Finite time blow-up in a 1D model of the incompressible porous media equation

Alexander Kiselev; Naji A. Sarsam

REVIEW 2 major objections 6 minor 1 cited by

Finite time blow-up in a 1D model of the incompressible porous media equation

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that smooth, bounded, even, nonnegative periodic data with a zero at the origin and monotone increase toward the boundary lose smoothness in finite time in a proposed 1D model of the incompressible porous media equation.

desk verdict A new 1D IPM boundary-layer model with a plausible finite-time blow-up proof, held back only by hand-verified monotonicity estimates that need to be made rigorous. read the letter →

arxiv 2412.16376 v2 pith:7I3OCU37 submitted 2024-12-20 math.AP

Alexander Kiselev , Naji A. Sarsam This is my paper · ORCID

classification math.AP MSC 76B0335Q3576S05

keywords one-dimensionalfluidmodelsingularityformationIPMequationfinite-timeblow-upactivescalarnonlocalvelocityCórdoba-Córdoba-Fontelosboundarylayer

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 proposes a one-dimensional transport equation with nonlocal velocity as a model for the boundary trace of a solution to the two-dimensional incompressible porous media (IPM) equation in a periodic half-plane. The velocity is given by a singular integral operator $H_a$ that interpolates between the zero operator and the Hilbert transform. The paper proves that smooth periodic data in a natural monotone class—even, nonnegative, vanishing at the origin, and increasing toward the boundary—lose smoothness in finite time under this model, even though the density itself stays bounded. The argument runs through a weighted inequality for $H_a$ that forces a certain weighted norm of the solution to blow up in finite time, and it also yields local well-posedness of the model.

What carries the argument

The central object is the singular integral operator $H_a$ with kernel $K_a(y)=\frac{a^2}{\pi y(y^2+a^2)}$, a regularized Hilbert transform that emerges from the boundary-layer Biot-Savart law for IPM. The proof of blow-up rests on the weighted inequality (4.2): for even nonnegative $f$ with $f(0)=0$ and $f'\ge 0$ on $[0,\pi)$, $$-\$int_0^{{\pi/2}}$ \frac{H_a f(x) f'(x)}{x^\$\sigma$}\, dx \ge C_{a,\$\sigma$}\$int_0^{{\pi/2}}$ \frac{f(x)^2}{$x^{{1+\sigma}}$}\, dx.$$ This is obtained by bounding $H_a f(x)$ through a kernel $G_a(x,y)$ defined in Lemma 4.5, whose monotonicity and sign properties are verified by hand and control the nonlocal interaction. Inserting this inequality into the evolution of $J(t)$ yields a Riccati-type lower bound and hence finite-time blow-up.

What would settle it

A numerical evaluation of $G_a(x,qx)-G_a(x,x/q)$ and $-G_a(x,qx)$ for a fixed $a>0$ and $1<q<2$ (say $a=1$, $q=1.5$) on $0<x\le \pi/2$ would settle the key step: if the difference is ever increasing or negative, or if $-G_a(x,qx)$ fails to be strictly decreasing and vanish at $x=2\pi/q$, then Proposition 4.3 and Theorem 1.2 lose their proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for any fixed $a,g>0$, if the initial density $\rho_0$ is smooth, bounded, nonnegative, even, non-identically zero on the circle, with $\rho_0(0)=0$ and $\rho_0'(x)\ge 0$ on $[0,\pi)$, then the unique local smooth solution to (1.2) satisfies $\lim_{t\to T^*}\int_0^t \|\partial_x \rho(\cdot,s)\|_\infty\, ds = \infty$ for some finite $T^*$, so $\rho$ cannot remain in $C^\infty(\mathbb{T})$ for all time. The proof couples a Beale-Kato-Majda type criterion (Proposition 4.1) with a weighted inequality for $H_a$ (Proposition 4.3) that gives $J'(t) \ge cJ(t)^2$ for $J(t)=\int_0^{\pi/2} \rho(x,t)/x^{1+\delta}\, dx$, forcing finite-time blow-up of this functional and hence of the derivative norm.

Load-bearing premise

The blow-up proof hinges on a hand-verified claim about the kernel comparison function $G_a$: the difference $G_a(x,qx)-G_a(x,x/q)$ is strictly decreasing and $-G_a(x,qx)$ is strictly decreasing with $-G_a(2\pi/q,2\pi)=0$; if any of these monotonicity or sign assertions fails, the weighted lower bound collapses.

Editorial extensions

If this is right

For the class of initial data in Theorem 1.2, the model (1.2) exhibits finite-time loss of smoothness for every choice of $a,g>0$, with the density remaining nonnegative and bounded up to the blow-up time.
Since $H_a$ approaches the Hilbert transform as $a\to\infty$, the blow-up result formally suggests finite-time blow-up for the CCF equation with the same initial-data profile (modulo a sign change).
A Beale-Kato-Majda type criterion holds for the model: the first blow-up time is characterized by divergence of $\int_0^t \|\partial_x \rho\|_\infty\, ds$, so singularity formation is driven purely by the derivative.
The same construction on the real line gives finite-time blow-up for smooth nonnegative even bounded data, and compactly supported modifications of such data should still blow up in finite time because the equation has finite speed of propagation.
The model preserves the symmetry class: evenness, nonnegativity, the zero at the origin, and monotonicity on the half-period all persist until blow-up, so the singularity forms within the same monotone boundary-layer profile.

Reading between the lines

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

Quantitative versions of the constant $C_{a,\sigma}$ in Proposition 4.3 would yield an explicit upper bound on the blow-up time, $T^* \le (g\,C_{a,\delta}\,J(0))^{-1}$, which numerical simulations of (1.2) could check directly.
The same weighted-inequality strategy may adapt to the finite periodic channel $\mathbb{T}\times(0,1)$, where the Green's function is more involved but approximates the half-plane Green's function near the boundary; proving the analogue would turn the authors' suspicion about a more confined geometry into a theorem.
Because the model keeps the $L^\infty$ norm of the density constant while derivatives blow up, it predicts that singularities in the physical IPM boundary layer can form without unbounded density—an observation that could guide numerical searches for blow-up in the two-dimensional IPM equation.
The formal limit $a\to\infty$ suggests a program to prove blow-up for the nonlocal transport equation with Hilbert kernel directly from a limiting version of the weighted inequality, removing the regularization parameter.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper to know: Kiselev and Sarsam introduce the first 1D model for the IPM boundary-layer evolution and prove finite-time blow-up for a natural class of initial data. The model (1.2) with kernel K_a is new; it interpolates between the zero operator and the Hilbert transform as a varies. The blow-up proof follows the CCF template but required real work: a new weighted inequality (4.2) for H_a, obtained through kernel analysis in Lemma 4.5 and Proposition 4.3. If the estimates hold, Theorem 1.2 is solid and the paper gives the field a useful testbed for IPM singularity formation.

The local well-posedness and BKM criterion parts are standard and clean. The discussion connecting the model to CCF and to the IPM boundary layer is thoughtful, and Remark 4.6 about the real line is a nice addition.

The soft spot is exactly where the stress-test note lands: Proposition 4.3. The decisive step is the assertion that G_a(x,qx) − G_a(x,x/q) is strictly decreasing and −G_a(x,qx) is strictly decreasing, with the latter vanishing at 2π/q. These are described as 'elementary (if somewhat lengthy) estimates' and 'one can verify by hand.' They are load-bearing: if any sign or monotonicity claim fails for some a > 0 or q ∈ (1,2), the lower bound on J'(t) collapses and the blow-up proof fails. This is a real verification gap, not a manufactured one. It is likely fixable with more detailed estimates or a computer-assisted check, but as written the proof is incomplete. The sign analysis in Lemma 4.5 has a similar flavor, though less severe.

Prop 4.1 is delegated to an 'immediate adaptation' of [18]; minor. No circularity or data fitting; the parameter a is not tuned. Citation pattern is proper; the borrowing from [26] is credited.

Overall, this is a well-motivated paper with a plausible core argument and an honestly stated limitation. It deserves a serious referee, not a desk reject. My recommendation: send it to peer review, and the referee should demand a complete proof of the monotonicity claims in Prop 4.3 before acceptance. For a reading group, it is a maybe — the architecture is instructive, but the hand-verified estimates make evaluation awkward without doing the computations yourself.

Referee Report

2 major / 6 minor

Summary. The paper introduces a one-dimensional transport equation, (1.2), with nonlocal velocity u = gH_aρ, as a model for the boundary trace of the 2D incompressible porous media (IPM) equation on the periodic half-plane. The kernel K_a interpolates between zero and the Hilbert kernel, and the authors argue that the resulting equation is a natural analogue of the Córdoba-Córdoba-Fontelos (CCF) model. The main results are local well-posedness in C^∞(T) (Theorem 1.1) and finite-time blow-up for smooth, even, nonnegative initial data with ρ0(0)=0 and ρ0′≥0 on [0,π) (Theorem 1.2). The blow-up proof combines a Beale-Kato-Majda type criterion, preservation of monotonicity, and a weighted inequality for H_a (Proposition 4.3) to derive a Gronwall-type lower bound on J(t)=∫_0^{π/2} ρ(x,t)x^{-1-δ}dx.

Significance. If Proposition 4.3 is fully justified, the paper provides a clean and interesting finite-time blow-up result for a new 1D model related to IPM. The model derivation is transparent, the local well-posedness argument is standard but competently executed, and the blow-up mechanism is elegant: a weighted functional naturally yields J′ ≥ cJ². The paper also gives an honest discussion of the heuristic nature of the boundary-layer ansatz and of the formal connection to the CCF equation. These are strengths. The main weakness is that the central weighted inequality rests on several monotonicity computations that are only described as hand-verified; moreover, one step in the proof of Theorem 1.2 appears not to follow from the stated hypotheses. Both issues are fixable but are load-bearing for the advertised result.

major comments (2)

[§4, proof of Theorem 1.2] The line "As J(0) > 0" is not a consequence of the hypotheses. For example, take a smooth, even, 2π-periodic, nonnegative function ρ0 that vanishes on [0,π/2], is strictly increasing on (π/2,π−ε), and is constant on a small neighborhood of π. Then ρ0 is non-identically zero, ρ0(0)=0, and ρ0′≥0 on [0,π), but J(0)=∫_0^{π/2} ρ0(x)x^{-1-δ}dx = 0. The Gronwall argument therefore cannot start for this admissible initial datum. Either add a hypothesis such as ρ0 positive on a set of positive measure in (0,π/2], or prove that J(t)>0 for every t>0 by exploiting the leftward drift of the flow; as written, the proof has a gap.
[§4, Proposition 4.3] The proof of the weighted inequality (4.2) is incomplete. After Lemma 4.5, the argument depends on three assertions that are verified only through phrases like "by hand", "elementary (if somewhat lengthy) estimates", and "one can verify by hand": (i) G_a(x,·) is increasing on (x,2x] and decreasing on [0,x); (ii) G_a(x,qx)−G_a(x,x/q) is strictly decreasing on (0,2π/(q−1)) and has a zero at x*; and (iii) −G_a(x,qx) is strictly decreasing on [0,2π/q] and vanishes at 2π/q. These statements are exactly what converts Lemma 4.5 into the positive lower bound (4.2); if any sign or monotonicity assertion fails, the lower bound on J′(t) collapses. The manuscript should supply the full computations, or provide a documented computer-assisted verification, for these claims.

minor comments (6)

[§4, Lemma 4.2] The assertion u(π,t)=0 should be justified explicitly by evenness and periodicity; the current sentence states it without explaining why H_aρ(π,t)=0 for even ρ.
[§4, Proposition 4.3] The estimate taken from the proof of Proposition 6.4 in [26] should be stated as a self-contained lemma with its precise hypotheses and constant, since the present application uses f on [0,π/2] while evaluating f(qx) and f(x/q), and the reader should be able to check that the conditions are met.
[§2.1 and Remark 2.1] The boundary-layer ansatz ρ(x,t)=ρ(x1,0,t)χ_[0,a](x2) is only heuristic, and Remark 2.1 acknowledges this; still, the introduction should state more prominently that the model is derived under a formal ansatz, not as an exact reduction of IPM.
[§2.3] The passage a→∞ is purely formal; since H_a converges to H in L² operator norm but not in norms relevant to the pointwise inequality, the sentence about obtaining a blow-up solution of CCF should be labeled explicitly as a heuristic.
[Various] There are minor typos: "well-possedness" in Remark 2.1, "maximal principle" should be "maximum principle" in Section 2.3, and "propogation" should be "propagation" in Remark 4.6.
[Figure 1] The axes of Figure 1 are not labeled; adding labels for x1 and the boundary trace would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the blow-up proof rests on new kernel estimates for Ha plus an independent published inequality from [26].

full rationale

The paper's central claim, Theorem 1.2, is derived from Proposition 4.3, whose proof is a new computation adapted to the operator Ha. Lemma 4.5 carries the main work by rewriting Ha f(x) as an integral against f'(y)Ga(x,y), and the subsequent monotonicity and sign properties of Ga are asserted to follow from direct estimates performed in the paper. The only imported ingredient is the weighted inequality involving f(qx)-f(x/q), cited from Proposition 6.4 of [26]. Although [26] is authored by one of the present authors, it is a separately published, parameter-free technical lemma about one-dimensional integrals; it does not assume finite-time blow-up for (1.2), and the inequality is not equivalent to the target conclusion. The model itself is derived in Section 2 from an explicit boundary-layer ansatz applied to the IPM Biot-Savart law, not defined in terms of the blow-up result. Proposition 4.1 is adapted from the external reference [18]. The hand-verified claims in Section 4 are potential verification gaps but not circularity: no equation is shown to be its own input, no fitted constant is renamed a prediction, and no uniqueness theorem is imported from the authors to force the argument. The derivation is therefore self-contained with respect to circularity concerns.

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

The central claim rests on the modeling ansatz, an adapted BKM criterion, and a cited integral inequality. No data-fitted constants appear; the parameters a, g, delta, and q are arbitrary and are not tuned to force the result.

assumptions (3)

ad hoc to paper Boundary-layer ansatz rho(x,t)=rho(x1,0,t) chi_[0,a](x2) for deriving the 1D model
Section 2.1. The ansatz is not an exact solution of IPM, as the authors state in Remark 2.1; it is a modeling assumption that selects the kernel H_a.
domain assumption BKM-type criterion for H_a (Proposition 4.1) adapted from Dong [18], Proposition 5.2
Section 4, Proposition 4.1. The criterion is stated and its proof is deferred to a minor adaptation of a published result; the blow-up conclusion depends on it.
standard math Integral inequality from Kiselev [26], Proposition 6.4: integral_0^{pi/2} (f(qx)-f(x/q)) f'(x)/x^sigma dx >= ~C integral f^2/x^{1+sigma}
Section 4, proof of Proposition 4.3, final step. This published lemma is used as a black box for the comparison function f.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Finite time blow-up in a 1D model of the incompressible porous media equation." pith.science (2026). https://pith.science/paper/7I3OCU37

@misc{pith2026241216376,
  author       = {Pith},
  title        = {Pith review of: Finite time blow-up in a 1D model of the incompressible porous media equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I3OCU37}},
  note         = {Machine review of arXiv:2412.16376}
}

read the original abstract

We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known C\'{o}rdoba-C\'{o}rdoba-Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data.

Figures

Figures reproduced from arXiv: 2412.16376 by the authors.

**Figure 1.** Example boundary trace of initial data suspected to lead to finite time blow-up in (1.1). The boundary trace is even, attains 0 at the origin, and is increasing from x1 = 0 to x1 = π (cf. Theorem 1.2). This suspected smooth blow-up data for the IPM equation parallels the proposed Hou-Luo scenario for the 3D axisymmetric Euler equation and 2D Boussinesq equation on domains with boundary [23, 24]. One avenue through … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations
math.AP 2025-07 conditional novelty 7.0 of 10

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.

Reference graph

Works this paper leans on

34 extracted references · 30 canonical work pages · cited by 1 Pith paper

[18]

Dong, Well-posedness for a transport equation with non-local vel ocity, J

H. Dong, Well-posedness for a transport equation with non-local vel ocity, J. Funct. Anal. 255 (2008), no. 11, 3070–3097

work page 2008
[26]

Kiselev, Regularity and blow up for active scalars, Math

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Nat. Phenom. 5 (2010), no. 4, 225–255 14 ALEXANDER KISELEV AND NAJI A. SARSAM

work page 2010
[1]

G. R. Baker, X. Li, and A. C. Morlet, Analytic structure of two 1D-transport equations with nonl ocal ﬂuxes, Physica D. 91 (1996), no. 1, 349–375

work page 1996
[2]

Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation

R. Bianchini, D. C´ ordoba, and L. Martn ´ ınez-Zoroa,Non existence and strong ill-posedness in H 2 for the stable IPM equation, preprint arXiv:2410.01297

work page Pith review arXiv
[3]

Bianchini, T

R. Bianchini, T. Crin-Barat, and M. Paicu, Relaxation approximation and asymptotic stability of stra tiﬁed solu- tions to the IPM equation, Arch. Ration. Mech. Anal. 248 (2024), no. 1, Paper No. 2, 35 pp

work page 2024
[4]

Castro and D

´A. Castro and D. C´ ordoba, Inﬁnite energy solutions of the surface quasi-geostrophic equation, Adv. Math. 225 (2010), no. 4, 1820–1829

work page 2010
[5]

Castro, D

´A. Castro, D. C´ ordoba, and D. Lear, Global existence of quasi-stratiﬁed solutions for the conﬁ ned IPM equation, Arch. Ration. Mech. Anal. 232 (2019), no. 1, 437–471

work page 2019
[6]

Chen and T

J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3 D Euler equations with smooth data I: Analysis, preprint arXiv:2210.07191

arXiv

Show all 34 references

[7]

Chen and T

J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3 D Euler equations with smooth data II: Rigorous numerics, preprint arXiv:2305.05660

arXiv
[8]

K. Choi, T. Y. Hou, A. Kiselev, G. Luo, V. ˇSver´ ak, and Y. Yao,On the ﬁnite-time blow up of a one-dimensional model for the three-dimensional axisymmetric Euler equati ons, Comm. Pure Appl. Math. 70 (2017), no. 11, 2218–2243

2017
[9]

K. Choi, A. Kiselev, and Y. Yao, Finite time blow up for a 1D model of 2D Boussinesq system, Comm. Math. Phys. 334 (2015), no. 3, 1667–1679

2015
[10]

Constantin, Analysis of hydrodynamic models, CBMS-NSF Regional Conference Series in Applied Mathemat- ics, 90, SIAM, Philadelphia, PA, 2017

P. Constantin, Analysis of hydrodynamic models, CBMS-NSF Regional Conference Series in Applied Mathemat- ics, 90, SIAM, Philadelphia, PA, 2017

2017
[11]

Constantin, P

P. Constantin, P. D. Lax, and A. J. Majda, A simple one-dimensional model for the three-dimensional v orticity equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724

1985
[12]

Constantin, A

P. Constantin, A. J. Majda, and E. G. Tabak, Singular front formation in a model for quasigeostrophic ﬂo w, Phys. Fluids 6 (1994), no. 1, 9–11

1994
[13]

C´ ordoba, D

A. C´ ordoba, D. C´ ordoba, and M. A. Fontelos, Formation of singularities for a transport equation with no nlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389

2005
[14]

C´ odorba, F

D. C´ odorba, F. Gancedo, and R. Orive, Analytical behavior of two-dimensional incompressible ﬂo w in porous media, J. Math. Phys. 48 (2007), no. 6, Paper No. 065206, 19 pp

2007
[15]

C´ ordoba and L

D. C´ ordoba and L. Mart ´ ınez-Zoroa,Finite time singularities of smooth solutions for the 2D inc ompressible porous media (IPM) equation with a smooth source, preprint arXiv:2410.22920

arXiv
[16]

De Gregorio On a one-dimensional model for the three-dimensional vorti city equation, J

S. De Gregorio On a one-dimensional model for the three-dimensional vorti city equation, J. Statist. Phys. 59 (1990), no. 5-6, 1251–1263

1990
[17]

T. Do, A. Kiselev, and X. Xu, Stability of blow up for a 1D model of axisymmetric 3D Euler eq uation, J. Nonlinear Sci. 28 (2018), no. 6, 2127–2152

2018
[19]

T. M. Elgindi, On the asymptotic stability of stationary solutions of the i nviscid incompressible porous medium equation, Arch. Ration. Mech. Anal. 225 (2017), no. 2, 573–599

2017
[20]

T. M. Elgindi and I.-J. Jeong, Symmetries and critical phenomena in ﬂuids, Comm. Pure Appl. Math. 73 (2020), no. 2, 257–316

2020
[21]

T. M. Elgindi and I.-J. Jeong, On the eﬀects of advection and vortex stretching, Arch. Ration. Mech. Anal. 235 (2020), no. 3, 1763–1817

2020
[22]

Hoang and M

V. Hoang and M. Radosz, Cusp formation for a nonlocal evolution equation, Arch. Ration. Mech. Anal. 224 (2017), no. 3, 1021–1036

2017
[23]

T. Y. Hou and G. Luo, Potentially singular solutions of the 3D axisymmetric Eule r equations, Proc. Natl. Acad. Sci. 111 (2014), no. 36, 12968–12973

2014
[24]

T. Y. Hou and G. Luo, Fromation of ﬁnite-time singularities in the 3D axisymmetr ic Euler equations: a numerics guided study, SIAM Rev. 61 (2019), no. 4, 793–835

2019
[25]

M. J. Jo and J. Kim, Quantitative asymptotic stability of the quasi-linearly s tratiﬁed densities in the IPM equation with sharp decay rates, J. Funct. Anal. 286 (2024), no. 11, Paper No. 110401, 66 pp

2024
[27]

Kiselev, F

A. Kiselev, F. Nazarov, and R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Diﬀer. Equ. 5 (2008), no. 3, 211–240

2008
[28]

Kiselev and Y

A. Kiselev and Y. Yao, Small scale formations in the incompressible porous media e quation, Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 1, 25 pp

2023
[29]

Li and J

D. Li and J. Rodrigo, Blow-up of solutions for a 1D transport equation with nonloc al velocity and supercritical dissipation, Adv. Math. 217 (2008), no. 6, 2563–2568

2008
[30]

A. C. Morlet, Further properties of a continuum of model equations with gl obally deﬁned ﬂux, J. Math. Anal. Appl. 221 (1998), no. 1, 132–160

1998
[31]

Okamoto, T

H. Okamoto, T. Sakajo, and M. Wunsch On a generalization of the Constantin-Lax-Majda equation, Nonlinearity 21 (2008), no. 10, 2447–2461

2008
[32]

Silvestre and V

L. Silvestre and V. Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188

2016
[33]

Zlatoˇ s, The 2D Muskat problem I: Local regularity on the half-plane, plane, and strips, preprint arXiv:2401.14659

A. Zlatoˇ s, The 2D Muskat problem I: Local regularity on the half-plane, plane, and strips, preprint arXiv:2401.14659

arXiv
[34]

Zlatoˇ s, The 2D Muskat problem II: Stable regime small data singulari ty on the half-plane, preprint arXiv:2401.14660 (A

A. Zlatoˇ s, The 2D Muskat problem II: Stable regime small data singulari ty on the half-plane, preprint arXiv:2401.14660 (A. Kiselev) Department of Mathematics, Duke University, Durham NC 2770 8, USA Email address : alexander.kiselev@duke.edu (N. A. Sarsam) Department of Math...

arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [18]

Dong, Well-posedness for a transport equation with non-local vel ocity, J

H. Dong, Well-posedness for a transport equation with non-local vel ocity, J. Funct. Anal. 255 (2008), no. 11, 3070–3097

work page 2008

[2] [26]

Kiselev, Regularity and blow up for active scalars, Math

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Nat. Phenom. 5 (2010), no. 4, 225–255 14 ALEXANDER KISELEV AND NAJI A. SARSAM

work page 2010

[3] [1]

G. R. Baker, X. Li, and A. C. Morlet, Analytic structure of two 1D-transport equations with nonl ocal ﬂuxes, Physica D. 91 (1996), no. 1, 349–375

work page 1996

[4] [2]

Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation

R. Bianchini, D. C´ ordoba, and L. Martn ´ ınez-Zoroa,Non existence and strong ill-posedness in H 2 for the stable IPM equation, preprint arXiv:2410.01297

work page Pith review arXiv

[5] [3]

Bianchini, T

R. Bianchini, T. Crin-Barat, and M. Paicu, Relaxation approximation and asymptotic stability of stra tiﬁed solu- tions to the IPM equation, Arch. Ration. Mech. Anal. 248 (2024), no. 1, Paper No. 2, 35 pp

work page 2024

[6] [4]

Castro and D

´A. Castro and D. C´ ordoba, Inﬁnite energy solutions of the surface quasi-geostrophic equation, Adv. Math. 225 (2010), no. 4, 1820–1829

work page 2010

[7] [5]

Castro, D

´A. Castro, D. C´ ordoba, and D. Lear, Global existence of quasi-stratiﬁed solutions for the conﬁ ned IPM equation, Arch. Ration. Mech. Anal. 232 (2019), no. 1, 437–471

work page 2019

[8] [6]

Chen and T

J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3 D Euler equations with smooth data I: Analysis, preprint arXiv:2210.07191

arXiv