On considère un modèle de Saint-Venant avec viscosité et terme de friction en dimension deux, pour lequel on obtient un résultat d'existence globale de solutions faibles. On montre également la convergence de ces solutions vers la solution forte globale des équations quasi-géostrophiques visqueuses avec terme de surface libre pour des données bien préparées.
We consider a two dimensional viscous shallow water model with friction term. The existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term is proven in the well prepared case.
Accepté le :
Publié le :
Didier Bresch 1 ; Benoı̂t Desjardins 2
@article{CRMATH_2002__335_12_1079_0, author = {Didier Bresch and Beno{\i}̂t Desjardins}, title = {Sur un mod\`ele de {Saint-Venant} visqueux et sa limite quasi-g\'eostrophique}, journal = {Comptes Rendus. Math\'ematique}, pages = {1079--1084}, publisher = {Elsevier}, volume = {335}, number = {12}, year = {2002}, doi = {10.1016/S1631-073X(02)02610-9}, language = {fr}, }
TY - JOUR AU - Didier Bresch AU - Benoı̂t Desjardins TI - Sur un modèle de Saint-Venant visqueux et sa limite quasi-géostrophique JO - Comptes Rendus. Mathématique PY - 2002 SP - 1079 EP - 1084 VL - 335 IS - 12 PB - Elsevier DO - 10.1016/S1631-073X(02)02610-9 LA - fr ID - CRMATH_2002__335_12_1079_0 ER -
Didier Bresch; Benoı̂t Desjardins. Sur un modèle de Saint-Venant visqueux et sa limite quasi-géostrophique. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1079-1084. doi : 10.1016/S1631-073X(02)02610-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02610-9/
[1] Global splitting in rotating shallow-water equations, European J. Mech. B Fluids, Volume 16 (1997), pp. 725-754
[2] D. Bresch, B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys. (2002), soumis
[3] D. Bresch, B. Desjardins, C.K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations (2002), à paraı̂tre
[4] Rotating fluid at high Rossby number driven by a surface stress: existence and convergence, Adv. Differential Equations, Volume 2 (1997) no. 5, pp. 715-751
[5] B. Desjardins, J.-Y. Chemin, I. Gallagher, E. Grenier (2001), livre en préparation
[6] Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geosphys. Astrophys. Fluid Dynamics, Volume 87 (1998), pp. 1-50
[7] Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations, Volume 21 (1996), pp. 619-658
[8] The energetically consistent shallow water equations, J. Atmos. Sci, Volume 50 (1993), pp. 1323-1325
[9] Derivation of viscous Saint-Venant system for laminar shallow water; Numerical results, Discrete Continuous Dynamical Systems Series B, Volume 1 (2001) no. 1, pp. 89-102
[10] Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, Volume 22 (1997) no. 5–6, pp. 953-975
[11] Mathematical Topics in Fluid Dynamics, Vol. 2. Compressible Models, Oxford University Press, 1998
[12] On the equations of the large scale Ocean, Nonlinearity, Volume 5 (1992), pp. 1007-1053
[13] Ekman layers of rotating fluids: the case of general initial data, Comm. Pure Appl. Math, Volume 53 (2000) no. 4, pp. 432-483
[14] Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal, Volume 130 (1995) no. 2, pp. 183-204
[15] Geophysical Fluid Dynamics, Springer-Verlag, 1987
[16] Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation, J. Differential Equations, Volume 68 (1987) no. 3, pp. 400-428
[17] Global existence for the Dirichlet problem for the viscous shallow water equations, J. Math. Anal. Appl, Volume 202 (1996) no. 1, pp. 236-258
Cité par Sources :
Commentaires - Politique