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Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework | | Raphaël Danchin
; Jiang Xu
; | | Date: |
3 May 2016 | | Abstract: | The global existence issue for the isentropic compressible Navier-Stokes
equations in the critical regularity framework has been addressed in [7] more
than fifteen years ago. However, whether (optimal) time-decay rates could be
shown in general critical spaces and any dimension d $ge$ 2 has remained an
open question. Here we give a positive answer to that issue not only in the L 2
critical framework of [7] but also in the more general L p critical framework
of [3, 6, 14]. More precisely, we show that under a mild additional decay
assumption that is satisfied if the low frequencies of the initial data are in
e.g. L p/2 (R d), the L p norm (the slightly stronger $dot B^0_{p,1}$ norm in
fact) of the critical global solutions decays like t --d(1 p -- 1 4) for t
$
ightarrow$ +$infty$, exactly as firstly observed by A. Matsumura and T.
Nishida in [23] 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. | | Source: | arXiv, 1605.0893 | | Services: | Forum | Review | PDF | Favorites | |
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