3.3.2 Vanishing viscosity approach: an example

Inviscid Burgers' equation: $u_t + uu_x=0$

Viscous Burgers' equation: $u_t + uu_x = \epsilon u_{xx}$

The correct physical behaviour is determined adopting the vanishing viscosity approach

The inviscid equation is a model of the viscous one valid only for small $\epsilon$ and smooth $u$

If the initial data is smooth and $\epsilon$ very small the term $\epsilon u_{xx}$ is negligible before the wave begins to break. The solutions to both PDEs look identical

As the wave begins to break the $u_{xx}$ term grows much faster that the $u_x$ term and the rhs begins to play a role

This term keeps the solution smooth for all time preventing the breakdown of solutions that occurs for the hyperbolic problem