Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure

Sofiane Martel (1, 2) , Julien Reygner (2)

1 LHSV - Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally Lipschitz continuous and have at most polynomial growth. Neither the flux nor the noise need to be non-degenerate. In a first part, we show the existence and uniqueness of a global solution in a strong sense. In a second part, we establish the existence and uniqueness of an invariant measure for this strong solution.

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Sofiane Martel, Julien Reygner. Viscous scalar conservation law with stochastic forcing: strong solution and invariant measure. 2019. ⟨hal-02133338v1⟩