{"id":"fa15560e-72f5-45db-8efe-555671ccbf8f","arxiv_id":"2507.17381","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Smooth small perturbations of the explicit IPM self-similar blow-up still blow up in the same self-similar form, with a sharp regularity threshold at C^2.","lead":"This paper proves that a known self-similar blow-up for a special class of solutions to the incompressible porous medium equations is stable under smooth perturbations, and unstable for rougher ones. The proof works by mapping the blow-up problem to the stability of steady states of the Proudman-Johnson equation.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharp-threshold claim is not established: Theorem 1.3 proves C^3 stability only, while the C^{2+ε} extension asserted in Remark 1.4 is unproved and does not follow from the cubic-weight argument of Proposition 3.1.","rationale":"I read the stability proof in good faith. The linear analysis (Lemma 4.1, Propositions 3.1 and 3.4) is coherent; the modulation choices in Lemma 4.5 correctly enforce ξ0(0)=ξ0'(0)=ξ0''(0)=0; the bootstrap estimates in Section 5 appear to close at the C^3 level. The omitted local well-posedness (Proposition 5.1) is a gap but likely fillable by the characteristics method sketched there. The more serious issue is the gap between what is proved and what is advertised. The phrase 'sharp regularity threshold' requires stability at every regularity above the threshold and instability below it. The paper proves stability exactly at C^3 (not, e.g., at C^{2+ε}) and instability at C^{2−ε}. The C^{2+ε} case is relegated to a remark with no proof, and the proof's cubic weight is a structural obstacle: the decay mechanism operates on functions vanishing to order three, and a C^{2+ε} perturbation only vanishes to order 2+ε. A weight with that exponent would give a much slower (order-ε) decay rate, and no calculation shows the nonlinear terms are then subcritical. Thus the central 'sharp threshold' claim is not supported as written. This matches the reader's identified weakest assumption, so I agree with the CONDITIONAL verdict and recommend no change.","tokens_in":30405,"tokens_out":33638,"duration_ms":318130,"concrete_test":"Formulate the C^{2+ε} analogue of Proposition 3.1 with weight W ~ |y|^{2+ε} and carry out the Section 5 bootstrap. Specifically, verify whether the forcing estimates for (2β+ξ)ξ, (∫η)∂yβ, and N1 in (5.7) are controlled by a decay factor e^{-ε s} and whether the modulation estimates (5.9)–(5.10) still close; if the estimates fail, the Remark 1.4 extension is false, and if they close, a written proof must be added before the sharp-threshold wording is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract advertises a sharp regularity threshold below which the steady states are unstable. What is proved is a dichotomy: C^3 perturbations converge (Theorem 1.3), while C^{2−ε} perturbations can be chosen not to converge (Theorem 1.5). The threshold itself requires the C^{2+ε} stability asserted only in Remark 1.4. That assertion is not a routine variant of the proof. Proposition 3.1 constructs a weight Wθ = O(|y|^3) and obtains decay because L0(y^p) = (2−p)y^p near the origin; the bootstrap in Section 5 needs the remainder ξ to vanish to third order at the maximum, which Lemma 4.5 achieves using C^3. For initial data merely C^{2+ε} with ε < 1, Taylor's theorem gives only ξ0 = O(|y|^{2+ε}) after the same modulation, so ∥ξ0/Wθ∥_{L∞} is infinite and the stated linear estimate (3.5) cannot even be applied. A natural fix is a weight O(|y|^{2+ε}), but then the linear decay rate is only ε s, and it is not shown that the nonlinear Duhamel terms in (5.6)–(5.8) close at that rate. The C^2 endpoint is also left open. Hence the paper does not yet identify a sharp threshold; the missing C^{2+ε} proof is load-bearing for the advertised conclusion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30712,"tokens_out":14700,"duration_ms":136314,"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":[{"comment":"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.","section":"Abstract and Remark 1.4"},{"comment":"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.","section":"Section 5.1, Proposition 5.1"},{"comment":"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.","section":"Section 5, Lemma 5.2"}],"minor_comments":[{"comment":"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.","section":"Theorems 1.1 and 1.2"},{"comment":"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.","section":"Section 5, Eq. (5.8)"},{"comment":"The abstract contains a typo: 'identity a sharp regularity threshold' should read 'identify a sharp regularity threshold'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.AP and the core C^3 stability proof appears to be a substantial contribution. The main obstacle is the discrepancy between the advertised 'sharp regularity threshold' and the actually proved C^3/C^{2-\\epsilon} dichotomy; this should be resolved by either proving the C^{2+\\epsilon} extension or narrowing the claims. The omitted proofs of Proposition 5.1 and Lemma 5.2 are also important for the self-containedness of the paper. With these points addressed, the paper would likely be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take. The core of this paper is real. The change of variables that maps the IPM blow-up family to the PJ steady states is genuinely new and makes the problem tractable. The stability proof for C^3 perturbations (Theorem 1.3) is substantial: the conserved quantity that selects μ*, the explicit eigenmodes of the linearized operator, and the bootstrap with the cubic weight all hang together as far as I can tell from a careful read. The instability construction for C^{2-ε} (Theorem 1.5) is also clean — a cusp perturbation that never returns to any stationary state. I would send this to a competent referee.\n\nBut the abstract oversells. It says a sharp regularity threshold is identified. What is proved is a dichotomy: C^3 perturbations converge (with exponential rate), and for any ε>0 there exist C^{2-ε} perturbations that do not converge to any stationary state. That leaves a gap, and the gap is not filled by Remark 1.4. The claim there that the arguments extend to C^{2+ε} looks dubious as stated, because Proposition 3.1 uses a weight Wθ ~ |y|^3 and requires the perturbation to vanish to third order at the maximum. For C^{2+ε} initial data, even after the modulation, you only get O(|y|^{2+ε}), so the quotient ξ0/Wθ is infinite. A weight O(|y|^{2+ε}) would give a decay rate of only ε, and the paper does not show the nonlinear Duhamel terms close at that slower rate. So the sharp threshold is not established; the C^2 endpoint is open.\n\nTwo other soft spots. The local well-posedness in Hölder spaces (Proposition 5.1) and the decay lemma for the derivative (Lemma 5.2) are stated without proof. The former is not a one-line remark — the stability theorem needs global existence, and while the characteristics argument is classical, it deserves to be written out or properly referenced. There are also equation-number typos in Theorems 1.1 and 1.2 (they say (1.7) where they mean (1.2)). Minor, but sloppy.\n\nMy bottom line: the main theorems are worthy of serious peer review, but the paper needs major revision before acceptance. The abstract should be rewritten to say what is actually proved, and the missing proofs should be supplied.\n\nI'd send it out. I would not cite the sharp threshold in the abstract, but I would cite the C^3 stability theorem and the conserved-quantity selection.","headline":"Solid C^3 stability theorem and a clean instability construction, but the advertised 'sharp regularity threshold' is not proved.","tokens_in":31261,"tokens_out":6925,"would_cite":true,"duration_ms":70663,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35B40","35B44","35Q31","76S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["incompressible porous medium equations","Proudman-Johnson equation","self-similar blow-up","asymptotic stability","sharp regularity threshold","infinite energy solutions","weighted linear damping","hydrostatic Euler equations"],"falsifier":"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.","tokens_in":30195,"feed_emoji":"🌊","tokens_out":11921,"duration_ms":113344,"temperature":0.7,"pith_summary":"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.","feed_headline":"Blow-up in porous media is stable above a sharp smoothness cutoff","feed_subtitle":"Small smooth perturbations keep the blow-up shape; cusp-shaped rough ones destroy it.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the 1D reduction (1.2) and the explicit blow-up solution (1.4) whose stability is the paper's target.","marker":"[9]"},{"why":"Defines the reduced one-dimensional equation (the Proudman-Johnson equation) and its derivation from stagnation-point flows, the setting of all main theorems.","marker":"[31]"},{"why":"Provides the Eulerian quantitative approach to blow-up stability in similar one-dimensional fluid models that this paper adapts to the Proudman-Johnson setting.","marker":"[12]"},{"why":"Establishes stable singularity formation for the inviscid primitive equations, the analogue that the present steady-state stability result extends.","marker":"[13]"},{"why":"Proves stability of the steady state $\\sin(\\theta)$ for the De Gregorio equation by spectral analysis and energy estimates, the closest existing model for the stability theorem.","marker":"[21]"},{"why":"Studies the role of initial curvature in solutions of the generalized Proudman-Johnson equation, the same feature that here determines the asymptotic amplitude $\\mu^*$.","marker":"[37]"}],"fun_headline_variants":["Porous media blow-up stable above sharp smoothness cutoff","Stable blow-up in porous media hinges on smoothness threshold","Sharp smoothness cutoff governs porous media blow-up stability","Infinite energy porous media blow-up: smooth stable, cusp unstable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Porous media blow-up stable above sharp smoothness cutoff","Stable blow-up in porous media hinges on smoothness threshold","Sharp smoothness cutoff governs porous media blow-up stability","Infinite energy porous media blow-up: smooth stable, cusp unstable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001164,"raw_usage":{"total_tokens":4893,"prompt_tokens":1093,"completion_tokens":3800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":3730}},"tokens_in":709,"tokens_out":3800,"duration_ms":26111,"temperature":1.0,"reasoning_tokens":3730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:50:10.114880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the 1D reduction (1.2) and the explicit blow-up solution (1.4) whose stability is the paper's target."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the reduced one-dimensional equation (the Proudman-Johnson equation) and its derivation from stagnation-point flows, the setting of all main theorems."},{"cited_title":"E., Ibrahim, S","cited_arxiv_id":null,"evidence_quote":"Provides the Eulerian quantitative approach to blow-up stability in similar one-dimensional fluid models that this paper adapts to the Proudman-Johnson setting."},{"cited_title":", & Lin, Q.Y","cited_arxiv_id":null,"evidence_quote":"Establishes stable singularity formation for the inviscid primitive equations, the analogue that the present steady-state stability result extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves stability of the steady state $\\sin(\\theta)$ for the De Gregorio equation by spectral analysis and energy estimates, the closest existing model for the stability theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Studies the role of initial curvature in solutions of the generalized Proudman-Johnson equation, the same feature that here determines the asymptotic amplitude $\\mu^*$."}],"review_version":1}