# American Institute of Mathematical Sciences

September  2014, 13(5): 2005-2038. doi: 10.3934/cpaa.2014.13.2005

## The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction

 1 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405, United States 2 The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, 47205

Received  August 2013 Revised  January 2014 Published  June 2014

In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle $(0,1)_x \times (0,1)_y$. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the $y$-direction, we propose the boundary conditions for the $x$-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.
Citation: Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2005-2038. doi: 10.3934/cpaa.2014.13.2005
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Series in Pure and Applied Mathematics, 65 (1975).   Google Scholar [2] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations,, Oxford University Press, (2007).   Google Scholar [3] L. Comtet, Advanced Combinatorics,, D. Reidel, (1978).   Google Scholar [4] J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations,, North-Holland Publishing Co., (1982).   Google Scholar [5] Faà F.di Bruno, Note sur une nouvelle formule de calcul differentiel,, vol. 1, (1857).   Google Scholar [6] K. O. Friedrichs, The identity of weak and strong extensions of differential operator,, Trans. Amer. Math. Soc. \textbf{55} (1944), 55 (1944), 132.   Google Scholar [7] Loukas Grafakos, Classical Fourier Analysis,, Second ed., (2008).   Google Scholar [8] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).   Google Scholar [9] A. Huang, M. Petcu, and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, \emph{Annals of the University of Bucharest (Mathematical Series)}, 2 (LX) (2011), 63.   Google Scholar [10] A. Huang, M. Petcu, and R. Temam, The nonlinear 2d supercritical inviscid shallow water equations in a rectangle,, submitted., ().   Google Scholar [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: boundary conditions and well-posedness,, \emph{Archive for Rational Mechanics and Analysis}, 211 (2014), 1027.  doi: 10.1007/s00205-013-0702-0.  Google Scholar [12] A. Huang and R. Temam, The linear hyperbolic initial boundary value problems in a domain with corners,, accepted by \emph{Discrete and Continuous Dynamical System - Series B}, ().   Google Scholar [13] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, \emph{Comm. Pure Appl. Math}, 23 (1970), 277.   Google Scholar [14] J. L. Lions, Problèmes aux Limites dans les Équations aux Dérivées Partielles, Montréal,, Presses de l'Universit\'e de Montr\'eal, (1965).   Google Scholar [15] Ya. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type,, \emph{Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A}, 668 (1970), 592.   Google Scholar [16] Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I,, \emph{Trans. Amer. Math. Soc.}, 176 (1973), 141.   Google Scholar [17] Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. II,, \emph{Trans. Amer. Math. Soc.}, 198 (1974), 155.   Google Scholar [18] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions,, \emph{Math. Meth. Appl. Sci.}, (2011).   Google Scholar [19] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems,, \emph{Trans. Amer. Math. Soc.}, 189 (1974), 303.   Google Scholar [20] 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,, \emph{J. Math. Pures Appl.}, 89 (2008), 297.  doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [21] S. Smale, Smooth solutions of the heat and wave equations,, \emph{Comment. Math. Helv.}, 55 (1980), 1.  doi: 10.1007/BF02566671.  Google Scholar [22] J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation,, \emph{Adv. Diff. Equations}, 15 (2010), 1001.   Google Scholar [23] M. E. Taylor, Partial Differential Equations. III Nonlinear Equations,, vol. 117, (1997).   Google Scholar [24] R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, \emph{J. Differential Equations}, 43 (1982), 73.  doi: 10.1016/0022-0396(82)90075-4.  Google Scholar [25] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001).   Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces,, Series in Pure and Applied Mathematics, 65 (1975).   Google Scholar [2] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations,, Oxford University Press, (2007).   Google Scholar [3] L. Comtet, Advanced Combinatorics,, D. Reidel, (1978).   Google Scholar [4] J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations,, North-Holland Publishing Co., (1982).   Google Scholar [5] Faà F.di Bruno, Note sur une nouvelle formule de calcul differentiel,, vol. 1, (1857).   Google Scholar [6] K. O. Friedrichs, The identity of weak and strong extensions of differential operator,, Trans. Amer. Math. Soc. \textbf{55} (1944), 55 (1944), 132.   Google Scholar [7] Loukas Grafakos, Classical Fourier Analysis,, Second ed., (2008).   Google Scholar [8] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985).   Google Scholar [9] A. Huang, M. Petcu, and R. Temam, The one-dimensional supercritical shallow-water equations with topography,, \emph{Annals of the University of Bucharest (Mathematical Series)}, 2 (LX) (2011), 63.   Google Scholar [10] A. Huang, M. Petcu, and R. Temam, The nonlinear 2d supercritical inviscid shallow water equations in a rectangle,, submitted., ().   Google Scholar [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: boundary conditions and well-posedness,, \emph{Archive for Rational Mechanics and Analysis}, 211 (2014), 1027.  doi: 10.1007/s00205-013-0702-0.  Google Scholar [12] A. Huang and R. Temam, The linear hyperbolic initial boundary value problems in a domain with corners,, accepted by \emph{Discrete and Continuous Dynamical System - Series B}, ().   Google Scholar [13] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, \emph{Comm. Pure Appl. Math}, 23 (1970), 277.   Google Scholar [14] J. L. Lions, Problèmes aux Limites dans les Équations aux Dérivées Partielles, Montréal,, Presses de l'Universit\'e de Montr\'eal, (1965).   Google Scholar [15] Ya. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type,, \emph{Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A}, 668 (1970), 592.   Google Scholar [16] Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I,, \emph{Trans. Amer. Math. Soc.}, 176 (1973), 141.   Google Scholar [17] Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. II,, \emph{Trans. Amer. Math. Soc.}, 198 (1974), 155.   Google Scholar [18] M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions,, \emph{Math. Meth. Appl. Sci.}, (2011).   Google Scholar [19] J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems,, \emph{Trans. Amer. Math. Soc.}, 189 (1974), 303.   Google Scholar [20] 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,, \emph{J. Math. Pures Appl.}, 89 (2008), 297.  doi: 10.1016/j.matpur.2007.12.001.  Google Scholar [21] S. Smale, Smooth solutions of the heat and wave equations,, \emph{Comment. Math. Helv.}, 55 (1980), 1.  doi: 10.1007/BF02566671.  Google Scholar [22] J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation,, \emph{Adv. Diff. Equations}, 15 (2010), 1001.   Google Scholar [23] M. E. Taylor, Partial Differential Equations. III Nonlinear Equations,, vol. 117, (1997).   Google Scholar [24] R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, \emph{J. Differential Equations}, 43 (1982), 73.  doi: 10.1016/0022-0396(82)90075-4.  Google Scholar [25] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001).   Google Scholar
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