June  2012, 17(4): 1113-1137. doi: 10.3934/dcdsb.2012.17.1113

Well-posedness of a model for water waves with viscosity

1. 

Department of Mathematics, Drexel University, Philadelphia, PA 19104

2. 

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States

3. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States

Received  January 2011 Revised  August 2011 Published  February 2012

The water wave equations of ideal free-surface fluid mechanics are a fundamental model of open ocean movements with a surprisingly subtle well-posedness theory. In consequence of both theoretical and computational difficulties with the full water wave equations, various asymptotic approximations have been proposed, analyzed and used in practical situations. In this essay, we establish the well-posedness of a model system of water wave equations which is inspired by recent work of Dias, Dyachenko, and Zakharov (Phys. Lett. A, 372:2008). The model in question includes dissipative effects and is weakly nonlinear. The present contribution is a first step in a larger program centered around the Dias-Dychenko-Zhakharov system.
Citation: David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113
References:
[1]

David M. Ambrose and Nader Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. doi: 10.1002/cpa.20085.

[2]

David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244 (electronic). doi: 10.1137/S0036141002403869.

[3]

Wooyoung Choi, Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth, Journal of Fluid Mechanics, 295 (1995), 381-394. doi: 10.1017/S0022112095002011.

[4]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis and analytic dependence, in "Pseudodifferential Operators and Applications" (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, (1985), 71-78.

[5]

Walter Craig and Catherine Sulem, Numerical simulation of gravity waves, Journal of Computational Physics, 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.

[6]

Walter Craig, Ulrich Schanz and Catherine Sulem, The modulation regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. Henri Poincaré, 14 (1997), 615-667.

[7]

F. Dias, A. I. Dyachenko and V. E. Zakharov, Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions, Phys. Lett. A, 372 (2008), 1297-1302. doi: 10.1016/j.physleta.2007.09.027.

[8]

Maria Kakleas and David P. Nicholls, Numerical simulation of a weakly nonlinear model for water waves with viscosity, Journal of Scientific Computing, 42 (2010), 274-290. doi: 10.1007/s10915-009-9324-y.

[9]

Horace Lamb, "Hydrodynamics," Reprint of the 1932 sixth edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993.

[10]

David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic). doi: 10.1090/S0894-0347-05-00484-4.

[11]

Y Matsuno, Nonlinear evolutions of surface gravity waves of fluid of finite depth, Physical Review Letters, 69 (1992), 609-611. doi: 10.1103/PhysRevLett.69.609.

[12]

Andrew J. Majda and Andrea L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.

[13]

D. Michael Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves, in "Proceedings of SPIE - The International Society for Optical Engineering" (San Diego, 1991), Vol. 1558, Int. Soc. for Optical Engineering, Bellingham, WA, (1991), 213-221.

[14]

David P. Nicholls and Fernando Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1411-1433. doi: 10.1017/S0308210500001463.

[15]

David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math., 94 (2003), 107-146. doi: 10.1007/s002110200399.

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. doi: 10.1007/s002220050177.

[17]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. doi: 10.1090/S0894-0347-99-00290-8.

[18]

Vladimir Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194. doi: 10.1007/BF00913182.

show all references

References:
[1]

David M. Ambrose and Nader Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. doi: 10.1002/cpa.20085.

[2]

David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211-244 (electronic). doi: 10.1137/S0036141002403869.

[3]

Wooyoung Choi, Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth, Journal of Fluid Mechanics, 295 (1995), 381-394. doi: 10.1017/S0022112095002011.

[4]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis and analytic dependence, in "Pseudodifferential Operators and Applications" (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, (1985), 71-78.

[5]

Walter Craig and Catherine Sulem, Numerical simulation of gravity waves, Journal of Computational Physics, 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.

[6]

Walter Craig, Ulrich Schanz and Catherine Sulem, The modulation regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. Henri Poincaré, 14 (1997), 615-667.

[7]

F. Dias, A. I. Dyachenko and V. E. Zakharov, Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions, Phys. Lett. A, 372 (2008), 1297-1302. doi: 10.1016/j.physleta.2007.09.027.

[8]

Maria Kakleas and David P. Nicholls, Numerical simulation of a weakly nonlinear model for water waves with viscosity, Journal of Scientific Computing, 42 (2010), 274-290. doi: 10.1007/s10915-009-9324-y.

[9]

Horace Lamb, "Hydrodynamics," Reprint of the 1932 sixth edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993.

[10]

David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic). doi: 10.1090/S0894-0347-05-00484-4.

[11]

Y Matsuno, Nonlinear evolutions of surface gravity waves of fluid of finite depth, Physical Review Letters, 69 (1992), 609-611. doi: 10.1103/PhysRevLett.69.609.

[12]

Andrew J. Majda and Andrea L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.

[13]

D. Michael Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves, in "Proceedings of SPIE - The International Society for Optical Engineering" (San Diego, 1991), Vol. 1558, Int. Soc. for Optical Engineering, Bellingham, WA, (1991), 213-221.

[14]

David P. Nicholls and Fernando Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1411-1433. doi: 10.1017/S0308210500001463.

[15]

David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math., 94 (2003), 107-146. doi: 10.1007/s002110200399.

[16]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. doi: 10.1007/s002220050177.

[17]

Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. doi: 10.1090/S0894-0347-99-00290-8.

[18]

Vladimir Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194. doi: 10.1007/BF00913182.

[1]

Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429

[2]

Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2415-2431. doi: 10.3934/cpaa.2022053

[3]

Anna Geyer, Ronald Quirchmayr. Weakly nonlinear waves in stratified shear flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2309-2325. doi: 10.3934/cpaa.2022061

[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]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

[6]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations and Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007

[7]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[8]

Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[9]

Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016

[10]

Luigi Roberti. The surface current of Ekman flows with time-dependent eddy viscosity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2463-2477. doi: 10.3934/cpaa.2022064

[11]

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

[12]

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

[13]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093

[14]

Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459

[15]

David Henry. Energy considerations for nonlinear equatorial water waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2337-2356. doi: 10.3934/cpaa.2022057

[16]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257

[17]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393

[18]

Pavel Krejčí, Elisabetta Rocca. Well-posedness of an extended model for water-ice phase transitions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 439-460. doi: 10.3934/dcdss.2013.6.439

[19]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

[20]

Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (7)

[Back to Top]