On multidimensional nonlocal conservation laws with BV kernels

TL;DR

This work establishes local-in-time well-posedness for multidimensional nonlocal conservation laws with rough kernels, proving existence for $\eta \in BV$ or $W^{1,p}$ and initial data in $L^\infty$ or $L^q$, respectively. It shows that finite-time blow-up can occur for rough kernels and that different smooth kernel approximations can yield distinct post-blow-up measure limits, ruling out natural continuation strategies. A Grönwall-type framework delivers uniqueness under precise regularity and decay assumptions on $\eta$ and $u_0$, with a stability control via characteristics. The authors also construct a concrete one-dimensional blow-up example and demonstrate nonuniqueness in the post-blow-up regime by introducing a family of kernels $\eta_n^\alpha$, whose limits concentrate on moving lines, thereby showing lack of a canonical selection after blow-up. These results have implications for pedestrian traffic and crowd dynamics models, highlighting critical differences between rough and smooth kernels in nonlocal conservation laws.

Abstract

We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumptions on the convolution kernel. In contrast to the case of a smooth kernel, in general the solution experiences finite-time blow-up. We provide an explicit example showing that solutions corresponding to different smooth approximations of the convolution kernel in general converge to different measures after the blow-up time. This rules out a fairly natural strategy for extending the notion of solution of the nonlocal conservation law after the blow-up time.

Figures (1)

• Figure 1: The function $\eta^\alpha_n$ defined as in \ref{['e:etaalpha']}

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['t:ex']}, item (i)