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

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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

Abstract
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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 and 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.
[2]	T. B. Benjamin, "Lectures on Nonlinear Wave Motion," Lectures in Applied Mathematics, Vol. 15, Amer. Math. Soc., Providence, R.I., 1974.
[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.
[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.
[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.
[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.
[7]	J. R. Cannon, "The One-dimensional Heat Equation," Encyclopedia of Mathematics and its Applications, 23, 1984.
[8]	N. Dunford and J. Schwartz, "Linear Operators. Part I," Pure and Applied Mathematics, Vol. VII, Interscience Publishers, New York, 1988.
[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.
[10]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, Englewood Cliffs, N.J. 1964.
[11]	P. D. Lax, "Shock Waves and Entropy," in Contributions to Non-Linear Functionnal Analysis, Academic Press, 1971.
[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.
[13]	W. Rudin, "Functionnal Analysis," McGraw-Hill, New York, 1973.
[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.
[15]	G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience, New-York-London-Sydney, 1974.

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.
[2]	T. B. Benjamin, "Lectures on Nonlinear Wave Motion," Lectures in Applied Mathematics, Vol. 15, Amer. Math. Soc., Providence, R.I., 1974.
[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.
[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.
[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.
[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.
[7]	J. R. Cannon, "The One-dimensional Heat Equation," Encyclopedia of Mathematics and its Applications, 23, 1984.
[8]	N. Dunford and J. Schwartz, "Linear Operators. Part I," Pure and Applied Mathematics, Vol. VII, Interscience Publishers, New York, 1988.
[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.
[10]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice Hall, Englewood Cliffs, N.J. 1964.
[11]	P. D. Lax, "Shock Waves and Entropy," in Contributions to Non-Linear Functionnal Analysis, Academic Press, 1971.
[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.
[13]	W. Rudin, "Functionnal Analysis," McGraw-Hill, New York, 1973.
[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.
[15]	G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience, New-York-London-Sydney, 1974.

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