Concentration of measure-valued solutions for semilinear parabolic equations

Charlie Lebarb\'e (LAAS); Didier Henrion (LAAS); \'Emilien Flayac; Michel Fourni\'e; Milan Korda (LAAS)

arxiv: 2605.23678 · v1 · pith:NMJJ2YXWnew · submitted 2026-05-22 · 🧮 math.OC

Concentration of measure-valued solutions for semilinear parabolic equations

Charlie Lebarb\'e (LAAS) , \'Emilien Flayac , Michel Fourni\'e , Didier Henrion (LAAS) , Milan Korda (LAAS) This is my paper

Pith reviewed 2026-05-25 04:13 UTC · model grok-4.3

classification 🧮 math.OC

keywords semilinear parabolic PDEsreaction-diffusion equationsmeasure-valued solutionsYoung measuresrelaxation gapmoment sum of squaresoptimal control

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The pith

Measure-valued solutions to semilinear parabolic PDEs concentrate on the unique classical solution.

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

The paper shows there is no relaxation gap when applying the moment-sum-of-squares hierarchy to scalar semilinear parabolic PDEs of reaction-diffusion type. Each solution of the linear measure equation produces an energy measure-valued solution using Young measures that obey energy identities. Any such energy measure-valued solution concentrates onto the classical nonlinear PDE solution whenever the latter exists and is unique. This establishes that the linear formulation recovers the physical solution under standard assumptions.

Core claim

Each solution to the linear measure equation gives rise to an energy measure-valued solution in the space of Young measures satisfying suitable energy identities. Any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. This proves the absence of a relaxation gap for these PDEs.

What carries the argument

energy measure-valued (emv) solutions: Young measures satisfying energy identities that bridge the linear measure formulation and the nonlinear PDE

If this is right

The moment-sum-of-squares hierarchy yields exact solutions without relaxation gap for these PDEs.
Linear measure formulations can be used to find or approximate the classical solutions reliably.
Optimal control problems governed by such PDEs can be solved exactly via convex relaxations.
Concentration holds conditionally on existence and uniqueness of the classical solution.

Where Pith is reading between the lines

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

This result suggests testing the hierarchy on specific equations like the Allen-Cahn or Fisher equation to observe the concentration numerically.
The method could potentially extend to other parabolic PDEs if uniqueness theorems are established first.
Connections to viscosity solutions or other weak solution concepts might be explored to broaden the applicability.

Load-bearing premise

The nonlinear PDE has a unique classical solution.

What would settle it

Constructing or finding a semilinear parabolic PDE with a unique classical solution but where some energy measure-valued solution fails to concentrate on it.

read the original abstract

The moment-sum-of-squares hierarchy provides a powerful framework for solving non-convex optimal control problems by constructing a sequence of convex semidefinite relaxations. However, when extending these methods to nonlinear partial differential equations (PDEs), a fundamental challenge is the potential existence of a relaxation gap, where the solution to the linear measure formulation using occupation measures fails to correspond to a classical physical solution of the original PDE. In this paper, we prove the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. We do so by showing that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities. We then prove that any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. To the best of our knowledge, this is the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

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 shows measure solutions concentrate to classical ones for scalar reaction-diffusion PDEs when a unique classical solution exists, via energy measure-valued solutions.

read the letter

The core result here is a concentration theorem: linear measure solutions to the relaxed problem yield energy measure-valued solutions that concentrate on the nonlinear PDE solution, but only if a classical solution exists and is unique. This directly targets the relaxation gap problem in moment-SOS approaches to PDE-constrained problems. It is new as the first explicit concentration statement for measure-valued solutions of reaction-diffusion equations. The construction via energy identities is a clean way to connect the linear measure equation to the nonlinear dynamics without extra assumptions beyond standard measure theory and PDE basics. The argument structure avoids circularity and states the conditional nature up front. The main limitation is that the result does not remove the need for a priori existence and uniqueness of the classical solution; in many applications that is the hard part, so the practical payoff depends on cases where that is already known. The abstract gives the proof outline but no detailed energy calculations or counterexample checks, which leaves the rigor to the full text. No invented entities or fitting issues appear in the stated claims. This work is aimed at people already using occupation measures or SOS hierarchies for parabolic control problems. Readers who need a theoretical justification for dropping the gap concern in the scalar semilinear case will get something concrete from it. It is worth sending to referees because the claim is focused, the conditional framing is honest, and the subfield has been waiting for exactly this kind of concentration result. If the derivations hold up under review, it strengthens the case for measure relaxations in this class of PDEs.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. It shows that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities, and that any such emv solution concentrates on the solution to the nonlinear PDE provided the latter exists and is unique. This is presented as the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

Significance. If the result holds, it is significant for extending the moment-sum-of-squares hierarchy to nonlinear PDEs by removing the possibility of a relaxation gap in this scalar semilinear parabolic setting. The approach relies on constructing emv solutions via energy identities and proving concentration under an explicit existence-uniqueness hypothesis for the classical solution; this conditional structure is clearly stated and aligns with standard tools from measure theory and PDEs. The introduction of emv solutions as an intermediate object is a notable technical contribution.

minor comments (2)

The abstract and introduction would benefit from a brief remark on the precise function space in which the classical solution is assumed to exist and be unique (e.g., C^{2,1} or appropriate Sobolev space), as this directly affects the scope of the concentration statement.
Notation for the energy identities satisfied by emv solutions should be introduced with an explicit equation number in the main text rather than only in the abstract, to improve readability for readers unfamiliar with the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its significance, and recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No circularity; conditional proof on external existence/uniqueness assumption

full rationale

The paper's central claim is a conditional concentration result: linear measure solutions yield emv solutions via energy identities, which then concentrate to the classical nonlinear PDE solution only when the latter exists and is unique. This assumption is stated explicitly as a hypothesis rather than derived internally. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The argument uses standard measure-theoretic constructions without reducing to its own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim depends on the introduction of the emv solution concept and standard measure theory axioms, with no data-fitted parameters.

axioms (2)

standard math Properties of Young measures and occupation measures
Used as the basis for the linear measure formulation in optimal control.
domain assumption Existence of suitable energy identities for emv solutions
Key step to connect the measure solution to the nonlinear PDE.

invented entities (1)

energy measure-valued (emv) solution no independent evidence
purpose: Represents solutions in Young measure space satisfying energy identities
Introduced in the paper as an intermediate concept for the concentration proof.

pith-pipeline@v0.9.0 · 5721 in / 1331 out tokens · 40695 ms · 2026-05-25T04:13:02.966594+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

[1]

Lasserre

J.B. Lasserre. Global optimization with polynomials and the problem of moments.SIAM Journal on Optimization 11(3), 796–817 (2001)

work page 2001
[2]

J. Nie. Moment and Polynomial Optimization . SIAM, Philadelphia (2023)

work page 2023
[3]

Theobald

T. Theobald. Real Algebraic Geometry and Optimization . AMS, Providence (2024)

work page 2024
[4]

Lasserre, D

J.B. Lasserre, D. Henrion, C. Prieur, and E. Tr´ elat. Nonlinear optimal control via occupation measures and LMI relaxations. SIAM Journal on Control and Optimization 47(4), 1643–1666 (2008)

work page 2008
[5]

L.C. Young. Lectures on the Calculus of Variations and Optimal Control Theory . Saun- ders, Philadelphia (1969)

work page 1969
[6]

Lewis and R.B

R.M. Lewis and R.B. Vinter. Relaxation of optimal control problems to equivalent convex programs. Journal of Mathematical Analysis and Applications 74(2), 475–493 (1980)

work page 1980
[7]

J. Rubio. Extremal points and optimal control theory. Annali di Matematica Pura ed Applicata 109(1), 165–176 (1976)

work page 1976
[8]

R.B. Vinter. Convex duality and nonlinear optimal control. SIAM Journal on Control and Optimization 31(2), 518–538 (1993)

work page 1993
[9]

Hern´ andez-Hern´ andez, O

D. Hern´ andez-Hern´ andez, O. Hern´ andez-Lerma, and M. Taksar. The linear program- ming approach to deterministic optimal control problems. Applied Mathematics 24, 17–33 (1996)

work page 1996
[10]

Henrion, M

D. Henrion, M. Korda, and J.B. Lasserre. The Moment-SOS Hierarchy. World Scientific, Singapore (2020)

work page 2020
[11]

Korda, D

M. Korda, D. Henrion, and J.B. Lasserre. Moments and convex optimization for analysis and control of nonlinear PDEs. In: E. Tr´ elat and E. Zuazua (eds.),Numerical Control: Part A, Handbook of Numerical Analysis, vol. 23, pp. 339–366. Elsevier, North-Holland (2022)

work page 2022
[12]

Fantuzzi and I

G. Fantuzzi and I. Tobasco. Sharpness and non-sharpness of occupation measure bounds for integral variational problems. arXiv:2207.13570 (2022)

work page arXiv 2022
[13]

Korda and R

M. Korda and R. R´ ıos-Zertuche. The gap between a variational problem and its occu- pation measure relaxation. ESAIM: Control, Optimisation and Calculus of Variations 32 (2026)

work page 2026
[14]

Henrion, M

D. Henrion, M. Korda, M. Kruˇ z´ ık, and R. R´ ıos-Zertuche. Occupation measure relax- ations in variational problems: the role of convexity. SIAM Journal on Optimization 34(2), 1708–1731 (2024)

work page 2024
[15]

Augier, M

N. Augier, M. Korda, and R. R´ ıos-Zertuche. Sufficient conditions for the absence of relaxation gaps in state-constrained optimal control. arXiv:2503.13780 (2025). 31

work page arXiv 2025
[16]

Palladino and R.B

M. Palladino and R.B. Vinter. Minimizers that are not also relaxed minimizers. SIAM Journal on Control and Optimization 52(4), 2164–2179 (2014)

work page 2014
[17]

Augier, M

N. Augier, M. Korda, and R. R´ ıos-Zertuche. Young measure relaxation gaps for con- trollable systems with smooth state constraints. Journal of Differential Equations 459, 114036 (2026)

work page 2026
[18]

S. Marx, T. Weisser, D. Henrion, and J.B. Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields 10(1), 113–140 (2020)

work page 2020
[19]

Cardo¨ en, S

C. Cardo¨ en, S. Marx, A. Nouy, and N. Seguin. A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws. Numerische Mathematik 156(4), 1289–1324 (2024)

work page 2024
[20]

Hopf and A

K. Hopf and A. J¨ ungel. Convergence of a finite-volume scheme and dissipative measure- valued–strong stability for a hyperbolic–parabolic cross-diffusion system. Numerische Mathematik 157(3), 951–992 (2025)

work page 2025
[21]

Bertsimas and C

D. Bertsimas and C. Caramanis. Bounds on linear PDEs via semidefinite optimization. Mathematical Programming 108(1), 135–158 (2006)

work page 2006
[22]

Goulart and S.I

P.J. Goulart and S.I. Chernyshenko. Global stability analysis of fluid flows using sum- of-squares. Physica D 241, 692–704 (2012)

work page 2012
[23]

Chernyshenko, P.J

S.I. Chernyshenko, P.J. Goulart, D. Huang, and A. Papachristodoulou. Polynomial sum of squares in fluid dynamics: A review with a look ahead. Philosophical Transactions of the Royal Society A 372 (2014)

work page 2014
[24]

Valmorbida, M

G. Valmorbida, M. Ahmadi, and A. Papachristodoulou. Stability analysis for a class of partial differential equations via semidefinite programming. IEEE Transactions on Automatic Control 61, 1649–1654 (2016)

work page 2016
[25]

Goluskin and G

D. Goluskin and G. Fantuzzi. Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming.Nonlinearity 32, 1705–1730 (2019)

work page 2019
[26]

Fuentes, D

F. Fuentes, D. Goluskin, and S.I. Chernyshenko. Global stability of fluid flows despite transient growth of energy. Physical Review Letters 128, 204502 (2022)

work page 2022
[27]

Magron and C

V. Magron and C. Prieur. Optimal control of linear PDEs using occupation measures and SDP relaxations. IMA Journal of Mathematical Control and Information 37(1), 159–174 (2020)

work page 2020
[28]

Chhatoi, D

S.P. Chhatoi, D. Henrion, S. Marx, and N. Seguin. Optimizing quasi-dissipative evo- lution equations with the moment-SOS hierarchy. Discrete and Continuous Dynamical Systems 45(11), 4139–4159 (2025)

work page 2025
[29]

Fjordholm, K

U.S. Fjordholm, K. Lye, S. Mishra, and F. Weber. Statistical solutions of hyperbolic sys- tems of conservation laws: numerical approximation. Mathematical Models and Methods in Applied Sciences 30(3), 539–609 (2020). 32

work page 2020
[30]

Henrion, M

D. Henrion, M. Infusino, S. Kuhlmann, and V. Vinnikov. Infinite-dimensional moment- SOS hierarchy for nonlinear partial differential equations. arXiv:2305.18768 (2023)

work page arXiv 2023
[31]

Henrion and A

D. Henrion and A. Rudi. Solving moment and polynomial optimization problems on Sobolev spaces. SIAM Journal on Optimization 35(2), 989–1003 (2025)

work page 2025
[32]

Adams and J.J.F

R.A. Adams and J.J.F. Fournier. Sobolev Spaces, 2nd edn. Pure and Applied Mathemat- ics, vol. 140. Academic Press, Amsterdam Boston (2003)

work page 2003
[33]

Quittner and P

P. Quittner and P. Souplet. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States . Birkha¨ user Advanced Texts. Birkh¨ auser, Basel (2007)

work page 2007
[34]

L.C. Evans. Partial Differential Equations . AMS, Providence, R.I. (2010)

work page 2010
[35]

A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems . Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkh¨ auser Basel, Basel (1995)

work page 1995
[36]

R.J. DiPerna. Measure-valued solutions to conservation laws. Archive for Rational Me- chanics and Analysis 88(3), 223–270 (1985)

work page 1985
[37]

Attouch, G

H. Attouch, G. Buttazzo, and G. Michaille. Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization . MPS-SIAM Series on Optimization . SIAM, Philadelphia (2006)

work page 2006
[38]

Kruˇ z´ ık and T

M. Kruˇ z´ ık and T. Roub´ ıˇ cek. Explicit characterization of Lp-Young measures.Journal of Mathematical Analysis and Applications 198(3), 830–843 (1996)

work page 1996
[39]

Schonbek

M.E. Schonbek. Convergence of solution to nonlinear dispersive equations. Communi- cations in Partial Differential Equations 7(8), 959–1000 (1982)

work page 1982
[40]

Gallou¨ et and R

T. Gallou¨ et and R. Herbin. Weak Solutions of Partial Differential Equations . Math´ ematiques et Applications, vol. 90. Springer, Cham (2025)

work page 2025
[41]

Arendt, I

W. Arendt, I. Chalendar, and R. Eymard. Lions’ representation theorem and applica- tions. Journal of Mathematical Analysis and Applications 522(2), 126946 (2023)

work page 2023
[42]

Lebarb´ e, E

C. Lebarb´ e, E. Flayac, M. Fourni´ e, D. Henrion, and M. Korda. Optimal control of 1D semilinear heat equations with moment-SOS relaxations. In: 2025 European Control Conference (ECC), pp. 731–736 (2025)

work page 2025
[43]

Hyt¨ onen, J

T. Hyt¨ onen, J. Neerven, M. Veraar, and L. Weis. Analysis in Banach Spaces. Volume I: Martingales and Littlewood-Paley Theory . Springer, Cham (2016)

work page 2016
[44]

R. Temam. Navier–Stokes Equations: Theory and Numerical Analysis . Studies in Math- ematics and Its Applications , vol. 2. North-Holland Publishing Company, Amsterdam (1977). 33

work page 1977

[1] [1]

Lasserre

J.B. Lasserre. Global optimization with polynomials and the problem of moments.SIAM Journal on Optimization 11(3), 796–817 (2001)

work page 2001

[2] [2]

J. Nie. Moment and Polynomial Optimization . SIAM, Philadelphia (2023)

work page 2023

[3] [3]

Theobald

T. Theobald. Real Algebraic Geometry and Optimization . AMS, Providence (2024)

work page 2024

[4] [4]

Lasserre, D

J.B. Lasserre, D. Henrion, C. Prieur, and E. Tr´ elat. Nonlinear optimal control via occupation measures and LMI relaxations. SIAM Journal on Control and Optimization 47(4), 1643–1666 (2008)

work page 2008

[5] [5]

L.C. Young. Lectures on the Calculus of Variations and Optimal Control Theory . Saun- ders, Philadelphia (1969)

work page 1969

[6] [6]

Lewis and R.B

R.M. Lewis and R.B. Vinter. Relaxation of optimal control problems to equivalent convex programs. Journal of Mathematical Analysis and Applications 74(2), 475–493 (1980)

work page 1980

[7] [7]

J. Rubio. Extremal points and optimal control theory. Annali di Matematica Pura ed Applicata 109(1), 165–176 (1976)

work page 1976

[8] [8]

R.B. Vinter. Convex duality and nonlinear optimal control. SIAM Journal on Control and Optimization 31(2), 518–538 (1993)

work page 1993

[9] [9]

Hern´ andez-Hern´ andez, O

D. Hern´ andez-Hern´ andez, O. Hern´ andez-Lerma, and M. Taksar. The linear program- ming approach to deterministic optimal control problems. Applied Mathematics 24, 17–33 (1996)

work page 1996

[10] [10]

Henrion, M

D. Henrion, M. Korda, and J.B. Lasserre. The Moment-SOS Hierarchy. World Scientific, Singapore (2020)

work page 2020

[11] [11]

Korda, D

M. Korda, D. Henrion, and J.B. Lasserre. Moments and convex optimization for analysis and control of nonlinear PDEs. In: E. Tr´ elat and E. Zuazua (eds.),Numerical Control: Part A, Handbook of Numerical Analysis, vol. 23, pp. 339–366. Elsevier, North-Holland (2022)

work page 2022

[12] [12]

Fantuzzi and I

G. Fantuzzi and I. Tobasco. Sharpness and non-sharpness of occupation measure bounds for integral variational problems. arXiv:2207.13570 (2022)

work page arXiv 2022

[13] [13]

Korda and R

M. Korda and R. R´ ıos-Zertuche. The gap between a variational problem and its occu- pation measure relaxation. ESAIM: Control, Optimisation and Calculus of Variations 32 (2026)

work page 2026

[14] [14]

Henrion, M

D. Henrion, M. Korda, M. Kruˇ z´ ık, and R. R´ ıos-Zertuche. Occupation measure relax- ations in variational problems: the role of convexity. SIAM Journal on Optimization 34(2), 1708–1731 (2024)

work page 2024

[15] [15]

Augier, M

N. Augier, M. Korda, and R. R´ ıos-Zertuche. Sufficient conditions for the absence of relaxation gaps in state-constrained optimal control. arXiv:2503.13780 (2025). 31

work page arXiv 2025

[16] [16]

Palladino and R.B

M. Palladino and R.B. Vinter. Minimizers that are not also relaxed minimizers. SIAM Journal on Control and Optimization 52(4), 2164–2179 (2014)

work page 2014

[17] [17]

Augier, M

N. Augier, M. Korda, and R. R´ ıos-Zertuche. Young measure relaxation gaps for con- trollable systems with smooth state constraints. Journal of Differential Equations 459, 114036 (2026)

work page 2026

[18] [18]

S. Marx, T. Weisser, D. Henrion, and J.B. Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control and Related Fields 10(1), 113–140 (2020)

work page 2020

[19] [19]

Cardo¨ en, S

C. Cardo¨ en, S. Marx, A. Nouy, and N. Seguin. A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws. Numerische Mathematik 156(4), 1289–1324 (2024)

work page 2024

[20] [20]

Hopf and A

K. Hopf and A. J¨ ungel. Convergence of a finite-volume scheme and dissipative measure- valued–strong stability for a hyperbolic–parabolic cross-diffusion system. Numerische Mathematik 157(3), 951–992 (2025)

work page 2025

[21] [21]

Bertsimas and C

D. Bertsimas and C. Caramanis. Bounds on linear PDEs via semidefinite optimization. Mathematical Programming 108(1), 135–158 (2006)

work page 2006

[22] [22]

Goulart and S.I

P.J. Goulart and S.I. Chernyshenko. Global stability analysis of fluid flows using sum- of-squares. Physica D 241, 692–704 (2012)

work page 2012

[23] [23]

Chernyshenko, P.J

S.I. Chernyshenko, P.J. Goulart, D. Huang, and A. Papachristodoulou. Polynomial sum of squares in fluid dynamics: A review with a look ahead. Philosophical Transactions of the Royal Society A 372 (2014)

work page 2014

[24] [24]

Valmorbida, M

G. Valmorbida, M. Ahmadi, and A. Papachristodoulou. Stability analysis for a class of partial differential equations via semidefinite programming. IEEE Transactions on Automatic Control 61, 1649–1654 (2016)

work page 2016

[25] [25]

Goluskin and G

D. Goluskin and G. Fantuzzi. Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming.Nonlinearity 32, 1705–1730 (2019)

work page 2019

[26] [26]

Fuentes, D

F. Fuentes, D. Goluskin, and S.I. Chernyshenko. Global stability of fluid flows despite transient growth of energy. Physical Review Letters 128, 204502 (2022)

work page 2022

[27] [27]

Magron and C

V. Magron and C. Prieur. Optimal control of linear PDEs using occupation measures and SDP relaxations. IMA Journal of Mathematical Control and Information 37(1), 159–174 (2020)

work page 2020

[28] [28]

Chhatoi, D

S.P. Chhatoi, D. Henrion, S. Marx, and N. Seguin. Optimizing quasi-dissipative evo- lution equations with the moment-SOS hierarchy. Discrete and Continuous Dynamical Systems 45(11), 4139–4159 (2025)

work page 2025

[29] [29]

Fjordholm, K

U.S. Fjordholm, K. Lye, S. Mishra, and F. Weber. Statistical solutions of hyperbolic sys- tems of conservation laws: numerical approximation. Mathematical Models and Methods in Applied Sciences 30(3), 539–609 (2020). 32

work page 2020

[30] [30]

Henrion, M

D. Henrion, M. Infusino, S. Kuhlmann, and V. Vinnikov. Infinite-dimensional moment- SOS hierarchy for nonlinear partial differential equations. arXiv:2305.18768 (2023)

work page arXiv 2023

[31] [31]

Henrion and A

D. Henrion and A. Rudi. Solving moment and polynomial optimization problems on Sobolev spaces. SIAM Journal on Optimization 35(2), 989–1003 (2025)

work page 2025

[32] [32]

Adams and J.J.F

R.A. Adams and J.J.F. Fournier. Sobolev Spaces, 2nd edn. Pure and Applied Mathemat- ics, vol. 140. Academic Press, Amsterdam Boston (2003)

work page 2003

[33] [33]

Quittner and P

P. Quittner and P. Souplet. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States . Birkha¨ user Advanced Texts. Birkh¨ auser, Basel (2007)

work page 2007

[34] [34]

L.C. Evans. Partial Differential Equations . AMS, Providence, R.I. (2010)

work page 2010

[35] [35]

A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems . Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkh¨ auser Basel, Basel (1995)

work page 1995

[36] [36]

R.J. DiPerna. Measure-valued solutions to conservation laws. Archive for Rational Me- chanics and Analysis 88(3), 223–270 (1985)

work page 1985

[37] [37]

Attouch, G

H. Attouch, G. Buttazzo, and G. Michaille. Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization . MPS-SIAM Series on Optimization . SIAM, Philadelphia (2006)

work page 2006

[38] [38]

Kruˇ z´ ık and T

M. Kruˇ z´ ık and T. Roub´ ıˇ cek. Explicit characterization of Lp-Young measures.Journal of Mathematical Analysis and Applications 198(3), 830–843 (1996)

work page 1996

[39] [39]

Schonbek

M.E. Schonbek. Convergence of solution to nonlinear dispersive equations. Communi- cations in Partial Differential Equations 7(8), 959–1000 (1982)

work page 1982

[40] [40]

Gallou¨ et and R

T. Gallou¨ et and R. Herbin. Weak Solutions of Partial Differential Equations . Math´ ematiques et Applications, vol. 90. Springer, Cham (2025)

work page 2025

[41] [41]

Arendt, I

W. Arendt, I. Chalendar, and R. Eymard. Lions’ representation theorem and applica- tions. Journal of Mathematical Analysis and Applications 522(2), 126946 (2023)

work page 2023

[42] [42]

Lebarb´ e, E

C. Lebarb´ e, E. Flayac, M. Fourni´ e, D. Henrion, and M. Korda. Optimal control of 1D semilinear heat equations with moment-SOS relaxations. In: 2025 European Control Conference (ECC), pp. 731–736 (2025)

work page 2025

[43] [43]

Hyt¨ onen, J

T. Hyt¨ onen, J. Neerven, M. Veraar, and L. Weis. Analysis in Banach Spaces. Volume I: Martingales and Littlewood-Paley Theory . Springer, Cham (2016)

work page 2016

[44] [44]

R. Temam. Navier–Stokes Equations: Theory and Numerical Analysis . Studies in Math- ematics and Its Applications , vol. 2. North-Holland Publishing Company, Amsterdam (1977). 33

work page 1977