study of time periodic solutions to the equations of gas dynamics

气体动力学方程时间周期解的研究

• 批准号: 17540161
• 负责人: MAKINO Tetu
• 金额: $ 1.15万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540161/
• 关键词: gas dynamics; quasilinear wave equations; periodic solutions; analysis; functional equations; fluid; 非線型波動方程式; 星の内部構造

Originally this study started with the problem of nonlinear stability of the equilibria governed by the spherically symmetric Euler-Poisson equation of barotropic gas provided that the adiabatic exponent is greater than 4/3. The conjecture is that there exist time periodic solutions around these equilibria.In order to clarify the essential points of the problem, we studied 1-dimensional movement of gas under a constant gravitatuional force, and proved that the linearized equation of the perturbations from the equilibria admits time periodic solutions which are described by the Besssel functions. Moreover we clarified a property of smooth periodic solutions, if exist, of the fully nonlinear equation. The existence of periodic solutions to the fully nonlinear equations is still open.The problem is a free boundary problem at the interface with the vacuum, we have great difficulty in the theoretical consideration,. So, we studied the 1-dimensional movement of gas without exterior forces. The equilibria are constant density, and the eqwuation of the perturbation from the equilibria is a quasilinear wave equation, whose coeeeficients are regular at both boundaries. We proved there are no smooth periodic solutions for this fully nonlinear equation. Special cases about time periodic solutions to quasilinear wave equations are done by Greenberg, Rascal et al., but the tipe of equations arising from the gasdynamicas has no results. But there are possibility to show the existence of periodic solutions to these quasilinear gave equations applyinf the discussion by Rabinowitz, Brezis, Coron, Nirebnberg, craig, Wayne on the semilinear wave equations. We are now in the step of the study.

最初，这项研究始于由面向对称的欧拉 - 波森方程（如果绝热指数大于4/3，则由球形对称的Euler-Poisson方程）开始。猜想是在这些平衡周围存在时间周期性解决方案。为了阐明问题的要点，我们研究了气体在恒定引力力下的一维运动，并证明了从平衡的扰动的线性方程承认贝塞尔函数描述的时间周期性解决方案。此外，我们阐明了完全非线性方程的平滑周期解决方案（如果存在）的属性。完全非线性方程的定期解决方案仍然是开放的。问题是与真空接口处的自由边界问题，在理论考虑中，我们有很大的困难。因此，我们研究了没有外部力的气体的一维运动。平衡是恒定的密度，而均衡的扰动的升级是一种准线性波方程，其卵性在两个边界上都是规则的。我们证明没有针对该完全非线性方程的平滑周期解决方案。 Greenberg，Rascal等人对准线性波方程的时间周期性解决方案进行了特殊情况，但是由Gasdynamicas产生的方程式的tipe是没有结果。但是，有可能显示这些准线性的周期性解决方案的存在，使方程在半线性波方程上对Rabinowitz，Brezis，Coron，Nirebnberg，Craig，Craig，Wayne进行了讨论。我们现在正在研究。

项目成果

The Lyapunov stability of the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation

Benjamin-Ono 方程 Lax 层次中 N-孤子解的 Lyapunov 稳定性

• 发表时间: 2006
• 期刊: J. Math. Phys 47
• 作者: Yoshihiro Shibata;Senjo Shimizu;T.Makino;T.Makino;Y.Matsuno;Y.Matsuno
• 通讯作者: Y.Matsuno

Cusp and loop soliton solutions of the short-wave models for the Camassa-Holn and Degasperis-Procesi equations

Camassa-Holn 和 Degasperis-Procesi 方程的短波模型的尖点和环孤子解

• 发表时间: 2006
• 期刊: Phys. Lett. A 359
• 作者: Yoshihiro Shibata;Senjo Shimizu;T.Makino;T.Makino;Y.Matsuno
• 通讯作者: Y.Matsuno

Cusp and loop soliton solutions of the short-wave medels for the Camassa-Holn and Degasperis-Procesi equations

Camassa-Holn 和 Degasperis-Procesi 方程的短波模型的尖点和环孤子解

• 发表时间: 2006
• 期刊: Phys.Lett.A 359
• 作者: Yoshihiro Shibata;Senjo Shimizu;T.Makino;T.Makino;Y.Matsuno;Y.Matsuno;T.Makino;Y.Matsuno;Y.Matsuno;T.Makino;Y.Matsuno
• 通讯作者: Y.Matsuno

The N-soliton solution of the Degasperis-Procesi equation

Degasperis-Procesi 方程的 N 孤子解

• 发表时间: 2005
• 期刊: Inverse Problem 21-6
• 作者: Yoshihiro Shibata;Senjo Shimizu;T.Makino;T.Makino;Y.Matsuno;Y.Matsuno;T.Makino;Y.Matsuno;Y.Matsuno;T.Makino;Y.Matsuno;Y.Matsuno;Y.Matsuno;Y.Matsuno;Y.Matsuno
• 通讯作者: Y.Matsuno

Smooth solutions to a class of squasilinear wave equations

一类拟线性波动方程的光滑解

• 发表时间: 2006
• 期刊: J.Diff.Eq. 224
• 作者: Yoshihiro Shibata;Senjo Shimizu;T.Makino;T.Makino;Y.Matsuno;Y.Matsuno;T.Makino
• 通讯作者: T.Makino