Onsager's Energy Conservation of Weak Solutions for a Compressible and Inviscid Fluid

被引:1
作者
Wu, Xinglong [1 ]
Zhou, Qian [1 ]
机构
[1] Wuhan Univ Technol, Ctr Math Sci, Sch Sci, Wuhan 430070, Peoples R China
关键词
conservation of energy; the isentropic compressible Euler equations; commutator estimate; regularity of the solutions; INCOMPRESSIBLE EULER; DISSIPATION; CONJECTURE;
D O I
10.3390/fractalfract7040324
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this article, two classes of sufficient conditions of weak solutions are given to guarantee the energy conservation of the compressible Euler equations. Our strategy is to introduce a test function phi(t)v(epsilon) to derive the total energy. The velocity field v needs to be regularized both in time and space. In contrast to the noncompressible Euler equations, the compressible flows we consider here do not have a divergence-free structure. Therefore, it is necessary to make an additional estimate of the pressure p, which takes advantage of an appropriate commutator. In addition, by using the weak convergence, we show that the energy equality is conserved in a point-wise sense.
引用
收藏
页数:17
相关论文
共 20 条
[1]   ENERGY CONSERVATION FOR THE COMPRESSIBLE EULER AND NAVIER-STOKES EQUATIONS WITH VACUUM [J].
Akramov, Ibrokhimbek ;
Debiec, Tomasz ;
Skipper, Jack ;
Wiedemann, Emil .
ANALYSIS & PDE, 2020, 13 (03) :789-811
[2]   Anomalous dissipation for 1/5-Holder Euler flows [J].
Buckmaster, Tristan ;
De Lellis, Camillo ;
Isett, Philip ;
Szekelyhidi, Laszlo, Jr. .
ANNALS OF MATHEMATICS, 2015, 182 (01) :127-172
[3]   Onsager's energy conservation for inhomogeneous Euler equations [J].
Chen, Robin Ming ;
Yu, Cheng .
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, 2019, 131 :1-16
[4]   Energy conservation and Onsager's conjecture for the Euler equations [J].
Cheskidov, A. ;
Constantin, P. ;
Friedlander, S. ;
Shvydkoy, R. .
NONLINEARITY, 2008, 21 (06) :1233-1252
[5]   Energy Conservation in Two-dimensional Incompressible Ideal Fluids [J].
Cheskidov, A. ;
Lopes Filho, M. C. ;
Nussenzveig Lopes, H. J. ;
Shvydkoy, R. .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2016, 348 (01) :129-143
[6]   h-Principles for the Incompressible Euler Equations [J].
Choffrut, Antoine .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2013, 210 (01) :133-163
[7]   ONSAGER CONJECTURE ON THE ENERGY-CONSERVATION FOR SOLUTIONS OF EULER EQUATION [J].
CONSTANTIN, P ;
TITI, ES ;
WEINAN, F .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1994, 165 (01) :207-209
[8]   Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation [J].
Cordoba, Diego ;
Faraco, Daniel ;
Gancedo, Francisco .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2011, 200 (03) :725-746
[9]   Dissipative Euler flows and Onsager's conjecture [J].
De Lellis, Camillo ;
Szekelyhidi, Laszlo .
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 2014, 16 (07) :1467-1505
[10]   Dissipative continuous Euler flows [J].
De Lellis, Camillo ;
Szekelyhidi, Laszlo, Jr. .
INVENTIONES MATHEMATICAE, 2013, 193 (02) :377-407