# American Institute of Mathematical Sciences

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] M. J. P. Cullen, Analysis of the semi-geostrophic shallow water equations, Phys. D, 237 (2008), 1461-1465. doi: doi:10.1016/j.physd.2008.03.014.  Google Scholar [2] C. D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515. doi: doi:10.1088/0951-7715/14/6/305.  Google Scholar [3] P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, 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] P.-L. Lions, B. Perthame and P. E. Souganidis, Weak stability of isentropic gas dynamics for $\gamma=5/3$, 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] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, in "Frontiers of the Mathematical Sciences: 1985," (New York, 1985), Comm. Pure Appl. Math., 39 (1986), S187-S220.  Google Scholar [6] A. J. Majda, "Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables," Springer-Verlag, New York, 1984.  Google Scholar [7] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge texts in applied mathematics, Cambridge University Press, New York, 2002.  Google Scholar [8] P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: doi:10.1007/BF00375155.  Google Scholar [9] D. Pritchard and L. Dickinson, The near-shore behaviour of shallow-water waves with localized initial conditions, J. Fluid Mech., 591 (2007), 413-436. doi: doi:10.1017/S002211200700835X.  Google Scholar [10] J. M. Rakotoson, R. Temam and J. Tribbia, Remarks on the nonviscous shallow water equations, Indiana University Mathematics Journal, 57 (2008), 2969-2998. doi: doi:10.1512/iumj.2008.57.3699.  Google Scholar [11] M. E. Taylor, Partial differential equations. III, 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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##### References:
 [1] M. J. P. Cullen, Analysis of the semi-geostrophic shallow water equations, Phys. D, 237 (2008), 1461-1465. doi: doi:10.1016/j.physd.2008.03.014.  Google Scholar [2] C. D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515. doi: doi:10.1088/0951-7715/14/6/305.  Google Scholar [3] P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, 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] P.-L. Lions, B. Perthame and P. E. Souganidis, Weak stability of isentropic gas dynamics for $\gamma=5/3$, 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] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, in "Frontiers of the Mathematical Sciences: 1985," (New York, 1985), Comm. Pure Appl. Math., 39 (1986), S187-S220.  Google Scholar [6] A. J. Majda, "Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables," Springer-Verlag, New York, 1984.  Google Scholar [7] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge texts in applied mathematics, Cambridge University Press, New York, 2002.  Google Scholar [8] P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: doi:10.1007/BF00375155.  Google Scholar [9] D. Pritchard and L. Dickinson, The near-shore behaviour of shallow-water waves with localized initial conditions, J. Fluid Mech., 591 (2007), 413-436. doi: doi:10.1017/S002211200700835X.  Google Scholar [10] J. M. Rakotoson, R. Temam and J. Tribbia, Remarks on the nonviscous shallow water equations, Indiana University Mathematics Journal, 57 (2008), 2969-2998. doi: doi:10.1512/iumj.2008.57.3699.  Google Scholar [11] M. E. Taylor, Partial differential equations. III, 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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