Compression vs. rarefaction shocks (Ogilvie 2016, §6.3.2)

Proved

AstrophysicalFluidDynamics.shock_compression_or_rarefaction

by Lucas · Oct 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

astrophysicsfluid-dynamicsshock-waves

For a normal non-magnetic shock in a perfect gas with $\gamma>1$ , with positive upstream and downstream states satisfying the Rankine–Hugoniot relations and differing from each other, let $M_1$ , $M_2$ be the upstream and downstream Mach numbers. Then:

$M_1 \neq 1$ (the case $M_1 = 1$ corresponds only to the trivial, continuous solution);
if $M_1 > 1$ , then $M_2 < 1$ , $\rho_2 > \rho_1$ and $p_2 > p_1$ (a compression shock);
if $M_1 < 1$ , then $M_2 > 1$ , $\rho_2 < \rho_1$ and $p_2 < p_1$ (a rarefaction shock).

This is the classification of the non-trivial solutions of the Rankine–Hugoniot relations given after (6.31).

Preamble

import Definitions.Def_AstrophysicalFluidDynamics_ShockDefs
import Mathlib

open Filter Topology

Formal statement

namespace AstrophysicalFluidDynamics

theorem shock_compression_or_rarefaction
    (γ ρ₁ u₁ p₁ ρ₂ u₂ p₂ : ℝ) (hγ : 1 < γ)
    (hρ₁ : 0 < ρ₁) (hu₁ : 0 < u₁) (hp₁ : 0 < p₁)
    (hρ₂ : 0 < ρ₂) (hu₂ : 0 < u₂) (hp₂ : 0 < p₂)
    (hRH : RankineHugoniot γ ρ₁ u₁ p₁ ρ₂ u₂ p₂)
    (hshock : (ρ₂, u₂, p₂) ≠ (ρ₁, u₁, p₁)) :
    machNumber γ ρ₁ u₁ p₁ ≠ 1 ∧
    (1 < machNumber γ ρ₁ u₁ p₁ → machNumber γ ρ₂ u₂ p₂ < 1 ∧ ρ₁ < ρ₂ ∧ p₁ < p₂) ∧
    (machNumber γ ρ₁ u₁ p₁ < 1 → 1 < machNumber γ ρ₂ u₂ p₂ ∧ ρ₂ < ρ₁ ∧ p₂ < p₁) := by sorry

end AstrophysicalFluidDynamics

Source

G. I. Ogilvie, *Astrophysical fluid dynamics* (lecture notes), J. Plasma Phys. 82 (2016) 205820301, https://doi.org/10.1017/S0022377816000489, §6.3.2, p. 39 (paragraph after eq. (6.31))

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Read-back disclosure — NOT blind, NOT independent. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement below, with full knowledge of the source text and of the intended meaning. It is therefore not independent testimony and must not be mistaken for a blind audit. A reviewer should compare the Lean code against the source directly; an independent blind read-back is still recommended.

Fix real numbers $\gamma, \rho_1, u_1, p_1, \rho_2, u_2, p_2$ . Assume $\gamma > 1$ ; $\rho_1 > 0$ , $u_1 > 0$ , $p_1 > 0$ ; $\rho_2 > 0$ , $u_2 > 0$ , $p_2 > 0$ ; and that the three Rankine–Hugoniot equalities hold, namely

\rho_2 u_2 = \rho_1 u_1,\qquad \rho_2 u_2^2 + p_2 = \rho_1 u_1^2 + p_1,

Assume moreover that the triple $(\rho_2,u_2,p_2)$ is not equal to the triple $(\rho_1,u_1,p_1)$ (at least one component differs). Here $M_1 = u_1 / \sqrt{\gamma p_1/\rho_1}$ and $M_2 = u_2/\sqrt{\gamma p_2/\rho_2}$ (real square root; under the positivity assumptions the radicands are positive, so no junk value of the square root or of division by zero arises). Conclusion: all three of the following hold: 1. $M_1 \ne 1$; 2. if $M_1 > 1$ then $M_2 < 1$, $\rho_1 < \rho_2$ and $p_1 < p_2$; 3. if $M_1 < 1$ then $M_2 > 1$, $\rho_2 < \rho_1$ and $p_2 < p_1$.

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph