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 & 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. doi: 10.1002/cpa.20085.

[2]

David M. Ambrose, Well-posedness of vortex sheets with surface tension,, SIAM J. Math. Anal., 35 (2003), 211. 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. doi: 10.1017/S0022112095002011.

[4]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis and analytic dependence,, in, 43 (1985), 71.

[5]

Walter Craig and Catherine Sulem, Numerical simulation of gravity waves,, Journal of Computational Physics, 108 (1993), 73. 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.

[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. 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. doi: 10.1007/s10915-009-9324-y.

[9]

Horace Lamb, "Hydrodynamics,", Reprint of the 1932 sixth edition, (1932).

[10]

David Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605. 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. 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, (2002).

[13]

D. Michael Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves,, in, (1991), 213.

[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. doi: 10.1017/S0308210500001463.

[15]

David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators,, Numer. Math., 94 (2003), 107. 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. 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. 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. 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. doi: 10.1002/cpa.20085.

[2]

David M. Ambrose, Well-posedness of vortex sheets with surface tension,, SIAM J. Math. Anal., 35 (2003), 211. 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. doi: 10.1017/S0022112095002011.

[4]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis and analytic dependence,, in, 43 (1985), 71.

[5]

Walter Craig and Catherine Sulem, Numerical simulation of gravity waves,, Journal of Computational Physics, 108 (1993), 73. 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.

[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. 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. doi: 10.1007/s10915-009-9324-y.

[9]

Horace Lamb, "Hydrodynamics,", Reprint of the 1932 sixth edition, (1932).

[10]

David Lannes, Well-posedness of the water-waves equations,, J. Amer. Math. Soc., 18 (2005), 605. 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. 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, (2002).

[13]

D. Michael Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves,, in, (1991), 213.

[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. doi: 10.1017/S0308210500001463.

[15]

David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators,, Numer. Math., 94 (2003), 107. 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. 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. 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. doi: 10.1007/BF00913182.

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