Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

Inviscid limit to the shock waves for the fractal Burgers equation

Pages: 1477 – 1491



Sona Akopian (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Moon-Jin Kang (Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea)

Alexis Vasseur (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)


We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence, for large initial perturbations.


fractal Burgers equation, fractional Laplacian, scalar conservation laws, shock waves, inviscid limit, large perturbation, relative entropy

2010 Mathematics Subject Classification

35L65, 35L67, 35S11, 47G20, 47G30

Received 5 August 2019

Accepted 6 January 2020

Published 4 November 2020