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.
Accepté le :
Publié le :
Dragoş Iftimie 1, 2
@article{CRMATH_2002__334_1_83_0,
author = {Drago\c{s} Iftimie},
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},
}
Dragoş Iftimie. Remarques sur la limite $ \mathbf{\alpha \to 0}$ pour les fluides de grade $ \mathrm{2}$. Comptes Rendus. Mathématique, Volume 334 (2002) no. 1, pp. 83-86. doi : 10.1016/S1631-073X(02)02187-8. https://comptes-rendus.academie-sciences.fr/mathematique/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
- Strong solutions for a stochastic model of two-dimensional second grade fluids driven by Lévy noise, Journal of Mathematical Analysis and Applications, Volume 471 (2019) no. 1-2, p. 126 | DOI:10.1016/j.jmaa.2018.10.068
- Random dynamics of two-dimensional stochastic second grade fluids, Stochastic Analysis and Applications, Volume 37 (2019) no. 5, p. 749 | DOI:10.1080/07362994.2019.1611448
- Viscosity limit and deviations principles for a grade-two fluid driven by multiplicative noise, Annali di Matematica Pura ed Applicata (1923 -), Volume 197 (2018) no. 5, p. 1547 | DOI:10.1007/s10231-018-0737-9
- Moderate deviations for stochastic models of two-dimensional second grade fluids, Stochastics and Dynamics, Volume 18 (2018) no. 03, p. 1850026 | DOI:10.1142/s0219493718500260
- Large Deviations for Stochastic Models of Two-Dimensional Second Grade Fluids, Applied Mathematics Optimization, Volume 75 (2017) no. 3, p. 471 | DOI:10.1007/s00245-016-9338-4
- Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity, Journal of Differential Equations, Volume 263 (2017) no. 5, p. 3027 | DOI:10.1016/j.jde.2017.04.022
- On the convergence of the two-dimensional second grade fluid model to the Navier–Stokes equation, Journal of Differential Equations, Volume 260 (2016) no. 3, p. 2557 | DOI:10.1016/j.jde.2015.10.019
- Exponential mixing for stochastic model of two-dimensional second grade fluids, Nonlinear Analysis, Volume 132 (2016), p. 196 | DOI:10.1016/j.na.2015.11.009
- Convergence of the 2D Euler-α to Euler equations in the Dirichlet case: Indifference to boundary layers, Physica D: Nonlinear Phenomena, Volume 292-293 (2015), p. 51 | DOI:10.1016/j.physd.2014.11.001
- Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type, Potential Analysis, Volume 38 (2013) no. 4, p. 1291 | DOI:10.1007/s11118-012-9316-7
- Dynamics of Second Grade Fluids: The Lagrangian Approach, Recent Trends in Dynamical Systems, Volume 35 (2013), p. 517 | DOI:10.1007/978-3-0348-0451-6_20
- Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations, Journal of Differential Equations, Volume 252 (2012) no. 6, p. 3695 | DOI:10.1016/j.jde.2011.10.015
- Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior, Nonlinear Analysis: Theory, Methods Applications, Volume 75 (2012) no. 11, p. 4251 | DOI:10.1016/j.na.2012.03.014
- Analyticity and Gevrey-Class Regularity for the Second-Grade Fluid Equations, Journal of Mathematical Fluid Mechanics, Volume 13 (2011) no. 4, p. 533 | DOI:10.1007/s00021-010-0032-z
- Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids, Boundary Value Problems, Volume 2010 (2010) no. 1, p. 636140 | DOI:10.1155/2010/636140
- On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations, Journal of Statistical Physics, Volume 138 (2010) no. 1-3, p. 305 | DOI:10.1007/s10955-009-9916-9
- The second grade fluid and averaged Euler equations with Navier-slip boundary conditions, Nonlinearity, Volume 16 (2003) no. 3, p. 1119 | DOI:10.1088/0951-7715/16/3/318
Cité par 17 documents. Sources : Crossref
Commentaires - Politique