Burgers' Equation with Vanishing Hyper-Viscosity

Fast Communication Burgers’ Equation with Vanishing Hyper-viscosity∗

We prove that bounded solutions of the vanishing hyper-viscosity equation, ut + f(u)x + (−1)sε∂2s x u = 0 converge to the entropy solution of the corresponding convex conservation law ut +f(u)x = 0, f ′′ > 0. The hyper-viscosity case, s > 1, lacks the monotonicity which underlines the Krushkov BV theory in the viscous case s = 1. Instead we show how to adapt the Tartar-Murat compensated compact...

Optimal Control and Vanishing Viscosity for the Burgers Equation

We revisit an optimization strategy recently introduced by the authors to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the one-dimensional (1-D) Burgers equation. This new descent strategy, called the alternating descent method, in the inviscid case, distinguishes and alternates descent di...

Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods

We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the co...

Adaptive control of Burgers' equation with unknown viscosity

In this paper, we propose a fortified boundary control law and an adaptation law for Burgers’ equation with unknown viscosity, where no a priori knowledge of a lower bound on viscosity is needed. This control law is decentralized, i.e., implementable without the need for central computer and wiring. Using the Lyapunov method, we prove that the closed-loop system, including the parameter estimat...

Numerical observers with vanishing viscosity for the 1d wave equation