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

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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

Aimin Huang ^1, and Roger Temam ^2,

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

Abstract
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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.

Keywords: existence and uniqueness., hyperbolic equations, Shallow water equations.

Mathematics Subject Classification: Primary: 76B03, 76B15; Secondary: 58J4.

Citation: Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure and 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, vol. 65, Academic Press, New York-London, 1975.
[2]	S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007.
[3]	L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1978.
[4]	J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam, 1982, Translated from French.
[5]	Faà F.di Bruno, Note sur une nouvelle formule de calcul differentiel, vol. 1, London: John W. Parker and Son, West Strand, 1857.
[6]	K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc. 55 (1944), 132-151.
[7]	Loukas Grafakos, Classical Fourier Analysis, Second ed., Graduate Texts in Mathematics, vol. 249, Springer, 2008.
[8]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985.
[9]	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) (2011), 63-82.
[10]	A. Huang, M. Petcu, and R. Temam, The nonlinear 2d supercritical inviscid shallow water equations in a rectangle, submitted.
[11]	A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063 (English). doi: 10.1007/s00205-013-0702-0.
[12]	A. Huang and R. Temam, The linear hyperbolic initial boundary value problems in a domain with corners, accepted by Discrete and Continuous Dynamical System - Series B, see also arXiv:1310.5757.
[13]	H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math, 23 (1970), 277-298.
[14]	J. L. Lions, Problèmes aux Limites dans les Équations aux Dérivées Partielles, Montréal, Presses de l'Université de Montréal, 1965.
[15]	Ya. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594.
[16]	Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-165.
[17]	Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175.
[18]	M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Math. Meth. Appl. Sci., (2011).
[19]	J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318.
[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, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.
[21]	S. Smale, Smooth solutions of the heat and wave equations, Comment. Math. Helv., 55 (1980), 1-12. doi: 10.1007/BF02566671.
[22]	J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.
[23]	M. E. Taylor, Partial Differential Equations. III Nonlinear Equations, vol. 117, Applied Mathematical Sciences (Springer-Verlag), 1997.
[24]	R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations, J. Differential Equations, 43 (1982), 73-92. doi: 10.1016/0022-0396(82)90075-4.
[25]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition.

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References:

[1]	R. A. Adams, Sobolev Spaces, Series in Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975.
[2]	S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007.
[3]	L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, 1978.
[4]	J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam, 1982, Translated from French.
[5]	Faà F.di Bruno, Note sur une nouvelle formule de calcul differentiel, vol. 1, London: John W. Parker and Son, West Strand, 1857.
[6]	K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc. 55 (1944), 132-151.
[7]	Loukas Grafakos, Classical Fourier Analysis, Second ed., Graduate Texts in Mathematics, vol. 249, Springer, 2008.
[8]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985.
[9]	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) (2011), 63-82.
[10]	A. Huang, M. Petcu, and R. Temam, The nonlinear 2d supercritical inviscid shallow water equations in a rectangle, submitted.
[11]	A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063 (English). doi: 10.1007/s00205-013-0702-0.
[12]	A. Huang and R. Temam, The linear hyperbolic initial boundary value problems in a domain with corners, accepted by Discrete and Continuous Dynamical System - Series B, see also arXiv:1310.5757.
[13]	H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math, 23 (1970), 277-298.
[14]	J. L. Lions, Problèmes aux Limites dans les Équations aux Dérivées Partielles, Montréal, Presses de l'Université de Montréal, 1965.
[15]	Ya. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594.
[16]	Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-165.
[17]	Stanley Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175.
[18]	M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Math. Meth. Appl. Sci., (2011).
[19]	J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318.
[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, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.
[21]	S. Smale, Smooth solutions of the heat and wave equations, Comment. Math. Helv., 55 (1980), 1-12. doi: 10.1007/BF02566671.
[22]	J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.
[23]	M. E. Taylor, Partial Differential Equations. III Nonlinear Equations, vol. 117, Applied Mathematical Sciences (Springer-Verlag), 1997.
[24]	R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations, J. Differential Equations, 43 (1982), 73-92. doi: 10.1016/0022-0396(82)90075-4.
[25]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition.

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