LAUSR

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $$\alpha $$ α at the left of the system and $$\beta $$ β at the right of the system. The strength of the reservoirs is ruled by $$\kappa N^{-\theta }>0$$ κ N - θ > 0 . Here N is the size of the system, $$\kappa >0$$ κ > 0 and $$\theta \in {{\mathbb {R}}}$$ θ ∈ R . Our results are valid for $$\theta \le 0$$ θ ≤ 0 . For $$\theta =0$$ θ = 0 , we obtain a collection of fractional reaction–diffusion equations indexed by the parameter $$\kappa $$ κ and with Dirichlet boundary conditions. Their solutions also depend on $$\kappa $$ κ . For $$\theta <0$$ θ < 0 , the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $$\theta > 0$$ θ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $$\theta =0$$ θ = 0 when we send the parameter $$\kappa $$ κ to zero. Indeed, we conjecture that the limiting profile when $$\kappa \rightarrow 0$$ κ → 0 is the one that we should obtain when taking small values of $$\theta >0$$ θ > 0 .