Weak asymptotic analysis approach for first order scalar conservation laws with nonlocal flux

Abstract

In this work, we expand on the weak asymptotic analysis originally proposed in Abreu et al. (2024) for the investigation of scalar equations and systems of conservation laws, extending it to encompass scalar equations with nonlocal fluxes. Subsequently, we apply this refined methodology to explore a specific class of nonlocal scalar conservation laws ${\partial }_t{\rho }_{\eta }+{\partial }_x\left({\rho }_{\eta }f\left({\rho }_{\eta }\right)V({\omega }_{\eta }\ast {\rho }_{\eta })\right)=0$, where $\eta >0$ and ${\omega }_{\eta }\left(\cdot \right)={\eta }^{-1}\omega (\cdot /\eta )$ represents a rescaled asymmetric convolution kernel. Essentially, the extension of the weak asymptotic analysis to nonlocal scalar conservation laws yields a family of approximate solutions that exhibit smoothness in time, local integrability, and essential boundedness in the spatial variable. This notable property facilitates the application of $L^p$-compactness arguments, leading to the convergence of a solution family. We further extend the concept of weak asymptotic solutions to a broader class of nonlocal scalar conservation laws by constructing a family of ordinary differential equations, providing a set $\left\{{\rho }_{\eta }(\cdot ,ϵ)\right\}$ of asymptotically approximated solutions. These solutions belong to the space $L^1\left(R\right)\cap L^{\infty }\left(R\right)$ and, in an asymptotic sense, adhere to Kruzhkov’s entropy inequalities. These characteristics, coupled with a suitable spatial and temporal modulus of continuity (which is independent of $ϵ$ but dependent on $\eta$, representing the horizon for capturing multiple scales of interactions in the nonlocal model), enable us to extract a subsequence converging to the weak and weak entropy solution of the nonlocal scalar conservation law (1). Furthermore, in scenarios where $f\left(\rho \right)=1$ in Eq. (1), we illustrate that the approximate solutions converge, in the weak asymptotic sense, to the weak asymptotic solution associated with the corresponding local scalar conservation law counterpart.