vanishing viscosity
Real gases and real fluids have a little internal friction (viscosity) that smears any would-be discontinuity into a thin but smooth transition layer. The idealised conservation law throws that friction away, which is why it allows infinitely thin, ambiguous shocks. Vanishing viscosity is the idea of putting a tiny bit of that friction back, solving the nicer smoothed equation, and then letting the friction shrink to zero — the limit is the physically correct solution.
Concretely, replace u_t + f(u)_x = 0 by the viscous equation u_t + f(u)_x = epsilon u_xx with a small epsilon > 0. The added diffusion term epsilon u_xx is a parabolic regulariser: it smooths instantly, so the viscous equation has a unique smooth solution u_epsilon for all time, with no shocks at all (shocks become thin layers of width of order epsilon). Now send epsilon -> 0. The smooth solutions u_epsilon converge to a limit u, and that limit is precisely the entropy solution of the original conservation law — sharp shocks reappear, but only the admissible ones, because the friction always picked the dissipative side. This is not just a trick: it explains WHERE the entropy condition comes from physically. The entropy inequality is exactly the inequality that survives the limit, inherited from the dissipation of the viscous term.
Why it matters: vanishing viscosity is the conceptual foundation of admissibility — it is the reason 'rarefaction shocks' are unphysical (they cannot arise as a viscous limit) and the reason the second-law-like entropy inequality holds. The same idea, applied to first-order Hamilton-Jacobi equations u_t + H(grad u) = 0, defines the viscosity solution of Crandall and Lions, which is the right notion of weak solution there. The honest limitation: proving that the limit exists and is unique is delicate, easiest for scalar laws (where it succeeds completely) and a deep, partly open story for general systems.
The viscous Burgers equation u_t + u u_x = epsilon u_xx has an exact traveling-wave shock profile, a smooth tanh-shaped step of width about epsilon connecting u_L to u_R and moving at the Rankine-Hugoniot speed (u_L + u_R)/2. As epsilon -> 0 the profile sharpens to the inviscid entropy shock — the inviscid jump is the shadow of this thin viscous layer.
A thin viscous layer of width ~epsilon smooths every shock; its zero-viscosity limit selects the entropy solution.
Adding artificial viscosity is also how numerical schemes stay stable, but real and numerical viscosity differ: too much numerical viscosity over-smears shocks, too little lets unphysical oscillations through. The vanishing-viscosity limit is the ideal that good shock-capturing schemes try to approximate.