the Euler equations of gas dynamics
/ OY-ler /
Strip a flowing gas of its tiny internal friction and heat conduction, and what governs it is the cleanest law of motion for a continuous fluid: conserve mass, conserve momentum, conserve energy. The Euler equations of gas dynamics are this trio of conservation laws — the founding example of a nonlinear hyperbolic system, and the place where shocks were first understood physically.
In one space dimension the conserved quantities are density rho, momentum rho u, and total energy E, and the system is rho_t + (rho u)_x = 0, (rho u)_t + (rho u^2 + p)_x = 0, E_t + (u(E + p))_x = 0, closed by an equation of state relating pressure p to density and energy (for an ideal gas, p = (gamma - 1)(E - rho u^2/2)). Written as U_t + f(U)_x = 0, the flux Jacobian f'(U) has three real eigenvalues — the characteristic speeds u - a, u, u + a, where a = sqrt(gamma p / rho) is the local speed of sound. So the system is hyperbolic (strictly so while a > 0). Its three wave families are beautifully different: the u +/- a fields are genuinely nonlinear and carry shocks and rarefactions (sound waves that can steepen into shocks), while the middle u field is linearly degenerate and carries a contact discontinuity, a slip surface across which density and temperature jump but pressure and velocity do not.
Why it matters: the Euler equations are the testbed and motivation for the entire theory — Riemann problem, Rankine-Hugoniot, entropy condition, and vanishing viscosity were all forged to understand gas shocks. They model supersonic flight, explosions, astrophysical jets and stellar winds. And they remain humbling: global existence of solutions for general large initial data in multiple dimensions is OPEN, and even in 1-D the large-data theory is delicate. Physical entropy here is a genuine convex entropy, which is why the second-law selection of admissible shocks has real thermodynamic content.
The Riemann problem for the 1-D Euler equations is the shock tube: a tube with high-pressure gas on the left and low-pressure gas on the right, separated by a membrane that bursts at t = 0. The solution has exactly three waves — a shock running into the low-pressure gas, a contact discontinuity behind it, and a rarefaction fan expanding back into the high-pressure gas — the canonical three-wave pattern u - a, u, u + a.
Three characteristic fields u - a, u, u + a give the shock-tube's shock, contact, and rarefaction.
The Euler equations are the inviscid (zero-viscosity) limit of the Navier-Stokes equations; the missing viscosity is exactly why shocks are infinitely thin here and why an entropy condition is needed to select them. Adding viscosity back gives Navier-Stokes, whose 3-D global regularity is itself a famous open (Millennium Prize) problem.