传播相边界：问题表述与通过Glimm格式的存在性
Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme
AI总结：
本文针对描述非单调应变-应力函数下弹性材料动力学的双曲-椭圆守恒律系统，提出了可容许弱解的概念，并利用Glimm格式证明了传播相边界的存在性与稳定性。
摘要：
本文考虑两个守恒律的双曲-椭圆系统，该系统描述了由非单调应变-应力函数控制的弹性材料的动力学。遵循Abeyaratne和Knowles，我们为该系统在有界变差函数类中提出了可容许弱解的概念。该公式包括一个熵不等式、一个动力学关系（沿任何亚声速相边界施加）和一个成核准则（用于新相边界的出现）。我们证明了Riemann问题解的L1连续依赖性。我们的主要结果给出了传播相边界的存在性和稳定性。证明基于Glimm格式，特别是基于Glimm-Lax的技术。为了处理动力学关系，我们证明了相边界的逐点收敛结果。
英文摘要：
In this paper we consider the hyperbolic-elliptic system of two conservation laws that describes the dynamics of an elastic material governed by a non-monotone strain-stress function. Following Abeyaratne and Knowles, we propose a notion of admissible weak solution for this system in the class of functions with bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary), and an nucleation criterion (for the appearance of new phase boundaries). We prove the L1 continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proof is based on Glimm's scheme and in particular on techniques going back to Glimm-Lax. In order to deal with the kinetic relation, we prove a result of pointwise convergence for the phase boundary.
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