01-07
Generalized Characteristics and the Uniqueness of Entropy Solutions to Zero-pressure Gas Dynamics
Preprint series: 01-07, Preprints
- MSC:
- 35Q35 Other equations arising in fluid mechanics
- 35D99 None of the above but in this section
- 35A05 General existence and uniqueness theorems
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
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.