带一般粘性一维等熵Navier-Stokes方程组解的存在性及粘性消失极限

The existence of solutions for Navier-Stokes equations has always being as one of core issues researched in Mathematics. Navier-Stokes equations are widely used in the fields of physics and engineering, so studying the existence of solutions for its Cauchy problem has important theoretical and practical values. The project researches the existence of solutions for Cauchy problem of 1D isentropic Navier-Stokes equations with genenral viscosity and prove the vanishing viscosity limit for solutions are weak solutions of 1D isentropic gas dynamics equations. Through improving the corresponding estimates of solutions for Navier-Stokes equations with special viscosity, we can get the existence of solutions for Navier-Stokes equations having genenral viscosity with Cauchy initial data. Based upon it, through establishing the better estimates for the weak entropy and entropy flux, we can prove the vanishing viscosity limit of solutions for Navier-Stokes equations are the weak solutions of isentropic gas dynamics equations using the compensated compactness theory, furthermore, are the entropy solutions. The research of the project contributes to be familiar with solutions of isentropic Navier-Stokes equations and the relation between them and solutions of isentropic gas dynamics equations deeply.

Navier-Stokes方程组解的存在性一直以来是数学界研究的核心问题之一。Navier-Stokes方程组被广泛应用于物理和工程等领域，因此研究此方程组Cauchy问题解的存在性具有重要理论和应用价值。本项目主要研究带一般粘性的一维等熵Navier-Stokes方程组Cauchy问题解的存在性并证明该解的粘性消失极限是一维等熵气体动力学方程组的弱解。通过改进具有特殊粘性的等熵Navier-Stokes方程组解的相应先验估计，得到带一般粘性的等熵Navier-Stokes方程组Cauchy问题解的存在性。在此基础上，通过建立弱熵和熵流量的精细估计式，利用补偿列紧方法证明等熵Navier-Stokes方程组解的收敛极限是等熵气体动力学方程组的弱解，进一步证明其为熵解。本项目的研究有助于进一步加深对等熵Navier-Stokes方程组解及其与等熵气体动力学方程组解之间的关系的理解和认识。

本项目对带一般粘性的一维等熵Navier-Stokes方程组Cauchy问题解的存在性及该解的粘性消失极限是一维等熵气体动力学方程组的弱解进行了研究。我们的研究成果主要由以下两部分组成：首先，我们研究了带一般粘性的一维等熵Navier-Stokes方程组Cauchy问题解的存在性。通过能量估计和熵估计等先验估计，我们得到了Navier-Stokes方程组Cauchy问题近似解的下界估计。利用下界估计，我们可以证明带一般粘性的一维等熵Navier-Stokes方程组Cauchy问题解的存在性。第二，利用补偿列紧理论，我们证明了带一般粘性的一维等熵Navier-Stokes方程组Cauchy问题解的粘性消失极限是一维等熵气体动力学方程组的弱解。此结果包含了前人关于常数粘性或者是密度的方次粘性的此类问题的研究结果。