Existence and uniqueness of entropy solutions, and nonlocal L1-stability to the local limit and numerical applications

In this work, we present a two-fold result: (1) the generalization of a nonlocal pair interaction model recently introduced in \cite{du2017nonlocal}  and (2) the construction of a novel semi-discrete approach by using the weak asymptotic method as introduced in \cite{abreu2016weak}. We ensure that the new semi-discrete scheme satisfies an entropy inequality that recovers the local entropy inequality associated to the corresponding nonlocal model, subject to any $C^1(\mathbb{R}^n)$  flux function with initial data $u_0 \in L^1(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$. We will also state and explain the main ideas of some results (this is a work in progress) that allows us to remove the boundedness of the initial data instead of assuming the more restrictive assumption Total Variation Bounded for the proposed new class of semi-discrete convergent schemes. Indeed, by using results of the work \cite{crandall1972semigroup}, we also ensure existence, uniqueness and $L^1(\mathbb{R}^n)$ stability with initial data belonging only to $L^1(\mathbb{R}^n)$. Some numerical results are presented and discussed in the efforts to justify the reliability, but also verifying the theory acquired.