Nonlinear wave steepening, characteristics, entropy condition, and viscous shock structure
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Burgers equation:∂u/∂t + u ∂u/∂x = ν ∂²u/∂x²
The nonlinear advection term u ∂u/∂x causes steeper regions to travel faster, leading to shock formation (wave breaking) at finite time t* = 1/max(−∂u/∂x₀). The viscous term regularizes the shock into a thin layer of width ~√(ν).
The entropy condition (Oleinik/Lax) selects the physically correct shock: characteristics must impinge on the shock from both sides. Inviscid shocks satisfy the Rankine-Hugoniot jump condition: shock speed = (u_L + u_R)/2.