G.M. Coclite, N. De Nitti, E. Zuazua (2023) Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave, CVGMT Nonlinearity, Vol. 36, No. 11, pp. 5998-6019, DOI: 10.1088/1361-6544/acf01d
Abstract. We study the long-time behavior of the unique weak solution of a nonlocal regularization of the (inviscid) Burgers’ equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the “N -wave” entropy solution of the Burgers’ equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.