Blow up Analysis for Anomalous Granular Gases

Thomas Rey (1)

1 ICJ - Institut Camille Jordan

We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is ''anomalous'', in the sense that energy dissipation increases when temperature decreases. This allows the gas to cool down in finite time. We study existence and uniqueness of blow up profiles for this model, together with the trend to equilibrium and the cooling law associated, generalizing the classical Haff's Law for granular gases. To this end, we investigate the asymptotic behaviour of the inelastic Boltzmann equation with and without drift term by introducing new strongly ''nonlinear'' self-similar variables.

Mots clés: Inelastic Boltzmann equation; hard spheres; spatially homogeneous; anomalous granular gases; cooling process; blow up profile; Haff's Law; self-similar solution; blow up profile.

Domaines: Equations aux dérivées partielles [math.AP]

Thomas Rey. Blow up Analysis for Anomalous Granular Gases. SIAM Journal on Mathematical Analysis, 2012, 44 (3), pp. 1544-1561. ⟨10.1137/110835645⟩. ⟨hal-00588970v3⟩