Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws

Abstract

We are interested in the one-dimensional scalar conservation law $${\partial }_{t}u(t,x)=\nu D^{\alpha }u(t,x)-{\partial }_{\mathrm{xA}}\left(u\right(t,x\left)\right)$$ with fractional viscosity operator D^αv(x) = F^-1(|ξ|^αF(v)(ξ))(x) is the cumulative distribution function of a signed measure on R. We associate a nonlinear martingale problem with the Fokker-Planck equation obtained by spatial differentiation of the conservation law. After checking uniqueness for both the conservation law and the martingale problem, we prove existence thanks to a propagation-of-chaos result for systems of interacting particles with fixed intensity of jumps related to ν. The empirical cumulative distribution functions of the particles converge to the solution of the conservation law. As a consequence, it is possible to approximate this solution numerically by simulating the stochastic differential equation which gives the evolution of particles. Finally, when the intensity of jumps vanishes (ν→0) as the number of particles tends to +∞, we obtain that the empirical cumulative distribution functions converge to the unique entropy solution of the inviscid (ν=0) conservation law.