February  2011, 4(1): 209-222. doi: 10.3934/dcdss.2011.4.209

The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity

1. 

Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil

2. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405

Received  June 2009 Revised  September 2009 Published  October 2010

In the present article we consider the nonviscous Shallow Water Equations in space dimension one with Dirichlet boundary conditions for the velocity and we show the locally in time well-posedness of the model.
Citation: Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
References:
[1]

Phys. D, 237 (2008), 1461-1465. doi: doi:10.1016/j.physd.2008.03.014.  Google Scholar

[2]

Nonlinearity, 14 (2001), 1493-1515. doi: doi:10.1088/0951-7715/14/6/305.  Google Scholar

[3]

Comm. Pure Appl. Math., 49 (1996), 599-638. doi: doi:10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar

[4]

in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), vol. 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.  Google Scholar

[5]

in "Frontiers of the Mathematical Sciences: 1985," (New York, 1985), Comm. Pure Appl. Math., 39 (1986), S187-S220.  Google Scholar

[6]

Springer-Verlag, New York, 1984.  Google Scholar

[7]

Cambridge texts in applied mathematics, Cambridge University Press, New York, 2002.  Google Scholar

[8]

Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: doi:10.1007/BF00375155.  Google Scholar

[9]

J. Fluid Mech., 591 (2007), 413-436. doi: doi:10.1017/S002211200700835X.  Google Scholar

[10]

Indiana University Mathematics Journal, 57 (2008), 2969-2998. doi: doi:10.1512/iumj.2008.57.3699.  Google Scholar

[11]

vol. 117 of Applied Mathematical Sciences, Springer-Verlag, New York, 1997, Nonlinear equations, Corrected reprint of the 1996 original.  Google Scholar

show all references

References:
[1]

Phys. D, 237 (2008), 1461-1465. doi: doi:10.1016/j.physd.2008.03.014.  Google Scholar

[2]

Nonlinearity, 14 (2001), 1493-1515. doi: doi:10.1088/0951-7715/14/6/305.  Google Scholar

[3]

Comm. Pure Appl. Math., 49 (1996), 599-638. doi: doi:10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar

[4]

in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), vol. 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.  Google Scholar

[5]

in "Frontiers of the Mathematical Sciences: 1985," (New York, 1985), Comm. Pure Appl. Math., 39 (1986), S187-S220.  Google Scholar

[6]

Springer-Verlag, New York, 1984.  Google Scholar

[7]

Cambridge texts in applied mathematics, Cambridge University Press, New York, 2002.  Google Scholar

[8]

Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: doi:10.1007/BF00375155.  Google Scholar

[9]

J. Fluid Mech., 591 (2007), 413-436. doi: doi:10.1017/S002211200700835X.  Google Scholar

[10]

Indiana University Mathematics Journal, 57 (2008), 2969-2998. doi: doi:10.1512/iumj.2008.57.3699.  Google Scholar

[11]

vol. 117 of Applied Mathematical Sciences, Springer-Verlag, New York, 1997, Nonlinear equations, Corrected reprint of the 1996 original.  Google Scholar

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