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Failure of the least action admissibility principle in the context of the compressible Euler equations

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Pith's one-line read For every final time T, the least action admissibility principle rules out the standard one-dimensional Euler solution.

desk verdict A targeted counterexample to the least action principle; the algebra checks, but the finite-time convex-integration existence theorem is the weak spot. read the letter →

arxiv 2502.09292 v1 pith:LE4UPOYX submitted 2025-02-13 math.AP

Simon Markfelder , Valentin Pellhammer This is my paper · ORCID
classification math.AP MSC 76N1035Q3135D3035L65
keywords compressibleEulerequationsleastactionadmissibilityprincipleRiemannproblemconvexintegrationadmissibleweaksolutionsnon-uniquenessselectioncriteriaenergyinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests the least action admissibility principle, a selection rule proposed for the multi-dimensional barotropic compressible Euler equations: among all admissible weak solutions with the same initial data, the physical one should minimize the action. The authors prove the principle fails. For initial data that are constant on each side of a horizontal line (a Riemann problem) and for every final time $T>0$, they exhibit another admissible weak solution, built by the solution-manufacturing technique of convex integration, whose action is strictly smaller than that of the standard one-dimensional shock solution. Thus the least action principle rules out the solution that is intuitively physical; the paper concludes that either intuition must be revised or the principle must be discarded.

What carries the argument

The load-bearing construction is Corollary 2.4's gluing recipe: take a convex integration solution on $[0,T_0]$ (with $\varrho=\varrho_1$ and $|u|^2=C_1$ on the wedge $\Gamma_1$), then continue with the classical solution from the trace data at $t=T_0$. The algebraic conditions of Proposition 2.1—Rankine–Hugoniot relations on the two interfaces, the subsolution inequality $C_1-(u_1)^2-(v_1)^2>0$, and the two entropy/energy inequalities at the interfaces—guarantee that infinitely many such solutions exist. The identity that does the work is the energy jump $\frac12\varrho_1C_1\mapsto\frac12\varrho_1|u_1|^2$ at $t=T_0$: because $C_1>|u_1|^2$, the jump is downward, which preserves the energy inequality and lowers the action density. The action functional here is $A[\varrho,u]=\int_0^T\int(\frac12\varrho|u|^2-P(\varrho))\,dy\,dx\,dt$, and the comparison reduces to quadrature of the piecewise-linear action density, yielding the $T^2$ coefficients in Lemma 3.1.

What would settle it

Compute the left-hand sides of the Rankine–Hugoniot and subsolution inequalities in Proposition 2.1 for the constants in (3.1) and Section 3.2, namely $\varrho_1=3$, $u_1=0$, $v_1=0$, $C_1=\frac{1121\sqrt{1281}}{20}+\frac{28037}{12}$, $\gamma_1=-\frac{1121\sqrt{1281}}{40}-\frac{28013}{24}$, $\delta_1=0$, and $\mu_0=-\frac{57\sqrt{35}}{20}-\frac{59\sqrt{915}}{60}$; if any required inequality is violated, the constructed competitor is not admissible. Alternatively, evaluate $K_{\mathrm{ex}}-K_{1\mathrm d}$ from Lemma 3.1 using $\varrho_M<94$ and $\sigma<11/10$; the theorem's claim is exactly that this number is negative.

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Extended reading notes

Core claim

The central claim is Theorem 1.4: for each $T>0$, there exist Riemann initial data $(\varrho_\pm,u_\pm)$ such that the one-dimensional (1-D) solution to (1.1) with pressure $p(\varrho)=\varrho^2$ does not fulfill the least action admissibility principle. The proof selects the explicit symmetric data (3.1), sets $T_0=T/2$, and uses convex integration on $[0,T_0]$ to form a competitor $(\varrho_{\mathrm{ex}},u_{\mathrm{ex}})$ that is constant, $\varrho_1=3$ with $|u|^2=C_1$, inside the wedge between two speeds $\mu_0t<y<\mu_1t$, then glues the classical two-shock 1-D solution on $[T_0,T]$. Lemma 3.1 gives the actions $A[\varrho_{\mathrm{ex}},u_{\mathrm{ex}}]=K_{\mathrm{ex}}T^2$ and $A[\varrho_{1\mathrm d},u_{1\mathrm d}]=K_{1\mathrm d}T^2$ with explicit constants, and since $K_{\mathrm{ex}}-K_{1\mathrm d}<0$, the competitor wins for every $T$. The competitor is an admissible weak solution in the sense of Definition 1.1; the key to admissibility is that the kinetic energy jumps downward, not upward, at the gluing time.

Load-bearing premise

Everything hangs on the finite-time convex integration existence theorem (Proposition 2.1) holding with the stated data on the interval $[0,T_0]$, and on the 'easily verified' algebraic inequalities for the explicit constants being true; if either fails, the competitor is not an admissible weak solution and the action comparison proves nothing.

Editorial extensions

If this is right

  • For every $T>0$ the 1-D solution fails the least action admissibility principle, so the principle does not single out the solution conventionally regarded as physical.
  • The failure is of the same type as for the global and local maximal dissipation criteria: in each case a convex-integration solution beats the 1-D solution on the criterion's own score.
  • Which solution is preferred depends on the final time: before $T_0$ the competitor is more expensive, and only after the downward kinetic-energy jump does its cumulative action fall below the 1-D solution's.
  • The downward energy jump at $t=T_0$ is the decisive mechanism; without it, the action comparison would go the other way.
  • Accepting the 1-D solution as physical forces one to discard the least action admissibility principle; retaining the principle forces one to revise what counts as the physical solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because all inequalities in Proposition 2.1 are strict, the explicit numbers are unlikely to be isolated: small perturbations of the Riemann data should still satisfy the conditions, giving an open set of counterexamples.
  • The construction is written for $p(\varrho)=\varrho^2$, but the downward-jump gluing mechanism itself is not tied to that exponent; a natural test is whether the same recipe refutes the principle for other polytropic pressures or for the full Euler system.
  • The horizon dependence is a structural warning: any selection rule defined by minimizing a time-integrated functional on a fixed horizon $[0,T]$ can be gamed by inserting an energy-decreasing jump after the interval where the competitor is more expensive.
  • Proponents of the principle could try to rescue it by restricting admissible competitors, for example by demanding structural symmetry or a bound on oscillation; the paper does not explore such restrictions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the least action admissibility principle proposed by Gimperlein, Grinfeld, Knops, and Slemrod as a selection criterion for admissible weak solutions of the two-dimensional barotropic compressible Euler equations. For the pressure law p(ρ)=ρ^2, the authors exhibit Riemann initial data with symmetric left and right states and, for every T>0, construct a competitor weak solution by gluing a convex-integration solution on [0,T/2] to an explicit two-shock solution on [T/2,T]. They compute the actions of both the self-similar 1-D solution and the competitor, obtaining the explicit constants K1d and Kex in Lemma 3.1, and show Kex<K1d using the bounds ρM<94 and σ<11/10. The conclusion is that the 1-D solution does not satisfy the least action principle, so the principle must either be rejected or the physical intuition reconsidered.

Significance. If the core existence input is valid, the theorem is a clean negative answer to a recently proposed admissibility criterion, and it complements earlier failures of the maximal dissipation criterion by the first author. The action comparison is explicit and self-contained, with concrete constants and no free parameters beyond the advertised ones. The main weakness is that the finite-time convex-integration Proposition 2.1 is quoted rather than proved; the result is conditional on that ingredient. The paper is a meaningful contribution to the ongoing assessment of selection criteria for multi-dimensional conservation laws.

major comments (3)
  1. [Section 2, Proposition 2.1 and Remark 2.2] Proposition 2.1 is the load-bearing existence theorem for the competitor on [0,T0], and in particular it prescribes terminal traces (2.2)-(2.4) so that the glued function (2.9) satisfies the weak formulation across t=T0. The proof is not given: the paper refers to [2, Props. 3.6 and 5.1], [3, Props. 3.1 and 4.1] and [12, Thm. 7.3.4 and Prop. 7.3.5], and Remark 2.2 asserts that the finite-interval version is easy to observe. The cited works are global-in-time theorems, and the trace condition at t=T0 is a stronger boundary-prescription property that is not obviously contained in them. Since the momentum trace in Γ1 is forced to the constant ρ1u1 while |u|^2=C1 a.e., this needs a proof or a precise reference with statement. Without it, the glued competitor is not known to be a weak solution and Lemma 3.1 does not refute the least action principle.
  2. [Section 2, Lemma 2.3] The proof of Lemma 2.3 invokes Glimm's theorem [10] for arbitrary one-dimensional initial data (2.5). Glimm's classical theorem requires sufficiently small total variation, while the data here has jumps of size 2 from density 1 to 3; the stated general form is not a consequence of [10]. For the specific data chosen in Section 3.2 an explicit two-shock solution is available and is in fact described there, so the gap is fixable, but the proof as written does not establish the lemma in the needed generality.
  3. [Section 3.2-3.3] Several load-bearing algebraic verifications are asserted without computation: the parameter values in (3.3) are said to satisfy all equations and inequalities of Proposition 2.1 (one can easily verify), the post-T0 solution is said to consist of two shock waves (one can easily verify), and in the proof of Theorem 1.4 the inequality Kex-K1d<0 is said to follow from (3.2) (which can easily be verified). These checks are central to the construction and to the final comparison; at least the explicit verification of the subsolution inequalities (2.1) and of the final numerical inequality should be included, for example in an appendix.
minor comments (5)
  1. [Section 1, Definition 1.1] The word 'followng' is a typo for 'following'.
  2. [Section 3.1] The bounds ρM<94 and σ<11/10 in (3.2) are stated as simple to verify; since they are used in the proof of Theorem 1.4, include the short derivation.
  3. [Section 4] The statements (4.1) and (4.2) about the action for t<T0 and for t>T0 are presented without proof; as they are not needed for Theorem 1.4, label them as heuristic remarks or provide supporting references.
  4. [Section 3.3, Lemma 3.1] In the proof of Lemma 3.1, the factor 4 in (3.4) and (3.5) is explained only implicitly; a sentence describing the symmetry in y and the x-periodicity would improve readability.
  5. [Section 2, Proposition 2.1] The notation u1 is used both for a two-dimensional velocity vector and for its horizontal component (u1,v1); an explicit notational remark would remove possible ambiguity for the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity: the competitor is built from independent convex-integration theorems, and the finite-time trace adaptation is an unproven gap rather than a circular step.

full rationale

The proof does not reduce to its own inputs. The target claim is that for explicit Riemann data (3.1), the 1-D solution has larger action than a constructed admissible weak solution. The constructed competitor is built by gluing a convex-integration existence result (Proposition 2.1) on [0,T0] to a classical Glimm solution on [T0,T] (Lemma 2.3). Proposition 2.1 is quoted from [2,3,12]; although [12] is the first author's monograph, [2] and [3] are independent sources, so the self-citation is not load-bearing. The action comparison in Lemma 3.1 is an explicit algebraic computation using the constants listed in Section 3.2, and the final inequality Kex < K1d is checked directly using (3.2). No parameter is fitted to the least-action principle, and the construction does not presuppose that the 1-D solution fails the principle. The genuine risks are omitted proofs: Remark 2.2 asserts the finite-time adaptation of the cited convex-integration method with prescribed terminal traces is easy to observe, and Section 3.2 states that the algebraic subsolution conditions and the two-shock fan can easily be verified. If those assertions fail, the glued function would not be an admissible weak solution and the counterexample would not be valid. However, this is a correctness gap, not circularity: it does not make the theorem equivalent to its assumptions. The self-citations to [12] and [13] provide background and one quoted theorem, but the central load-bearing theorem is also supported by independent references. Score 2 reflects the presence of minor self-citation without circular reduction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim needs (i) the convex integration theorem, quoted from the authors' own book and from Chiodaroli-De Lellis-Kreml; (ii) classical 1-D Riemann solvability; (iii) the hand-picked constants above. No new physical entities are introduced. The paper-specific choices are all explicit, so an independent reader can re-check them.

free parameters (5)
  • Riemann state normal velocity v_- (and v_+ = -v_-) = 57√35/10 + 59√915/30
    Hand-chosen initial data; the proof needs the 1-D shock to have ρM<94 and σ<11/10, and this value drives the final action inequality sign.
  • Middle density ϱ1 in the convex integration strip Γ1 = 3
    Hand-picked; sets the action density awild = (3/2)C1 - 9 and enters all action terms in Γ1.
  • Kinetic energy level C1 in Γ1 = 1121√1281/20 + 28037/12
    Hand-picked; must make the subsolution condition (2.1) hold, i.e. C1 > |u1|^2 = 0, and determines the downward kinetic energy jump at t=T0.
  • Internal momentum parameter γ1 = -1121√1281/40 - 28013/24
    Hand-picked to satisfy the algebraic Rankine-Hugoniot and positivity conditions in Proposition 2.1; enters awild and the interface entropy inequalities.
  • Left interface speed μ0 (μ1 = -μ0) = -57√35/20 - 59√915/60
    Hand-picked; fixes the cone Γ1, the glue time boundary y0=μ0T0, and L2=-y0; used in the action formulas.
assumptions (4)
  • standard math Glimm's theorem on existence of admissible weak solutions to 1-D systems of conservation laws
    Used in Lemma 2.3 to solve the 1-D Riemann problem on (T0,T); cited to [10].
  • domain assumption Convex integration subsolution existence (Proposition 2.1)
    Quoted from [2,3,12]; supplies the competitor with prescribed density and kinetic energy on [0,T0]. This is the key existence input and is not reproved.
  • domain assumption The 1-D Riemann problem for p(ρ)=ρ² has the shock-fan solution stated in Section 3.2 (two shocks per initial discontinuity with states ρ2,ρ3 and speeds μ2,...,μ5)
    The post-T0 evolution and the action formula At>T0 rely on this fan; the paper says it is easy to verify without proof.
  • standard math Admissible weak solutions satisfy the entropy inequality with energy η=1/2ρ|u|²+P(ρ) and the Rankine-Hugoniot conditions at interfaces
    Foundational definition used in Definition 1.1 and Proposition 2.1.

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Pith. "Pith review of Failure of the least action admissibility principle in the context of the compressible Euler equations." pith.science (2026). https://pith.science/paper/LE4UPOYX

@misc{pith2026250209292,
  author       = {Pith},
  title        = {Pith review of: Failure of the least action admissibility principle in the context of the compressible Euler equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LE4UPOYX}},
  note         = {Machine review of arXiv:2502.09292}
}
read the original abstract

Finding a proper solution concept for the multi-dimensional barotropic compressible Euler equations and related systems is still an unsolved problem. As revealed by convex integration, the classical notion of an admissible weak solutions (also known as weak entropy solutions) does not lead to uniqueness and allows for solutions which do not seem to be physical. For this reason, people have studied additional criteria in view of their ability to rule out the counterintuitive solutions generated by convex integration. Recently, in [H.~Gimperlein, M.~Grinfeld, R.~J.~Knops and M.~Slemrod: The least action admissibility principle, arXiv: 2409.07191 (2024)] it was suggested that the least action admissibility principle serves as the desired selection criterion. In this paper, however, we show that the least action admissibility principle rules out the solution which is intuitively the physically relevant one. Consequently, one either has to reconsider one's intuition, or the least action admissibility principle must be discarded.

Figures

Figures reproduced from arXiv: 2502.09292 by the authors.

Figure 1
Figure 1. The solution (ϱex, uex) (black), which is constructed in Section 3.2, and the 1-D solution (ϱ1d, u1d) (red) in the y − t plane for T0 = 1 2 . 2For the classical solution of one-dimensional Riemann problems to the Euler equations (1.1), we again refer to [3, Lemma 2.4], [12, Prop. 7.1.1] or standard textbooks on hyperbolic conservation laws like [5]. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The functions A[ϱ, u] and Ae[ϱ, u] for the solutions (ϱ1d, u1d) (red) and (ϱex, uex) (black), respectively, where T = 1. Let us next consider the action on the time interval [0, t] as a function dependent on the final time, Ae[ϱ, u](t) := ˆ t 0 A[ϱ, u](s) ds, so that A[ϱ, u] = Ae[ϱ, u](T). Since t 7→ A[ϱex, uex](t) is piecewise linear, the function t 7→ Ae[ϱex, uex](t) is piecewise quadratic as shown in Figure 2b. F… view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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  2. A unified duality framework for barotropic, quantum and Korteweg fluids

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