# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5327-5345. doi: 10.3934/dcds.2013.33.5327

## An interface problem: The two-layer shallow water equations

 1 Laboratoire de Mathématiques et applications, Univ. de Poitiers, Teleport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex, France 2 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405

Received  September 2011 Revised  April 2012 Published  May 2013

The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
Citation: Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
##### References:
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Anal., 82 (1983), 27-70. doi: 10.1007/BF00251724.  Google Scholar [11] B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. USA, 74 (1977), 1765-1766. doi: 10.1073/pnas.74.5.1765.  Google Scholar [12] B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math., 4 (1988), 21-45. doi: 10.1016/S0168-9274(88)80004-7.  Google Scholar [13] D. Givoli and B. Neta, High-order nonre ecting boundary conditions for the dispersive shallow water equations, J. Comput. Appl. Math., 158 (2003), 49-60. Selected Papers from the Conference on Computational and Math- Ematical Methods for Science and Engineering (Alicante, 2002). doi: 10.1016/S0377-0427(03)00462-X.  Google Scholar [14] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique. (French) [The characteristic quasilinear hyperbolic mixed problem], Comm. 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Souganidis, Weak stability of isentropic gas dynamics for $\gamma=5/3$, in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.  Google Scholar [20] 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: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar [21] P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.  Google Scholar [22] T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985.  Google Scholar [23] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2002.  Google Scholar [24] A. J. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504.  Google Scholar [25] A. McDonald, Transparent boundary conditions for the shallow water equa- tions: Testing in a nested environment, Mon. Wea. Rev., 131 (2003), 698-705. Google Scholar [26] J. Nycander and K. Döös, Open boundary conditions for barotropic waves, Journal of Geophysical Research, 108 (2003). Google Scholar [27] J. Nycander, A. McC. Hogg and L. M. Frankcombe, Open boundary conditions for nonlinear channel flow, Ocean Modelling, 24 (2008), 108-121. Google Scholar [28] I. M. Navon, B. Neta and M. Y. Hussaini, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review, 132 (2004), 1369-1378. doi: 10.1175/1520-0493(2004)132<1369:APMLAT>2.0.CO;2.  Google Scholar [29] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Applied Math., 35 (1978), 419-446. doi: 10.1137/0135035.  Google Scholar [30] P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.  Google Scholar [31] 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: 10.1017/S002211200700835X.  Google Scholar [32] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences, (2011). doi: 10.1002/mma.1482.  Google Scholar [33] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar [34] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861.  Google Scholar [35] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary Conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [36] A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domains, in computational methods for the atmosphere and the oceans, Special Volume of the Handbook of Numerical Analysis, XIV, Roger M. Temam and Joseph J. Tribbia, Guest Editors, P. G. Ciarlet Editor, Elsevier, Amsterdam, (2008). Google Scholar [37] R. Salmon, Numerical solution of the two-layer shallow water equation with bottom topography, Journal of Marine Research, 60 (2002), 605-638. doi: 10.1357/002224002762324194.  Google Scholar [38] M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116 (2011). doi: 10.1029/2010JC006315.  Google Scholar [39] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci. 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.  Google Scholar [40] B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, (New York). A Wiley-Interscience Publication. John Wiley & Sons Inc., New York 1999. doi: 10.1002/9781118032954.  Google Scholar [41] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.  Google Scholar [42] T. Yanagisawa, The initial boundary value problem for equations of ideal magneto-hydrodynamics, Hakkaido Math. Jour., 16 (1987), 295-314.  Google Scholar

show all references

##### References:
 [1] E. Audusse, A multilayer Saint-Venant system: Derivation and numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189-214. doi: 10.3934/dcdsb.2005.5.189.  Google Scholar [2] A. Bousquet, M. Petcu, M.-C. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited area models, Commun. Comput. Phys., 14 (2013), no. 3, 664-702. doi: 10.4208/cicp.070312.061112a.  Google Scholar [3] S. Benzoni-Gavage and D. Serre, "Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications," Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar [4] F. Bouchut and V. Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations, Discrete Cont. Dyn. Syst. Ser. B, 13 (2010), 739-758. doi: 10.3934/dcdsb.2010.13.739.  Google Scholar [5] G.-Q. Chen and P. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98. doi: 10.1007/s00205-002-0229-2.  Google Scholar [6] Q. Chen, M.-C. Shiue and R. Temam, The barotropic mode for the primitive equations, Special issue in memory of David Gottlieb, Journal of Scientific Computing, SpringerLink, 2009. doi: 10.1007/s10915-009-9343-8.  Google Scholar [7] Q. Chen, M.-C. Shiue, R. Temam and J. Tribbia, Numerical approximation of the inviscid 3D Primitive equations in a limited domain, Math. Mod. and Num. Anal., (M2AN), 46 (2012), no. 3, 619-646. doi: 10.1051/m2an/2011058.  Google Scholar [8] M. J. P. Cullen, Analysis of the semi-geostrophic shallow water equations, Phys. D, 237 (2008), 1461-1465. doi: 10.1016/j.physd.2008.03.014.  Google Scholar [9] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30. doi: 10.1007/BF01206047.  Google Scholar [10] R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), 27-70. doi: 10.1007/BF00251724.  Google Scholar [11] B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. USA, 74 (1977), 1765-1766. doi: 10.1073/pnas.74.5.1765.  Google Scholar [12] B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math., 4 (1988), 21-45. doi: 10.1016/S0168-9274(88)80004-7.  Google Scholar [13] D. Givoli and B. Neta, High-order nonre ecting boundary conditions for the dispersive shallow water equations, J. Comput. Appl. Math., 158 (2003), 49-60. Selected Papers from the Conference on Computational and Math- Ematical Methods for Science and Engineering (Alicante, 2002). doi: 10.1016/S0377-0427(03)00462-X.  Google Scholar [14] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique. (French) [The characteristic quasilinear hyperbolic mixed problem], Comm. Partial Differential Equations, 15 (1990), 595-645. doi: 10.1080/03605309908820701.  Google Scholar [15] R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47 (1986), 437-459. doi: 10.2307/2008166.  Google Scholar [16] R. L. Higdon, Numerical absorbing boundary conditions for the wave equation, Math. Comput., 49 (1987), 65-90. doi: 10.1090/S0025-5718-1987-0890254-1.  Google Scholar [17] A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography, Annals of the University of Bucharest, Mathematical Series 2 (LX), no. 1 (2011), 63-82.  Google Scholar [18] R. G. Keys, Absorbing boundary conditions for acoustic media, Geophysics, 50 (1985), 892-902. doi: 10.1190/1.1441969.  Google Scholar [19] 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), 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.  Google Scholar [20] 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: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar [21] P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.  Google Scholar [22] T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985.  Google Scholar [23] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2002.  Google Scholar [24] A. J. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504.  Google Scholar [25] A. McDonald, Transparent boundary conditions for the shallow water equa- tions: Testing in a nested environment, Mon. Wea. Rev., 131 (2003), 698-705. Google Scholar [26] J. Nycander and K. Döös, Open boundary conditions for barotropic waves, Journal of Geophysical Research, 108 (2003). Google Scholar [27] J. Nycander, A. McC. Hogg and L. M. Frankcombe, Open boundary conditions for nonlinear channel flow, Ocean Modelling, 24 (2008), 108-121. Google Scholar [28] I. M. Navon, B. Neta and M. Y. Hussaini, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review, 132 (2004), 1369-1378. doi: 10.1175/1520-0493(2004)132<1369:APMLAT>2.0.CO;2.  Google Scholar [29] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Applied Math., 35 (1978), 419-446. doi: 10.1137/0135035.  Google Scholar [30] P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.  Google Scholar [31] 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: 10.1017/S002211200700835X.  Google Scholar [32] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences, (2011). doi: 10.1002/mma.1482.  Google Scholar [33] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.  Google Scholar [34] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861.  Google Scholar [35] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary Conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [36] A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domains, in computational methods for the atmosphere and the oceans, Special Volume of the Handbook of Numerical Analysis, XIV, Roger M. Temam and Joseph J. Tribbia, Guest Editors, P. G. Ciarlet Editor, Elsevier, Amsterdam, (2008). Google Scholar [37] R. Salmon, Numerical solution of the two-layer shallow water equation with bottom topography, Journal of Marine Research, 60 (2002), 605-638. doi: 10.1357/002224002762324194.  Google Scholar [38] M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116 (2011). doi: 10.1029/2010JC006315.  Google Scholar [39] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci. 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.  Google Scholar [40] B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, (New York). A Wiley-Interscience Publication. John Wiley & Sons Inc., New York 1999. doi: 10.1002/9781118032954.  Google Scholar [41] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.  Google Scholar [42] T. Yanagisawa, The initial boundary value problem for equations of ideal magneto-hydrodynamics, Hakkaido Math. Jour., 16 (1987), 295-314.  Google Scholar
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