On considère la limite α→0 dans l'équation des fluides de grade 2. On montre la convergence faible des solutions vers une solution faible de l'équation de Navier–Stokes, en supposant que les données initiales convergent faiblement dans L2.
We consider the limit α→0 for the equation of the second grade fluids. We prove that weak convergence of the solutions to a weak solution of the Navier–Stokes equation holds under the assumption that the initial data weakly converges in L2.
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@article{CRMATH_2002__334_1_83_0, author = {Iftimie, Drago\c{s}}, title = {Remarques sur la limite $ \mathbf{\alpha \to 0}$ pour les fluides de~grade $ \mathrm{2}$}, journal = {Comptes Rendus. Math\'ematique}, pages = {83--86}, publisher = {Elsevier}, volume = {334}, number = {1}, year = {2002}, doi = {10.1016/S1631-073X(02)02187-8}, language = {fr}, url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02187-8/} }
TY - JOUR AU - Iftimie, Dragoş TI - Remarques sur la limite $ \mathbf{\alpha \to 0}$ pour les fluides de grade $ \mathrm{2}$ JO - Comptes Rendus. Mathématique PY - 2002 SP - 83 EP - 86 VL - 334 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02187-8/ DO - 10.1016/S1631-073X(02)02187-8 LA - fr ID - CRMATH_2002__334_1_83_0 ER -
%0 Journal Article %A Iftimie, Dragoş %T Remarques sur la limite $ \mathbf{\alpha \to 0}$ pour les fluides de grade $ \mathrm{2}$ %J Comptes Rendus. Mathématique %D 2002 %P 83-86 %V 334 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02187-8/ %R 10.1016/S1631-073X(02)02187-8 %G fr %F CRMATH_2002__334_1_83_0
Iftimie, Dragoş. Remarques sur la limite $ \mathbf{\alpha \to 0}$ pour les fluides de grade $ \mathrm{2}$. Comptes Rendus. Mathématique, Tome 334 (2002) no. 1, pp. 83-86. doi : 10.1016/S1631-073X(02)02187-8. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02187-8/
[1] Busuioc V., On second grade fluids with vanishing viscosity, Portugal. Math. (à paraı̂tre)
[2] Chemin J.-Y., Méthodes mathématiques en mécanique des fluides, I, Cours de DEA et Preprint Laboratoire d'Analyse Numérique A97004, 1997
[3] Weak and classical solutions of a family of second grade fluids, Internat. J. Non-Linear Mech., Volume 32 (1997) no. 2, pp. 317-335
[4] Existence and uniqueness for fluids of second grade, Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, Vol. VI (Paris, 1982/1983), Boston, MA, Pitman, 1984, pp. 178-197
[5] Navier–Stokes Equations, University of Chicago Press, Chicago, 1988
[6] Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal., Volume 56 (1974), pp. 191-252
[7] Existence and uniqueness of classical solutions of the equations of motion for second-grade fluids, Arch. Rational Mech. Anal., Volume 124 (1993) no. 3, pp. 221-237
[8] Further existence results for classical solutions of the equations of a second-grade fluid, Arch. Rational Mech. Anal., Volume 128 (1994) no. 4, pp. 297-312
[9] Iftimie D., Remarques sur la limite α→0 pour les fluides de grade 2, Nonlinear Partial Differential Equations and Their Applications, Séminaire du Collège de France (à paraı̂tre)
[10] Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934), pp. 193-248
[11] Marsden J.E., Ratiu T.S., Shkoller S., A nonlinear analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal. (à paraı̂tre)
[12] The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal., Volume 10 (2000) no. 3, pp. 582-599
[13] The vortex blob method as a second-grade non-Newtonian fluid, Comm. Partial Differential Equations, Volume 26 (2001) no. 1–2, pp. 295-314
[14] Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., Volume 4 (1955), pp. 323-425
[15] Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations, Appl. Math. Lett., Volume 14 (2001) no. 5, pp. 539-543
[16] Partial Differential Equations. III, Springer-Verlag, New York, 1997
[17] Navier–Stokes Equations, North-Holland, Amsterdam, 1984
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