{"id":"0cb759f7-58ea-4d88-81ea-441b27066898","arxiv_id":"2505.05165","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Asymptotic stability of the incompressible porous media equation holds in H^k for every k>2, with L2 convergence to the measure-preserving stratification at rate t^{-k/2}, and the threshold is sharp.","lead":"This paper proves that small perturbations of a stably stratified density in a porous medium flow stay bounded and converge to a specific stratified state at a precise algebraic rate, whenever the initial perturbation has slightly more than two Sobolev derivatives. It closes a gap: below two derivatives the same configuration is known to be unstable, so the result identifies the sharp smoothness threshold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.2 rests on a mis-stated imported ODE lemma; the intended decay bound is likely recoverable by a direct Jensen argument, but as written the rate proof has a gap.","rationale":"The stress-test pass confirms the reader's identification of the weakest point: Proposition 5.2, the bridge between the variational inequality (2.31) and the t^{-k} decay of the potential energy, uses an imported lemma that is both false as stated and inapplicable to the differential inequality at hand. The energy decay is then used to obtain the averaged L2 decay (5.6), which feeds into Proposition 5.3 and the final bootstrap; if the decay step were false, the main theorem would not follow. This is a genuine gap in the written proof, not a mere stylistic issue. However, the concern is not fatal to the underlying mathematics: a standard ODE comparison via G = E^{-1/(k-1)} combined with Jensen's inequality converts the a priori bound ∫||u||^2_{H^k} ≤ δ into the claimed E(t) ≤ Cδ/t^k. Therefore the central claim is likely true, but the manuscript must be corrected to replace Lemma 2.1 with a valid argument (or state and prove a corrected lemma). Other weaknesses noted in passing, such as the footnote in Proposition 4.2 admitting an omitted commutator proof and the boundary terms in Lemma 5.1, are less load-bearing; they are either standard or affect only technical uniformity. The verdict CONDITIONAL remains appropriate: the proof is in outline credible and the flaw is localized and fixable, but as printed the rate argument is not rigorous.","tokens_in":35146,"tokens_out":16765,"duration_ms":159711,"concrete_test":"Re-derive Proposition 5.2 without invoking Lemma 2.1. From E' ≤ -C E^{k/(k-1)} ||u||_{H^k}^{-2/(k-1)} and ∫_0^t ||u||^2_{H^k} ds ≤ δ, set G = E^{-1/(k-1)} and apply Jensen's inequality to bound G(t) from below by c t^{k/(k-1)} δ^{-1/(k-1)}, yielding E(t) ≤ Cδ t^{-k}. If this derivation succeeds for all t in [0,T], the rate claim stands and the issue reduces to an erroneous citation; if it fails because the support or integrability of ||u||_{H^k} prevents the Jensen lower bound, then the bootstrap in Section 6 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rate estimate E(t) ≤ Cδ t^{-k} in Proposition 5.2 is proved by invoking Lemma 2.1, a lemma imported from [17] without proof. As printed, Lemma 2.1 states: if f' ≤ -a - α f^n, then f(t) ≤ C A^{α/(n-1)} t^{-(α+1)/(n-1)} with A = ∫_0^t a. This lemma is false when a ≡ 0, since then it would imply f(t) = 0 for a solution of f' ≤ -α f^n. Moreover, the actual differential inequality derived in Proposition 5.2 is E' ≤ -C E^{k/(k-1)} ||u||_{H^k}^{-2/(k-1)}, which is not of the form f' ≤ -a - α f^n for a(t) = C||u(t)||^2_{H^k}. Thus the application does not match the lemma's hypotheses, even setting aside the a = 0 issue. This is load-bearing because Proposition 5.2 supplies both the pointwise energy decay (5.5) and the averaged decay (5.6), which are used in Proposition 5.3 and in the bootstrap in Section 6. If the intended ODE comparison cannot convert ∫_0^t ||u||^2_{H^k} ≤ δ into pointwise t^{-k} decay of E, the main convergence rate and the bootstrap collapse. That said, the conclusion appears recoverable: setting G = E^{-1/(k-1)}, the differential inequality gives G' ≥ c ||u||_{H^k}^{-2/(k-1)}, and Jensen's inequality yields G(t) ≥ c t^{k/(k-1)} δ^{-1/(k-1)}, hence E(t) ≤ Cδ/t^k. So the gap is likely repairable, but the manuscript as written does not contain this argument and instead cites an invalid lemma.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the incompressible porous media equation on the periodic strip T×R near a stably stratified steady state ρs satisfying (1.4). The main result, Theorem 1.1, asserts asymptotic stability in H^k for every real k>2: small H^k perturbations lead to global unique solutions with a uniform H^k bound, and the solution converges to the measure-preserving stratification ρ*_0 of the initial data at the rate ||ρ(t)-ρ*_0||_{L^2} ≤ Cε t^{-k/2}. Remarks 1.2-1.3 interpret the rate and the regularity threshold as sharp, using an explicit linear lower bound in Appendix A and the known H^2 ill-posedness. The proof combines a potential-energy functional E(t) with decay analysis in Section 5, anisotropic commutator estimates in Section 4, and a bootstrap argument in Section 6.","tokens_in":35501,"tokens_out":35018,"duration_ms":321575,"significance":"If the result is correct, it closes the regularity gap for this problem, improving the H^3 threshold of [17] to the H^{2+} endpoint and identifying the sharp algebraic decay rate. The two main ingredients—the non-coercive potential-energy inequality (2.31) and the commutator estimates valid for all real k>2—are genuine contributions, and the linear sharpness construction in Appendix A is explicit. The paper is clearly written and most estimates are presented in detail. However, the proof as written contains a gap in the central decay Proposition 5.2, caused by an invalid imported ODE lemma. The gap appears repairable by a direct Jensen argument, so the overall result is likely salvageable, but the manuscript in its current form does not establish the advertised rate.","major_comments":[{"comment":"Lemma 2.1 is false as printed: with a≡0 it asserts f(t)≤0 for every nonnegative solution of f'≤-α f^n, which is impossible for positive initial data. Moreover, the differential inequality actually derived in Proposition 5.2, namely E' ≤ -C E^{k/(k-1)} ||u||_{H^k}^{-2/(k-1)}, is not of the form f'≤-a-α f^n: the coefficient of E^{k/(k-1)} is time-dependent and involves a negative power of ||u||_{H^k}, not a constant α plus an additive -a(t). The application of Lemma 2.1 with a(t)=C||u(t)||^2_{H^k}, n=k/(k-1), α=1/(k-1) therefore does not yield E(t)≤Cδ/t^k. This gap is load-bearing because Proposition 5.2 supplies both (5.5) and (5.6), which are used in Proposition 5.3 and in the Section 6 bootstrap. The conclusion appears recoverable: setting G=E^{-1/(k-1)}, the inequality gives G' ≥ c ||u||_{H^k}^{-2/(k-1)}; integrating and applying Jensen's inequality with ∫_0^t ||u||^2_{H^k}≤δ yields G(t) ≥ c δ^{-1/(k-1)} t^{k/(k-1)}, hence E(t)≤Cδ/t^k. I recommend replacing the invalid lemma by a correct ODE comparison or integrating this Jensen argument directly, and removing the citations to Lemma 2.1 in Proposition 5.2.","section":"Section 2, Lemma 2.1, and Section 5, Proposition 5.2"},{"comment":"The displayed estimate for g_2 has an algebraic slip in the power of δ. Since g_2 contains a factor ||θ||_{H^k} and (5.1) gives ||θ||_{H^k}≤√δ, the factor outside the first term on the right-hand side should be √δ, not δ. The resulting bound is C M^{1/2}δ^{3/2}/t^k, rather than C M^{1/2}δ^2/t^k. This does not destroy the contradiction argument because for δ≤1/M the final right-hand side can still be absorbed into Cδ/t^{k-1}, but the displayed chain of inequalities as written is not correct and should be fixed.","section":"Section 5, Proposition 5.3, equation (5.20)"}],"minor_comments":[{"comment":"The statement of Lemma 2.3 writes ∫_T^t f(t)ds, which should read ∫_t^T f(s)ds; the proof makes the intended meaning clear.","section":"Section 2, Lemma 2.3"},{"comment":"In the definition of |D_2|^k f, the Fourier transform variable is written as ξ_1; it should be ξ to match the notation used elsewhere.","section":"Section 2, equation (2.2)"},{"comment":"\"Threfore\" should be \"Therefore\".","section":"Section 3, after (3.3)"},{"comment":"\"diffierentiating\" should be \"differentiating\".","section":"Section 2.3, proof of Lemma 2.5"},{"comment":"\"Soboelv\" should be \"Sobolev\" in the two occurrences near equations (4.45)-(4.51).","section":"Section 4, proof of Lemma 4.1"},{"comment":"When Proposition 5.2 is invoked, the constant in E(t)≤Cε^2/t^k should first be written as E(t)≤C M ε^2/t^k before absorbing M into the generic constant, since the a priori size in (6.4) is Mε^2.","section":"Section 6, after (6.11)"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized to Lemma 2.1/Proposition 5.2, and I am confident it is repairable by the Jensen argument sketched in the report. The rest of the proof appears coherent after the minor corrections listed. I would not recommend rejection, but the current version does not establish the advertised decay rate as written. Since Lemma 2.1 is imported from the coauthor's preprint [17], the authors should also check and correct the lemma in that source."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real result—sharp threshold k>2 for IPM stability—and the variational-energy strategy is the right idea. But the written proof of Proposition 5.2 leans on a lemma that, as stated, is false, and the application doesn't match the lemma's hypotheses. The gap is repairable by a direct Jensen argument, but it needs to be fixed before I'd call the proof complete.\n\nWhat's new: Theorem 1.1 is a genuine improvement over Park's H^3, reaching every real k>2, which is optimal given the H^2 ill-posedness. The level-set decomposition using the measure-preserving stratification and the potential energy functional is elegant and gives quantitative L2 decay to the actual long-time limit, not just to some stratified state. The commutator estimates in Section 4, especially the anisotropic bound involving only ∇u2, look careful and are the technical heart; I didn't find a fatal issue there. The linear rate in Appendix A is consistent with the claimed sharpness.\n\nWhere it's soft: the energy-decay proposition. Lemma 2.1 is imported from [17] without proof and, as printed, is false when a≡0—the RHS would force f to vanish. Worse, the actual inequality E' ≤ -C E^{k/(k-1)} ||u||^{-2/(k-1)}_{H^k} is not of the form covered, and even a naive application of the lemma would produce t^{-k/(k-1)}, not t^{-k}. The fix is short: let G=E^{-1/(k-1)}, get G' ≥ c||u||^{-2/(k-1)}_{H^k}, then Jensen gives G(t) ≥ c t^{k/(k-1)} δ^{-1/(k-1)}, hence E(t) ≤ Cδ/t^k. The authors should replace the citation with this argument. It's a genuine gap in the written proof, but not a deep one, and I don't think it sinks the theorem.\n\nMinor: the boundary term in Lemma 5.1 relies on x2 u2 decay; the authors dismiss it quickly, but it seems fine given the weighted estimates from Proposition 3.1. The bootstrapping in Section 6 is a bit dense, but the log-scale stopping time trick checks out.\n\nWho it's for: anyone working on long-time behavior of transport equations, stratified fluids, or optimal regularity for IPM. It deserves a serious referee—send it out. If the authors patch the ODE lemma, I'd accept without major hesitation.","headline":"Sharp k>2 stability for IPM is a genuine result, but Proposition 5.2's decay proof cites a false imported lemma; the gap is repairable and the theorem likely holds.","tokens_in":36114,"tokens_out":3057,"would_cite":true,"duration_ms":27572,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76S05","35Q35","34D05","76B03"],"pacs":[],"model":"deepseek-v4-flash","headline":"Near a stably stratified density, the incompressible porous media equation is asymptotically stable in $H^k$ for every real $k>2$, with convergence to the measure-preserving stratification of the initial data at rate $t^{-k/2}$; the…","keywords":["asymptotic stability","incompressible porous media equation","stratified density","Sobolev regularity threshold","potential energy","measure-preserving rearrangement","sharp decay rate","ill-posedness"],"falsifier":"The decisive check is to compare the inequality available at (2.31), $dE/dt \\le -C E^{k/(k-1)} \\|u\\|_{H^k}^{-2/(k-1)}$, with the hypotheses of [17, Lemma 2.1]: the lemma as printed requires $f'\\le -a(t)-\\alpha f^n$, and with $a=0$ its conclusion would assert $f(t)\\le 0$ for positive $f$, which is false. If the intended inequality can be rewritten as $dE/dt \\le -C\\|u\\|_{H^k}^2 - C' E^{k/(k-1)}$ or another admissible form, the decay step is sound; otherwise the model solution of $f'=-C f^{k/(k-1)}$, which decays like $t^{-(k-1)}$ rather than $t^{-k}$, shows that the printed lemma alone cannot deliver the claimed rate.","tokens_in":34884,"feed_emoji":"🌊","tokens_out":11916,"duration_ms":102400,"temperature":0.7,"pith_summary":"The paper establishes that stably stratified steady states of the incompressible porous media equation are asymptotically stable against perturbations in the Sobolev space $H^k$ for any real $k>2$, with the density converging to the measure-preserving stratification of its initial data at the rate $t^{-k/2}$. This closes the gap left by earlier results that required $H^3$ or smoother data, while $H^2$ perturbations are known to be ill-posed. The proof works by showing that a non-coercive potential energy functional, which measures how far the density is from its measure-preserving stratification, decays polynomially in time, and by controlling higher Sobolev norms through refined commutator estimates. The decay rate matches the linearized equation, and an explicit linear example shows that no faster uniform rate is possible.","feed_headline":"Porous-media flow near a stable layer is stable in H^k for every k>2","feed_subtitle":"Perturbations settle to the initial rearrangement at rate t^{-k/2}; no lower regularity can work.","key_machinery":"The central object is the potential energy $E(\\rho(t))=\\lim_{s\\to\\infty}\\left(\\int_{\\{-s<\\rho<s\\}}\\rho x_2\\,dx - \\int_{\\{-s<\\rho_0^*<s\\}}\\rho_0^* x_2\\,dx\\right)$, which stays nonnegative, satisfies $dE/dt=-\\|u\\|_{L^2}^2$, and is comparable to $\\|\\rho-\\rho_0^*\\|_{L^2}^2$; the measure-preserving stratification $\\rho_0^*$ is its unique minimizer. Because $E$ is not coercive against $\\|u\\|_{L^2}$, the argument uses a level-set decomposition to prove the interpolation inequality $\\|u\\|_{L^2}^2 \\ge C E^{k/(k-1)} \\|u\\|_{H^k}^{-2/(k-1)}$, which turns the energy identity into a differential inequality with polynomial decay $E(t)\\le C\\delta t^{-k}$. The second load-bearing ingredient is a family of commutator estimates for the transport term that lose no derivatives and control the evolution of $\\|\\theta\\|_{H^k}$ and of second derivatives of $\\theta$ using only $\\|\\nabla u_2\\|_{L^\\infty}$, uniformly in $k>2$.","core_discovery":"Theorem 1.1 asserts that if $\\rho_s$ is a stratified density with $\\inf(-\\partial_2\\rho_s)>0$ and $\\partial_2\\rho_s \\in C^{k+1}$ for some $k>2$, and if $\\|\\rho_0-\\rho_s\\|_{H^k}\\le \\varepsilon$, then the IPM equation has a unique global solution with $\\|\\rho(t)-\\rho_s\\|_{H^k}\\le C\\varepsilon$ for all $t$, and $\\|\\rho(t)-\\rho_0^*\\|_{L^2}\\le C\\varepsilon t^{-k/2}$, where $\\rho_0^*$ is the measure-preserving stratification of the initial density. The regularity condition $k>2$ is optimal because the stratified state is strongly ill-posed in $H^2$, and the algebraic rate is sharp because a solution of the linearized equation can be chosen whose $L^2$ norm decays no faster than $t^{-k/2-\\epsilon}$.","pith_inferences":["Editorial inference: the same potential-energy comparison, with the interpolation exponent $k/(k-1)$, plausibly sets the optimal regularity threshold and algebraic rate for the damped 2D Boussinesq analogue mentioned in the paper, although the details are not worked out here.","Editorial inference: replacing the imported ODE comparison step by a self-contained lemma for $f'\\le -C f^{k/(k-1)} A(t)^{-1/(k-1)}$ with $\\int_0^T A \\le \\delta$ would make the decay proof independent of the exact hypotheses of [17, Lemma 2.1] and would follow directly from (2.31) together with the time-averaged bound.","Editorial inference: a numerical run of the IPM equation near $\\rho_s=-x_2$ with $H^k$ data for $k$ just above 2 should show $\\|\\rho(t)-\\rho_0^*\\|_{L^2}\\sim t^{-k/2}$; observing a slower power would locate the obstruction in the energy-decay step rather than in the commutator estimates."],"forward_implications":["For every $k>2$, a small $H^k$ perturbation of a stable stratified density exists globally and settles to the unique measure-preserving rearrangement of its initial data, with a quantitative $t^{-k/2}$ rate in $L^2$.","The threshold $k>2$ is final: no stability result of this type can hold in $H^2$, in view of the known ill-posedness there, and perturbations in $H^{2-\\epsilon}$ can grow.","The nonlinear decay rate equals the linearized decay rate, so the nonlinearity does not degrade the sharp linear prediction.","The corollary $\\int_0^T \\|\\nabla u_2\\|_{L^\\infty}dt \\le \\sqrt{\\delta}$ independent of $T$ supplies the smallness that closes the bootstrap, yielding global existence in addition to decay."],"supporting_citations":[{"why":"Supplies the energy-structure approach and the ODE comparison lemma used in Proposition 5.2, and provides the previous $H^3$ stability result that this paper improves.","marker":"[17]"},{"why":"Introduced the potential-energy functional and its monotonicity along the flow, the starting point for the variational argument.","marker":"[11]"},{"why":"Establishes strong ill-posedness in $H^2$, which makes $k>2$ the optimal regularity threshold.","marker":"[2]"},{"why":"Shows long-time growth of Sobolev norms and instability in $H^{2-\\epsilon}$, motivating the criticality of $H^2$.","marker":"[15]"},{"why":"Provides local well-posedness in $H^k$ for $k>2$, the baseline theory on which the global bootstrap is built.","marker":"[8]"},{"why":"Introduced the measure-preserving stratification via level-set decomposition, used here to identify the long-time limit $\\rho_0^*$.","marker":"[10]"},{"why":"Quantified sharp decay rates for quasi-linearly stratified densities, serving as a comparison point for the sharp rate obtained here.","marker":"[12]"}],"fun_headline_variants":["IPM stability sharp: H^k for k>2, ill-posed in H^2","Sharp stability for porous media: H^k settles at t^{-k/2} for k>2","Porous media flow: stable above H^2, sharp decay t^{-k/2}","Incompressible porous media: H^k stability threshold k=2 sharp","Porous media: H^k decay t^{-k/2} for any k>2, optimal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ODE comparison lemma imported from [17] can be applied to $dE/dt \\le -C E^{k/(k-1)} \\|u\\|_{H^k}^{-2/(k-1)}$ to yield $E(t)\\le C\\delta t^{-k}$, even though the lemma as printed assumes $f'\\le -a(t)-\\alpha f^n$ and gives no bound when $a=0$; if that comparison cannot be justified, the sharp rate and the bootstrap do not follow from the written argument.","fun_headline_variants_meta":{"raw":{"variants":["IPM stability sharp: H^k for k>2, ill-posed in H^2","Sharp stability for porous media: H^k settles at t^{-k/2} for k>2","Porous media flow: stable above H^2, sharp decay t^{-k/2}","Incompressible porous media: H^k stability threshold k=2 sharp","Porous media: H^k decay t^{-k/2} for any k>2, optimal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000323,"raw_usage":{"total_tokens":1783,"prompt_tokens":881,"completion_tokens":902,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":784}},"tokens_in":497,"tokens_out":902,"duration_ms":7654,"temperature":1.0,"reasoning_tokens":784,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:13:12.704697+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is to compare the inequality available at (2.31), $dE/dt \\le -C E^{k/(k-1)} \\|u\\|_{H^k}^{-2/(k-1)}$, with the hypotheses of [17, Lemma 2.1]: the lemma as printed requires $f'\\le -a(t)-\\alpha f^n$, and with $a=0$ its conclusion would assert $f(t)\\le 0$ for positive $f$, which is false. If the intended inequality can be rewritten as $dE/dt \\le -C\\|u\\|_{H^k}^2 - C' E^{k/(k-1)}$ or another admissible form, the decay step is sound; otherwise the model solution of $f'=-C f^{k/(k-1)}$, which decays like $t^{-(k-1)}$ rather than $t^{-k}$, shows that the printed lemma alone cannot deliver the claimed rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the potential-energy functional and its monotonicity along the flow, the starting point for the variational argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows long-time growth of Sobolev norms and instability in $H^{2-\\epsilon}$, motivating the criticality of $H^2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides local well-posedness in $H^k$ for $k>2$, the baseline theory on which the global bootstrap is built."},{"cited_title":"Long-time behavior of the Stokes-transport system in a channel","cited_arxiv_id":"2306.00780","evidence_quote":"Introduced the measure-preserving stratification via level-set decomposition, used here to identify the long-time limit $\\rho_0^*$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Quantified sharp decay rates for quasi-linearly stratified densities, serving as a comparison point for the sharp rate obtained here."}],"review_version":1}