Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

Vanishing viscosity limit to the 3D Burgers equation in Gevrey class

Pages: 1723 – 1738

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a13


Ridha Selmi (Department of Mathematics, Northern Border University, Arar, Saudi Arabia; Department of Mathematics, University of Gabes, Tunisia; Department of Mathematics, University of Tunis El Manar, Tunis, Tunisia)

Abdelkerim Chaabani (Department of Mathematics, University of Tunis El Manar, Tunis, Tunisia)


We consider the Cauhcy problem to the 3D diffusive periodic Burgers equation. We prove that a unique solution exists on time interval independent of the viscosity and tends, as the viscosity vanishes, to the solution of the limiting equation, the inviscid periodic three-dimensional Burgers equation, in Gevrey–Sobolev spaces. Compared to Navier–Stokes equations, the main difficulties come from the lack of the divergence-free condition which is essential to handle the nonlinear term. Our alternative tool will be to use a change of functions to estimate nonlinearities. Fourier analysis and compactness methods are widely used.


existence and uniqueness, vanishing viscosity limit

2010 Mathematics Subject Classification

Primary 35A01, 35A02. Secondary 35B45, 35B99.

The authors gratefully acknowledge the approval and the support of this research study by the grant number SCI-2017-1-7-F-6901 from the Deanship of Scientific Research at Northern Border University, Arar, K. S. A.

Received 9 September 2019

Accepted 24 October 2019

Published 17 February 2021