Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data
Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data
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DOI：
10.1006/jdeq.1995.1111
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发表时间：
1995-07
影响因子：
2.4
通讯作者：
D. Hoff
中科院分区：
文献类型：
--
作者：
D. Hoff
本文证明了二维和三维空间可压缩等温流的N-S方程的整体弱解的存在性，当初始密度在L 2和L∞中接近一个常数，初速在L 2中很小且有界在L 2 n中(在二维中L 2范数必须略微加权)。获得了大量关于该解的定性信息。例如，我们证明了速度和涡度在正时间是相对平滑的，有效粘性通量F也是如此，有效粘性通量F是速度的散度减去一定倍数的压力。我们发现，F在整个分析中起着至关重要的作用，特别是在接近所需的能量估计、了解初始层附近的正则化速率，以及最重要的是，获得与时间无关的密度的逐点界限。
Abstract We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, isothermal flow in two and three space dimensions when the initial density is close to a constant in L 2 and L ∞ , and the initial velocity is small in L 2 and bounded in L 2 n (in two dimensions the L 2 norms must be weighted slightly). A great deal of qualitative information about the solution is obtained. For example, we show that the velocity and vorticity are relatively smooth in positive time, as is the "effective viscous flux" F , which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a crucial role in the entire analysis, particularly in closing the required energy estimates, understanding rates of regularization near the initial layer, and most important, obtaining time-independent pointwise bounds for the density.