zbMATH — the first resource for mathematics

Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity. (English) Zbl 1202.35002
Global well-posedness for compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium are proved. This result allows authors to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Full Text: DOI arXiv
[1] Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoam. 23 pp 537– (2007) · Zbl 1175.35099 · doi:10.4171/RMI/505
[2] Bony, Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 pp 209– (1981)
[3] Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana 13 pp 515– (1997) · Zbl 0897.35061 · doi:10.4171/RMI/229
[4] Cannone, Séminaire sur les Équations aux Dérivées Partielles, 1993-1994, Exp. No. VIII, 12 pp. École Polytechnique (1994)
[5] Cannone, On the inviscid limit of the two-dimensional Navier-Stokes equations with fractional diffusion, Adv. Math. Sci. Appl. 18 pp 607– (2008) · Zbl 1387.35446
[6] Chemin, Perfect incompressible fluids (1998)
[7] Chemin, Phase space analysis of partial differential equations I. pp 53– (2004)
[8] Chemin, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data, Ann. Sci. École Norm. Sup. (4) 39 pp 679– (2006) · Zbl 1124.35052
[9] Chemin, J.-Y.; Gallagher, I.; Paicu, M. Global regularity for some classes of large solutions to the Navier-Stokes equations. arXiv:0807.1265, July 2008. · Zbl 1229.35168
[10] Chemin, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys. 272 pp 529– (2007) · Zbl 1132.35068
[11] Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math. 141 pp 579– (2000)
[12] Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations 26 pp 1183– (2001) · Zbl 1007.35071
[13] Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A 133 pp 1311– (2003) · Zbl 1050.76013
[14] Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl. 12 pp 111– (2005)
[15] Danchin, Uniform estimates for transport-diffusion equations, J. Hyperbolic Differ. Equ. 4 pp 1– (2007) · Zbl 1117.35012
[16] Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations 32 pp 1373– (2007) · Zbl 1120.76052
[17] Fujita, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 pp 269– (1964)
[18] Hmidi, Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces, Adv. Math. 214 pp 618– (2007) · Zbl 1119.76070
[19] Hmidi, Incompressible viscous flows in borderline Besov space, Arch. Ration. Mech. Anal. 189 pp 283– (2008) · Zbl 1147.76014
[20] Hoff, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 pp 603– (1995) · Zbl 0842.35076
[21] Kato, Strong Lp -solutions of the Navier-Stokes equation in \(\mathbb{R}\)m with applications to weak solutions, Math. Z. 187 pp 471– (1984) · Zbl 0545.35073
[22] Lions, Mathematical topics in fluid mechanics 2 (1998)
[23] Matsumura, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 pp 337– (1979) · Zbl 0447.76053
[24] Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France 90 pp 487– (1962)
[25] Paicu, M.; Zhang, Z. Global regularity for the Navier-Stokes equations with large, slowly varying initial data in the vertical direction. arXiv:0903.5194, March 2009.
[26] Weissler, The Navier-Stokes initial value problem in Lp, Arch. Rational Mech. Anal. 74 pp 219– (1980) · Zbl 0454.35072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.