Mathematics Colloquia and Seminars

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We consider the following scalar equation: u_t + uu_x − α^2 u_{txx} − α^2 uu_{xxx} = 0, (1) with α > 0. We may rewrite (1) as v_t + uv_x = 0, (2) where v = u − α^2 u_{xx} , (3) One can think of the equation (2) as the inviscid Burgers equation, v_t + v v_x = 0, where the convective velocity in the nonlinear term is replaced by a smoother velocity field u. This idea goes back to Leray (1934) who employed it in the context of the incompressible Navier-Stokes equation. Leray’s program consisted in proving existence of solutions for his modified equations and then showing that these solutions converge, as α ↓ 0, to solutions of Navier-Stokes.

We apply Leray’s ideas in the context of Burgers equation. We show strong analytical and numerical indication that (2)-(3) (or equivalently, (1)) represent a valid regularization of the Burgers equation. That is, we claim that solutions uα (x, t) of (1) converge strongly, as α → 0, to unique entropy solutions of the inviscid Burgers equation. Interestingly, for all α > 0, the regularized equation possesses a Hamiltonian structure.

We also study the stability of the traveling waves for equation (1). These traveling waves consist of ”fronts,” which are monotonic profiles that connect a left state to a right state. The front stability results show that the regularized equation (1) mirrors the physics of rarefaction and shock waves in the Burgers equation.