Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

Naoki Tsuge

REVIEW 3 major objections 4 minor 18 references

Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A time-periodic entropy weak solution exists for the compressible Euler equations with a time-periodic outer force on a bounded interval.

desk verdict A genuinely new existence strategy for time-periodic Euler flows, but the theorem as stated is stronger than the fixed-point proof delivers and the key convergence lemmas are deferred to earlier papers. read the letter →

arxiv 1908.03120 v1 pith:JERPIQCY submitted 2019-08-08 math.AP math-phmath.MP

Naoki Tsuge This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35L0335L6535Q3176N1076N1535A0135B3535B50

keywords compressibleEulerequationstimeperiodicsolutionouterforcegeneralizedinvariantregionmodifiedLax-FriedrichsschemecompensatedcompactnessRiemanninvariantssupersonicflow

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 proves that the one-dimensional compressible Euler equations with a time-periodic outer force admit a time-periodic entropy weak solution on the bounded interval that the flow occupies for one period. This matters because, although time-periodic solutions are standard for many evolution equations, almost nothing was known for systems of conservation laws. The proof combines a generalized invariant region built from Riemann invariants, a modified Lax-Friedrichs scheme whose one-period map is continuous, and a Brouwer fixed point argument that forces the solution after one period to equal the initial data. The resulting solution stays inside the same linear-in-x bounds on the Riemann invariants for the whole period.

What carries the argument

The central object is the generalized invariant region $\Delta_x = \{\rho \ge 0,\ L - Kx \le z \le w \le M + Kx\}$, expressed in the Riemann invariants $w = v + \rho^\theta/\theta$, $z = v - \rho^\theta/\theta$, where $\theta = (\gamma - 1)/2$, with bounds that shift linearly in $x$. The main mechanism is the modified Lax-Friedrichs scheme: approximate solutions are built cell-by-cell from steady-state profiles and Riemann solutions, and the cell averages obey the explicit recurrence (5.1). These averages define a continuous map from the vector of initial Riemann invariants to the vector after one period; the invariant region keeps the map inside a bounded convex set, and Brouwer's fixed point theorem gives initial data whose time-1 state coincides with itself. A compensated compactness step then extracts an almost-everywhere convergent subsequence whose limit is the periodic entropy weak solution.

What would settle it

Take $F(x,t) = K\cos(2\pi t)$ with $K>0$ and any $M \ge L \ge 1+K$, discretize the modified Lax-Friedrichs recurrence (5.1)-(5.3), and compute the period map on the invariant region; if for some mesh size the image leaves $\{L - Kx - \delta \le z,\ w \le M + Kx + \delta\}$ by more than $o(\Delta x)$, or the Brouwer fixed point fails to appear, then estimate (4.1) is false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for a $C^1$ outer force $F(x,t)$ with period 1, if $F$ is bounded by $K$ and the constants satisfy $M \ge L \ge 1 + K$, and if the initial data satisfy $0 \le \rho_0$, $L - Kx \le z(u_0)$, $w(u_0) \le M + Kx$ while the left boundary data satisfy the analogous inequalities, then the initial-boundary value problem has a time-periodic entropy weak solution. Moreover, the solution obeys $L - Kx \le z(u(x,t))$ and $w(u(x,t)) \le M + Kx$ for $(x,t) \in (0,1) \times (0,1)$, which implies that both characteristic speeds are positive. In the smooth case the Riemann invariants obey $z_t + \lambda_1 z_x = F$ and $w_t + \lambda_2 w_x = F$; the space-dependent change of variables $\tilde z = z + Kx$, $\tilde w = w - Kx$ makes the source terms push the solution back into the invariant region on the two sides of the triangle. The paper makes this formal argument rigorous for weak solutions by an approximate scheme.

Load-bearing premise

The theorem rests on the claim that the $L^\infty$ estimates for the near-vacuum case and the compensated compactness and periodicity arguments, which are omitted and referred to earlier work, carry over unchanged when the source term is time-periodic and the interval is bounded.

Editorial extensions

If this is right

If Theorem 1.1 is correct, supersonic isentropic gas flow through a bounded interval with a time-periodic body force can be exactly periodic in time with the same period as the force.
The invariant-region bounds give uniform $L^\infty$ control: density and velocity remain bounded in terms of $L$, $M$, and $K$ throughout the period.
The proof constructs the periodic solution as a fixed point of a finite-dimensional period map, so the solution is obtained as a limit of explicitly computable approximate solutions.
The method is restricted to monotone space-dependent bounds, so periodic boundary conditions and reflecting boundary conditions on a bounded interval are explicitly left open in Section 6.

Reading between the lines

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

The proof as written relies on stated carry-over claims: the near-vacuum $L^\infty$ estimates and the compensated compactness and periodicity-of-limit arguments are omitted and referred to earlier work, so a self-contained verification in the present periodic-source, bounded-interval setting would remove the main open technical point.
A natural numerical test is to solve the recurrence (5.1)-(5.3) for a concrete periodic force such as $F(x,t) = K\sin(2\pi t)$ and check whether the discrete period map has a fixed point inside the invariant region as the mesh size tends to zero.
The success of the space-dependent invariant region suggests that similar existence results could hold for any hyperbolic system whose Riemann invariants satisfy diagonal equations with a common forcing term controlled by the same slope $K$.
The paper's open problems indicate that genuinely periodic boundary conditions require non-monotone invariant bounds; constructing such bounds would be a natural next step beyond this paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here is my read on Tsuge's arXiv:1908.03120. The genuinely new thing is the combination of a generalized invariant region with a Brouwer fixed-point argument to get a time-periodic solution of the isentropic Euler system with a periodic source. That is a real first for systems, and the paper correctly notes how little has been done on this problem. The formal invariant-region computation in Section 1.1 is clean, and the modified Lax-Friedrichs scheme is a natural extension of the author's established program. I see no circularity or data fitting; the reliance on previous work is explicit.

The main soft spot is logical. The fixed-point map in Section 5 produces an admissible initial grid datum whose time-one image is within o(1) of itself, and that datum is then fed back as the initial data. So the proof establishes that there exists some u0 in the admissible class for which a periodic solution exists. Theorem 1.1, read in the standard way, quantifies over every u0 satisfying (1.8). The proof does not show that every such u0 is the initial trace of a periodic solution. This is not cosmetic; the theorem should be restated existentially, or the proof must show that the time-one map contracts every admissible point, which it does not do.

The second soft spot is the heavy deferral. The near-vacuum L∞ bounds in Section 3 and Appendix A are explicitly omitted, and Proposition 5.1 and Theorem 5.2, which are the compensated compactness convergence and periodicity of the limit, are asserted to follow from [11]–[13]. If those carry over, the compactness argument is likely fine; if they do not, even the existential claim is unsupported. The paper is honest about this, but as it stands the main theorem is not proved in the preprint.

There is also a minor presentation issue: the boundary procedure and the recurrence formula in Section 5 are dense, and the relationship between the cut-off in (3.1) and the o(Δx) error in (4.1) is sketched rather than fully checked. I do not think those are fatal, but they contribute to the sense that the proof is not yet self-contained.

Who should read this? Anyone working on hyperbolic conservation laws with source terms, or on periodic solutions of Euler-type systems, should know about the approach. It points to the right difficulties and states the open problems clearly. But as a proof of Theorem 1.1 as printed, it is incomplete. I would send it to a serious referee: the idea is worth engaging, and a careful referee could help the author decide whether to restate the theorem or strengthen the argument.

Referee Report

3 major / 4 minor

Summary. The paper claims the existence of a time-periodic entropy weak solution to the one-dimensional isentropic compressible Euler equations with a time-periodic outer force on the bounded interval (0,1), under the assumption that both characteristic speeds are nonnegative so that no boundary condition is needed at x=1. The proof strategy combines an x-dependent generalized invariant region of the form L - Kx <= z(u) and w(u) <= M + Kx with a modified Lax-Friedrichs scheme whose recurrence formula defines a continuous time-one map on admissible discrete initial data; a Brouwer fixed point then yields a periodic discrete trajectory, and compensated compactness is invoked to pass to the limit and obtain a periodic entropy weak solution. The main theorem is stated for every initial datum u0 satisfying the invariant-region inequalities (1.8).

Significance. If the proof were complete, the paper would be a valuable contribution: time-periodic solutions for systems of conservation laws with a periodic source on a bounded interval have not been treated before, and the use of an x-dependent invariant region combined with a fixed-point argument is a natural and promising approach. The approximate scheme is carefully designed so that the source terms point inward on the sides of the invariant region, and the recurrence formula is explicit enough that continuity of the discrete time-one map is plausible. The paper also correctly identifies the two main difficulties, namely proving that the solution set maps into itself over one period and constructing a continuous finite-dimensional map. However, as written, the proof establishes less than the theorem states, and several load-bearing technical steps are deferred to previous papers by the author; these gaps must be addressed before the result can be considered proved.

major comments (3)

[Section 5, Eq. (5.6), and Theorem 1.1] Theorem 1.1 is stated for every initial datum u0 satisfying (1.8), and Definition 1.2 incorporates u0 into the weak formulation through the terms involving phi(x,0)-phi(x,1). However, the proof defines the map F on admissible discrete initial data and then applies Brouwer's fixed point theorem, obtaining a fixed point (tilde u^0_j)^* which is then supplied as the initial data u^0_j for the approximate scheme. This establishes only that there exists some admissible discrete initial datum whose time-one image is within o(1) of itself; it does not show that an arbitrary prescribed u0 satisfying (1.8) is the initial trace of a time-periodic solution. This is not a cosmetic discrepancy, because the fixed-point method cannot deliver the universal reading. The theorem should be weakened to an existential statement over u0, or the authors need an additional argument, such as uniqueness or a strong stability property, to pass from the fixed point to arbitrary admissible initial data.
[Section 3, Appendix A, Proposition 5.1, and Theorem 5.2] The L-infinity estimates for the near-vacuum case are explicitly omitted: the text says 'we omit the L-infinity estimates for the case in this paper. The detail can be found in [11].' In addition, Proposition 5.1 and Theorem 5.2, which are the results that justify the compensated compactness convergence and the periodicity of the limit, are only asserted to follow 'in the same manner to [11]-[13].' These are not routine localizations: the present problem has a time-periodic source and a bounded interval, so the interaction of the source with the x-dependent invariant region, the boundary Riemann problems, and the periodicity of the limit must be checked explicitly. Without these estimates and proofs, the fixed point obtained in Section 5 is not shown to converge to a time-periodic entropy weak solution. Please include the missing near-vacuum estimates and either prove Proposition 5.1 and Theorem 5.2 in detail or provide a careful verification that the arguments of [11]-[13] apply unchanged to the periodic-source bounded-interval setting.
[Section 4, Eq. (4.2), and final paragraph of Section 5] The proof of Theorem 4.1 is carried out under the strict margin M >= (1+K)+epsilon in (4.2), and the paper concludes by saying that since epsilon is arbitrary, Theorem 4.1 holds under (1.7). For M = 1+K, which is allowed by (1.7), no positive epsilon exists, and the argument in Estimate 2 uses the epsilon-margin to absorb the O(sqrt(Delta x)) error in (4.21). The passage from the strict-inequality case to the equality case needs an explicit limiting argument, for example by applying the result with M+epsilon and L-epsilon and then letting epsilon tend to zero with uniform bounds. As written, the equality case M = 1+K is not proved.

minor comments (4)

[Section 2] The numbering of Lemma 2.1 and the subsequent Lemma 2.3 is inconsistent: after Lemma 2.1 the text says 'Lemma 2.3 can be found in [2, Lemma 3.3]', and a second statement labeled Lemma 2.3 appears after Theorem 2.2. Please correct the numbering and cross-references.
[Section 4] In the sentence 'As a result, from (4.20), we drive (4.1)1', the word 'drive' should read 'derive'.
[Section 3] The notation u^Delta(x,-0) is used without explanation; since the approximate solution is piecewise constant in time and may jump at each time level, the notation should be defined explicitly as the appropriate one-sided limit.
[Section 5, Eq. (5.5)] The choice of delta(Delta x) in (5.5) is asserted to exist from (5.1)-(5.3), but it should be clarified that the o(Delta x) remainder in (5.1) is absorbed uniformly over all admissible discrete initial data, since the fixed-point argument requires a single compact convex set that is mapped into itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the periodic solution is produced by a fixed-point argument, and the same-author citations supply independent technical estimates rather than restating the conclusion.

full rationale

The derivation chain does not reduce any conclusion to its own input. The invariant-region bound (1.10) is obtained from the maximum-principle-type estimates in Section 4; the projection step (3.1) truncates cell values into the candidate triangle, but the text tracks the truncation as o(∆x) and shows that it disappears in the limit, so the bound is not simply assumed. The periodic solution is found as a Brouwer fixed point of the discrete time-one map (5.6), not by fitting parameters to the solution; the constructed initial grid data are part of the existence argument. The same-author citations are load-bearing for technical estimates: Section 3 defers near-vacuum L∞ estimates to [11], Section 4.1 cites [11, Appendix A] for the quasi-steady-state properties, and Section 5 states that Proposition 5.1 and Theorem 5.2 'can be proved in the same manner to [11]–[13]'. These are prior external results for related problems rather than restatements of the present theorem, so they do not make the argument circular; at worst they leave proof gaps if the carry-over fails. The prescribed-u0 reading of Theorem 1.1 versus the fixed-point-constructed initial data is a logical quantifier issue, not a self-referential reduction. Therefore no circularity step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted constants, no new physical entities, and no new free parameters beyond the hypotheses K, L, and M that are part of the theorem statement. The main external inputs are standard Riemann-solution theory and the author's earlier technical estimates.

assumptions (3)

standard math Unique Riemann solutions and invariant regions for the homogeneous isentropic Euler system, cited to Chen [2].
Used in Section 2 (Lemma 2.1, Theorem 2.2) as background; the paper does not reprove them.
domain assumption The modified Lax-Friedrichs scheme estimates, including the omitted near-vacuum L-infinity bound, hold as in [11, Appendix A] and [12]-[17].
Sections 3 and 4 rely on these; the text explicitly says the L-infinity estimates are omitted and details are in [11]. This is load-bearing for the invariant-region bounds.
ad hoc to paper The compactness theorem (Proposition 5.1) and convergence to a time-periodic entropy weak solution (Theorem 5.2) follow by the same argument as [11]-[13].
Section 5 asserts these results without proof. If the analogous arguments do not carry over to the periodic-source, two-boundary setting, the fixed-point limit need not be periodic.

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Pith. "Pith review of Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force." pith.science (2026). https://pith.science/paper/JERPIQCY

@misc{pith2026190803120,
  author       = {Pith},
  title        = {Pith review of: Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JERPIQCY}},
  note         = {Machine review of arXiv:1908.03120}
}

read the original abstract

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.

Figures

Figures reproduced from arXiv: 1908.03120 by the authors.

**Figure 1.** The invariant region in (z, w)-plane hold on the side AC, we find that λ2 ≧ 1 + θ 2 M + 1 − θ 2 L + θKx. (1.16) Therefore, we obtain w˜t + λ2w˜x =F(x, t) − Kλ2 ≦ − K(λ2 − 1) (from (1.6)) ≦ − K 1 + θ 2 M + 1 − θ 2 L + θKx − 1 (from (1.16)) ≦0 (from (1.7)). We thus conclude that the source term of (1.13)1 is positive on the side AB and the source term of (1.13)2 is negative on the side AC. We apply the maximum pri… view at source ↗

**Figure 2.** The rarefaction curves, the shock curves and the inverse rarefaction curves in (z, w)-plane Remark 2.1. Assume that there exists C > 1 such that 1/C ≦ ρ/ρ0 ≦ C. Then, considering w along S1(u0), we have w|S1(v0) = v0 − s 1 ρρ0 p(ρ) − p(ρ0) ρ − ρ0 (ρ − ρ0) + ρ θ θ = w(v0) + O(1)(ρ0) γ−7 2 (ρ − ρ0) 3 , where O(1) depends only on C. Considering z along S2(u0), we similarly have z|S2(v0) = v0 + s 1 ρρ0 p(ρ) − p(ρ0) ρ −… view at source ↗

**Figure 3.** The approximate solution in the case where a 1- rarefaction and a 2-shock arise in the cell. We denote this approximate Riemann solution, which consists of (3.6), by u ∆(x, t). The validity of the above construction is demonstrated in [11, Appendix A]. Remark 3.2. u ∆(x, t) satisfies the Rankine–Hugoniot conditions at the middle time of the cell, tM := (n + 1/2)∆t. Remark 3.3. The approximate solution u ∆(x, t) is p… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The invariant region in (z, w)-plane 5. Recurrence formula From Remark 3.3, u ∆ satisfy (u ∆)t + f(u ∆)x − g(x, t, u∆) = O(∆x) on the divided part in the cell where u ∆ are smooth. Moreover, u ∆ satisfy an entropy condition (see [11, Lemma 5.1–Lemma 5.4]) along discont…

**Figure 5.** Figure 5: Case 1.1: The approximate solution ¯u ∆ in the cell. Case 1.2 ρL ≦ (∆x) β (i) z(uL) ≧ Lj In this case, we define u ∆(x, t) as a Riemann solution (uL, uR). (ii) z(uL) < Lj In this case, recalling z(uL) = z(u n j ) ≧ L − K((j − 1)∆x), we can choose x (4) such that (j − 1…

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Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Tsuge, N.: Global L∞ solutions of the compressible Euler equations with spheric al symmetry. J. Math. Kyoto Univ. 46, 457–524 (2006)

work page 2006

[2] [13]

Tsuge, N.: Isentropic gas ﬂow for the compressible Eule r equation in a nozzle. Arch. Ration. Mech. Anal. 209, 365400 (2013)

work page 2013

[3] [1]

Acta Mathematica Scientia 6, 75–120 (1986)

Chen, G.-Q.: Convergence of the Lax–Friedrichs scheme f or isentropic gas dynamics (III). Acta Mathematica Scientia 6, 75–120 (1986)

work page 1986

[4] [2]

MSRI preprint 00527-91, Berkeley, 1990

Chen, G.-Q.: The compensated compactness method and the system of isentropic gas dy- namics. MSRI preprint 00527-91, Berkeley, 1990

work page 1990

[5] [3]

DiPerna, R.J.: Convergence of the viscosity method for i sentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)

work page 1983

[6] [4]

Acta Mathematica Scientia 5, 415–432, 433–472 (1985)

Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax–Fr iedrichs scheme for isentropic gas dynamics (I)–(II). Acta Mathematica Scientia 5, 415–432, 433–472 (1985)

work page 1985

[7] [5]

Ding, X., Chen, G.-Q., Luo, P.: Convergence of the fracti onal step Lax–Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics . Commun. Math. Phys. 121, 63—84 (1989)

work page 1989

[8] [6]

Murat, F.: Compacit´ e per compensation. Ann. Scuola Norm. Sup. Pisa Sci. Math. 5, 489– 507 (1978); II, In: De Giorgi, E., Magenes, E., and Mosco, U. ( eds.) Proc. Int. Meeting on Recent Methods on Nonlinear Analysis. Pitagora, Bologna, 1 979; III, Ann. Scuola. Norm. Sup. Pisa Sci. Math. 8, 69–102 (1981) THE COMPRESSIBLE EULER EQUATIONS WITH A TIME PERIOD...

work page 1978