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Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip

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Pith's one-line read The paper constructs the missing case of least total curvature steady Euler flows in a strip and uses the same method to give a stable semilinear solution with non-convex superlevel sets.

desk verdict Solid completion of the least-total-curvature classification and a neat counterexample to quasiconcavity-from-stability in a strip; send to review. read the letter →

arxiv 2507.11837 v1 pith:VJVIORYQ submitted 2025-07-16 math.AP

Changfeng Gui , David Ruiz , Chunjing Xie , Huan Xu This is my paper · ORCID
classification math.AP MSC 35Q3135Q3535J6176B03
keywords SteadyEulerflowsemilinearellipticequationsleasttotalcurvatureflowsheteroclinicsolutionsnonconvexsuperlevelsetsstripdomainstable
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

This paper proves existence of the last missing type of least total curvature steady Euler flow in the strip $\mathbb{R}\times(0,1)$. In the classification of bounded steady flows by streamline direction, such flows are the non-shear extremal cases of a sharp total-curvature bound; the two previously known types have boundary velocity signs that both change or neither change, and this paper constructs the intermediate case where exactly one boundary changes sign. The proof builds a strictly $x_1$-monotone solution of the semilinear equation $-\Delta u=f(u)$ with $u=0$ and $u=c$ on the two sides of the strip, connecting two one-dimensional energy minimizers, and then designs a nonlinearity $f$ for which exactly two such ordered minimizers exist. Using the same machinery with $c=0$, the paper constructs a positive, monotone (hence stable) solution with a non-convex superlevel set, which contradicts the generalized quasiconcavity-from-semi-stability conjecture in unbounded convex domains.

What carries the argument

The engine is the variational construction of ordered heteroclinic connections. Starting from two ordered global minimizers of the 1D energy (Assumption 2.1), the authors minimize the full 2D action on truncated strips with Dirichlet data $\phi,\varphi$, truncate minimizers to keep the solution between $\phi$ and $\varphi$, and pass to the limit with a fixed reference point and the Hamiltonian identity to force the limits to be exactly the two minimizers. The essential design step is the one-parameter family $I_\lambda$ built on $\chi(s)^3-\lambda\chi(s)^4$; at the threshold $\lambda^*$ a second global minimizer appears above the trivial one, and a minimal-minimizer selection argument shows the minimizer set is totally ordered, so Assumption 2.1 holds. For the $c=0$ case the two minimizers are both strictly positive and symmetric, which makes the level-set contradiction work: a convex superlevel set would have to be a full horizontal strip $\mathbb{R}\times(a,b)$, whose asymptotic limits would force $\phi(a)=\varphi(a)$ and $\phi(b)=\varphi(b)$, impossible because $\phi<\varphi$.

What would settle it

Numerically solve the one-dimensional minimization $I_\lambda$ on $H_{0,c}(0,1)$ for the explicit family $F_\lambda$ at $\lambda=\lambda^*$; if a third global minimizer strictly between $\phi$ and $\varphi$ is found for either $c=1$ or $c=0$, Assumption 2.1 fails and the heteroclinic construction collapses. Alternatively, directly test the boundary sign pattern of the flow from Theorem 1.2: if $v_1(\cdot,1)$ does not change sign exactly once (positive for $x_1>0$, negative for $x_1<0$), the flow is not a case (c) least total curvature flow.

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Extended reading notes

Core claim

The central discovery is a variational existence theorem for monotone heteroclinic solutions in the strip: if the one-dimensional energy $I(\psi)=\int_0^1(\tfrac12|\psi'|^2-F(\psi))dx_2$ has two global minimizers $\phi<\varphi$ with no minimizer of $I$ lying strictly between them, then the boundary-value problem (5) has a solution $u\in C^{2,\alpha}$ with $\partial_{x_1}u>0$, converging to $\phi$ as $x_1\to-\infty$ and to $\varphi$ as $x_1\to+\infty$. Applied to a family of nonlinearities $F_\lambda(s)=\chi(s)^3-\lambda\chi(s)^4$, this yields the first least total curvature flow in case (c): a bounded smooth steady Euler flow in the strip with $v_2>0$, $v_1(\cdot,0)\le v_{1,-}<0$, $v_1(\cdot,1)>0$ on $(0,\infty)$ and $v_1(\cdot,1)<0$ on $(-\infty,0)$. With $c=0$ and a different nonlinearity, the same construction yields a positive, $x_1$-monotone solution of $-\Delta u=f(u)$ whose superlevel set $\{u>\alpha\}$ is non-convex for some $\alpha>0$; since monotone positive solutions are stable, this is a negative answer to the generalized quasiconcavity-from-semi-stability question of [27] in the unbounded convex strip.

Load-bearing premise

The load-bearing premise is Assumption 2.1: the one-dimensional energy $I$ has two global minimizers, one strictly below the other, with no minimizer between them; the paper's verification of this premise is the technical heart, and parts of it (notably the minimal-minimizer selection in Proposition 4.1) are only sketched or left to analogy with [16], so a gap in those steps would undermine both theorems.

Editorial extensions

If this is right

  • The classification of least total curvature steady flows in the strip is complete: cases (a), (b), and (c) are each realized by a $C^\infty$ bounded flow.
  • Any nonlinearity satisfying Assumption 2.1 yields a strictly $x_1$-monotone heteroclinic solution with prescribed limits $\phi$ and $\varphi$, so the construction is a general existence principle rather than a one-off example.
  • The semilinear solution of Theorem 1.3 is positive, monotone, and therefore stable, so semi-stability of positive solutions in unbounded convex domains does not force convexity of superlevel sets.
  • The proof shows that $\{u>\alpha\}$ is non-convex for every $\alpha\in(0,\|\phi\|_{L^\infty}]$, so the constructed solution has a whole continuum of non-convex superlevel sets, not an isolated one.
  • The case-(c) Euler flow fixes the missing streamline diagram in which one boundary layer changes sign exactly once while the other remains of one sign.

Reading between the lines

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

  • Editor's inference: the same two-minimizer construction should work for any symmetric double-well potential with an ordered pair of global minimizers, so monotone heteroclinic solutions in strips are likely generic rather than tied to the specific $F_\lambda$ family.
  • Editor's inference: the counterexample leaves open whether bounded convex domains can host stable solutions with nonconvex superlevel sets; the strip's unbounded direction may be essential, since the convexity contradiction uses the translation invariance of the domain.
  • Editor's inference: a direct computation of the total curvature integral (4) for the new case-(c) flow would test whether it is an exact extremal of the sharp lower bound, as in cases (a) and (b), and would make the 'least total curvature' designation quantitative.
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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

0 major / 5 minor

Summary. The paper studies steady incompressible Euler flows in the strip Ω∞ = R × (0,1) that belong to case (III) of the Hamel–Nadirashvili classification, the so-called least total curvature flows. It constructs heteroclinic solutions to a semilinear elliptic equation with prescribed boundary values, assuming two ordered global minimizers of the associated one-dimensional energy with no minimizer between them (Assumption 2.1), following and extending the framework of [16]. The authors then engineer nonlinearities so that Assumption 2.1 holds. For Theorem 1.2, connecting an increasing minimizer to a concave non-monotone minimizer yields a bounded smooth Euler flow with v2 > 0 and boundary signs corresponding to the previously missing case (c). For Theorem 1.3, two positive symmetric minimizers with c = 0 yield a positive, x1-monotone (hence stable) solution with a nonconvex superlevel set, giving a negative answer to the generalized quasiconcavity-from-semi-stability question in [27] in the unbounded convex strip. The proofs combine variational minimization on truncated boxes, principal-eigenvalue maximum principles, Hamiltonian identities, and explicit cutoff nonlinearities.

Significance. If the results hold, they complete the existence picture for least-total-curvature flows in a strip by treating the missing case (c), complementing the constructions in [23] and [16]. The more striking contribution is Theorem 1.3: it shows that in an unbounded convex domain, positivity and monotonicity (which imply stability) do not force convex superlevel sets for semilinear elliptic equations, answering a generalized version of an open problem in [27] negatively. The paper is self-contained in its verification of Assumption 2.1, and the variational and heteroclinic arguments are standard and rigorous. No machine-checked code or numerical data is involved, but the constructions are sufficiently explicit that the key identities—energy comparisons, Hamiltonian conservation, nondegeneracy of the trivial minimizer—can be checked by hand.

minor comments (5)
  1. [Section 3, Step 4 and Section 4, Step 4] The assertion that mλ* = 1/2 follows 'by the definition of λ*' is not immediate from the monotonicity of mλ alone; one should add the standard argument that if mλ* < 1/2 with a minimizer φ*, then using φ* as a test function for λ > λ* close to λ* gives mλ < 1/2, contradicting the definition of λ*. The same remark applies to the hat-functional in Proposition 4.1.
  2. [Section 2, Lemma 2.2] In the proof of Lemma 2.2, the line 'By maximum principle, ∂x1w < 0' should read ∂x1w > 0, because w is a limit of functions that are strictly increasing in x1 and has nonnegative boundary data; the subsequent Hopf-lemma sentence also appears to contain a sign typo and an undefined symbol v (probably w̃). These are local typos and do not affect the contradiction argument.
  3. [Section 3, Step 3] In the sentence 'take any w ∈ H^1_0(0,1), v > 0, and define ψµ = ϕ + µw', the symbol v is undefined; presumably it should read 'w > 0' or 'take any w ∈ H^1_0(0,1) with w > 0'.
  4. [Section 4, Steps 4 and 5] The references to 'Step 3' and 'Step 4' of Proposition 3.1 are off by one: the analogous arguments are Proposition 3.1's Steps 4 and 5, respectively.
  5. [Proof of Theorem 1.3] The statement that u(x) > ∥ϕ∥L∞ on the line R × {1/2} relies on the fact that ϕ(1/2) = ∥ϕ∥L∞ for ϕ(t) = t(1−t); this identity is not explicitly noted and could be added for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the heteroclinic construction and both existence theorems rest on self-contained variational arguments; self-citations to [16] and [23] are used for framing/classification and are not load-bearing.

full rationale

No load-bearing circularity is present. The derivation chain is linear: Assumption 2.1 postulates two ordered global minimizers of the 1D energy I with no minimizer between; Proposition 2.1 proves the heteroclinic by finite-cylinder minimization, truncation between phi and varphi, monotonicity from the principal eigenvalue of -Delta - f'(u_n), the Hamiltonian identity, and a minimal-minimizer argument. Remark 2.3 notes the case phi=0 is [16, Theorem 1.1], but the proof is actually carried out in the text, so [16] is not load-bearing. Assumption 2.1 is then verified, not assumed: Proposition 3.1 builds f_lambda with phi(t)=t as a nondegenerate local minimizer, proves E={lambda:m_lambda<1/2} is nonempty and bounded, and at lambda* obtains a second minimal minimizer varphi above phi via compactness and total ordering of the minimizer set; Proposition 4.1 repeats this for c=0 with phi(t)=t(1-t). Theorems 1.2 and 1.3 follow from the heteroclinic plus Hopf/convexity arguments. Self-citations to [23] (classification of steady Euler flows, used to frame 'case (c)' and identify the constructed flow as type III) and to [16] are prior theorems with independent proofs, not assumptions of the target conclusions. There is no fitted parameter renamed as a prediction and no definition in terms of the result being proved. The only mild point is that the 'least total curvature' label for the constructed flow is imported from [23] rather than re-derived; this is a self-containedness/verification concern, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The construction rests on standard elliptic theory plus an explicitly engineered nonlinearity; the only 'input' from outside are the prior classification results [16,23] used for motivation. No numbers are fitted to data.

free parameters (1)
  • lambda* = sup {lambda > 0 : m_lambda < 1/2} in Prop 3.1; sup {lambda > 0 : m_hat_lambda < -1/6} in Prop 4.1
    Auxiliary threshold tuned so that the 1D energy functional has exactly two ordered global minimizers; not a physical constant, not fitted to any external data.
assumptions (5)
  • standard math Elliptic regularity and maximum principle theory, including Berestycki-Nirenberg-Varadhan [3], used in Prop 2.1 Step 2 and Theorem 1.2.
    Standard background results invoked without proof for monotonicity and boundary regularity.
  • standard math Hamiltonian identity of Gui [22] along solutions in the strip, used in Lemma 2.2 and Prop 2.1 Step 4 to identify the asymptotic limits.
    Known identity for semilinear elliptic equations; the paper applies it to the limiting half-strip solutions.
  • standard math Monotone positive solutions of elliptic equations on strips are stable; cited as known in the discussion of Theorem 1.3.
    Standard fact used to connect monotonicity to stability without proof.
  • domain assumption The domain is the infinite strip Omega_infinity = R x (0,1) with Dirichlet data u = 0 and u = c on the two boundary components.
    The entire paper is set on this unbounded convex strip; the boundary conditions are part of the problem.
  • ad hoc to paper The nonlinearities are engineered: f_lambda = chi'(3 chi^2 - 4 lambda chi^3) with the cutoff chi satisfying (13)-(14); in Prop 4.1 the functional includes an extra -2 psi term to make both minimizers positive.
    This is the constructed example, not an external constraint; the existence results depend on these choices.

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Pith. "Pith review of Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip." pith.science (2026). https://pith.science/paper/VJVIORYQ

@misc{pith2026250711837,
  author       = {Pith},
  title        = {Pith review of: Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJVIORYQ}},
  note         = {Machine review of arXiv:2507.11837}
}
read the original abstract

This paper focuses on establishing the existence of a class of steady solutions, termed least total curvature solutions, to the incompressible Euler system in a strip. The solutions obtained in this paper complement the least total curvature solutions already known. Our approach employs a minimization procedure to identify a monotone heteroclinic solution for a conveniently chosen semilinear elliptic PDE. This method also enables us to construct positive and monotone (and consequently stable) solutions to semilinear elliptic PDEs with non-convex superlevel sets in a strip domain. This can be regarded as a negative answer to a generalized problem raised in [27].

Figures

Figures reproduced from arXiv: 2507.11837 by the authors.

Figure 1
Figure 1. Flows of type I, v = (x2 − 1 2 , 0). The key of the proof for Theorem 1.1 lies in the estimate of the total curvature Z Ω |v1∇v2 − v2∇v1| 2 |v| 2 dx, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Flows of type II, v = (−π sin x1 cos(πx2), cos x1 sin(πx2)). where the integrand is understood as 0 in the stagnation set {v = 0}. The estimate for the total curvature is based on the delicate energy estimate for the equation v · ∇ω = div(v1∇v2 − v2∇v1) = 0. Clearly, a flow is a parallel shear flow if and only if its total curvature is zero. Furthermore, a type (III) flow is characterized as the non-shear extremal f… view at source ↗
Figure 3
Figure 3. The least total curvature flows in case (a). Our first theorem is precisely stated as follows. Theorem 1.2. There exists a bounded solution v ∈ C∞(Ω∞) to (1)-(2) satisfying the following properties [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The least total curvature flows in case (b). (i) v2 > 0 in Ω∞; (ii) v1(·, 0) ≤ v1,− < 0 on R, v1(·, 1) > 0 on (0, ∞), and v1(·, 1) < 0 on (−∞, 0). See [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The least total curvature flows in case (c). To prove Theorem 1.2, we shall establish the existence of a monotone (in the x1-direction) solution to the following boundary value problem of semilinear elliptic equation    − ∆u = f(u), x ∈ Ω∞, u(x1, 0) = 0, x1 ∈ R,…
Figure 6
Figure 6. Figure 6: Nonconvex streamlines. We have the following remarks on Theorem 1.3. Remark 1.1. In [33, Remark 3 in page 268], P. L. Lions writes that, in a bounded convex domain Ω, “We believe that ... for general f, the (super)level sets of any (positive solution of semilinear elli…
Figure 7
Figure 7. Figure 7: The graphs of the functions ϕ and φ. We start by pointing out that mλ∗ = 1/2 by the definition of λ ∗ . Consider a sequence {λn}n∈N ⊆ E such that λn → λ ∗ as n → +∞. Then there exists a sequence {φn}n∈N ⊆ H0,1(0, 1) such that Iλn (φn) = mλn < 1/2 for all n ∈ N. Now, ta…
Figure 8
Figure 8. Figure 8: The graphs of the functions ϕ and φ. lim x1→−∞ u(x1, x2) = ϕ(x2), lim x1→+∞ u(x1, x2) = φ(x2) uniformly in x2 ∈ [0, 1]. Take now α ∈ (0, ∥ϕ∥L∞]. Observe that u(x) > ∥ϕ∥L∞ on the line L = R×{1/2} by monotonicity, and hence the superlevel set U α = {x ∈ Ω∞ : u(x) > α} co…

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

  1. Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets

    math.AP 2026-07 accept novelty 8.0 of 10

    Minimal strictly stable solutions of -Δu=f(u) in smooth uniformly convex planar domains can have nonconvex superlevel sets for f(u)=e^u or f(u)=(a+u)^p, answering Brezis's open question negatively.

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Works this paper leans on

39 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [16]

    De Regibus and D

    F. De Regibus and D. Ruiz, Monotone heteroclinic solutions to semilinear PDEs in cylinders and applications, Calc. Var. Partial Differential Equations 64 (2025), no. 4, Paper No. 111, 18 pp

    work page 2025

  2. [23]

    C. Gui, C. Xie, and H. Xu, On a classification of steady solutions to two-dimensional Euler equations , arXiv:2405.15327, 2024

    arXiv 2024

  3. [27]

    Hamel, N

    F. Hamel, N. Nadirashvili, and Y. Sire, Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples , Amer. J. Math. 138 (2016), no. 2, 499–527

    work page 2016

  4. [1]

    Alama, L

    S. Alama, L. Bronsard, C. Gui, Stationary layered solutions in R2 for an Allen-Cahn systems with multiple-well potentials, Calc. Var. Partial Differential Equations, 5 (1997), 359–390

    work page 1997

  5. [2]

    C. O. Alves, Existence of a heteroclinic solution for a double well potential equation in an infinite cylinder of RN , Adv. Nonlinear Stud. 19 (2019), no. 1, 133–147

    work page 2019

  6. [3]

    Berestycki, L

    H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second- order elliptic operators in general domains , Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92

    work page 1994

  7. [4]

    Berestycki, L

    H. Berestycki, L. Caffarelli, and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), no. 4, 69–94

    work page 1997

  8. [5]

    M. T. Barlow, R. F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi , Comm. Pure Appl. Math., 53 (2000), 1007-1038

    work page 2000

Show all 39 references
  1. [6]

    H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Pr´ ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , J. Functional Analysis 22 (1976), no. 4, 366–389

    1976

  2. [7]

    H. R. Brezis, Is there failure of the inverse function theorem?, in Morse theory, minimax theory and their applications to nonlinear differential equations , 23–33, New Stud. Adv. Math., 1, Int. Press, Somerville, MA

  3. [8]

    Byeon, P

    J. Byeon, P. Montecchiari, and P. H. Rabinowitz, A double well potential system , Anal. PDE, 9 (2016), no. 7, 1737–1772

    2016

  4. [9]

    Cabr´ e and S

    X. Cabr´ e and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), no. 1, 1–10

    1998

  5. [10]

    D. Cao, B. Fan, and W. Zhan, Free boundary problems for the two-dimensional Euler equations in exterior domains, arXiv:2406.16134, 2024

    2024 arXiv

  6. [11]

    Castro and D

    ´A. Castro and D. Lear, Time periodic solutions for the 2D Euler equation near Taylor-Couette flow, Calc. Var. Partial Differential Equations 63 (2024), no. 9, Paper No. 224, 89 pp

    2024

  7. [12]

    Castro and D

    ´A. Castro and D. Lear, Traveling waves near Poiseuille flow for the 2d Euler equation , arXiv:2404.19034, 2024

    2024 arXiv

  8. [13]

    K. Choi, I. Jeong, and Y. Sim, On Existence of Sadovskii Vortex Patch: A Touching Pair of Symmetric Counter- Rotating Uniform Vortices, Ann. PDE, 11 (2025), 18

    2025

  9. [14]

    D Drivas, and D

    P Constantin, T. D Drivas, and D. Ginsberg, Flexibility and rigidity in steady fluid motion , Comm. Math. Phys., 385 (2021), 521–563

    2021

  10. [15]

    Coti Zelati, T

    M. Coti Zelati, T. M Elgindi, and K. Widmayer, Stationary structures near the Kolmogorov and Poiseuille flows in the 2d Euler equations , Arch. Ration. Mech. Anal., 247 (2023), 12

    2023

  11. [17]

    T. D. Drivas and D. Ginsberg, Islands in stable fluid equilibria, Proc. Amer. Math. Soc. 152 (2024), no. 11, 4855–4863 STEADY EULER FLOW WITH LEAST TOTAL CUR V ATURE 21

    2024

  12. [18]

    T. D. Drivas and M. Nualart, A geometric characterization of steady laminar flow, arXiv:2410.18946, 2024

    2024 arXiv

  13. [19]

    T. M. Elgindi, Y. Huang, A. R. Said, and C. Xie, A classification theorem for steady Euler flows , arXiv:2408.14662, 2024

    2024

  14. [20]

    Enciso, A

    A. Enciso, A. J. Fern´ andez, and D. Ruiz, Smooth nonradial stationary Euler flows on the plane with compact support, arXiv:2406.04414, 2024

    2024 arXiv

  15. [21]

    G´ omez-Serrano, J

    J. G´ omez-Serrano, J. Park, J. Shi, and Y. Yao.Symmetry in stationary and uniformly rotating solutions of active scalar equations, Duke Math. J., 170 (2021), 2957–3038

    2021

  16. [22]

    Gui, Hamiltonian identities for elliptic partial differential equations, J

    C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 (2008), no. 4, 904– 933

    2008

  17. [24]

    Hamel and N

    F. Hamel and N. Nadirashvili, Shear flows of an ideal fluid and elliptic equations in unbounded domains , Comm. Pure Appl. Math., 70 (2017), 590–608

    2017

  18. [25]

    Hamel and N

    F. Hamel and N. Nadirashvili, A Liouville theorem for the Euler equations in the plane , Arch. Ration. Mech. Anal., 233 (2019), 599–642

    2019

  19. [26]

    Hamel and N

    F. Hamel and N. S. Nadirashvili, Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems , J. Eur. Math. Soc. (JEMS) 25 (2023), no. 1, 323–368

    2023

  20. [28]

    Keady, The power concavity of solutions of some semilinear elliptic boundary value problems , Bull

    G. Keady, The power concavity of solutions of some semilinear elliptic boundary value problems , Bull. Austral. Math. Soc. 31 (1985), no. 2, 181–184

    1985

  21. [29]

    Kirchg¨ assner, Wave-solutions of reversible systems and applications , J

    K. Kirchg¨ assner, Wave-solutions of reversible systems and applications , J. Differential Equations 45 (1982), no. 1, 113–127

    1982

  22. [30]

    C. Li, Y. Lv, H. Shahgholian, and C. Xie. Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle , arXiv:2203.08375, 2022

    2022 arXiv

  23. [31]

    Lin and C

    Z. Lin and C. Zeng. Inviscid dynamical structures near Couette flow , Arch. Ration. Mech. Anal., 200 (2011), 1075–1097

    2011

  24. [32]

    L. G. Makar-Limanov, The solution of the Dirichlet problem for the equation ∆u = −1 in a convex region , Mat. Zametki 9 (1971), 89–92

    1971

  25. [33]

    P. L. Lions, Two geometrical properties of solutions of semilinear problems, Applicable Anal., 12 (1981), 267–272

    1981

  26. [34]

    M. Nualart. On zonal steady solutions to the 2d Euler equations on the rotating unit sphere , Nonlinearity, 36 (2023), 4981–5006

    2023

  27. [35]

    P. H. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations, J. Math. Sci. Univ. Tokyo 1 (1994), no. 3, 525–550

    1994

  28. [36]

    P. H. Rabinowitz, Spatially heteroclinic solutions for a semilinear elliptic P.D.E. , ESAIM Control Optim. Calc. Var. 8 (2002), 915–931

    2002

  29. [37]

    P. H. Rabinowitz, A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 507–515

    2004

  30. [38]

    Ruiz, Symmetry results for compactly supported steady solutions of the 2D Euler equations, Arch

    D. Ruiz, Symmetry results for compactly supported steady solutions of the 2D Euler equations, Arch. Ration. Mech. Anal., 247(3): Paper No. 40, 25, 2023

    2023

  31. [39]

    Tian and K

    Z. Tian and K. Cui, A study of streamline geometries in subsonic and supersonic regions of compressible flow fields, J. Fluid Mech., 987 (2024), A26. 22 CHANGFENG GUI, DA VID RUIZ, CHUNJING XIE, AND HUAN XU Department of Mathematics, F aculty of Science and Technology, Univers...

    2024

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