Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. (English) Zbl 0852.35097
Summary: 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 \(2\times 2\) systems of conservation 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.
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.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35Q72 | Other PDE from mechanics (MSC2000) |
References:
| [1] | [BG] Brenier, Y., Grenier, E.: On the model of pressureless gases with sticky particles. Preprint, 1995 |
| [2] | [CPY] Carnevale, G.F., Pomeau, Y., Young, W.R.: Statistics of ballistic agglomeration. Phys. Rev. Lett.,64, no. 24, 2913 (1990) · doi:10.1103/PhysRevLett.64.2913 |
| [3] | [D] Dafermos, C.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J.26, 1097–1119 (1977) · Zbl 0377.35051 · doi:10.1512/iumj.1977.26.26088 |
| [4] | [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, 1991 · Zbl 0860.76002 |
| [5] | [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) |
| [6] | [L] Lax, P.D.: Hyperbolic systems of conservation laws: II. Comm. Pure. Appl. Math.10, 537–556 (1957) · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [7] | [O] Oleinik, O.A.: Discontinuous solutions of nonlinear differential equatons. Uspekhi Mat. Nauk.12, 3–73 (1957) |
| [8] | [P] Peebles, P.J.E.: The Large Scale Structures of the Universe. Princeton, NJ: Princeton University Press, 1980 |
| [9] | [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) · doi:10.1103/RevModPhys.61.185 |
| [10] | [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) · Zbl 0755.60104 · doi:10.1007/BF02096551 |
| [11] | [S] Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys.148, 601–622 (1992) · Zbl 0755.60105 · doi:10.1007/BF02096550 |
| [12] | [VDFN] Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers’ equation, devil’s staircases and the mass distribution function for large-scale structures. Astron & Astrophys289, 325–356 (1994) |
| [13] | [V] Volpert, A.I.: The space BV and quasilinear equations. Math. USSR-Sbornik.2, 225–267 (1967) · Zbl 0168.07402 · doi:10.1070/SM1967v002n02ABEH002340 |
| [14] | [Z] Zeldovich, Ya.B.: Gravitational instability: An approximate theory for large density perturbations. Astron & Astrophys.5, 84–89 (1970) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
