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.