[20]

D. Henry and B. V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974.   Google Scholar

[21]

D. Henry and B. V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and parabolic equations, Springer Proc. Math. Stat., Springer, Cham, 119 (2015), 159-177.  doi: 10.1007/978-3-319-12547-3_7.  Google Scholar

[22]

D. Henry and S. Sastre-Gomez, Steady Periodic Waver Waves Bifurcating for Fixed-Depth Rotational Flows with Discontinuous Vorticity, Differential and Integral Equations, 2017. Google Scholar

[23] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar
[24]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.  doi: 10.1017/S0022112008002371.  Google Scholar

[25]

M. J. Lighthill, Waves in fluids, View issue TOC, 20 (1967), 267-293.  doi: 10.1002/cpa.3160200204.  Google Scholar

[26]

C. I. Martin and B. V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114.  doi: 10.1016/j.jde.2014.01.036.  Google Scholar

[27]

C. I. Martin and B. V. Matioc, Steady periodic water waves with unbounded vorticity: Equivalent formulations and existence results, J. Nonlinear Sci., 24 (2014), 633-659.  doi: 10.1007/s00332-014-9201-1.  Google Scholar

[28]

A. V. Matioc and B. V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886.  doi: 10.1007/s00220-014-1918-z.  Google Scholar

[29]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, Journal of Fluid Mechanics, 66 (1974), 625-640.  doi: 10.1017/S0022112074000413.  Google Scholar

[30]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, gravity waves in water of finite depth, Advances in Fluid Mechanics, 10 (1997), 215-319.   Google Scholar

[31]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.   Google Scholar

[32]

E. Varvaruca, On the existence of extreme waves and the {S}tokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.  doi: 10.1016/j.jde.2008.12.018.  Google Scholar

[33]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized {S}tokes conjectures, Acta Math., 206 (2011), 363-403.  doi: 10.1007/s11511-011-0066-y.  Google Scholar

[34]

E. Varvaruca and A. Zarnescu, Equivalence of weak formulations of the steady water waves equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1703-1719.  doi: 10.1098/rsta.2011.0455.  Google Scholar

[35]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943.   Google Scholar

[36]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.   Google Scholar

[37]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.  doi: 10.1137/080721583.  Google Scholar

show all references

References:
[1]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 81 (2011), xii+315.  doi: 10.1137/1.9781611971873.  Google Scholar

[2]

A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406.  doi: 10.3934/cpaa.2012.11.1397.  Google Scholar

[3]

A. Constantin, The flow beneath a periodic travelling surface water wave, J. Phys. A: Math. Theor., 48 (2015), 1397-143001(25pp).  doi: 10.1088/1751-8113/48/14/143001.  Google Scholar

[4]

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.  Google Scholar

[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.  Google Scholar

[6]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  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), 33-67.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, Journal de Mathématiques Pures et Appliquées, 13 (1934), 217-291.   Google Scholar

[11]

J. Escher and B. V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function, Differential Integral Equations, 27 (2014), 217-232.   Google Scholar

[12]

J. EscherA. V. Matioc and B. V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations, 251 (2011), 2932-2949.  doi: 10.1016/j.jde.2011.03.023.  Google Scholar

[13]

D. Henry, Dispersion relations for steady periodic water waves of fixed mean-depth with an isolated bottom vorticity layer, J. Nonlinear Math. Phys., 19 (2012), 1240007(14 pp).  doi: 10.1142/S1402925112400074.  Google Scholar

[14]

D. Henry, Regularity for steady periodic capillary water waves with vorticity, Philos. Trans. R. Soc. Lond. A., 370 (2012), 1616-1628.  doi: 10.1098/rsta.2011.0449.  Google Scholar

[15]

D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043.  doi: 10.1016/j.nonrwa.2012.08.015.  Google Scholar

[16]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037.  doi: 10.1080/03605302.2012.734889.  Google Scholar

[17]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487.  doi: 10.1090/S0033-569X-2013-01293-8.  Google Scholar

[18]

D. Henry and A. V. Matioc, Global bifurcation of capillary-gravity-stratified water waves, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 775-786.  doi: 10.1017/S0308210512001990.  Google Scholar

[19]

D. Henry and B. V. Matioc, On the regularity of steady periodic stratified water waves, Comm. Pure Appl. Anal., 11 (2012), 1453-1464.   Google Scholar

[20]

D. Henry and B. V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 955-974.   Google Scholar

[21]

D. Henry and B. V. Matioc, Aspects of the mathematical analysis of nonlinear stratified water waves, Elliptic and parabolic equations, Springer Proc. Math. Stat., Springer, Cham, 119 (2015), 159-177.  doi: 10.1007/978-3-319-12547-3_7.  Google Scholar

[22]

D. Henry and S. Sastre-Gomez, Steady Periodic Waver Waves Bifurcating for Fixed-Depth Rotational Flows with Discontinuous Vorticity, Differential and Integral Equations, 2017. Google Scholar

[23] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar
[24]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215.  doi: 10.1017/S0022112008002371.  Google Scholar

[25]

M. J. Lighthill, Waves in fluids, View issue TOC, 20 (1967), 267-293.  doi: 10.1002/cpa.3160200204.  Google Scholar

[26]

C. I. Martin and B. V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, J. Differential Equations, 256 (2014), 3086-3114.  doi: 10.1016/j.jde.2014.01.036.  Google Scholar

[27]

C. I. Martin and B. V. Matioc, Steady periodic water waves with unbounded vorticity: Equivalent formulations and existence results, J. Nonlinear Sci., 24 (2014), 633-659.  doi: 10.1007/s00332-014-9201-1.  Google Scholar

[28]

A. V. Matioc and B. V. Matioc, Capillary-gravity water waves with discontinuous vorticity: Existence and regularity results, Comm. Math. Phys., 330 (2014), 859-886.  doi: 10.1007/s00220-014-1918-z.  Google Scholar

[29]

O. M. Phillips and M. L. Banner, Wave breaking in the presence of wind drift and swell, Journal of Fluid Mechanics, 66 (1974), 625-640.  doi: 10.1017/S0022112074000413.  Google Scholar

[30]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, gravity waves in water of finite depth, Advances in Fluid Mechanics, 10 (1997), 215-319.   Google Scholar

[31]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.   Google Scholar

[32]

E. Varvaruca, On the existence of extreme waves and the {S}tokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.  doi: 10.1016/j.jde.2008.12.018.  Google Scholar

[33]

E. Varvaruca and G. S. Weiss, A geometric approach to generalized {S}tokes conjectures, Acta Math., 206 (2011), 363-403.  doi: 10.1007/s11511-011-0066-y.  Google Scholar

[34]

E. Varvaruca and A. Zarnescu, Equivalence of weak formulations of the steady water waves equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1703-1719.  doi: 10.1098/rsta.2011.0455.  Google Scholar

[35]

E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943.   Google Scholar

[36]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.   Google Scholar

[37]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.  doi: 10.1137/080721583.  Google Scholar

[1]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[2]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[3]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[4]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[5]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[6]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[7]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[8]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[9]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[10]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[12]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[13]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[14]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[16]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[17]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[18]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[19]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (53)
  • Cited by (4)

Other articles
by authors

[Back to Top]