Division of Applied Mathematics, Brown University, , Providence, RI 02912, USA.
Philos Trans A Math Phys Eng Sci. 2013 Nov 18;371(2005):20120344. doi: 10.1098/rsta.2012.0344. Print 2013 Dec 28.
The paper discusses systems of conservation laws endowed with involutions and contingent entropies. Under the assumption that the contingent entropy function is convex merely in the direction of a cone in state space, associated with the involution, it is shown that the Cauchy problem is locally well posed in the class of classical solutions, and that classical solutions are unique and stable even within the broader class of weak solutions that satisfy an entropy inequality. This is on a par with the classical theory of solutions to hyperbolic systems of conservation laws endowed with a convex entropy. The equations of elastodynamics provide the prototypical example for the above setting.
本文讨论了具有对合和余熵的守恒律系统。在余熵函数仅在状态空间中的一个锥方向上是凸的假设下，证明了在经典解类中局部适定柯西问题，并且即使在满足余熵不等式的更广泛的弱解类中，经典解也是唯一稳定的。这与具有凸余熵的双曲型守恒律系统的经典解理论相当。弹性动力学方程为上述设定提供了原型示例。
