TY - JOUR
T1 - The Lp energy methods and decay for the compressible Navier-Stokes equations with capillarity
AU - Kawashima, Shuichi
AU - Shibata, Yoshihiro
AU - Xu, Jiang
N1 - Funding Information:
The authors would like to thank anonymous referees for their valuable comments, which greatly improve the presentation of this manuscript. The work is partially supported by Top Global University Project and Toyota Central Research Institute Joint Research Fund . S. Kawashima is partially supported by JSPS KAKENHI Grant Numbers JP18H01131 , JP19H05597 and JP20H00118 . Y. Shibata is partially supported by JSPS KAKENHI Grant Number JP17H0109 . J. Xu is partially supported by the National Natural Science Foundation of China ( 11871274 , 12031006 ) and the China Scholarship Council ( 201906835023 ). He would like to thank R. Danchin for his warm communication when visiting the LAMA in UPEC.
Publisher Copyright:
© 2021 Elsevier Masson SAS
PY - 2021/10
Y1 - 2021/10
N2 - We consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effect. Referring to those studies in the non-capillary case, the purpose of this paper is to investigate the dissipation effect of Korteweg tensor with the density-dependent capillarity κ(ϱ). It is observed by the pointwise estimate that the linear third-order capillarity behaves like the heat diffusion of density fluctuation, which allows to develop the Lp energy methods (independent of spectral analysis). As a result, the time-decay estimates of Lq-Lr type regarding this system can be established. The treatment of nonlinear capillarity depends mainly on new Besov product estimates and the elaborate use of Sobolev embeddings and interpolations. Our results can be also applied to the quantum Navier-Stokes system, since it is a special choice of capillarity κ(ϱ)=κ/ϱ.
AB - We consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effect. Referring to those studies in the non-capillary case, the purpose of this paper is to investigate the dissipation effect of Korteweg tensor with the density-dependent capillarity κ(ϱ). It is observed by the pointwise estimate that the linear third-order capillarity behaves like the heat diffusion of density fluctuation, which allows to develop the Lp energy methods (independent of spectral analysis). As a result, the time-decay estimates of Lq-Lr type regarding this system can be established. The treatment of nonlinear capillarity depends mainly on new Besov product estimates and the elaborate use of Sobolev embeddings and interpolations. Our results can be also applied to the quantum Navier-Stokes system, since it is a special choice of capillarity κ(ϱ)=κ/ϱ.
KW - Critical Besov spaces
KW - L energy methods
KW - Navier-Stokes-Korteweg equations
KW - Time-decay rates
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U2 - 10.1016/j.matpur.2021.08.009
DO - 10.1016/j.matpur.2021.08.009
M3 - Article
AN - SCOPUS:85113373940
SN - 0021-7824
VL - 154
SP - 146
EP - 184
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
ER -