Sharp decay characterization for the compressible Navier-Stokes equations

The low-frequency L 1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible NavierStokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical L p framework. Precisely, it is proved that the Besov space boundedness condition (with d 2 - 2dp dp = s 1 < d 2 - 1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B ? s 1 2,8. , 8 . To the best of our knowledge, our B ? s 1 2,8- , 8- work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.