ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Burgers shock entropy and Lax speeds

Example

Assume the Axiom of Countable Choice. Let $T > 0$ , let $a, b \in R$ with $a > b$ , and set $F (z) = z^{2} / 2$ and $s = (a + b) / 2$ . For $t > 0$ define $u (t, x) = a$ for $x < s t$ and $u (t, x) = b$ for $x > s t$ ; assign any value at $x = s t$ . Let $u_{0} (x) = a$ for $x < 0$ and $u_{0} (x) = b$ for $x > 0$ , with an arbitrary value at $x = 0$ . Then $u$ satisfies the initial-value weak identity for $u_{t} + \partial_{x} F (u) = 0$ . For every convex $η \in C^{2} (R)$ and every $q \in C^{1} (R)$ with $q^{'} = η^{'} F^{'}$ , it satisfies the initial-value entropy inequality with the convention that the distributional entropy residual is nonpositive. It also satisfies every Kružkov inequality for $η_{k} (z) = ∣ z - k ∣$ and $q_{k} (z) = sgn (z - k) (F (z) - F (k))$ , where $sgn (0) = 0$ . Its shock speed obeys $F^{'} (b) < s < F^{'} (a)$ .

Example references

strictly convex scalar Riemann construction

published Burgers shock and rarefaction theorem

initial weak and entropy formulations

Axiom of Countable Choice

Facts & Assumptions

Given: Countable Choice; the states $a > b$ ; the flux, profile, initial data, and entropy pairs in the Example.

[F1]

Strictly convex scalar Riemann construction: for a $C^{2}$ scalar flux with $F^{''} > 0$ between the two states, the stated shock profile is an initial-value weak solution, satisfies every compatible convex $C^{2}$ entropy inequality and every Kružkov inequality, and has the scalar Lax speed inequalities.

[F2]

Riemann shock and rarefaction waves for $u_{t} + (u^{2} / 2)_{x} = 0$ are weak solutions, and the shock is an entropy solution: assuming Countable Choice, for $F (z) = z^{2} / 2$ and left state greater than right state, the same shock profile satisfies the Lebesgue weak formulation and all stated convex $C^{1}$ and Kružkov entropy inequalities.

[F3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the choice assumption used by the imported Lebesgue weak formulation.

[F4]

Distributional systems and entropy flux pairs: for smooth tests compactly supported in $[0, T) \times R$ , the scalar initial weak identity is $\int_{0}^{T} ⁣ \int_{R} (u φ_{t} + F (u) φ_{x}) d x d t + \int_{R} u_{0} φ (0, x) d x = 0$ ; the entropy inequality replaces $(u, F, u_{0})$ by $(η, q, η (u_{0}))$ , requires a nonnegative test, and requires the whole integral to be nonnegative.

[A1]

$a > b$ .

Proof

1.1givenF1F2F3F4A1

The Burgers flux is $C^{2}$ and $F^{''} (z) = 1 > 0$ on $[b, a]$ . Therefore [F1] applies to these states and this profile, giving the initial weak identity, the compatible convex entropy inequalities, the Kružkov inequalities, and the scalar Lax pattern in the formulations fixed by [F4]. The imported formulation in [F2] has the same profile and explicitly assumes the Countable Choice recorded in [F3].

2.1givenF2A1step 1.1algebra∎

Directly, $s = (a^{2} - b^{2}) / (2 (a - b)) = (a + b) / 2$ , while $F^{'} (b) = b$ and $F^{'} (a) = a$ ; hence $F^{'} (b) < s < F^{'} (a)$ . For $b \leq k \leq a$ , put $C (z) = F (b) + s (z - b)$ , the chord of $F$ joining the endpoint fluxes. Then $F (k) - C (k) = \frac{1}{2} (k - a) (k - b)$ , so the Kružkov interface coefficient is $[q_{k}] - s [η_{k}] = 2 (F (k) - C (k)) = (k - a) (k - b) \leq 0$ ; for $k \notin [b, a]$ , $η_{k}$ is affine on the state interval and $q_{k}$ is a constant-sign multiple of $F - F (k)$ , so the coefficient is zero by the Rankine–Hugoniot equality. This agrees with the nonpositive residual convention.

Burgers shock entropy and Lax speeds

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources