학술논문
Long-time convergence of a nonlocal Burgers' equation towards the local $N$-wave.
Document Type
Journal
Author
Coclite, Giuseppe Maria (I-PBAR-MMM) AMS Author Profile; De Nitti, Nicola (D-ERL) AMS Author Profile; Keimer, Alexander (D-ERL) AMS Author Profile; Pflug, Lukas (D-ERL) AMS Author Profile; Zuazua, Enrique (D-ERL) AMS Author Profile
Source
Subject
35 Partial differential equations -- 35L Hyperbolic equations and systems
35L65Conservation laws
35L65
Language
English
Abstract
In this paper the authors study the large time behaviour of solutions of a non-local regularisation of the inviscid Burgers equation. They prove that the long-time asymptotic profile of the solution is the $N$-wave entropy solution of the inviscid Burgers equation, when the initial datum is non-negative, bounded and integrable. The convergence is done in $L^p$ norm for $1\leq p<\infty$. The main ingredients of the proof are a scaling argument and a non-local Oleinik-type estimate. The result is also illustrated by several numerical simulations. The concluding section contains several open problems.