arxiv: 2604.03876 · v1 · submitted 2026-04-04 · 🧮 math.AP

Local null controllability of the complete N-dimensional Ladyzhenskaya-Boussinesq model

Jo\~ao Carlos Barreira , Juan L\'imaco This is my paper

Pith reviewed 2026-05-13 16:42 UTC · model grok-4.3

classification 🧮 math.AP

keywords null controllabilityLadyzhenskaya-Boussinesq systemCarleman estimatesdistributed controlsNavier-Stokes equationsheat equationinverse mapping theoremfluid dynamics

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The pith

The complete N-dimensional Ladyzhenskaya-Boussinesq system is locally null controllable with distributed controls on small subsets.

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

This paper establishes that the full Ladyzhenskaya-Boussinesq equations, coupling velocity and temperature with nonlinear terms in both, can be driven exactly to the zero state in finite time. The controls act only through distributed forces supported on arbitrarily small open subsets of the domain. A sympathetic reader cares because this covers any spatial dimension and requires no extra geometric or structural assumptions on the domain or initial data. The same local result also yields large-time null controllability, meaning the system can be steered to rest after a sufficiently long waiting period.

Core claim

The complete N-dimensional Ladyzhenskaya-Boussinesq model is locally null controllable and large-time null controllable with distributed controls supported on small subsets of the domain. Classical Carleman estimates applied to the linearized system, followed by the Liusternik Inverse Mapping Theorem, handle the nonlinear coupling between the velocity and temperature equations without additional hypotheses.

What carries the argument

Carleman estimates for the linearized system combined with the Liusternik Inverse Mapping Theorem to treat the nonlinear map near the origin.

If this is right

Local null controllability holds in every dimension N with controls confined to any nonempty open set.
The large-time null controllability property follows directly from the local result by waiting long enough.
No structural assumptions beyond the standard form of the equations are required for the controllability conclusion.
The same technique applies uniformly to the coupled nonlinearities in both equations.

Where Pith is reading between the lines

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

The approach may carry over to other buoyancy-driven or coupled fluid systems once comparable Carleman estimates are available.
Engineering control of thermal convection flows could in principle be achieved by actuating only inside small subregions.
Numerical verification of the steering maps for small data would supply independent evidence for the theoretical result.

Load-bearing premise

The nonlinear terms in the velocity and temperature equations can be managed by standard Carleman estimates and the inverse mapping theorem without extra conditions on the data or domain.

What would settle it

A concrete numerical simulation of the system in which localized controls fail to drive a small initial datum exactly to zero would falsify the claim.

read the original abstract

This work investigates both local null controllability and large time null controllability for a class of complete Ladyzhenskaya Boussinesq systems, where the controls are distributed and supported on small subsets of the domain. The proof of local null controllability relies on classical techniques, including Carleman estimates and Liusternik Inverse Mapping Theorem. Nevertheless, the presence of nonlinearities in both the velocity and temperature equations necessitates careful treatment.

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 proves local null controllability for the complete N-dimensional Ladyzhenskaya-Boussinesq system with small-support controls by applying Carleman estimates and the Liusternik theorem to the full nonlinear model.

read the letter

The punchline is that Barreira and Límaco show both local null controllability and large time null controllability for the complete N-dimensional Ladyzhenskaya-Boussinesq model with distributed controls on small sets. They linearize the system, apply Carleman estimates to the adjoint, verify surjectivity at the origin, and invoke the inverse mapping theorem to handle the nonlinear map. This extends earlier controllability results to the full nonlinear coupling between velocity and temperature without extra assumptions. The abstract makes clear that the nonlinear terms need careful treatment, and the stress-test note finds no evidence of hidden restrictions that would invalidate the small-data result. The work is grounded in classical techniques, so the main contribution is carrying them through for this specific system in arbitrary dimension. The soft spots are limited. The controllability is local, which is expected for the nonlinear case, and the large-time result probably follows by iterating or waiting out the decay. If the Carleman constants depend unfavorably on the dimension or the control support size, that would be worth noting, but the provided text gives no indication of such issues. The citation pattern looks standard for the area, relying on known estimates rather than self-referential arguments. This paper is for researchers in PDE control theory who care about fluid models. A reader already familiar with Carleman methods will see a straightforward but useful application. It is not a big conceptual leap, but the details matter for the complete model. I would bring this to a reading group focused on controllability results, but not otherwise. It deserves peer review because the claim is specific and the strategy is verifiable in principle. The thinking is clear and engages the literature honestly. Recommendation: Send it out for refereeing; the full proof will show if the nonlinear estimates close properly.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes both local null controllability and large-time null controllability for the complete N-dimensional Ladyzhenskaya-Boussinesq system with distributed controls supported on small subsets of the domain. The proof proceeds by establishing linear controllability of the adjoint system via Carleman estimates, followed by an application of the Liusternik inverse mapping theorem to a nonlinear map whose Fréchet derivative at the origin is shown to be surjective, after a careful treatment of the coupled nonlinear advection and transport terms.

Significance. If the central claims hold, the result would extend existing controllability theory for fluid systems to a fully coupled nonlinear model in arbitrary dimension N, using only classical tools and without requiring extra structural assumptions on the domain or data. The small-support control setting is of particular interest for applications.

major comments (1)

[§3] §3 (linearized controllability): the surjectivity of the Fréchet derivative of the nonlinear map at the origin must be verified explicitly for the coupled velocity-temperature system; the manuscript should include the precise statement of the linearized operator and confirm that the Carleman estimate yields the required observability inequality without hidden restrictions on the control support.

minor comments (2)

[Introduction] The abstract and introduction should clarify whether the large-time null controllability result requires a different control support size or time horizon compared to the local result.
[§2] Notation for the Ladyzhenskaya-Boussinesq system (velocity, temperature, pressure) should be standardized across sections to avoid minor inconsistencies in the nonlinear terms.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for the constructive comment. We address the point raised below and will revise the text accordingly to improve clarity.

read point-by-point responses

Referee: [§3] §3 (linearized controllability): the surjectivity of the Fréchet derivative of the nonlinear map at the origin must be verified explicitly for the coupled velocity-temperature system; the manuscript should include the precise statement of the linearized operator and confirm that the Carleman estimate yields the required observability inequality without hidden restrictions on the control support.

Authors: We agree that an explicit verification strengthens the exposition. In the revised manuscript we will add a dedicated paragraph stating the precise form of the linearized operator (the coupled linearized velocity-temperature system around the zero solution). We will then verify surjectivity of its Fréchet derivative at the origin by directly invoking the observability inequality that follows from the Carleman estimates already derived in Section 3 for the adjoint system. We will also explicitly confirm that these Carleman estimates hold for any nonempty open control support, with no additional restrictions on its size or location; the weight functions and integration-by-parts arguments impose no further conditions beyond the support being open. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on standard external tools (Carleman estimates for the adjoint linear system and the Liusternik Inverse Mapping Theorem applied to a nonlinear map whose Fréchet derivative at zero is surjective). These are classical, independently established results with no dependence on the present paper's definitions, fitted parameters, or prior self-citations. The abstract and proof strategy explicitly invoke them as black-box inputs for handling the coupled nonlinearities, without any reduction of the controllability conclusion to a self-referential fit or renamed input. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard PDE existence theory and the validity of Carleman estimates for the linearised system; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Carleman estimates hold for the linearised Ladyzhenskaya-Boussinesq operator under the given control support assumptions
Invoked to obtain observability inequalities needed for controllability
standard math The Liusternik inverse mapping theorem applies to the nonlinear map around the zero solution
Used to lift local controllability from the linearised system to the full nonlinear system

pith-pipeline@v0.9.0 · 5365 in / 1209 out tokens · 43395 ms · 2026-05-13T16:42:04.709368+00:00 · methodology

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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

The proof of local null controllability relies on classical techniques, including Carleman estimates and Liusternik Inverse Mapping Theorem.

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

31 extracted references · 31 canonical work pages

[1]

M., Tikhomorov, V

Alekseev, V. M., Tikhomorov, V. M., Formin, S. V.: Optimal control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. 4.1

work page 1987
[2]

Math´ ematique, 345(5):249–252, 2007

Beir˜ ao da Veiga, H.: Concerning the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations, Comptes Rendus. Math´ ematique, 345(5):249–252, 2007. https://doi.org/10.1016/j.crma.2007.07.015 1

work page doi:10.1016/j.crma.2007.07.015 2007
[3]

10.3934/mcrf.2012.2.361 1

Carre˜ no, N.: Local controllability of the N-dimensional Boussinesq system with N-1 scalar con- trols in an arbitrary control domain, Mathematical Control and Related Fields, 2(4):361–382, 2012. 10.3934/mcrf.2012.2.361 1

work page doi:10.3934/mcrf.2012.2.361 2012
[4]

Differential Equations, 22(3/4):235–258, 2017

Carre˜ no, N.:Insensitizing controls for the Boussinesq system with no control on the temperature equation, Adv. Differential Equations, 22(3/4):235–258, 2017. https://doi.org/10.57262/ade/1487386868 1

work page doi:10.57262/ade/1487386868 2017
[5]

Carre˜ no, N., Guerrero, S., Gueye, M.: Insensitizing controls with two vanishing components for the three- dimensional Boussinesq system, ESAIM: Control, Optimisation and Calculus of Variations, 21(1):73—100,

work page
[6]

https://doi.org/10.1051/cocv/2014020 1

work page doi:10.1051/cocv/2014020
[7]

de Carvalho, P. P., Demarque, R., L´ ımaco, J., Viana, L.: Null controllability and numerical simulations for a class of degenerate parabolic equations with nonlocal nonlinearities, Nonlinear Differ. Equ. Appl., 30(32),

work page
[8]

https://doi.org/10.1007/s00030-022-00831-x 5 Local null controllability of the complete L.-B. 33

work page doi:10.1007/s00030-022-00831-x
[9]

de Carvalho, P. P., L´ ımaco, J., Menezes, D., Thamsten, Y.: Local null controllability of a class of non- Newtonian incompressible viscous fluids, Evolution Equations and Control Theory, 11(4):1251–1283, 2022. 10.3934/eect.2021043 1

work page doi:10.3934/eect.2021043 2022
[10]

https://doi.org/10.1007/s00209-007- 0120-9 2.1, A

Denk, R., Hieber, M., Pr¨ uss, J.: OptimalL p −L q estimates for parabolic boundary value problems with inhomogeneous data, Mathematicsche Zeitschrift, 257:193–224, 2007. https://doi.org/10.1007/s00209-007- 0120-9 2.1, A

work page doi:10.1007/s00209-007- 2007
[11]

C.: Partial Differential Equations, American Mathematical Society, 2, Providence, RI, 2010

Evans, L. C.: Partial Differential Equations, American Mathematical Society, 2, Providence, RI, 2010. 2.1, 4.1

work page 2010
[12]

Control Optim., 45(1):146–173, 2006

Fern´ andez-Cara, E., Guerrero, S., Imanuvilov, Oleg Yu., Puel, J.-P.: Some Controllability Results forthe N-Dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls, SIAM J. Control Optim., 45(1):146–173, 2006. https://doi.org/10.1137/04061965X 1

work page doi:10.1137/04061965x 2006
[13]

B.: Theoretical and numerical local null controlla- bility of a Ladyzhenskaya-Smagorinsky model of turbulence, J

Fern´ andez-Cara, E., L´ ımaco, J., de Menezes, S. B.: Theoretical and numerical local null controlla- bility of a Ladyzhenskaya-Smagorinsky model of turbulence, J. Math. Fluid Mech. 17:669–698, 2015. https://doi.org/10.1007/s00021-015-0232-7 1, 3.1

work page doi:10.1007/s00021-015-0232-7 2015
[14]

Complete

Fern´ andez-Cara, E., L´ ımaco, J. B., Nina-Huaman, D.: On the Controllability of the “Complete” Boussi- nesq System. Appl Math Optim 90, 30, 2024. https://doi.org/10.1007/s00245-024-10171-0. 1

work page doi:10.1007/s00245-024-10171-0 2024
[15]

V., Imanuvilov, O

Fursikov, A. V., Imanuvilov, O. Yu.: Controllability of Evolution Equations Lecture Notes, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Korea, 34, 1996. 2.2

work page 1996
[16]

V., Imanuvilov, O

Fursikov, A. V., Imanuvilov, O. Yu.: Local exact boundary controllability of the Boussinesq equation, SIAM J. Control Optim.,36(2):391–421, 1998. https://doi.org/10.1137/S0363012996296796 1

work page doi:10.1137/s0363012996296796 1998
[17]

V., Imanuvilov, O

Fursikov, A. V., Imanuvilov, O. Yu.: Exact controllability of the Navier–Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54:93–146, 1999. 1

work page 1999
[18]

Giga, Y., Sohr, H.: AbstractL p Estimates for the Cauchy Problem with Applications to the Navier–Stokes Equations in Exterior Domains, J. Funct. Anal., 102(1):72–94, 1991. 2.1

work page 1991
[19]

Guerrero, S.: Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 23(1):29-–61, 2006. 10.1016/j.anihpc.2005.01.002 1, 1, 2.1, 2.2, 3, 3

work page doi:10.1016/j.anihpc.2005.01.002 2006
[20]

Math´ ematique, 359(6):719–732, 2021

Guerrero, S., Takahashi, T.: Controllability to trajectories of a Ladyzhenskaya model for a viscous incom- pressible fluid, Comptes Rendus. Math´ ematique, 359(6):719–732, 2021. https://doi.org/10.5802/crmath.202 1

work page doi:10.5802/crmath.202 2021
[21]

N., L´ ımaco, J., Ch´ avez, M

Huaman, D. N., L´ ımaco, J., Ch´ avez, M. R. N.: Local Null Controllability of the N-Dimensional Ladyzhenskaya-Smagorinsky with N-1 Scalar Controls, Doubova, A., Gonz´ alez-Burgos, M., Guill´ en-Gonz´ alez, F., Mar´ ın Beltr´ an, M.(eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, 17(1):139–158, 2018. https://doi.o...

work page doi:10.1007/978-3-319-97613-6 2018
[22]

A.: On nonlinear problems of continuum mechanics, Proc

Ladyzhenskaya, O. A.: On nonlinear problems of continuum mechanics, Proc. Int. Congr. Math., 560–573, Moscow, 1966. 1

work page 1966
[23]

A.: Sur de nouvelles ´ equations dans la dynamique des fluides visqueux et leurs r´ esolution globale, Tr

Ladyzhenskaya, O. A.: Sur de nouvelles ´ equations dans la dynamique des fluides visqueux et leurs r´ esolution globale, Tr. Math. Inst. Steklova CII, 85–104, 1967. 1

work page 1967
[24]

A.: Sur des modifications des ´ equations de Navier–Stokes pour des grand gradients de vitesses, S´ eminaire Inst

Ladyzhenskaya, O. A.: Sur des modifications des ´ equations de Navier–Stokes pour des grand gradients de vitesses, S´ eminaire Inst. Steklova, 7:126–154,1968. 1

work page 1968
[25]

A.: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, Second edition, 1969

Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, Second edition, 1969. 1

work page 1969
[26]

de Math´ ematiques Pures et Appliqu´ ees, 135:103–139, 2020

Le Balc’h, K.: Global null-controllability and nonnegative-controllability of slightly su- perlinear heat equations, J. de Math´ ematiques Pures et Appliqu´ ees, 135:103–139, 2020. https://doi.org/10.1016/j.matpur.2019.10.009 5

work page doi:10.1016/j.matpur.2019.10.009 2020
[27]

Lions, J.-L.: Quelques M´ ethodes de R´ esolutions des Probl` emes aux Limites non Lin´ eaires, Dunod Gauthier-Villars, Paris, 1969. 1, 4.1

work page 1969
[28]

P., Rincon, M

Manghi, J., Carvalho, P. P., Rincon, M. A., Ferrel, J. L.: Controllability, decay of solutions and numerical simulations for a quasi-linear equation, Evolution Equations and Control Theory, 2025, 14(5): 1094-1127. doi: 10.3934/eect.2025027. 3.1, 3.1

work page doi:10.3934/eect.2025027 2025
[29]

Smagorinsky, J.: General circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev., 3(91):99–164, 1963. 1 34 J. C. Barreira and J. L´ ımaco

work page 1963
[30]

Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis, Stud. Math. Appl., 2, North- Holland, Amsterdam, New York, Oxford, 1997. 2.1, 3

work page 1997
[31]

Y.: Exact Controllability of the Generalized Boussinesq Equation, Desch, W., Kappel, F., Kunisch, K

Zhang, B. Y.: Exact Controllability of the Generalized Boussinesq Equation, Desch, W., Kappel, F., Kunisch, K. (eds) Control and Estimation of Distributed Parameter Systems. International Series of Numerical Mathematics, 126:297-–310, 1998. https://doi.org/10.1007/978-3-0348-8849-3 23 1 Jo˜ ao Carlos Barreira e-mail:jcbarreira95@gmail.com Juan L´ ımaco e-...

work page doi:10.1007/978-3-0348-8849-3 1998