Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum
Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum
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DOI：
10.1002/cpa.22015
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发表时间：
2020-02
影响因子：
3
通讯作者：
Jinkai Li;Z. Xin
中科院分区：
文献类型：
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作者：
Jinkai Li;Z. Xin
在真空存在下，多变气体的物理熵表现出奇异性，因此研究其动力学是一个挑战。本文证明，只要初始真空只存在于初始密度衰减足够慢的远场，熵的有界性可以传播到任意有限时间。更准确地说，对于一维热传导可压缩Navier-Stokes方程的Cauchy问题，只要初始密度仅在远场以不超过O 1x 2的速率为零，就建立了强解的整体适定性和相应熵的一致有界性。证明熵的一致有界性的主要工具是为热传导可压缩Navier-Stokes方程精心设计的一些奇异加权能量估计和为某些退化抛物方程精心设计的De Giorgi型迭代技术。对不同的方程进行De Giorgi型迭代，建立熵的上下界。© 2021 Wiley Periodicals LLC.
In the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one‐dimensional heat conductive compressible Navier‐Stokes equations, the global well‐posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than O1x2 . The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier‐Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi–type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy. © 2021 Wiley Periodicals LLC.