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

TL;DR

The paper studies scalar nonlocal conservation laws where the velocity depends on the downstream density via an $L^p$-norm of a weighted density. It establishes existence, uniqueness, and a maximum principle for data bounded away from zero, analyzes the singular limit as the kernel concentrates (\(\eta\to 0\)) to the local entropy solution, and develops total-variation estimates and Oleĭnik-type bounds to support compactness. It extends the classical $p=1$ results to general $p\in(0,\infty)$, including the challenging case $p<1$, and investigates the limit $p\to 0$ to a novel nonlocal formulation, with numerical experiments illustrating the theory in traffic-flow contexts. The findings demonstrate the robustness of local conservation-law approximations by nonlocal models and provide a foundation for future work on zero-density data, $p\to\infty$ behavior, and broader kernel classes.

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.

Figures (4)

• Figure 1: The figures show numerical approximations of solutions to \ref{['eq:pnorm_problem']} for the exponential kernel (first and second) constant kernel (third and fourth), with $\eta = 0.5$, for various values of $p$ and two choices of initial data: first and third: monotonically increasing, $q_0(x)=0.5+0.5\chi_{\mathbb R_{>0}}(x)$; second and fourth: monotonically decreasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{<0})}(x)$. The first two figures, i.e., the once with the exponential kernel, illustrate the monotonicity preservation for all $p$ (see \ref{['eq:monotonicity_preserving']}); the third and last figure, i.e., the once with the constant kernel illustrate monotonicity preservation in the increasing case for all $p\in[1,\infty)$, and in the decreasing case for all $p\in (0,1]$.
• Figure 2: The figures show numerical approximations of solutions to \ref{['eq:pnorm_problem']} for the constant kernel, i.e., \ref{['eq:nonlocal_operator_2']} with $\eta = 0.5$, for $p\rightarrow0$ and two choices of initial data: left: monotonically increasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{>0})}(x)$; right: monotonically decreasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{<0})}(x)$. The results clearly show the proven convergence fpr $p\rightarrow 0$ to the equation with nonlinear nonlocality as in \ref{['lem:L_p_p=0']}.
• Figure 3: The figures show numerical approximations of solutions to \ref{['eq:pnorm_problem']} for the constant kernel, i.e., \ref{['eq:nonlocal_operator_2']} with $\eta = 0.5$, for $p\rightarrow \infty$ at final time $T=0.5$ and two choices of initial data: left: monotonically increasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{>0})}(x)$; right: monotonically decreasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{<0})}(x)$. These results show, on the one hand, for mononically inceasing datum
• Figure 4: The figures show numerical approximations of solutions to \ref{['eq:pnorm_problem']} for the constant kernel, i.e., \ref{['eq:nonlocal_operator_2']} with $\eta = 0.5$, for different $p\in\{1,4,16\}$ over the time-horizon $[0,0.5]$ and two choices of initial data: top row: monotonically increasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{>0})}(x)$; bottom row: monotonically decreasing, $q_0(x)=0.5+0.5\chi_{(\mathbb R_{<0})}(x)$ where $p=1$ in the first column, $p=4$ in the second column and $p=16$ in the last column.

Theorems & Definitions (55)

• Definition 2.1: The model class considered
• Remark 2.2: Kernel with $p-$th power
• Remark 2.4: The meaning of $q_{\min}$, $V$, $\gamma$
• Definition 2.5: Weak solution for \ref{['defi:model_class']}
• Theorem 2.6: Existence, uniqueness, maximum principle and continuity w.r.t. initial datum
• proof
• Lemma 3.1: Approximation of bounded functions with bounded total variation on $\mathbb R$
• proof
• Lemma 3.3: A PDE entirely in the nonlocal term
• proof