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Existence of global strong solutions in critical spaces for barotropic viscous fluids. (English) Zbl 1427.76230
Summary: This paper is dedicated to the study of viscous compressible barotropic fluids in dimension \(N \geqq 2\). We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of F. Charve and R. Danchin [Arch. Ration. Mech. Anal. 198, No. 1, 233–271 (2010; Zbl 1229.35167)] and Q. Chen et al. [Commun. Pure Appl. Math. 63, No. 9, 1173–1224 (2010; Zbl 1202.35002)] with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in [B. Haspot, “Well-posedness in critical spaces for barotropic viscous fluids”, Preprint, arXiv:0903.0533]. Second, we improve the results of Charve and Danchin [loc. cit.] and Chen et al. [loc. cit.] by adding as in [Charve and Danchin, loc. cit.] some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of D. Hoff [Arch. Ration. Mech. Anal. 139, No. 4, 303–354 (1997; Zbl 0904.76074); Commun. Pure Appl. Math. 55, No. 11, 1365–1407 (2002; Zbl 1020.76046); J. Math. Fluid Mech. 7, No. 3, 315–338 (2005; Zbl 1095.35025)] and those of Charve and Danchin [loc. cit.] and Chen et al. [loc. cit.]. We conclude by generalizing these results for general viscosity coefficients.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
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