可压缩等温Navier-Stokes方程整体适定性问题研究

As the model of describing the motion of fluids, the Navier–Stokes equations have been extensively studied by many physicists, meteorologists and mathematicians. This project is devoted to studying the global classical solutions of compressible isothermal Navier-Stokes equations in high dimensional space. Utilizing the stability of velocity and density at infinity, some results about global regularity to the compressible isentropic Navier-Stokes have been obtained. However, similar results to the compressible isothermal Navier-Stokes equations with vacuum at infinity are few due to the restriction of energy inequality. Therefore, we will investigate global well-posedness to the compressible isothermal Navier-Stokes equations through piecewise estimation and some time-depending a priori estimates. Under the condition that the initial total mass is small, we will obtain uniform upper bounds for density. We will also study the compressible isothermal Navier-Stokes equations with the Navier-slip boundary conditon. By establishing elliptic systems for effective viscous flux and vorticity, the global regularity will be investigated under the condition that the initial energy is small.

Navier-Stokes方程作为描述流体运动的模型，已被流体力学家、气象学家和数学家进行了广泛而深入的研究。本项目旨在研究高维可压缩等温Navier-Stokes方程的整体古典解。利用无穷远场的稳定性，可压缩等熵Navier-Stokes方程Cauchy问题整体解存在唯一性的研究已经取得了一定进展。但是，由于能量不等式的限制，无穷远密度为0的可压缩等温Navier-Stokes方程类似研究还没有任何结果。申请人计划利用分段估计的技巧和一类依赖于时间的新的先验估计来研究可压缩等温Navier-Stokes方程Cauchy问题的整体古典解。在质量小的条件下，我们将证明密度一致有界。项目还拟研究具有Navier-slip边值条件的可压缩等温Navier-Stokes方程，通过建立有效粘性通量和旋度的椭圆估计，在初始能量小的条件下研究解的整体正则性。

本项目主要研究了可压缩等温Navier-Stokes方程组、变粘性非等熵可压缩Navier-Stokes方程组和气-液两相流模型强解的整体存在性。对于可压缩等温Navier-Stokes方程组的Cauchy问题，根据Zlotnik不等式的特点，我们利用一类依赖于时间的先验估计，在初始质量充分小的条件下得到密度的一致上界。当速度场和温度场满足Dirichlet边界条件时，利用对时间加权的先验估计，我们研究了粘性系数依赖于密度和温度的非等熵可压缩Navier-Stokes方程组强解的整体存在性。对于气-液两相流模型，我们克服了压力梯度产生的奇性，成功去掉了前人结果中对初始气体质量和液体质量之间的约束条件。