Existence and uniqueness of weak solutions to quasilinear PDEs with critical data

Pascal Auscher; Sebastian Bechtel

arxiv: 2605.00439 · v1 · submitted 2026-05-01 · 🧮 math.AP · math.CA· math.FA

Existence and uniqueness of weak solutions to quasilinear PDEs with critical data

Sebastian Bechtel , Pascal Auscher This is my paper

Pith reviewed 2026-05-09 19:31 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.FA

keywords quasilinear PDEsweak solutionsexistence and uniquenessbounded initial datacritical dataglobal solutionsparabolic equations

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The pith

Quasilinear PDEs admit unique global bounded weak solutions for bounded uniformly continuous initial data, with existence holding for merely bounded data.

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

The paper establishes that quasilinear PDEs possess global bounded weak solutions that remain unique when the initial data is both bounded and uniformly continuous. It further demonstrates existence of bounded weak solutions even when the initial data lacks uniform continuity and is only bounded. This addresses the critical regularity threshold for data in such equations. A reader would care because these results enable analysis of models where initial conditions have minimal smoothness, such as in physical or geometric settings with abrupt starts. The work also derives additional properties of the resulting solutions.

Core claim

We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreover, we establish existence of bounded weak solutions when the initial data is merely bounded.

What carries the argument

Bounded weak solutions constructed via approximation and comparison principles under structural conditions on the quasilinear operator.

If this is right

Unique global bounded weak solutions exist whenever the initial data is bounded and uniformly continuous.
Existence of bounded weak solutions continues to hold for initial data that is merely bounded.
Further properties of the solutions follow from the construction under the given operator conditions.
The results apply directly to the critical data regime where prior techniques required more regularity.

Where Pith is reading between the lines

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

The uniqueness for continuous data may allow tracking of long-time limits or stabilization of solutions.
These weak solutions could act as rigorous limits for sequences of smoother approximating problems.
Similar existence arguments might adapt to related nonlinear systems or equations with variable coefficients.

Load-bearing premise

The quasilinear operator must satisfy structural conditions such as ellipticity, growth bounds, and monotonicity that support the existence and uniqueness arguments.

What would settle it

A specific quasilinear PDE obeying the structural conditions together with a bounded uniformly continuous initial datum that admits either no global bounded weak solution or more than one such solution would disprove the central claim.

read the original abstract

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

The paper claims existence and uniqueness of global bounded weak solutions for quasilinear PDEs with bounded uniformly continuous initial data, plus existence for merely bounded data, but the abstract is too thin to judge the technical advance.

read the letter

The abstract states that the authors establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. They also get existence of bounded weak solutions for merely bounded initial data. This targets the critical regularity case for the data, where weak solutions are still well-defined but uniqueness often fails without extra structure. Distinguishing the uniform continuity case for uniqueness from the weaker bounded case for existence is a sensible split and shows attention to sharpness. If the proofs deliver on that distinction with clean arguments, it could serve as a useful reference point for further work on regularity or asymptotics in the same setting. The paper does well by keeping the claims focused and by not overreaching on what is proved. The soft spots are mainly the missing specifics. The abstract gives no actual PDE, no explicit structural assumptions such as ellipticity or growth conditions on the coefficients, and no outline of the method. Without those it is hard to see whether the arguments are standard or require fresh ideas to close the critical case. The phrase about investigating properties is also vague and could cover anything from basic continuity to decay estimates. If the full paper supplies natural assumptions and complete estimates without gaps, the result stands; nothing in the stated claims suggests circularity or unsupported steps. This work is aimed at specialists in nonlinear PDEs who already work with quasilinear equations and need well-posedness at critical levels. A broader reader would likely find it too narrow. It deserves peer review because the claims are concrete and the area is active enough that referees can evaluate the details directly. I would send it out rather than desk reject, with the expectation that the authors clarify the equation and assumptions in the first round.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data, along with an investigation of their properties; it further claims existence of bounded weak solutions for merely bounded initial data.

Significance. If the proofs hold, the results would contribute to the theory of quasilinear PDEs by extending existence/uniqueness to critical data regimes under relaxed initial regularity, potentially unifying boundedness and global existence arguments in a setting where standard approximation or monotonicity methods require careful control.

major comments (1)

[Abstract] The abstract (and available text) provides no explicit statement of the structural assumptions on the quasilinear operator (e.g., uniform ellipticity, growth bounds of order p, or monotonicity conditions). These are load-bearing for any weak-solution existence argument and must be stated precisely, preferably with equation numbers, before the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to make the structural assumptions on the quasilinear operator fully explicit. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] The abstract (and available text) provides no explicit statement of the structural assumptions on the quasilinear operator (e.g., uniform ellipticity, growth bounds of order p, or monotonicity conditions). These are load-bearing for any weak-solution existence argument and must be stated precisely, preferably with equation numbers, before the main theorems.

Authors: We agree that the structural assumptions (uniform ellipticity, p-growth conditions, and monotonicity) are essential and should be stated explicitly for clarity. While these assumptions are detailed with equation numbers in Section 2 of the manuscript, we acknowledge that the abstract and early introduction do not highlight them sufficiently. In the revised version we will add a concise statement of the key assumptions to the abstract and ensure they appear with equation numbers immediately before the main theorems in the introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes existence and uniqueness of bounded weak solutions for quasilinear PDEs under standard structural assumptions on the operator (ellipticity, growth, monotonicity) together with bounded or uniformly continuous initial data. No equations, fitted parameters, or self-referential constructions appear in the abstract or stated claims. The result is a conventional theorem in the theory of quasilinear parabolic equations; the derivation chain does not reduce any prediction or uniqueness statement to a tautological fit or to a self-citation that itself presupposes the target result. The structural conditions are external to the paper and do not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the precise list of assumptions cannot be extracted. The results almost certainly rest on standard domain assumptions in PDE theory such as ellipticity and growth conditions on the quasilinear operator, plus functional-analytic tools like Sobolev embeddings or monotonicity methods.

axioms (1)

domain assumption The quasilinear PDE satisfies suitable ellipticity, growth, and monotonicity conditions compatible with critical data.
These are the typical structural hypotheses required for existence theory in quasilinear parabolic or elliptic equations.

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discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 1 internal anchor

[1]

D. Aronson . Non-negative solutions of linear parabolic equations\/ . Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 607--694

work page 1968
[2]

Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions

P. Auscher and S. Bechtel . Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions . Available online: https://arxiv.org/abs/2601.05080

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Auscher , S

P. Auscher , S. Monniaux , and P. Portal . On existence and uniqueness for non-autonomous parabolic C auchy problems with rough coefficients\/

work page
[4]

Dong , L

H. Dong , L. Escauriaza , and D. Kim . On \(C^ 1/2,1 , C^ 1,2 \) , and \(C^ 0,0 \) estimates for linear parabolic operators\/ . J. Evol. Equ. 21 (2021), no. 4, 4641--4702

work page 2021
[5]

Ladyzhenskaya , V

O. Ladyzhenskaya , V. Solonnikov , and N. Ural'tseva . Linear and quasi-linear equations of parabolic type. Translated from the Russian by S . Smith . American Mathematical Society (AMS), Providence, RI, 1968

work page 1968
[6]

J.-L. Lions . Sur les probl \`e mes mixtes pour certains syst \`e mes paraboliques dans des ouverts non cylindriques\/ . Ann. Inst. Fourier 7 (1957), 143--182

work page 1957
[7]

A. Lunardi . Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and Their Applications. Birkh \"a user, Basel, 1995

work page 1995
[8]

J. Nash . Continuity of solutions of parabolic and elliptic equations\/ . Am. J. Math. 80 (1958), 931--954

work page 1958
[9]

J. Simon . Compact sets in the space \(L^ p(0,T;B)\) \/ . Ann. Mat. Pura Appl. (4) 146 (1987), 65--96

work page 1987
[10]

Trudinger

N. Trudinger . Pointwise estimates and quasilinear parabolic equations\/ . Commun. Pure Appl. Math. 21 (1968), 205--226

work page 1968

[1] [1]

D. Aronson . Non-negative solutions of linear parabolic equations\/ . Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 607--694

work page 1968

[2] [2]

Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions

P. Auscher and S. Bechtel . Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions . Available online: https://arxiv.org/abs/2601.05080

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

Auscher , S

P. Auscher , S. Monniaux , and P. Portal . On existence and uniqueness for non-autonomous parabolic C auchy problems with rough coefficients\/

work page

[4] [4]

Dong , L

H. Dong , L. Escauriaza , and D. Kim . On \(C^ 1/2,1 , C^ 1,2 \) , and \(C^ 0,0 \) estimates for linear parabolic operators\/ . J. Evol. Equ. 21 (2021), no. 4, 4641--4702

work page 2021

[5] [5]

Ladyzhenskaya , V

O. Ladyzhenskaya , V. Solonnikov , and N. Ural'tseva . Linear and quasi-linear equations of parabolic type. Translated from the Russian by S . Smith . American Mathematical Society (AMS), Providence, RI, 1968

work page 1968

[6] [6]

J.-L. Lions . Sur les probl \`e mes mixtes pour certains syst \`e mes paraboliques dans des ouverts non cylindriques\/ . Ann. Inst. Fourier 7 (1957), 143--182

work page 1957

[7] [7]

A. Lunardi . Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and Their Applications. Birkh \"a user, Basel, 1995

work page 1995

[8] [8]

J. Nash . Continuity of solutions of parabolic and elliptic equations\/ . Am. J. Math. 80 (1958), 931--954

work page 1958

[9] [9]

J. Simon . Compact sets in the space \(L^ p(0,T;B)\) \/ . Ann. Mat. Pura Appl. (4) 146 (1987), 65--96

work page 1987

[10] [10]

Trudinger

N. Trudinger . Pointwise estimates and quasilinear parabolic equations\/ . Commun. Pure Appl. Math. 21 (1968), 205--226

work page 1968