# Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

- 238 Downloads
- 168 Citations

## Abstract

We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

## Keywords

Entropy Weak Solution Variational Principle Initial Velocity Quantum Computing## Preview

Unable to display preview. Download preview PDF.

## References

- [BG] Brenier, Y., Grenier, E.: On the model of pressureless gases with sticky particles. Preprint, 1995Google Scholar
- [CPY] Carnevale, G.F., Pomeau, Y., Young, W.R.: Statistics of ballistic agglomeration. Phys. Rev. Lett.,
**64**, no. 24, 2913 (1990)Google Scholar - [D] Dafermos, C.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J.
**26**, 1097–1119 (1977)Google Scholar - [GMS] Gurbatov, S.N., Malakhov, A.N., Saichev, A.I.: Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays and Particles. Manchester: Manchester University Press, 1991Google Scholar
- [KPS] Kofman, L., Pogosyan, D., Shandarin, S.: Structure of the universe in the two-dimensional model of adhesion. Mon. Nat. R. Astr. Soc.
**242**, 200–208 (1990)Google Scholar - [L] Lax, P.D.: Hyperbolic systems of conservation laws: II. Comm. Pure. Appl. Math.
**10**, 537–556 (1957)Google Scholar - [O] Oleinik, O.A.: Discontinuous solutions of nonlinear differential equatons. Uspekhi Mat. Nauk.
**12**, 3–73 (1957)Google Scholar - [P] Peebles, P.J.E.: The Large Scale Structures of the Universe. Princeton, NJ: Princeton University Press, 1980Google Scholar
- [SZ] Shandarin, S.F., Zeldovich, Ya.B.: The large-scale structures of the universe: Turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys.
**61**, 185–220 (1989)Google Scholar - [SAF] She, Z.S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys.
**148**, 623–641 (1992)Google Scholar - [S] Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys.
**148**, 601–622 (1992)Google Scholar - [VDFN] Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers' equation, devil's staircases and the mass distribution function for large-scale structures. Astron & Astrophys
**289**, 325–356 (1994)Google Scholar - [V] Volpert, A.I.: The space BV and quasilinear equations. Math. USSR-Sbornik.
**2**, 225–267 (1967)Google Scholar - [Z] Zeldovich, Ya.B.: Gravitational instability: An approximate theory for large density perturbations. Astron & Astrophys.
**5**, 84–89 (1970)Google Scholar