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2001-07

Generalized Characteristics and the Uniqueness of Entropy Solutions to Zero-pressure Gas Dynamics

by Li, J., Warnecke, G..


Series: 2001-07, Preprints

MSC:
35Q35 PDEs in connection with fluid mechanics
35D99 None of the above, but in this section

Abstract:
The system of zero-pressure gas dynamics conservation laws
describes the dynamics of free particles sticking under
collision while mass and momentum are conserved both at the
discrete and continuous levels. The existence of such
solutions was established in [CLZ, Science in China 40,
1997; ERS, Comm. Math. Phys. 177, 1996]. In this paper we
are concerned with the uniqueness of entropy solutions. We
first introduce additionally to the Oleinik entropy
condition a cohesion condition. Both conditions together
form our extended concept of an admissibility condition for
solutions to the system. The cohesion condition is
automatically satisfied by the solutions obtained in the
existence results mentioned above. Further, we regularize
such a given admissible solution so that generalized
characteristics are well-defined. Through limiting
procedures the concept of generalized characteristics is
then extended to a very large class of admissible solutions
containing vacuum states and singular measures. Next we use
the generalized characteristics and the dynamics of the
center of mass in order to prove that all entropy solutions
are equal in the sense of distributions.

Keywords:
zero-pressure gas dynamics, uniqueness, entropy condition, cohesion condition, generalized characteristics