Abstract
We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(α2) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial data, the α > 0 equation has classical solutions globally in time. Furthermore, in the zero-α limit, solutions of the regularized equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-α limit satisfies the Oleinik entropy inequality. For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure. We prove the Jacobi identity for this generalized Hamiltonian structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bhat, H., Fetecau, R. A Hamiltonian Regularization of the Burgers Equation. J Nonlinear Sci 16, 615–638 (2006). https://doi.org/10.1007/s00332-005-0712-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-005-0712-7