PDE 4: Burgers' equation

In 1D: \[u_t + u u_x = \varepsilon u_{x x} \] with initial condition \(u(x,0)\). Consider e.g. Dirichlet boundary conditions \(u(0,t) = g_1(t), u(1,t) = g_2(t).\) What happens when \(\varepsilon \rightarrow 0?\)

Google: (inviscid/viscous) Burgers' equation, shock, diffusion.

Challenges: nonlinearity.