Exterior problem for nonlinear wave equations

非线性波动方程的外问题

基本信息

批准号：
13440049
负责人：
NAKAO Mitsuhiro
金额：
$ 7.3万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

The main purpose of this research is concerned with the exterior problem for the quasi-linear wave equations. For this problem we have been successful in proving the global existence of smooth solutions under the effect of localized dissipation. We have achieved the results through two ways; one is based on the local energy decay and L^p estimates of solutions for linear equation, and the other one is the method to utilize total energy decay for the llinearized equation. Both ways are intended to make the effects of dissipation as weaker as possible, but, we have made no geometrical conditions on the shape of the boundary.Concerning another problem on the energy decay for the equation with nonlinear dissipations we introduced anew concept ‘Half linear' and has been successful in deriving very delicate decay estimates of energy and applied them to the existence of global solutions for the equations with a nonlinear source term.As related problems we have considered the existence and stability of periodic solutions for the nonlinear wave equations in bounded domains with some nonlinear localized dissipations. Further, we have considered the Kirchhoff type nonlinear wave equations in exterior domains. Under a nonlinear dissipations we have proved various results on global solutions. For the wave equation in exterior domains with a Neumann type boundary dissipation we have derived a new energy decay estimate.Investigator Kawashima has derived many interesting results concerning Boltzman equations and hyperbolic conservation equations. Investigator Shibata has derived by the method of spectral analysis, many interesting results concerning the exterior problem for the compressive Navier-Stokes equations. Investigator Ogawa has proved precise estimates of solutions concerning behaviors and regularities of solutions for the nonlinear wave equations, nonlinear Shroadinger equations and some harmonic evolution equation.

这项研究的主要目的与准线性波方程的外部问题有关。对于这个问题，我们已经成功地在局部耗散的效果下提供了平滑解决方案的全球存在。我们通过两种方式实现了结果。一个基于线性方程的溶液的局部能量衰减和l^p估计，另一种是利用lllLinerearized方程的总能量衰减的方法。这两种方式均旨在使耗散的影响尽可能弱，但是，我们没有在边界形状上做出几何条件。通过非线性耗散来构造方程式的另一个问题，我们引入了“半线性”的概念，并且在能源的范围内成功地将它们衍生出来，并在其上既不存在了，并且都将其置于了与众不同的情况下，并且存在着与众不同的依据。具有某些非线性局部耗散的有限域中非线性波方程的周期性解决方案的稳定性。此外，我们考虑了外部域中的Kirchhoff型非线性波方程。在非线性耗散的情况下，我们为全球解决方案提供了各种结果。对于具有Neumann类型边界耗散的外部结构域中的波方程，我们得出了一个新的能量衰减估计。InvestigatorKawashima得出了许多有关Boltzman方程和多重方便方程的有趣结果。研究者Shibata通过光谱分析的方法得出了许多有趣的结果，这些结果涉及压缩Navier-Stokes方程的外部问题。研究人员的大瓦证明了有关非线性波方程，非线性shroadinger方程和某些谐波演化方程的解决方案的解决方案和规律性解决方案的精确估计。