Optimal Time-decay Estimates for the Compressible Navier–Stokes Equations in the Critical Lp Framework

By Raphaël Danchin, Jiang Xu. 2016

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The global existence issue for the isentropic compressible Navier–Stokes equations in the critical regularity framework was addressed in Danchin (Invent Math 141(3):579–614, 2000) more than 15 years ago. However, whether (optimal) time-decay rates could be shown in critical spaces has remained an open question. Here we give a positive answer to that issue not only in the L2 critical framework of Danchin (Invent Math 141(3):579–614, 2000) but also in the general Lp critical framework of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010), Chen et al. (Commun Pure Appl Math 63(9):1173–1224, 2010), Haspot (Arch Ration Mech Anal 202(2):427–460, 2011): we show that under a mild additional decay assumption that is satisfied if, for example, the low frequencies of the initial data are in $${L^{p/2}(\mathbb{R}^{d})}$$Lp/2(Rd), the Lp norm (the slightly stronger $${\dot B^0_{p,1}}$$B˙p,10 norm in fact) of the critical global solutions decays like $${t^{-d(\frac 1p-\frac14)}}$$t-d(1p-14) for $${t\to+\infty,}$$t→+∞, exactly as firstly observed by Matsumura and Nishida in (Proc Jpn Acad Ser A 55:337–342, 1979) in the case p = 2 and d = 3, for solutions with high Sobolev regularity.Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6ccf7e0a647ea0b61df2da5a5e91a47316594b86