Stability of nonlinear waves for hyperbolic conservation laws will viscosity

双曲守恒定律非线性波的稳定性将粘性

• 批准号: 13640223
• 负责人: NISHIHARA Kenji
• 金额: $ 0.64万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640223/
• 关键词: viscous shock wave; rarefaction wave; diffusion wave; stability; damped wave equation; critical exponent; nonlinear stability; p-system

Our aim in this research is to investigate the asymptoic behavior of time-global solutions for one dimensional compressible flow with viscosity due to the Newton viscosity or the friction. The system is written by the hyperbolic conservation laws with viscosity, which has the nonlinear waves like viscous shock wave, rare faction wave, diffusion wave and the wave corresponding to contact discontinuity.For the compressible Navier-Stokes equation the global stability of strong rare faction wave is shown, whose method is applied to the Jin-Xin relaxation model for p-system. Also, in the inflow problem on half-line the solution is shown to tend the superposition of viscous shock wave and boundary layer under some conditions, in which case the problem was open.On the other hand, the p-system with friction is modeled by the compressible flow in porous media. The solution was shown by Hsiao-Liu to approach to the solution of the corresponding parabolic system due to the Darcy law. Through the precise consideration of the approach we have reached to the fact that the damped wave equation of second order is closely related to the corresponding heat equation in one and three dimensional space, which is applied to show the existence of time-global solution or the blow-up of solution in a finite time for the semilinear damped wave equation. The critical exponent is same as that in the semilinear heat equation, which is reasonably understood by the fact obtained. It is also seen in the abstract setting. So, our result may give some suggestions in the investigation on the damped wave equation and related problem.

本研究的目的是研究一维可压缩流的时间整体解的渐近性态，其粘性是由牛顿粘性或摩擦引起的。该方程组由具有粘性的双曲守恒律方程组构成，其中包含粘性激波、稀分数波、扩散波和接触间断波等非线性波，对于可压缩Navier-Stokes方程，证明了强稀分数波的全局稳定性，并将其方法应用于p-方程组的Jin-Xin松弛模型.在半直线上的入流问题中，在一定条件下，解趋于粘性激波和边界层的叠加，此时问题是开放的。另一方面，将多孔介质中的可压缩流动模拟为带摩擦的p-系统。Hsiao-Liu证明，由于达西定律，该解接近于相应抛物方程组的解。通过对该方法的精确考虑，我们得到了二阶阻尼波动方程与一维和三维空间中相应的热方程密切相关的事实，并应用于半线性阻尼波动方程的时间整体解的存在性或解在有限时间内的爆破.其临界指数与半线性热方程中的临界指数相同，这一事实得到了合理的理解。它也出现在抽象的背景中。因此，我们的结果可以为阻尼波动方程及相关问题的研究提供一些参考。

项目成果

P.Marcati, K.Nishihara: "The L^p-L^q estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media"J.Differential Equations. 191. 445-469 (2003)

P.Marcati、K.Nishihara：“一维阻尼波动方程解的 L^p-L^q 估计及其在多孔介质可压缩流动中的应用”J.微分方程。

K.Nishihara: "L^p-L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application"Math.Z.. 244. 631-649 (2003)

K.Nishihara：“3 维空间中阻尼波动方程解的 L^p-L^q 估计及其应用”Math.Z.. 244. 631-649 (2003)

K.Nishihara: "L^p-L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application"Math. Z.. 244. 631-649 (2003)

K.Nishihara：“3维空间中阻尼波动方程解的L^p-L^q估计及其应用”数学。

K.Nishihara, T.Yang, H.Zhao: "Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations"SIAM J.Math.Appl.. (未定). (2004)

K.Nishihara、T.Yang、H.Zhao：“可压缩纳维-斯托克斯方程的强稀疏波的非线性稳定性”SIAM J.Math.Appl.. (TBD)。

K.Nishihara: "Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media"Proc. Roy. Soc. Edinburgh. 133A. 177-196 (2003)

K.Nishihara：“通过多孔介质的一维可压缩流的扩散波渐进”Proc。