Pith. sign in

REVIEW 2 major objections 4 minor 34 references

A priori bounds for stochastic porous media equations via regularity structures

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Stochastic porous media equations with rough multiplicative noise satisfy explicit a priori L¹-Hölder-α space-time bounds, uniform in the smoothing, and a stronger modelledness estimate to order β.

desk verdict Serious and original a priori bounds for the first genuinely singular-and-degenerate regime; the proof is conditional on partly deferred model verification, but the core argument is sound and deserves a careful referee. read the letter →

arxiv 2507.03575 v2 pith:YHUK3JF2 submitted 2025-07-04 math.AP math.PR

Markus Tempelmayr , Hendrik Weber This is my paper · ORCID
classification math.APmath.PR MSC 60H1735K6560L3035B45
keywords stochasticporousmediumequationdegeneratesingularSPDEregularitystructureskineticformulationaprioriboundsmodellednessrenormalizedenergyinequalityL1Besov
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

The paper proves the first positive regularity estimates for a stochastic PDE that is simultaneously singular (noise too rough for classical products) and degenerate (the diffusion coefficient a(u)=M|u|^{M−1} vanishes where u=0): the stochastic porous media equation with multiplicative noise of space-time regularity α−2, α∈(2/3,1). The main result, Theorem 1.4, is an a priori bound: the solution of the regularized, renormalized equation satisfies ∫_{D_y}|u(x+y)−u(x)| ≲ ||y||_s^α, with the constant independent of the qualitative smoothness of the coefficients, noise, and initial data. Theorem 1.5 strengthens this to higher-order modelledness over the solution of the corresponding linear stochastic heat equation, to order β ∈ (2−α, 2α/(1+(M−1)α)) > 1. A sympathetic reader should care because uniform a priori bounds of this kind are the ingredient needed to pass from regularized problems to a genuine solution theory in the singular regime, and because the proof introduces a new renormalized energy inequality plus an L¹-based reconstruction machinery that should apply to other degenerate singular SPDEs.

What carries the argument

The argument runs on four connected mechanisms. The kinetic formulation: the indicator function χ(t,x,v)=1_{(−∞,u(t,x))}(v)−1_{(−∞,0)}(v) converts the degenerate equation into a linear transport equation (1.14), and the splitting u=u_<+u_> at a(u)≍δ separates the trivially controlled small-velocity part from the effectively non-degenerate large-velocity part. The regularity-structures model Π — the noise ξ, its heat-flow integral, the two renormalized singular products ξ·(integral) and |∇(integral)|², and the first-order symbol X — encodes the renormalized singular products σ(u)ξ and a(u)|∇u|²; the counterterms C̄_ā and C̄̄_ā enter through the closeness conditions (1.10)–(1.11). The renormalized energy inequality of Proposition 4.4 tests (1.1) with |u|^ε and bounds ∫ g′(u)a(u)|ν|², the good-sign quantity controlling the renormalized kinetic measure, using the fact that the counterterm for u(σ(u)ξ−σ′σC_a) and for aσ²|∇Π|² coincide up to sign and an integrable time blow-up. Finally, the reconstruction and integration lemmas of Section 3 (Lemmas 3.2–3.5) are L¹-based versions of reconstruction that track the blow-up in the diffusion coefficient a, and Lemma 5.1 extends real interpolation to basepoint-dependent seminorms, which is what turns the two-parameter δ-balancing into the final exponent β.

What would settle it

Evaluate the integrals in (1.10)–(1.11) for space white noise on T² using the explicit counterterms of Example 1.11 with σ(v)=|v|^M (violating N≥M+1) at the boundary case of (1.12): the paper's balancing inequality (II) fails at that point, so the integrals' divergence would confirm the restriction on σ is load-bearing, while their finiteness would show the admissible class is wider than stated.

Watch

Extended reading notes

Core claim

The central claim is that in the first singular regime α∈(2/3,1), under the restriction M < 1+(3α−2)/(α(2−α)) and the enhanced-noise and counterterm-closeness assumptions of Assumption 1.2, the classical solution u of the regularized renormalized equation (1.1) satisfies the modelledness estimate of Theorem 1.5, hence the L¹ space-time regularity estimate ∫_{D_y}|u(x+y)−u(x)| dx ≲ ||y||_s^α of Theorem 1.4, with the optimal exponent α and with constants uniform in the qualitative smoothness of a, ξ, u0. The proof splits the solution into small and large velocities through the kinetic formulation, shows through the cancellation in (1.16) that the two singular products — the forcing σ(u)ξ and the kinetic measure a(u)|∇u|² — require the same counterterm up to an integrable blow-up at t=0, and controls the renormalized kinetic measure by a new energy inequality (Proposition 4.4) obtained by testing the equation with |u|^ε. Real interpolation, extended to basepoint-dependent modelledness seminorms (Lemma 5.1), combines the small-velocity decay $δ^{{1/(M−1)}}$ with the large-velocity blow-up $δ^{{−α−ε}}$, yielding the order 2α/(1+(M−1)(α+ε)), and a buckling argument proves modelledness of σ(u)ξ and hence, after integration, of u itself; the two requirements are compatible exactly when (1.12) holds.

Load-bearing premise

The load-bearing premise is that the renormalization corrections, which carry a time dependence from starting the auxiliary heat flow at time zero, differ from their steady counterparts by an integrable-in-time amount (Assumption 1.2(ii), equations (1.10)–(1.11)) — if those differences diverge with the wrong rate or sign, the cancellation that keeps the singular products finite, and with it the entire energy estimate, collapses.

Editorial extensions

If this is right

  • The a priori bounds pass to kinetic solutions constructed by approximation: by Remark 1.6 the estimate transmits immediately when u is obtained as a limit of classical solutions, giving the same L¹-Hölder-α control for the weaker notion of solution.
  • The space-time regularity exponent α is optimal for noise of regularity α−2, so the bound cannot be improved merely by refining the argument at fixed noise regularity.
  • The modelledness order β exceeds 2−α, hence exceeds one, meaning the solution is captured to first order by the linear stochastic heat flow plus a correction ν·y — a much finer description than a plain Hölder bound.
  • The two canonical noises — space white noise on the torus T² and white-in-time, spatially coloured noise — fall under Assumption 1.2, so the theorems give explicit, uniform a priori estimates for those cases.
  • The real-interpolation lemma for basepoint-dependent seminorms is a reusable device for turning regularity-structures estimates into L¹-Besov regularity in degenerate settings.

Reading between the lines

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

  • The a priori bound appears to be the missing ingredient for a convergence theory: one expects that the same estimates, combined with compactness in L¹-Besov spaces, would produce kinetic solutions of the unrenormalized degenerate singular equation at α>2/3 — the paper stops at the bound, so the limit passage is not yet written down.
  • The counterterm cancellation between the forcing and the kinetic measure is plausibly a general design principle: any degenerate singular SPDE whose noise intensity satisfies σ²(v)≍a(v) near the degeneracy should admit an analogous renormalized energy inequality, suggesting the method transfers to degenerate p-Laplacian or thin-film type flows.
  • The integrable blow-up in conditions (1.10)–(1.11) is an artifact of starting the auxiliary heat flow at time zero; choosing a stationary model would make both differences vanish, so a stationary formulation could yield the same theorems with cleaner constants.
  • A numerical check is available: on T² with space white noise and a(u)=M|u|^{M−1}, the uniform L¹-Besov seminorm should stay bounded below the threshold (1.12) and blow up above it, which would directly test whether the restriction on M is sharp.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proves a priori Hölder-type regularity estimates in L1 for solutions of the regularized renormalized stochastic porous medium equation (1.1) with multiplicative noise of space-time Hölder regularity α−2 for α∈(2/3,1). Under Assumptions 1.1–1.3, Theorem 1.4 states that the solution u satisfies ∫_{D_y}|u(x+y)−u(x)| ≲ ||y||_s^α, and Theorem 1.5 states a higher-order modelledness estimate with respect to the solution of the linear stochastic heat equation. The proof uses the kinetic formulation, a splitting into small and large velocities, new reconstruction/integration lemmas for kernels composed with rough solutions, a renormalized energy inequality (Proposition 4.4), and a real interpolation argument for basepoint-dependent seminorms. The theorems are conditional on an enhanced model satisfying Assumption 1.2, which is verified for two examples (space white noise on T^2 and white-in-time coloured-in-space noise).

Significance. If correct, these are the first a priori estimates establishing positive space-time regularity and modelledness for SPDEs that are both degenerate and singular. The proof strategy is novel, combining kinetic formulations with the theory of regularity structures, and the paper is largely self-contained from Section 2 onward. The abstract results in Section 3, especially Lemmas 3.2–3.5, are stated with full proofs and carefully track the dependence on the diffusion coefficient. The main estimates are derived from explicit assumptions rather than fitted to the conclusion, so there is no circularity. The main caveat is that the verification of Assumption 1.2 for the two examples relies on the PhD thesis [dLF24] for several central bounds.

major comments (2)
  1. [§1.2, Examples 1.11 and 1.12] The verification of Assumption 1.2(ii) for the two examples is incomplete within the paper: the bound (1.8) for the renormalized symbols and the derivative-in-ā estimates with m=1 are deferred to the PhD thesis [dLF24] (e.g., 'we refer the reader to [dLF24, Section 3.3.1] for further details'). These bounds are load-bearing because the cancellations in (1.16) and the renormalized energy inequality in Proposition 4.4 rely on the exact ā-dependence and ε-uniformity of the enhanced model. Without them, the theorem has no fully verified instance in the singular regime. Please include the missing estimates or state precisely which results of [dLF24] are used and how they imply (1.8)–(1.9) with the required uniformity.
  2. [§5, proof of Theorem 1.4] The estimate of ∥ν∥_{L1} in the proof of Theorem 1.4 uses a uniform-in-time bound on ∥u(t)∥_{L1(T^d)}, citing the proof of Proposition 4.4. However, Proposition 4.4 as stated only gives an estimate for the final-time energy ∫ G(u(T)) dz. Please clarify that the energy identity in the proof of Proposition 4.4 is applied on an arbitrary interval [0,t] to obtain the sup_t bound needed here.
minor comments (4)
  1. [§4.3 / Proposition 4.4] The notation for the two counterterms C_{\bar a}(s) and C^{\bar a}(s) is easy to confuse in equations such as (1.16); consider renaming them or adding a small table that fixes their definitions and homogeneities.
  2. [§4, equation (4.5)] The abuse of notation concerning the constant [ ]_{2α−2} in (4.5) is acknowledged but the precise combination [ ]_{2α−2} + [ ]_α[ ]_{α−2} could be written out in the display to help the reader track the provenance of constants.
  3. [§1.2, Example 1.12] For the white-in-time coloured-in-space example, the verification of (1.9) for time increments and for derivatives in ā is only sketched; a few more details, or a precise pointer to the corresponding pages in [dLF24], would improve readability.
  4. [Remark 1.8] Since inequalities (I)–(XVI) are used throughout Section 4, it would be helpful to collect them in an appendix or table rather than leaving the reader to reconstruct them from the text, especially because Remark 1.8 states that Assumption 1.1 is a simplification of these conditions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the a priori estimates are conditional on explicit model assumptions and are not fitted to the conclusion.

full rationale

The derivation is a genuine conditional theorem: Theorem 1.4 and Theorem 1.5 assume Assumptions 1.1–1.3, including the enhanced-model bounds (1.8)–(1.9) and the counterterm closeness conditions (1.10)–(1.11), and then prove space-time regularity and modelledness for the solution u of the regularized equation (1.1). No quantity in the conclusion is used to define the model or the counterterms; the constants R, T, and [u]B^α_{1,∞} are estimated in terms of the data, not chosen to match the target. The proof chain—small/large velocity splitting, Propositions 2.5–2.7, the reconstruction and integration lemmas of Section 3, the renormalized energy inequality Proposition 4.4, and the interpolation bootstrap in Section 5—is self-contained and traces the dependence on the data explicitly. The bootstrap in Proposition 2.3 and Lemma 2.8, where the right-hand side contains the same seminorm [U]B^β_{1,∞}, is a standard small-R, small-T absorption argument rather than circular. The verification of Assumption 1.2 for the two examples delegates some estimates to the PhD thesis [dLF24]; this is external evidence, not a self-citation, and any gap there would affect the verification of the examples, not make the theorem's derivation circular. Self-references [Tem24], [LOTT24], and [LOT23] appear only in the related-literature overview and as general tools, not as load-bearing premises. Remark 1.8 openly notes that Assumption 1.1 is a simplification of a longer list of conditions (I)–(XVI); this is an acknowledged limitation or readability compromise, not a circular reduction. Hence no circular step can be exhibited.

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

No free parameters are fitted to data; the quantitative constants appearing in the theorem are data, not parameters. The hypotheses are explicit domain assumptions; no new physical entities are introduced.

assumptions (5)
  • domain assumption Assumption 1.1: a∈C^2(R), a≥a0>0 qualitatively; quantitatively |v|^{M−1}≲a(v), |a'|≲|v|^{M−2}, |a''|≲|v|^{M−3}, and σ∈C^2(R) compactly supported with |σ|≲|v|^N, |σ'|≲|v|^{N−1}, |σ''|≲|v|^{N−2}, N≥M+1.
    These balance conditions on the diffusion and the noise are the hypotheses of the main theorem; they are not derived. M is restricted by (1.12), N≥M+1 is imposed for simplicity per Remark 1.8.
  • domain assumption Assumption 1.2: existence of a model Π for the symbols {Ξ, I, J, X} with bounds (1.8), (1.9) and counterterm closeness (1.10), (1.11).
    The model encodes the renormalized singular products; the closeness conditions guarantee that the counterterms in the equation and in the model differ only by an integrable error. Verified for the examples in Section 1.2.
  • domain assumption Assumption 1.3: u0∈L^p(T^d), p>1.
    Initial condition with finite p-th moment; used in the interpolation step and energy inequality.
  • standard math Kinetic formulation equivalence (Appendix A.1) for classical solutions of (1.1).
    Justifies the representation u=∫χ and the mild formulation used throughout.
  • standard math Reconstruction theorem in L1 (Theorem A.3), real interpolation (Bergh-Löfström), heat kernel bounds (Lemma A.1–A.2).
    Background tools that are either proved in the appendix or classical.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A priori bounds for stochastic porous media equations via regularity structures." pith.science (2026). https://pith.science/paper/YHUK3JF2

@misc{pith2026250703575,
  author       = {Pith},
  title        = {Pith review of: A priori bounds for stochastic porous media equations via regularity structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHUK3JF2}},
  note         = {Machine review of arXiv:2507.03575}
}
abstract

We prove a priori bounds for solutions of singular stochastic porous media equations with multiplicative noise in their natural $L^1$-based regularity class. We consider the first singular regime, i.e.~noise of space-time regularity $\alpha-2$ for $\alpha\in(2/3,1)$, and prove modelledness of the solution in the sense of regularity structures with respect to the solution of the corresponding linear stochastic heat equation. The proof relies on the kinetic formulation of the equation and a novel renormalized energy inequality. A careful analysis allows to balance the degeneracy of the diffusion coefficient against sufficiently strong damping of the multiplicative noise for small values of the solution.

Figures

Figures reproduced from arXiv: 2507.03575 by the authors.

Figure 1
Figure 1. Visualization of aε of Example 1.10 for a(v) = M|v|M−1 with M = 3/2. Example 1.11 (White noise on T 2 ). Consider space white noise on T 2 , which we realize as the random Fourier series ξ = X k∈(2πZ) 2 ek ˆξk , where ek(x) = e ik·x and ˆξk are independent (up to the usual restriction that the complex conjugate of ˆξk coincides with ˆξ−k) complex-valued centered Gaussians with variance 1. In this example k is always… view at source ↗
Figure 2
Figure 2. One can then check that φx¯,m defined by 32 [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 2
Figure 2. Visualization of the dyadic decomposition [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 15 canonical work pages

  1. [1]

    Bruned and V

    Y. Bruned and V. Dotsenko. Chain rule symmetry for singular spdes. Preprint arXiv:2403.17066, 2024. URL: https://arxiv.org/abs/2403.17066

    arXiv 2024

  2. [2]

    Bailleul, A

    I. Bailleul, A. Debussche, and M. Hofmanov\' a . Quasilinear generalized parabolic A nderson model equation. Stoch. Partial Differ. Equ. Anal. Comput. , 7(1):40--63, 2019. https://doi.org/10.1007/s40072-018-0121-1 doi:10.1007/s40072-018-0121-1

    doi:10.1007/s40072-018-0121-1 2019

  3. [3]

    Barbu, G

    V. Barbu, G. Da Prato, and M. R\"ockner. Stochastic porous media equations , volume 2163 of Lecture Notes in Mathematics . Springer, [Cham], 2016. https://doi.org/10.1007/978-3-319-41069-2 doi:10.1007/978-3-319-41069-2

    doi:10.1007/978-3-319-41069-2 2016

  4. [4]

    Bruned, M

    Y. Bruned, M. Gerencsér, and U. Nadeem. Quasi-generalised kpz equation. Preprint arXiv:2401.13620, 2024. URL: https://arxiv.org/abs/2401.13620

    arXiv 2024

  5. [5]

    Bruno, B

    S. Bruno, B. Gess, and H. Weber. Optimal regularity in time and space for stochastic porous medium equations. Ann. Probab. , 50(6):2288--2343, 2022. https://doi.org/10.1214/22-aop1583 doi:10.1214/22-aop1583

    work page doi:10.1214/22-aop1583 2022

  6. [6]

    Bailleul, M

    I. Bailleul, M. Hoshino, and S. Kusuoka. Regularity structures for quasilinear singular SPDE s. Arch. Ration. Mech. Anal. , 248(6):Paper No. 127, 87, 2024. https://doi.org/10.1007/s00205-024-02069-6 doi:10.1007/s00205-024-02069-6

    doi:10.1007/s00205-024-02069-6 2024

  7. [7]

    Bergh and J

    J. Bergh and J. L\"ofstr\"om. Interpolation spaces. A n introduction , volume No. 223 of Grundlehren der Mathematischen Wissenschaften . Springer-Verlag, Berlin-New York, 1976

    work page 1976

  8. [8]

    Broux and D

    L. Broux and D. Lee. Besov reconstruction. Potential Anal. , 59(4):1875--1912, 2023. https://doi.org/10.1007/s11118-022-10028-7 doi:10.1007/s11118-022-10028-7

    work page doi:10.1007/s11118-022-10028-7 1912

Show all 34 references
  1. [9]

    Bailleul and A

    I. Bailleul and A. Mouzard. Paracontrolled calculus for quasilinear singular PDE s. Stoch. Partial Differ. Equ. Anal. Comput. , 11(2):599--650, 2023. https://doi.org/10.1007/s40072-022-00239-9 doi:10.1007/s40072-022-00239-9

    2023 doi

  2. [10]

    Dareiotis, M

    K. Dareiotis, M. Gerencs\'er, and B. Gess. Entropy solutions for stochastic porous media equations. J. Differential Equations , 266(6):3732--3763, 2019. https://doi.org/10.1016/j.jde.2018.09.012 doi:10.1016/j.jde.2018.09.012

    2019 doi

  3. [11]

    Dareiotis, M

    K. Dareiotis, M. Gerencs\'er, and B. Gess. Porous media equations with multiplicative space-time white noise. Ann. Inst. Henri Poincar\'e Probab. Stat. , 57(4):2354--2371, 2021. https://doi.org/10.1214/20-aihp1139 doi:10.1214/20-aihp1139

    2021 doi

  4. [12]

    Dareiotis, B

    K. Dareiotis, B. Gess, and P. Tsatsoulis. Ergodicity for stochastic porous media equations with multiplicative noise. SIAM J. Math. Anal. , 52(5):4524--4564, 2020. https://doi.org/10.1137/19M1278521 doi:10.1137/19M1278521

    2020 doi

  5. [13]

    Debussche, M

    A. Debussche, M. Hofmanov\'a, and J. Vovelle. Degenerate parabolic stochastic partial differential equations: quasilinear case. Ann. Probab. , 44(3):1916--1955, 2016. https://doi.org/10.1214/15-AOP1013 doi:10.1214/15-AOP1013

    1916 doi

  6. [14]

    de Lima Feltes

    G. de Lima Feltes. Large scale behaviour of infinite dimensional stochastic systems. PhD Thesis, 2024. URL: https://doi.org/10.17879/53928488946

    2024 doi

  7. [15]

    Furlan and M

    M. Furlan and M. Gubinelli. Paracontrolled quasilinear SPDE s. Ann. Probab. , 47(2):1096--1135, 2019. https://doi.org/10.1214/18-AOP1280 doi:10.1214/18-AOP1280

    2019 doi

  8. [16]

    Gerencs\' e r

    M. Gerencs\' e r. Nondivergence form quasilinear heat equations driven by space-time white noise. Ann. Inst. H. Poincar\' e Anal. Non Lin\' e aire , 37(3):663--682, 2020. https://doi.org/10.1016/j.anihpc.2020.01.003 doi:10.1016/j.anihpc.2020.01.003

    2020 doi

  9. [17]

    B. Gess. Optimal regularity for the porous medium equation. J. Eur. Math. Soc. (JEMS) , 23(2):425--465, 2021. https://doi.org/10.4171/jems/1014 doi:10.4171/jems/1014

    2021 doi

  10. [18]

    Gess and M

    B. Gess and M. Hofmanov\'a. Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE . Ann. Probab. , 46(5):2495--2544, 2018. https://doi.org/10.1214/17-AOP1231 doi:10.1214/17-AOP1231

    2018 doi

  11. [19]

    Gerencs\' e r and M

    M. Gerencs\' e r and M. Hairer. A solution theory for quasilinear singular SPDE s. Comm. Pure Appl. Math. , 72(9):1983--2005, 2019. https://doi.org/10.1002/cpa.21816 doi:10.1002/cpa.21816

    1983 doi

  12. [20]

    Gubinelli, P

    M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singular PDE s. Forum Math. Pi , 3:e6, 75, 2015. https://doi.org/10.1017/fmp.2015.2 doi:10.1017/fmp.2015.2

    2015 doi

  13. [21]

    B. Gess, J. Sauer, and E. Tadmor. Optimal regularity in time and space for the porous medium equation. Anal. PDE , 13(8):2441--2480, 2020. https://doi.org/10.2140/apde.2020.13.2441 doi:10.2140/apde.2020.13.2441

    2020 doi

  14. [22]

    M. Hairer. A theory of regularity structures. Invent. Math. , 198(2):269--504, 2014. https://doi.org/10.1007/s00222-014-0505-4 doi:10.1007/s00222-014-0505-4

    2014 doi

  15. [23]

    N. V. Krylov and B. L. Rozovski . Stochastic evolution equations. In Current problems in mathematics, V ol. 14 ( R ussian) , Itogi Nauki i Tekhniki, pages 71--147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1979

    1979

  16. [24]

    Linares, F

    P. Linares, F. Otto, and M. Tempelmayr. The structure group for quasi-linear equations via universal enveloping algebras. Comm. Amer. Math. Soc. , 3:1--64, 2023. https://doi.org/10.1090/cams/16 doi:10.1090/cams/16

    2023 doi

  17. [25]

    Linares, F

    P. Linares, F. Otto, M. Tempelmayr, and P. Tsatsoulis. A diagram-free approach to the stochastic estimates in regularity structures. Invent. Math. , 237(3):1469--1565, 2024. https://doi.org/10.1007/s00222-024-01275-z doi:10.1007/s00222-024-01275-z

    2024 doi

  18. [26]

    Lions, B

    P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. , 7(1):169--191, 1994. https://doi.org/10.2307/2152725 doi:10.2307/2152725

    1994 doi

  19. [27]

    O. A. Lady z enskaja, V. A. Solonnikov, and N. N. Ural'ceva. Linear and quasilinear equations of parabolic type , volume 23 of Translations of Mathematical Monographs . American Mathematical Society, Providence, RI, 1968. Translated from the Russian by S. Smith

    1968

  20. [28]

    A. Lunardi. Interpolation theory , volume 16 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] . Edizioni della Normale, Pisa, third edition, 2018. https://doi.org/10.1007/978-88-7642-638-4 doi:10.1007/978...

    2018 doi

  21. [29]

    F. Otto, J. Sauer, S. Smith, and H. Weber. Parabolic equations with rough coefficients and singular forcing. Preprint arXiv:1803.07884, 2018. URL: https://arxiv.org/abs/1803.07884

    2018 arXiv

  22. [30]

    F. Otto, J. Sauer, S. Smith, and H. Weber. A priori bounds for quasi-linear SPDE s in the full subcritical regime. J. Eur. Math. Soc. (JEMS) , 27(1):71--118, 2025. https://doi.org/10.4171/jems/1574 doi:10.4171/jems/1574

    2025 doi

  23. [31]

    Otto and H

    F. Otto and H. Weber. Quasilinear SPDE s via rough paths. Arch. Ration. Mech. Anal. , 232(2):873--950, 2019. https://doi.org/10.1007/s00205-018-01335-8 doi:10.1007/s00205-018-01335-8

    2019 doi

  24. [32]

    E. Pardoux. \'equations aux d\'eriv\'ees partielles stochastiques de type monotone. In S\'eminaire sur les \'Equations aux D \'eriv\'ees P artielles (1974--1975), III , pages Exp. No. 2, 10. Coll\`ege de France, Paris, 1975

    1974

  25. [33]

    Tempelmayr

    M. Tempelmayr. Characterizing models in regularity structures: a quasilinear case. Probab. Theory Relat. Fields , 2024. https://doi.org/10.1007/s00440-024-01292-2 doi:10.1007/s00440-024-01292-2

    2024 doi

  26. [34]

    Tadmor and T

    E. Tadmor and T. Tao. Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDE s. Comm. Pure Appl. Math. , 60(10):1488--1521, 2007. https://doi.org/10.1002/cpa.20180 doi:10.1002/cpa.20180

    2007 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.