A Hamiltonian Regularization of the Burgers Equation

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.

Authors and Affiliations

1. Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

   H.S. Bhat

2. Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA

   R.C. Fetecau

Corresponding author

Correspondence to H.S. Bhat.

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