Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow

Abstract

In this contribution, we study scalar nonlocal conservation laws with the $p$-norm. Here, 'nonlocal' means that the velocity of the conservation law depends on an integral term in space. Typically, the nonlocal term consists of integrating the solution in $L^{1}$, whereas here we will study the case when the solution is integrated in the $L^{p}$-norm. We consider even the case of the $L^{p}$ metric when $p\in (0,1)$ and establish, for an initial datum which is uniformly bounded away from zero, the existence and uniqueness of weak solutions. We then demonstrate that there are also solutions to the initial datum being zero under more restrictive assumptions. Furthermore, we investigate the singular limit, i.e., what happens when the nonlocal kernel converges to a Dirac distribution. Indeed, for the one-sided exponential kernel, we recover the (entropy) solution of the corresponding local conservation law for all $p\in(0,\infty)$ with further restrictions for $p\in(0,1)$. This generalizes the celebrated singular limit result for nonlocal conservation laws for $p=1$ significantly and showcases the robustness of the approximation of local conservation laws by nonlocal ones. We investigate also the monotonicity of the solution when assuming that the initial datum is monotone. Finally, we prove the convergence of solutions for $p\rightarrow 0$ on a small time horizon, resulting in a different kind of nonlocal conservation law. Numerical studies showcasing the effect of $p$ on the singular limit convergence and more conclude the contribution.