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January  2011, 29(1): 25-49. doi: 10.3934/dcds.2011.29.25

Existence of solutions for a Boussinesq system on the half line and on a finite interval

Karine Adamy 1,

1. 

Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, CNRS UMR 8628, 91405 Orsay, France

Received  January 2009 Revised  February 2010 Published  September 2010

The initial boundary-value problem for a Boussinesq system is studied on the half line and on a finite interval. Global existence of weak solutions satisfying the boundary conditions is proven and uniqueness for solutions in a suitable class is studied. A proof of the persistence of finite regularity for solutions in the whole space is also presented.
Keywords: Boussinesq systems, boundary value problems, hyperbolic systems, regularity of solutions..
Mathematics Subject Classification: 35A01, 35F61, 35Q35, 35B6.
Citation: Karine Adamy. Existence of solutions for a Boussinesq system on the half line and on a finite interval. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 25-49. doi: 10.3934/dcds.2011.29.25
References:
[1]

C. J. Amick, Regularity and uniqueness of solutions to the boussinesq system of equations, J. Diff. Eq., 54 (1984), 231-247.  Google Scholar

[2]

T. B. Benjamin, "Lectures on Nonlinear Wave Motion," Lectures in Applied Mathematics, Vol. 15, Amer. Math. Soc., Providence, R.I., 1974.  Google Scholar

[3]

J. Bona and R. Smith, A model for the two-way propagation of water waves in a channel, Math. Proc. Cambridge Philos., Soc., 79 (1976), 167-182. doi: doi:10.1017/S030500410005218X.  Google Scholar

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J. Bona and V. Dougalis, An initial and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal Appl., 75 (1980), 503-522. doi: doi:10.1016/0022-247X(80)90098-0.  Google Scholar

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J. Bona, M. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: doi:10.1007/s00332-002-0466-4.  Google Scholar

[6]

J. Bona, M. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media II: The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: doi:10.1088/0951-7715/17/3/010.  Google Scholar

[7]

J. R. Cannon, "The One-dimensional Heat Equation," Encyclopedia of Mathematics and its Applications, 23, 1984. Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators. Part I," Pure and Applied Mathematics, Vol. VII, Interscience Publishers, New York, 1988.  Google Scholar

[9]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys., Anal. Geom., 8 (2005), 59-96. doi: doi:10.1007/s11040-004-1650-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, Englewood Cliffs, N.J. 1964.  Google Scholar

[11]

P. D. Lax, "Shock Waves and Entropy," in Contributions to Non-Linear Functionnal Analysis, Academic Press, 1971.  Google Scholar

[12]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker Inc., New York, 1991.  Google Scholar

[13]

W. Rudin, "Functionnal Analysis," McGraw-Hill, New York, 1973.  Google Scholar

[14]

M. E. Schonbek, Existence of solutions for the Boussinesq system of equations, J. Differential Equations, 42 (1981), 325-352. doi: doi:10.1016/0022-0396(81)90108-X.  Google Scholar

[15]

G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience, New-York-London-Sydney, 1974.  Google Scholar

show all references

References:
[1]

C. J. Amick, Regularity and uniqueness of solutions to the boussinesq system of equations, J. Diff. Eq., 54 (1984), 231-247.  Google Scholar

[2]

T. B. Benjamin, "Lectures on Nonlinear Wave Motion," Lectures in Applied Mathematics, Vol. 15, Amer. Math. Soc., Providence, R.I., 1974.  Google Scholar

[3]

J. Bona and R. Smith, A model for the two-way propagation of water waves in a channel, Math. Proc. Cambridge Philos., Soc., 79 (1976), 167-182. doi: doi:10.1017/S030500410005218X.  Google Scholar

[4]

J. Bona and V. Dougalis, An initial and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal Appl., 75 (1980), 503-522. doi: doi:10.1016/0022-247X(80)90098-0.  Google Scholar

[5]

J. Bona, M. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: doi:10.1007/s00332-002-0466-4.  Google Scholar

[6]

J. Bona, M. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media II: The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: doi:10.1088/0951-7715/17/3/010.  Google Scholar

[7]

J. R. Cannon, "The One-dimensional Heat Equation," Encyclopedia of Mathematics and its Applications, 23, 1984. Google Scholar

[8]

N. Dunford and J. Schwartz, "Linear Operators. Part I," Pure and Applied Mathematics, Vol. VII, Interscience Publishers, New York, 1988.  Google Scholar

[9]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys., Anal. Geom., 8 (2005), 59-96. doi: doi:10.1007/s11040-004-1650-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, Englewood Cliffs, N.J. 1964.  Google Scholar

[11]

P. D. Lax, "Shock Waves and Entropy," in Contributions to Non-Linear Functionnal Analysis, Academic Press, 1971.  Google Scholar

[12]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker Inc., New York, 1991.  Google Scholar

[13]

W. Rudin, "Functionnal Analysis," McGraw-Hill, New York, 1973.  Google Scholar

[14]

M. E. Schonbek, Existence of solutions for the Boussinesq system of equations, J. Differential Equations, 42 (1981), 325-352. doi: doi:10.1016/0022-0396(81)90108-X.  Google Scholar

[15]

G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience, New-York-London-Sydney, 1974.  Google Scholar

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