Existence and regularity of solutions to parabolic-elliptic nonlinear systems

Marco Picerni

arxiv: 2604.16100 · v2 · pith:CNNFGHD3new · submitted 2026-04-17 · 🧮 math.AP

Existence and regularity of solutions to parabolic-elliptic nonlinear systems

Marco Picerni This is my paper

Pith reviewed 2026-05-22 09:57 UTC · model grok-4.3

classification 🧮 math.AP

keywords parabolic-elliptic systemexistence of solutionsregularitysummabilityL1 datadiscontinuous coefficientsnonlinear coupling

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The pith

Solutions to the coupled parabolic-elliptic system exist for L1 data and acquire higher integrability despite the coupling term belonging only to L2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence of weak solutions to a parabolic equation coupled to an elliptic equation that determines a potential ψ from the power |u|^θ of the unknown. It establishes that these solutions gain summability, belonging to L^s(Ω_T) for s > 1 and to L^q(0,T; W^{1,q}_0(Ω)) for q > 1, even though the convective term u M ∇ψ lies only in L2(Ω_T). The argument relies on the known L^p-regularity theory for linear parabolic equations with measurable coefficients. A reader would care because the result covers data that are merely integrable and coefficients that may jump, extending classical theory to this nonlinear coupling under a smallness restriction on θ.

Core claim

For the system with uniformly elliptic measurable coefficients A(x,t) and M(x), and with 0 < θ < 2/N, there exist solutions u for every f in L1(Ω_T) such that u satisfies the parabolic equation in the weak sense and obeys the summability u ∈ L^s(Ω_T) ∩ L^q(0,T; W^{1,q}_0(Ω)) for suitable s > 1 and q > 1; this follows by controlling the contribution of the coupling term and invoking the Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina regularity theorems for the parabolic operator.

What carries the argument

The coupling that expresses ∇ψ through the elliptic equation -div(M ∇ψ) = |u|^θ, which supplies the structural control needed to absorb the low-regularity drift term u M ∇ψ into the estimates for the parabolic equation.

If this is right

Existence holds for merely integrable forcing terms f.
The solution u obtains positive gradient integrability in the Sobolev sense.
The same summability conclusions apply to the linear parabolic equation with an additional drift of limited regularity.
The smallness restriction on θ keeps the nonlinear source controllable in the a priori estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may adapt to systems in which M also depends on time or on u itself.
Numerical schemes for such coupled problems could exploit the gained integrability to justify convergence.
Similar regularity might hold for related models in which the elliptic equation arises from a steady-state approximation to a parabolic component.

Load-bearing premise

The coefficients A and M must be uniformly elliptic and bounded while θ stays small enough that the nonlinear source |u|^θ does not overpower the integrability gains coming from the parabolic regularity theory.

What would settle it

An explicit example with the same structure but θ ≥ 2/N (or with coefficients that violate uniform ellipticity) for which an L1 forcing produces a solution u that fails to belong to any L^s with s > 1.

read the original abstract

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t - \operatorname{div}(A(x, t) \nabla u) = -\operatorname{div}(u M(x) \nabla \psi) + f(x, t) & \text{in } \Omega_T, \\ -\operatorname{div}(M(x) \nabla \psi) = |u|^\theta & \text{in } \Omega_T, \\ \psi(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, 0) = 0 & \text{in } \Omega. \end{cases} \end{equation*} Here, $\Omega$ is an open and bounded subset of $\mathbb R^N$, $N>2$, $\theta\in(0,\frac{2}{N})$, $0<T<+\infty$ and $\Omega_T=\Omega\times(0,T)$. We prove existence results for data $f\in L^1(\Omega_T)$ and a corresponding increase in summability that obeys the $L^p$-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall'Aglio-Gallou\"et-Orsina. In particular, despite the term $u M(x)\nabla\psi$ not being regular enough (since it only belongs to $L^2(\Omega_T)$), the solution $u$ belongs to $L^s(\Omega_T)\cap L^q(0,T;W^{1, q}_0(\Omega))$ for suitable $s>1$ and $q>1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gets existence plus L^s and W^{1,q} regularity for u in a parabolic-elliptic system with L1 data and discontinuous coefficients by keeping the coupling weak, but the L2 drift term needs explicit justification against the cited theorems.

read the letter

The main takeaway is that the authors prove existence of solutions to this coupled system and recover improved integrability for u even when f is only L1 and the coefficients are merely measurable. They do this by restricting θ below 2/N so that the elliptic equation for ψ produces a controllable drift in the parabolic equation for u. That combination for this exact setup looks new, and they correctly flag that the coupling term sits only in L2 yet still claim the classical summability results apply. Credit is due for working with discontinuous coefficients and L1 data without extra assumptions that would make the problem easier. The setup is clean and the exponent range is the natural one that lets the estimates close. The soft spot is the handling of the extra first-order term -div(u M ∇ψ). Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina are stated for equations without drift or with drifts in better spaces, so the paper must either absorb the term via truncation and approximation or prove a small variant. The abstract states the conclusion but does not sketch how the test-function argument or bootstrap is adjusted; if the full proof supplies that step without hidden gaps, the claim is fine, otherwise it needs tightening. No other red flags appear in the abstract or the stated assumptions. This is for readers already working on regularity for parabolic equations with lower-order terms or coupled systems. Someone who knows the two cited papers will see the incremental extension clearly. It is not broad enough for me to cite in the next year, but the technical point on the drift is worth checking in detail. I would send it to a serious referee rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes existence of solutions to the parabolic-elliptic system with discontinuous coefficients A(x,t) and M(x) for f ∈ L¹(Ω_T), together with improved summability: the solution u belongs to L^s(Ω_T) ∩ L^q(0,T; W^{1,q}_0(Ω)) for suitable s>1, q>1. The proof invokes the classical L^p-regularity theorems of Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina, asserting that the smallness condition 0<θ<2/N renders the coupling term -div(u M ∇ψ) (which lies only in L²(Ω_T)) controllable despite its limited integrability.

Significance. If the central claim holds, the work shows that standard L^p-regularity theory extends to coupled parabolic-elliptic systems with measurable coefficients and a drift of only L² integrability. This is of interest for models with nonlinear couplings under minimal data assumptions. A strength is the direct reliance on established theorems rather than new ad-hoc constructions, provided the necessary adaptations for the drift are rigorously justified.

major comments (2)

[Regularity estimates / bootstrap] In the section containing the regularity estimates and bootstrap argument: the central claim that u gains L^s and W^{1,q} integrability rests on applying the Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina theorems to the parabolic equation that includes the additional first-order term -div(u M ∇ψ) ∈ L²(Ω_T). These theorems are classically stated for equations without such a drift or with drifts satisfying stronger integrability; the manuscript must therefore specify whether the L² term is absorbed via a modified test-function or truncation procedure, or whether a variant of the cited results is proved. The condition θ < 2/N is asserted to make the term controllable, but the explicit exponent relations that close the bootstrap are not displayed and must be provided to verify that the summability improvement is obtained.
[Existence construction] In the existence proof (likely §3 or §4): the construction of approximate solutions and passage to the limit must be checked for compatibility with the discontinuous coefficients and the specific form of the coupling; any truncation or approximation scheme should preserve the uniform ellipticity and boundedness assumptions on A and M while controlling the |u|^θ source term.

minor comments (2)

[Introduction] The definition of Ω_T and the precise range of admissible exponents s and q should be stated explicitly at the first appearance rather than only in the abstract.
A brief comparison with related results on parabolic-elliptic systems (e.g., those with different coupling exponents or regular coefficients) would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional details.

read point-by-point responses

Referee: In the section containing the regularity estimates and bootstrap argument: the central claim that u gains L^s and W^{1,q} integrability rests on applying the Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina theorems to the parabolic equation that includes the additional first-order term -div(u M ∇ψ) ∈ L²(Ω_T). These theorems are classically stated for equations without such a drift or with drifts satisfying stronger integrability; the manuscript must therefore specify whether the L² term is absorbed via a modified test-function or truncation procedure, or whether a variant of the cited results is proved. The condition θ < 2/N is asserted to make the term controllable, but the explicit exponent relations that close the bootstrap are not displayed and must be provided to verify that the summability improvement is obtained.

Authors: We appreciate this observation. The term -div(u M ∇ψ) is treated as a divergence-form lower-order contribution whose L² integrability follows from the assumption θ < 2/N together with the elliptic regularity for ψ. This term is absorbed by adapting the truncation and test-function arguments already present in the proofs of Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina, which accommodate drifts of this type under smallness conditions on the coupling. To make the argument fully transparent we will insert a new subsection that records the explicit exponent relations: starting from the L¹ datum and the source |u|^θ, the bootstrap yields u ∈ L^s(Ω_T) for s = 1 + δ(N,θ) > 1 and u ∈ L^q(0,T; W^{1,q}_0(Ω)) for q = 2 - ε(N,θ) > 1, with the precise δ and ε chosen so that the L² drift remains controllable and the estimates close. These calculations will be displayed in full. revision: yes
Referee: In the existence proof (likely §3 or §4): the construction of approximate solutions and passage to the limit must be checked for compatibility with the discontinuous coefficients and the specific form of the coupling; any truncation or approximation scheme should preserve the uniform ellipticity and boundedness assumptions on A and M while controlling the |u|^θ source term.

Authors: We agree that additional verification is useful. The approximate solutions are constructed by mollifying the measurable coefficients A and M at a fixed scale while retaining the uniform ellipticity and boundedness constants; the mollified coefficients remain measurable and satisfy the same structural assumptions. The nonlinear source |u|^θ is controlled by introducing a truncation operator on the approximate solutions at each step, which preserves non-negativity and yields uniform L¹ bounds. Passage to the limit then follows from the a priori L^s ∩ W^{1,q} estimates already obtained and standard weak-convergence arguments. We will expand the existence section with these explicit compatibility checks and a short paragraph confirming that the truncation does not violate the ellipticity or boundedness hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity: central regularity claim rests on external classical theorems

full rationale

The derivation invokes the Aronson-Serrin and Boccardo-Dall'Aglio-Gallouët-Orsina L^p-regularity theorems as independent external support for the claimed summability of u despite the L^2 drift term. These are classical results by other authors, not self-citations or self-definitional steps. No fitted parameters are renamed as predictions, no ansatz is smuggled via prior work by the same author, and the existence/summability conclusion does not reduce by construction to the paper's own inputs. The θ < 2/N condition and uniform ellipticity are standard structural assumptions that do not create a definitional loop. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard assumptions for elliptic and parabolic operators with measurable coefficients. No free parameters or invented entities are introduced in the abstract. The key domain assumptions are ellipticity/boundedness of A and M and the smallness of θ.

axioms (2)

domain assumption A(x,t) and M(x) are measurable, bounded, and uniformly elliptic.
Required for the divergence-form operators to be well-defined and for the cited regularity theorems to apply; implicit in the system statement.
domain assumption θ ∈ (0, 2/N) ensures the source term |u|^θ produces a controllable coupling.
Stated explicitly in the abstract to make the L2 bound on the interaction term sufficient for the regularity lift.

pith-pipeline@v0.9.0 · 5878 in / 1572 out tokens · 35417 ms · 2026-05-22T09:57:26.582474+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove existence results for data f∈L¹(Ω_T) and a corresponding increase in summability that obeys the L^p-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall'Aglio-Gallouët-Orsina. In particular, despite the term u M(x)∇ψ not being regular enough (since it only belongs to L²(Ω_T)), the solution u belongs to L^s(Ω_T)∩L^q(0,T;W^{1,q}_0(Ω))
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

System (1.1) is loosely related to the minimal version of the Keller-Segel system... with discontinuous coefficients and added source term f≥0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

D. G. Aronson and J. Serrin,Local behavior of solutions of quasilinear parabolic equations, Archive for Rational Mechanics and Analysis25(1967), no. 1, 81–122

work page 1967
[2]

B´ enilan, L

P. B´ enilan, L. Boccardo, T. Gallou¨ et, R. Gariepy, M. Pierre, and J. L. Vazquez,AnL1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze22(1995), no. 2, 241–273

work page 1995
[3]

Boccardo,Dirichlet problems with singular convection terms and applications, Journal of Differential Equa- tions258(2015), 2290–2314

L. Boccardo,Dirichlet problems with singular convection terms and applications, Journal of Differential Equa- tions258(2015), 2290–2314

work page 2015
[4]

Boccardo and T

L. Boccardo and T. Gallou¨ et,Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis87(1989), no. 1, 149–169

work page 1989
[5]

Boccardo, A

L. Boccardo, A. Dall’Aglio, T. Gallou¨ et, and L. Orsina,Nonlinear parabolic equations with measure data, Journal of Functional Analysis147(1997), no. 1, 237–258

work page 1997
[6]

Boccardo and L

L. Boccardo and L. Orsina,Sublinear elliptic systems with a convection term, Communications in Partial Differential Equations45(2020), no. 7, 690–713

work page 2020
[7]

Boccardo, L

L. Boccardo, L. Orsina, and A. Porretta,Some noncoercive parabolic equations with lower order terms in divergence form, Journal of Evolution Equations3(2003), 407–418

work page 2003
[8]

Boccardo, L

L. Boccardo, L. Orsina, and M. M. Porzio,Existence and supercontractive estimates for parabolic–elliptic systems, Nonlinear Analysis227(2023), 113170

work page 2023
[9]

L. C. Evans,Partial Differential Equations, American Mathematical Society, 2010

work page 2010
[10]

Hillen and K

T. Hillen and K. J. Painter,A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology 58(2009), 183–217

work page 2009
[11]

J. L. Lions,Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires,´Etudes math´ ematiques, Dunod, 1969

work page 1969
[12]

Porretta,Existence results for nonlinear parabolic equations via strong convergence of truncations, Annali di Matematica Pura ed Applicata177(1999), 143–172

A. Porretta,Existence results for nonlinear parabolic equations via strong convergence of truncations, Annali di Matematica Pura ed Applicata177(1999), 143–172

work page 1999
[13]

Prignet,Existence and uniqueness of “entropy” solutions of parabolic problems withL 1 data, Nonlinear Analysis: Theory, Methods & Applications28(1997), no

A. Prignet,Existence and uniqueness of “entropy” solutions of parabolic problems withL 1 data, Nonlinear Analysis: Theory, Methods & Applications28(1997), no. 12, 1943–1954

work page 1997
[14]

Simon,Compact sets in the spaceL p(0, T;B), Annali di Matematica Pura ed Applicata146(1986), 65–96

J. Simon,Compact sets in the spaceL p(0, T;B), Annali di Matematica Pura ed Applicata146(1986), 65–96

work page 1986
[15]

Stampacchia,Le probl` eme de Dirichlet pour les ´ equations elliptiques du second ordre ` a coefficients discon- tinus, Annales de l’Institut Fourier15(1965), no

G. Stampacchia,Le probl` eme de Dirichlet pour les ´ equations elliptiques du second ordre ` a coefficients discon- tinus, Annales de l’Institut Fourier15(1965), no. 1, 189–257. Email address:mpicerni@sissa.it SISSA, via Bonomea 265, 34136, Trieste, Italy

work page 1965

[1] [1]

D. G. Aronson and J. Serrin,Local behavior of solutions of quasilinear parabolic equations, Archive for Rational Mechanics and Analysis25(1967), no. 1, 81–122

work page 1967

[2] [2]

B´ enilan, L

P. B´ enilan, L. Boccardo, T. Gallou¨ et, R. Gariepy, M. Pierre, and J. L. Vazquez,AnL1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze22(1995), no. 2, 241–273

work page 1995

[3] [3]

Boccardo,Dirichlet problems with singular convection terms and applications, Journal of Differential Equa- tions258(2015), 2290–2314

L. Boccardo,Dirichlet problems with singular convection terms and applications, Journal of Differential Equa- tions258(2015), 2290–2314

work page 2015

[4] [4]

Boccardo and T

L. Boccardo and T. Gallou¨ et,Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis87(1989), no. 1, 149–169

work page 1989

[5] [5]

Boccardo, A

L. Boccardo, A. Dall’Aglio, T. Gallou¨ et, and L. Orsina,Nonlinear parabolic equations with measure data, Journal of Functional Analysis147(1997), no. 1, 237–258

work page 1997

[6] [6]

Boccardo and L

L. Boccardo and L. Orsina,Sublinear elliptic systems with a convection term, Communications in Partial Differential Equations45(2020), no. 7, 690–713

work page 2020

[7] [7]

Boccardo, L

L. Boccardo, L. Orsina, and A. Porretta,Some noncoercive parabolic equations with lower order terms in divergence form, Journal of Evolution Equations3(2003), 407–418

work page 2003

[8] [8]

Boccardo, L

L. Boccardo, L. Orsina, and M. M. Porzio,Existence and supercontractive estimates for parabolic–elliptic systems, Nonlinear Analysis227(2023), 113170

work page 2023

[9] [9]

L. C. Evans,Partial Differential Equations, American Mathematical Society, 2010

work page 2010

[10] [10]

Hillen and K

T. Hillen and K. J. Painter,A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology 58(2009), 183–217

work page 2009

[11] [11]

J. L. Lions,Quelques m´ ethodes de r´ esolution des probl` emes aux limites non lin´ eaires,´Etudes math´ ematiques, Dunod, 1969

work page 1969

[12] [12]

Porretta,Existence results for nonlinear parabolic equations via strong convergence of truncations, Annali di Matematica Pura ed Applicata177(1999), 143–172

A. Porretta,Existence results for nonlinear parabolic equations via strong convergence of truncations, Annali di Matematica Pura ed Applicata177(1999), 143–172

work page 1999

[13] [13]

Prignet,Existence and uniqueness of “entropy” solutions of parabolic problems withL 1 data, Nonlinear Analysis: Theory, Methods & Applications28(1997), no

A. Prignet,Existence and uniqueness of “entropy” solutions of parabolic problems withL 1 data, Nonlinear Analysis: Theory, Methods & Applications28(1997), no. 12, 1943–1954

work page 1997

[14] [14]

Simon,Compact sets in the spaceL p(0, T;B), Annali di Matematica Pura ed Applicata146(1986), 65–96

J. Simon,Compact sets in the spaceL p(0, T;B), Annali di Matematica Pura ed Applicata146(1986), 65–96

work page 1986

[15] [15]

Stampacchia,Le probl` eme de Dirichlet pour les ´ equations elliptiques du second ordre ` a coefficients discon- tinus, Annales de l’Institut Fourier15(1965), no

G. Stampacchia,Le probl` eme de Dirichlet pour les ´ equations elliptiques du second ordre ` a coefficients discon- tinus, Annales de l’Institut Fourier15(1965), no. 1, 189–257. Email address:mpicerni@sissa.it SISSA, via Bonomea 265, 34136, Trieste, Italy

work page 1965