On généralise aux fluides incompressibles à densité variable un certain nombre de résultats bien connus pour les équations de Navier-Stokes et d’Euler incompressibles.
@article{SEDP_2002-2003____A11_0, author = {Danchin, Rapha\"el}, title = {Fluides incompressibles \`a densit\'e variable}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:11}, pages = {1--16}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2002-2003}, zbl = {1063.35132}, mrnumber = {2030706}, language = {fr}, url = {http://archive.numdam.org/item/SEDP_2002-2003____A11_0/} }
TY - JOUR AU - Danchin, Raphaël TI - Fluides incompressibles à densité variable JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:11 PY - 2002-2003 SP - 1 EP - 16 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2002-2003____A11_0/ LA - fr ID - SEDP_2002-2003____A11_0 ER -
%0 Journal Article %A Danchin, Raphaël %T Fluides incompressibles à densité variable %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:11 %D 2002-2003 %P 1-16 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2002-2003____A11_0/ %G fr %F SEDP_2002-2003____A11_0
Danchin, Raphaël. Fluides incompressibles à densité variable. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2002-2003), Exposé no. 11, 16 p. http://archive.numdam.org/item/SEDP_2002-2003____A11_0/
[1] S. Antontsev, A. Kazhikhov and V. Monakhov : Boundary value problems in mechanics of nonhomogeneous fluids. Translated from the Russian. Studies in Mathematics and its Applications, 22. North-Holland Publishing Co., Amsterdam, 1990. | MR | Zbl
[2] J. Beale, T. Kato and A. Majda : Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communications in Mathematical Physics, 94, pages 61–66 (1984). | MR | Zbl
[3] J. Beale and A. Majda : Rates of convergence for viscous splitting of the Navier-Stokes equations, Mathematics of Computation, 37, 243-259, (1981). | MR | Zbl
[4] J.-M. Bony : Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales Scientifiques de l’école Normale Supérieure, 14, pages 209-246 (1981). | Numdam | Zbl
[5] J.-Y. Chemin : Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, Journal d’Analyse Mathématique, 77, pages 25–50 (1999). | Zbl
[6] P. Constantin and C. Foias : Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press (1988). | MR | Zbl
[7] R. Danchin : A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14, pages 953–988 (2001). | MR | Zbl
[8] R. Danchin : Density-dependent incompressible fluids in critical spaces, to appear in the Proceedings of the Royal Society of Edinburgh. | MR
[9] R. Danchin : Local and global well-posedness results for flows of inhomogeneous viscous fluids, preprint. | MR
[10] R. Danchin : Local theory in critical spaces for compressible viscous and heat-conductive gases, Communications in Partial Differential Equation, 26, pages 1183–1233 (2001). | MR | Zbl
[11] R. Danchin : The inviscid limit for density dependent incompressible fluids, preprint. | Numdam | MR
[12] B. Desjardins : Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space, Diff. and Integral Equations, 10, pages 587–598 (1997). | MR | Zbl
[13] H. Fujita and T. Kato : On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16, pages 269-315 (1964). | MR | Zbl
[14] S. Itoh : On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incompressible fluid, Glasgow Journal of Mathematics, 36, pages 123–129 (1994). | MR | Zbl
[15] S. Itoh and A. Tani : Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo Journal of Mathematics, 22, pages 17–42 (1999). | MR | Zbl
[16] T. Kato and G. Ponce : Commutator estimates and the Euler and Navier-Stokes equations, Communications on Pure and Applied Mathematics, 41, pages 891–907 (1988). | MR | Zbl
[17] O. Ladyzhenskaya and V. Solonnikov : The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9, pages 697–749 (1978). | Zbl
[18] P.-L. Lions : Mathematical Topics in Fluid Dynamics, Vol. Incompressible Models, Oxford University Press (1996). | MR | Zbl
[19] J. Marsden : Well-posedness of the equations of a non-homogeneous perfect fluid, Communications in Partial Differential Equations, 1, pages 215–230 (1976). | MR | Zbl
[20] H. Okamoto : On the equation of nonstationary stratified fluid motion : uniqueness and existence of the solutions, J. Fac. Sci. of the Univ. of Tokyo, 30, pages 615–643 (1984). | MR | Zbl
[21] M. Vishik : Hydrodynamics in Besov spaces, Archiv for Rational Mechanics and Analysis, 145, pages 197–214 (1998). | MR | Zbl