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Totally dissipative dynamical processes and their uniform global attractors
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 |
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. |
show all references
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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