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A global existence result for the compressible Navier–Stokes equations in the critical \(L ^{p }\) framework. (English) Zbl 1229.35167
The paper investigates the global well-posedness problem for the barotropic compressible Navier-Stokes system in the whole space \({\mathbb{R}}^d\), \(d\geq 2\). Specifically, the case of a critical framework is considered. This critical framework is not related to the energy space. Global existence is obtained for small perturbations of a stable equilibrium state in the sense of suitable \(L^p\)-type Besov norms. Thus one may exhibit a large highly oscillating initial velocity fields for which global well-posedness holds.

35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Full Text: DOI
[1] Bahouri, H., Chemin J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (to appear) · Zbl 1227.35004
[2] Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14(4), 209–246 (1981)
[3] Bresch D., Desjardins B.: On the existence of global weak solutions to the Navier– Stokes equations for viscous compressible and heat conducting fluids. Journal de Mathématiques Pures et Appliquées 87(1), 57–90 (2007) · Zbl 1122.35092 · doi:10.1016/j.matpur.2006.11.001
[4] Cannone M.: A generalization of a theorem by Kato on Navier–Stokes equations. Revista Matemática Iberoamericana 13(3), 515–541 (1997) · Zbl 0897.35061 · doi:10.4171/RMI/229
[5] Chemin J.-Y.: Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel. Journal d’Analyse Mathématique 77, 25–50 (1999) · Zbl 0938.35125 · doi:10.1007/BF02791256
[6] Chemin, J.-Y., Gallagher, I., Paicu, M.: Global regularity for some classes of large solutions to the Navier–Stokes equations. Ann. Math. (to appear) · Zbl 1229.35168
[7] Danchin R.: Global existence in critical spaces for compressible Navier–Stokes equations. Inventiones Mathematicae 141(3), 579–614 (2000) · Zbl 0958.35100 · doi:10.1007/s002220000078
[8] Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26, 1183–1233 (2001) and 27, 2531–2532 (2002) · Zbl 1007.35071
[9] Danchin R.: On the uniqueness in critical spaces for compressible Navier–Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 12(1), 111–128 (2005) · Zbl 1125.76061 · doi:10.1007/s00030-004-2032-2
[10] Danchin R.: Uniform estimates for transport-diffusion equations. J. Hyperbolic Differ. Equ. 4(1), 1–17 (2007) · Zbl 1117.35012 · doi:10.1142/S021989160700101X
[11] Danchin R.: Well-posedness in critical spaces for barotropic viscous fluids with truly nonconstant density. Commun. Partial Differ. Equ. 32, 1373–1397 (2007) · Zbl 1120.76052 · doi:10.1080/03605300600910399
[12] Danchin R., Desjardins B.: Existence of solutions for compressible fluid models of Korteweg type. Annales de l’IHP, Analyse Non Linéaire 18, 97–133 (2001) · Zbl 1010.76075
[13] Fujita H., Kato T.: On the Navier–Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964) · Zbl 0126.42301 · doi:10.1007/BF00276188
[14] Germain, P.: Weak-strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. (in press) · Zbl 1270.35342
[15] Giga Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986) · Zbl 0577.35058 · doi:10.1016/0022-0396(86)90096-3
[16] Kato T.: Strong L p -solutions of the Navier–Stokes equation in $${\(\backslash\)mathbb{R}\^{m} }$$ , with applications to weak solutions. Math. Z. 187(4), 471–480 (1984) · Zbl 0545.35073
[17] Haspot, B.: Well-posedness in critical spaces for barotropic viscous fluids, preprint · Zbl 1238.35100
[18] Hmidi T.: Régularité höldérienne des poches de tourbillon visqueuses. Journal de Mathématiques Pures et Appliquées 84(11), 1455–1495 (2005) · Zbl 1095.35024 · doi:10.1016/j.matpur.2005.01.004
[19] Kobayashi T., Shibata Y.: Remark on the rate of decay of solutions to linearized compressible Navier–Stokes equations. Pac. J. Math. 207(1), 199–234 (2002) · Zbl 1060.35104 · doi:10.2140/pjm.2002.207.199
[20] Lions P.-L.: Mathematical Topics in Fluid Dynamics: Compressible Models, vol 2. Oxford University Press, Oxford (1998)
[21] Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980) · Zbl 0429.76040
[22] Mucha P.B.: The Cauchy problem for the compressible Navier–Stokes equations in the L p -framework. Nonlinear Anal. 52(4), 1379–1392 (2003) · Zbl 1048.35065 · doi:10.1016/S0362-546X(02)00270-5
[23] Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bulletin de la Société Mathématique de France 90, 487–497 (1962)
[24] Planchon F.: Asymptotic behavior of global solutions to the Navier–Stokes equations in $${\(\backslash\)mathbb{R}\^3}$$ . Revista Matemática Iberoamericana 14(1), 71–93 (1998) · Zbl 0910.35096 · doi:10.4171/RMI/235
[25] Vishik M.: Hydrodynamics in Besov spaces. Arch. Ration. Mech. Anal. 145, 197–214 (1998) · Zbl 0926.35123 · doi:10.1007/s002050050128
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