Burgers Equation: Shock Formation
u_t + u·u_x = ν·u_xx — steepening, shocks, and viscous regularization
Inviscid (ν=0): characteristics cross → multivalued → shock forms in finite time t* = 1/max(-u₀').
Viscous case: shock width ∝ ν. Hopf-Cole transform: u = −2ν(∂lnΦ/∂x) linearizes to heat equation.