论文标题
大型BV溶液在颗粒流中的指数稳定性
Exponential Stability of Large BV Solutions in a Model of Granular flow
论文作者
论文摘要
我们考虑一个$ 2 \ times 2 $在一个空间维度上的双曲线平衡定律系统，它描述了具有缓慢侵蚀和沉积的颗粒状材料的演变。动力学是根据顶部和底部的站立层的移动层的厚度表示的。该系统沿着相平面的两条直线进行线性退化，而在受此类线限制的子域中确实非线性。特别是，第一个特征家族的特征速度严格在线性堕落线上的区域中严格增加，并且在该线以下的区域中严格降低。非耗散源项是用两个不同特征速度运输的两个数量的乘积。 在假设移动层的初始高度足够小的假设下，Amadori和Shen的全球熵弱解决方案的全局弱解决方案是由Amadori和Shen建立了具有有界但可能很大的总变化的初始数据。 在本文中，我们建立了Lipschitz $ {\ bf l^1} $ - 解决方案对初始数据的连续依赖性使用Lipschitz常数，该常数会及时生长。 The proof of the ${\bf L^1}$-stability of solutions is based on the construction of a Lyapunov like functional equivalent to the ${\bf L^1}$-distance, in the same spirit of the functional introduced by Liu and Yang and then developed by Bressan, Liu, Yang for systems of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields.
We consider a $2\times 2$ system of hyperbolic balance laws, in one-space dimension, that describes the evolution of a granular material with slow erosion and deposition. The dynamics is expressed in terms of the thickness of a moving layer on top and of a standing layer at the bottom. The system is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. In particular, the characteristic speed of the first characteristic family is strictly increasing in the region above the line of linear degeneracy and strictly decreasing in the region below such a line. The non dissipative source term is the product of two quantities that are transported with the two different characteristic speeds. The global existence of entropy weak solutions of the Cauchy problem for such a system was established by Amadori and Shen for initial data with bounded but possibly large total variation, under the assumption that the initial height of the moving layer be sufficiently small. In this paper we establish the Lipschitz ${\bf L^1}$-continuous dependence of the solutions on the initial data with a Lipschitz constant that grows exponentially in time. The proof of the ${\bf L^1}$-stability of solutions is based on the construction of a Lyapunov like functional equivalent to the ${\bf L^1}$-distance, in the same spirit of the functional introduced by Liu and Yang and then developed by Bressan, Liu, Yang for systems of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields.
