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Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $$L^{p}$$ framework. (English) Zbl 1366.35126
This work is devoted to the derivation of the (optimal) time decay rates of solutions for the isentropic compressible Navier-Stokes equations in the norms of critical spaces. The existence of global in time solutions has been discussed previously in $$L^2$$ [R. Danchin, Invent. Math. 141, No. 3, 579–614 (2000; Zbl 0958.35100)] and $$L^p$$ frameworks [F. Charve and R. Danchin, Arch. Ration. Mech. Anal. 198, No. 1, 233–271 (2010; Zbl 1229.35167)], [Q. Chen et al., Commun. Pure Appl. Math. 63, No. 9, 1173–1224 (2010; Zbl 1202.35002)], [B. Haspot, Arch. Ration. Mech. Anal. 202, No. 2, 427–460 (2011; Zbl 1427.76230)]. The method of proof relies on refined time weighted inequalities in the Fourier space.

MSC:
 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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References:
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