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Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations

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Pith's one-line read The paper proves that the explicit self-similar blow-up in the incompressible porous medium equations is stable under $C^3$ perturbations but destroyed by $C^{2-\epsilon}$ data, by converting blow-up into global stability of…

desk verdict Solid C^3 stability theorem and a clean instability construction, but the advertised 'sharp regularity threshold' is not proved. read the letter →

arxiv 2507.17381 v1 pith:MMXSPX7J submitted 2025-07-23 math.AP

Charles Collot , Christophe Prange , Jin Tan This is my paper · ORCID
classification math.AP MSC 35Q3535B4035B4435Q3176S05
keywords incompressibleporousmediumequationsProudman-Johnsonequationself-similarblow-upasymptoticstabilitysharpregularitythresholdinfiniteenergysolutionsweightedlineardampinghydrostaticEuler
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 that the explicit self-similar blow-up found for a special class of infinite-energy solutions of the two-dimensional incompressible porous medium equations is stable under smooth perturbations, and that the stability disappears below a sharp regularity threshold. The key move is a change of variables that converts the finite-time blow-up problem into the global-in-time stability of the stationary states $\mu \cos(x)$ of the Proudman-Johnson equation, a one-dimensional nonlocal transport equation. For any small mean-free $C^3$ perturbation of $\mu\cos(x)$, the solution converges to a nearby stationary state with exponential rate, the final amplitude being fixed by the initial datum through $\mu^* = -\partial_x^2 a_0(x_0^*)$, the second derivative at the initial maximum. Undoing the change of variables gives IPM blow-up with leading profile $\cos(x)/(\tau^*-\tau)$ and error $O((\tau^*-\tau)^{-3/4})$. The same stability transfers to special classes of two-dimensional Euler and inviscid primitive equation solutions that reduce to the same one-dimensional equation.

What carries the argument

The load-bearing object is the change of variables (1.6), which maps the 1D porous-medium equation (1.2) to the Proudman-Johnson equation $\partial_t a + (\int_{-\pi}^{x} a)\,\partial_x a - a^2 + \tfrac{1}{\pi}\int_{-\pi}^{\pi} a^2 = 0$, and maps the finite-time blow-up profiles (1.4) onto the stationary states $\mu\cos(x)$. Around $\cos(x)$ the linearized operator has three explicit eigenmodes $\varphi_{-1}, \varphi_0=\cos, \varphi_1$ with $L\varphi_l = l\varphi_l$; modulation in that basis makes the renormalized perturbation vanish to third order at the moving maximum point, so that the linearized semigroup sees a weight $W_\theta(x)$ of order $|x|^3$ near the origin. The main decay estimate is a comparison-principle bound $\|\xi/W_\theta\|_{L^\infty} \le e^{-(1-\theta')s}\|\xi_0/W_\theta\|_{L^\infty}$ for the quasilinearized problem on a moving interval, with the nonlocal term handled by an exponentially weighted auxiliary function. The apparently unstable $\varphi_1$ mode is then neutralized by the mean-free condition, which expresses its coefficient through the remainder $\xi$; this is why smooth mean-free perturbations decay exponentially instead of growing.

What would settle it

Integrate the Proudman-Johnson equation (1.7) from the even cusp datum $a_0(x)=\cos(x)+|x|^{2-\epsilon/2}$ near the origin and track the characteristic gap $a(t,0)-a(t,z(t))$: Theorem 1.5 predicts this gap grows like $e^{(2\mu-\gamma)t}|z_0|^{2-\epsilon/2}$, forcing $\|a(t,\cdot)\|_{L^\infty}\to\infty$, while convergence to any $\mu\cos(x)$ would require it to tend to zero; observing the predicted growth numerically would confirm the sharp-threshold claim, whereas boundedness would refute it.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: for every $\mu>0$ there is a $\delta>0$ such that any mean-free $a_0$ with $\|a_0 - \mu\cos(\cdot)\|_{C^3} \le \delta\mu$ generates a unique global $C^3$ solution of the Proudman-Johnson equation (1.7) satisfying $\|a(t,\cdot) - \mu^*\cos(\cdot)\|_{C^1} \lesssim \sigma e^{-\mu t/2}$, where $\mu^* = -\partial_x^2 a_0(x_0^*)$ is determined by the initial maximum point $x_0^*$. Through the change of variables (1.6) this becomes Theorem 1.1: IPM solutions starting close to $\mu\cos(x)$ blow up in finite time with $b(\tau,x) = \cos(x)/(\tau^*-\tau) + O((\tau^*-\tau)^{-3/4})$. The companion results Theorems 1.2 and 1.5 show the threshold is sharp: for every $\epsilon>0$ there are $C^{2-\epsilon}$ data arbitrarily close to $\cos(x)$ whose solutions never converge to any stationary state, because a cusp-like maximum of the form $|x|^{2-\epsilon/2}$ at the origin is amplified along characteristics into unlimited $L^\infty$ growth. The paper also classifies all steady states of (1.7) as $\mu\cos(kx)$ and $\mu\sin((2k+1)x/2)$ and identifies the mean-free condition as the mechanism that suppresses the one naively unstable linear mode.

Load-bearing premise

The proof requires the perturbation to be at least $C^3$ near the maximum, because then the renormalized perturbation vanishes to third order at that point; if only $C^{2-\epsilon}$ regularity is available, the cubic vanishing fails, a cusp-like perturbation is amplified exponentially, and the whole stability conclusion breaks down.

Editorial extensions

If this is right

  • Small $C^3$ mean-free perturbations of $\mu\cos(x)$ in the Proudman-Johnson equation converge to a nearby steady state $\mu^*\cos(x)$ at rate $e^{-\mu t/2}$, and the limiting amplitude is read off from the initial datum alone.
  • The corresponding IPM solutions blow up in finite time with the self-similar leading term $\cos(x)/(\tau^*-\tau)$ and a $C^1$ error of size $O((\tau^*-\tau)^{-3/4})$.
  • The stability transfers to the special classes of two-dimensional Euler and inviscid primitive equation solutions that reduce to (1.7), so their steady states of the form $\mu\cos(x)$ with $\mu>0$ are asymptotically stable as well.
  • The threshold is sharp: for every $\epsilon>0$ there are $C^{2-\epsilon}$ data arbitrarily close to the steady state whose solutions never settle on any $\mu\cos(x)$ stationary state.
  • The mean-free condition acts as a stabilizing mechanism: it cancels the one genuinely growing linear mode, converting apparent exponential instability into exponential decay.

Reading between the lines

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

  • A direct test of the conjectured instability of the other classified steady states $\mu\cos(kx)$ and $\mu\sin((2k+1)x/2)$ would be to run the same linearized analysis around them; the paper's classification suggests their maxima being attained away from a single interior point is what removes stability.
  • Because the asymptotic state is selected by the local second derivative at the initial maximum, one expects that in related nonlocal transport models the late-time state is fixed by the local shape of the datum at its extremum rather than by its global profile; this is an extension the authors do not pursue.
  • The $C^2$ endpoint is left open; a numerical simulation of (1.7) starting from a datum with a pure $|x|^2$ cusp at the maximum would show whether convergence to $\mu^*\cos(x)$ is merely slow or fails, settling whether the threshold is exactly $C^2$ or strictly above it.
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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 / 3 minor

Summary. The paper studies the stability of an explicit self-similar blow-up solution for a class of infinite-energy solutions to the 2D incompressible porous medium equations (IPM). Via a change of variables, the IPM blow-up problem is transformed into the asymptotic stability of the family of steady states \mu\cos(x) for the Proudman--Johnson equation (1.7). The main result, Theorem 1.3, proves that for any mean-free C^3 initial datum a0 satisfying \|a0 - \mu\cos\|_{C^3} \le \delta\mu, the solution is global in time and converges to \mu^*\cos(x) in C^1 with exponential rate e^{-\mu t/2}, where \mu^* = -\partial_x^2 a0(x_0^*) is selected by a conservation law along characteristics. Theorem 1.1 translates this into stable self-similar blow-up b(\tau,x) = \cos(x)/(\tau_*-\tau) + O((\tau_*-\tau)^{-3/4}) for IPM. Theorems 1.5 and 1.2 construct C^{2-\epsilon} perturbations that do not converge to any steady state, giving a dichotomy between C^3 and C^{2-\epsilon} behaviour. The proof combines weighted linear damping estimates (Propositions 3.1 and 3.4), a modulation decomposition in the eigenbasis of the linearized operator (Lemma 4.1), and a bootstrap argument in Section 5.

Significance. If the claims are appropriately adjusted, this would be a significant contribution to the rigorous understanding of singularity formation in the IPM equations and to the stability theory of the Proudman--Johnson equation. The determination of the asymptotic amplitude \mu^* by a conserved quantity is elegant and well supported. The linear analysis is detailed and the comparison-principle weights are carefully constructed. The explicit computation of the linearized spectrum and the stabilizing role of the mean-free condition are notable strengths. However, the advertised 'sharp regularity threshold' in the abstract is not established by the proven statements, and the local well-posedness proposition and Lemma 5.2 are stated without proofs. The paper is therefore not yet ready for acceptance as it stands.

major comments (3)
  1. [Abstract and Remark 1.4] The abstract claims that the paper identifies a sharp regularity threshold below which the blow-up is unstable. What is proved is stability for C^3 perturbations (Theorem 1.3) and instability for C^{2-\epsilon} perturbations (Theorem 1.5), with the intermediate C^{2+\epsilon} cases untreated. Remark 1.4 asserts that the C^{2+\epsilon} extension is 'only a technical extension', but this is not a routine variant of the proof: Proposition 3.1 constructs a weight W_\theta = O(|y|^3) near the origin, and Lemma 4.5 uses the full C^3 assumption to impose \xi_0(0)=\xi_0'(0)=\xi_0''(0)=0. For initial data merely in C^{2+\epsilon}, Taylor's theorem gives only \xi_0 = O(|y|^{2+\epsilon}) after the same modulation, so \xi_0/W_\theta is unbounded and the linear estimate (3.5) cannot be applied. A weight O(|y|^{2+\epsilon}) would yield a linear decay rate of order \epsilon s, and it is not shown that the nonlinear Duhamel terms in (5.6)--(5.8) close at that rate. The sharp threshold is therefore not established; either the C^{2+\epsilon} stability must be proved, or the abstract and Remark 1.4 must be revised to present only the proved C^3/C^{2-\epsilon} dichotomy.
  2. [Section 5.1, Proposition 5.1] The local well-posedness of (1.7) in C^3(\Omega) is stated with the remark that the argument is 'quite classical' and the details are omitted. This is a load-bearing existence statement for Theorem 1.3, since the bootstrap is performed on solutions whose existence is asserted by Proposition 5.1, and the regularity criterion (5.4) is used to extend the local solution globally. The manuscript should include a complete proof of the C^3 local well-posedness, or provide a precise reference with the statement adapted to the nonlocal term and boundary conditions, and justify the bound (5.4) as written.
  3. [Section 5, Lemma 5.2] Lemma 5.2, which gives the exponential decay in the weighted norm \omega = \sin^2(y/2) for the linearized derivative equation, is stated without proof ('the proof is similar'). This lemma is essential for the C^1 convergence in (1.20). The comparison-principle argument for the equation \partial_s u + (\sin y + \int_0^y \eta)\partial_y u - (\eta + \cos y)u = 0 is not literally the same as that for Lemma 3.5, because the zero-order coefficient is -(\eta + \cos y) and the weight is different. The proof should be written out in the paper or a complete reference should be supplied.
minor comments (3)
  1. [Theorems 1.1 and 1.2] In the statements of Theorems 1.1 and 1.2, the solution b is said to solve 'equation (1.7)', but b is a solution of the IPM-reduced equation (1.2); equation (1.7) is the Proudman--Johnson equation, which is used only in the change of variables. Please correct the equation numbers.
  2. [Section 5, Eq. (5.8)] The displayed inequality preceding (5.8) appears to contain a sign error in the exponential in the first integrand: it reads C_1\delta e^{(1-\theta')s} \|\xi(s,\cdot)/W_\theta\|_{L^\infty} inside the integral after factoring e^{(1-\theta')s}. If taken literally, the Grönwall argument would give growth of order \exp(C\delta e^{(1-\theta')s}), not the bound stated in (5.8). The intended estimate is presumably C_1\delta e^{-(1-\theta')s} \|\xi/W_\theta\|; please correct the exponent.
  3. [Abstract] The abstract contains a typo: 'identity a sharp regularity threshold' should read 'identify a sharp regularity threshold'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic amplitude is selected by an invariant proved in the paper, and all decay estimates are derived from self-contained linear and nonlinear arguments.

full rationale

The derivation chain in this paper is self-contained, and no claim reduces to its own input by construction or by load-bearing self-citation. In Theorem 1.3, the limiting amplitude mu* = -d^2_x a0(x*_0) is not a fitted parameter: it is fixed by the conserved second derivative along the transported maximum (Lemmas 2.3-2.5 and Corollary 2.6), and the proof then verifies mu(s) = mu0 through the modulation equation (4.19) and Lemma 4.3. The initial choice mu0 = -d^2_x a0(x*_0) is a normalization that makes the perturbation vanish cubically at the origin (Lemma 4.5), not a determination of the answer by assumption. The linear decay estimates of Propositions 3.1 and 3.4 are obtained from an explicitly constructed weight W_theta = O(|y|^3) via comparison principles, and the nonlinear bootstrap closes through the Duhamel estimates (5.6)-(5.8) and Gronwall's lemma, without importing the theorem's conclusion. The only external inputs are the explicit solution class and reduction of [9], and the authors' prior works [12,13] appear only as motivation ('in the continuation of [12,13]') and are not premises of Theorems 1.1-1.5. No uniqueness theorem or ansatz is imported from the authors' own prior work. The advertised sharp threshold is not fully proven because Remark 1.4 merely asserts that a C^{2+epsilon} extension would be technical, while the paper proves C^3 stability (Theorem 1.3) and C^{2-epsilon} instability (Theorem 1.5); however, this is a completeness and correctness gap, not an instance of circular reasoning. Accordingly, no circular step is present.

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

The central claim rests on the classical comparison-principle machinery for transport equations, on the explicit steady-state profile cos(x), and on the domain restriction to the special infinite-energy solution class; no free parameters are fitted to data, and the only new mathematical objects are analytical tools (eigenfunctions, weights) rather than postulated physical entities.

assumptions (5)
  • domain assumption The infinite-energy solutions of (IPM) reduce to the 1D equation (1.2) via the stream function ansatz psi = y f(tau,x), as established in [9].
    Invoked in Section 1.1 to set up the problem; the stability results apply only within this special class.
  • domain assumption Local well-posedness of (1.7) in Hoelder spaces C^3 and the regularity criterion (5.4) hold.
    Assumed in Proposition 5.1 with proof sketched and details omitted; required for the bootstrap and global existence.
  • standard math The comparison principle for linear transport equations with smooth coefficients applies and preserves inequalities for subsolutions and supersolutions.
    Used throughout Section 3 (Lemmas 3.2, 3.5, Proposition 3.4) and in Lemma 2.3.
  • domain assumption Initial data are mean-free and C^3 (or C^{2-epsilon} for the instability side).
    The theorems are stated under these conditions; the mean-free condition is preserved and is essential for killing the unstable mode (Remark 4.2 and Lemma 4.3).
  • domain assumption The characteristics flow for (1.7) is globally well-defined for the constructed solutions.
    Used in Lemmas 2.3-2.5, Lemma 6.1, and Section 5; follows from local well-posedness and the bootstrap bounds, but is not separately proved.

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Pith. "Pith review of Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations." pith.science (2026). https://pith.science/paper/MMXSPX7J

@misc{pith2026250717381,
  author       = {Pith},
  title        = {Pith review of: Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMXSPX7J}},
  note         = {Machine review of arXiv:2507.17381}
}
read the original abstract

We consider a special class of infinite energy solutions to the inviscid incompressible porous medium equations (IPM), introduced in Castro-C\'ordoba-Gancedo-Orive [9]. The (IPM) equations then reduce to a one-dimensional nonlocal nonlinear equation, for which an explicit self-similar blow-up solution is found in [9]. We show the stability of this explicit blow-up solution by smooth enough perturbations, and identity a sharp regularity threshold below which it is unstable. The heart of our proof is a change of variables that transforms the study of finite time blow-up solutions, to the study of global-in-time solutions to the Proudman-Johnson equation, which is a reduced equation that appears for special classes of solutions of the two-dimensional Euler equations, and of the inviscid primitive equations (or hydrostatic Euler equations). Our main result is in fact the asymptotic stability with decay estimates for a family of steady states of this reduced equation. It thus also implies the corresponding stability of steady states for the associated classes of solutions to the Euler equations and inviscid primitive equations.

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