VACUUM BEHAVIORS AROUND RAREFACTION WAVES TO 1D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY
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AbstractIn this paper, we study the large time asymptotic behavior toward rarefaction waves for solutions to the one-dimensional compressible Navier-Stokes equations with density-dependent viscosities for general initial data whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. First, a global-in-time weak solution around the rarefaction wave is constructed by approximating the system and regularizing the initial data with general perturbations, and some a priori uniform-in-time estimates for the energy and entropy are obtained. Then it is shown that the density of any weak solution satisfying the natural energy and entropy estimates will converge to the rarefaction wave connected to vacuum with arbitrary strength in sup-norm time-asymptotically. Our results imply, in particular, that the initial vacuum at far field will remain for all the time, which is in sharp contrast to the case of nonvacuum rarefaction waves studied in [Q. S. Jiu, Y. Wang, and Z. P. Xin, Comm. Partial Differential Equations, 36 (2011), pp. 602-634], where all the possible vacuum states will vanish in finite time. Finally, it is proved that the weak solution becomes regular away from the vacuum region of the rarefaction wave.
All Author(s) ListJiu QS, Wang Y, Xin ZP
Journal nameSIAM Journal on Mathematical Analysis
Year2013
Month1
Day1
Volume Number45
Issue Number5
PublisherSIAM PUBLICATIONS
Pages3194 - 3228
ISSN0036-1410
eISSN1095-7154
LanguagesEnglish-United Kingdom
Keywordscompressible Navier-Stokes equations; density-dependent viscosity; rarefaction wave; stability; vacuum; weak solution
Web of Science Subject CategoriesMathematics; Mathematics, Applied; MATHEMATICS, APPLIED

Last updated on 2020-29-03 at 01:18