论文标题
一维热导电可压缩的Navier-Stokes方程的熵结合的解决方案具有远场真空
Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier-Stokes equations with far field vacuum
论文作者
论文摘要
在存在真空的情况下，多变态气体的物理熵表现得很单独，因此研究其动力学是一个挑战。本文显示，只要初始真空才能在远场上呈现出足够慢的初始密度衰减，就可以在任何有限的时间内传播到熵的界限。更确切地说，对于一维热导电可压缩的Navier-Stokes方程的库奇问题，只要初始密度仅在远处消失，而初始密度仅消失，则建立了强溶液的全球良好性和相应熵的均匀界限。证明熵统一的界限的主要工具是针对热导电可压缩的Navier-Stokes方程和精心设计的一些奇特的加权能量估计，以及针对某些退化抛物面方程的精心设计的De Giorgi型迭代技术。在建立熵的下限和上限时，将De Giorgi型迭代进行到不同的方程式。
In the presence of vacuum, the physical entropy for polytropic gases behaves 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 $O(\frac{1}{x^2})$. 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.
