CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The expansion shock is weak but not entropic

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the analytic prerequisites used below.

Let $f (u) = \frac{1}{2} u^{2}$ and let the Riemann data be $u_{0} (x) = 0$ for $x < 0$ and $u_{0} (x) = 1$ for $x > 0$ . The increasing-state jump with left state $u^{-} = 0$ , right state $u^{+} = 1$ , and speed $s = \frac{1}{2}$ is $u (t, x) = {\begin{cases} 0, & x < t / 2, \\ 1, & x > t / 2. \end{cases}$ It is a distributional weak solution with those data because the Rankine--Hugoniot condition holds. It is not a Kruzhkov entropy solution: for the convex entropy $η (u) = \frac{1}{2} u^{2}$ with flux $q (u) = \frac{1}{3} u^{3}$ , the jump production is $[q] - s [η] = \frac{1}{3} - \frac{1}{4} = \frac{1}{12} > 0$ . It also fails the Kruzhkov test $k = \frac{1}{2}$ : $q_{k} (0) = \frac{1}{8}$ , $q_{k} (1) = \frac{3}{8}$ , and $[η_{k}] = 0$ , so $[q_{k}] - s [η_{k}] = \frac{1}{4} > 0$ . For the same Riemann data, The Burgers rarefaction Riemann solution gives an entropy solution, so the Rankine--Hugoniot condition alone admits both the expansion shock and the entropy rarefaction (The Rankine--Hugoniot jump condition in space--time normal form, Kruzhkov entropy solutions, Distributional weak solutions of the Cauchy problem).

Facts & Assumptions

Given: Countable Choice, the flux $f (u) = \frac{1}{2} u^{2}$ , the Riemann datum $u_{0} = 1_{(0, \infty)}$ , the expansion-shock profile $u$ with speed $s = \frac{1}{2}$ , and a test function $φ \in C_{c}^{\infty} (Π_{T})$ .

[F1]

Interface computation: for a single jump with traces $u^{-}$ on the left and $u^{+}$ on the right of the ray $x = s t$ , the weak residual against a test concentrated near the ray is proportional to $[f] - s [u]$ with $[h] = h (u^{+}) - h (u^{-})$ ; the Rankine--Hugoniot condition $s [u] = [f]$ makes it vanish, so a jump profile with that condition is a distributional weak solution (The Rankine--Hugoniot jump condition in space--time normal form, Distributional weak solutions of the Cauchy problem).

[F2]

Entropy production at a jump: for an entropy pair $(η, q)$ the distribution $\partial_{t} η (u) + \partial_{x} q (u)$ equals $([q] - s [η]) δ_{Γ}$ , so the entropy inequality holds iff $[q] - s [η] \leq 0$ ; this is the general chord condition, and for the Kruzhkov pairs $η_{k} (s) = ∣ s - k ∣$ , $q_{k} (s) = sgn (s - k) (f (s) - f (k))$ the same test applies (The convex entropy condition for a single shock is the chord condition, Kruzhkov entropy solutions, Convex entropy--entropy flux pairs).

[F3]

For the same Riemann data the Burgers rarefaction $u (t, x) = 0$ for $x \leq 0$ , $x / t$ for $0 < x < t$ , $1$ for $x \geq t$ is the entropy solution (The Burgers rarefaction Riemann solution).

Proof

technique · direct

1.1F1

The shock is a weak solution. With $u^{-} = 0$ , $u^{+} = 1$ : $[u] = 1$ and $[f] = \frac{1}{2}$ , so $s = \frac{1}{2}$ satisfies $s [u] = \frac{1}{2} = [f]$ . By [F1] the interface coefficient of the weak residual vanishes, so $u$ is a distributional weak solution of $u_{t} + \partial_{x} (\frac{1}{2} u^{2}) = 0$ with datum $u_{0}$ ; the strong local $L^{1}$ trace is immediate because $u (t, \cdot)$ equals the step datum except on the interval $(0, t / 2)$ of length $t / 2$ .

1.2F2

Failure of the convex entropy inequality. Take $η (u) = \frac{1}{2} u^{2}$ and $q (u) = \frac{1}{3} u^{3}$ , so that $q^{'} (u) = u^{2} = η^{'} (u) f^{'} (u)$ and $(η, q)$ is a convex entropy pair. The jump coefficients are $[η] = \frac{1}{2} (1^{2} - 0^{2}) = \frac{1}{2}$ and $[q] = \frac{1}{3} (1^{3} - 0^{3}) = \frac{1}{3}$ , so by [F2] the entropy production is $[q] - s [η] = \frac{1}{3} - \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{3} - \frac{1}{4} = \frac{1}{12} > 0$ . The entropy inequality fails strictly at the jump.

1.3F2

Failure in the Kruzhkov family. For $k = \frac{1}{2}$ , $η_{k} (0) = ∣ 0 - \frac{1}{2} ∣ = \frac{1}{2} = η_{k} (1)$ , so $[η_{k}] = 0$ ; and $q_{k} (s) = sgn (s - \frac{1}{2}) (\frac{1}{2} s^{2} - \frac{1}{8})$ gives $q_{k} (0) = (- 1) (- \frac{1}{8}) = \frac{1}{8}$ and $q_{k} (1) = \frac{1}{2} - \frac{1}{8} = \frac{3}{8}$ , so $[q_{k}] = \frac{3}{8} - \frac{1}{8} = \frac{1}{4}$ . Hence $[q_{k}] - s [η_{k}] = \frac{1}{4} > 0$ , directly violating a Kruzhkov entropy inequality that every entropy solution must satisfy. Equivalently, the chord through $(0, 0)$ and $(1, \frac{1}{2})$ lies above the convex parabola, so the one-sided chord condition of [F2] fails for the upward jump $u^{-} = 0 < u^{+} = 1$ .

2.1F2F3step 1.1step 1.2step 1.3∎

Conclusion. The expansion shock is a distributional weak solution with the prescribed data but fails the entropy condition, while the rarefaction of [F3] is the entropy solution of the same Riemann problem. Thus the Rankine--Hugoniot condition and the weak formulation alone admit non-entropic solutions, and an entropy selection principle is needed.

Depends on

Used by

Rankine--Hugoniot alone does not give uniqueness Counterexample

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. A. Chechkin and A. Yu. Goritsky (translated by B. Andreianov), “S. N. Kruzhkov’s lectures on first-order quasilinear PDEs,” in Analytical and Numerical Aspects of PDEs, de Gruyter 2009, complete lecture-notes text (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations, University of Toronto, current complete 415-page PDF (standard reference, not scraped)