A Determining Form for the 2D Rayleigh-B\'enard Problem

Edriss S. Titi; Michael S. Jolly; Yu Cao

arxiv: 1907.00387 · v1 · pith:VDA77OFJnew · submitted 2019-06-30 · 🧮 math.AP

A Determining Form for the 2D Rayleigh-B\'enard Problem

Yu Cao , Michael S. Jolly , Edriss S. Titi This is my paper

Pith reviewed 2026-05-25 12:31 UTC · model grok-4.3

classification 🧮 math.AP

keywords determining formRayleigh-Bénard convectionglobal attractorvelocity trajectories2D fluid dynamicsODE reductionno-slip boundary conditionsstress-free boundary conditions

0 comments

The pith

An ODE on velocity trajectories alone captures the long-time dynamics of the 2D Rayleigh-Bénard system and determines the temperature field.

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

The paper constructs a determining form for the two-dimensional Rayleigh-Bénard convection equations in a strip with solid horizontal boundaries. This form is an ordinary differential equation in a Banach space whose solutions are trajectories of the velocity field. The steady states of the ODE correspond to the long-time behavior of the full system. The velocity trajectories determine the temperature trajectories, so the entire dynamics can be recovered from velocity alone. Solutions on the global attractor can be identified by checking zeros of a scalar equation that the ODE reduces to for each initial trajectory.

Core claim

We construct a determining form for the 2D Rayleigh-Bénard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.

What carries the argument

The determining form: an ODE in a Banach space of velocity trajectories whose steady states encode the long-time dynamics of the Rayleigh-Bénard system and determine the associated temperature evolution.

If this is right

The long-time dynamics of the full RB system, including temperature, reduce to an ODE whose solutions are velocity trajectories.
Steady states of this ODE correspond exactly to the global attractor of the RB system.
For each initial trajectory the ODE reduces to a scalar equation whose zeros single out solutions on the attractor.
The construction holds for both no-slip and stress-free boundary conditions in the strip geometry.

Where Pith is reading between the lines

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

If the velocity-to-temperature determination holds, computations could evolve only the velocity field and recover the full convection state.
The same reduction technique may apply to other coupled systems in which one variable determines the evolution of another.
Checking the scalar equation for specific initial data offers a direct test of attractor membership without integrating the full PDE system.
The approach suggests exploring whether similar velocity-only determining forms exist for three-dimensional or other parameter regimes of convection.

Load-bearing premise

Velocity field trajectories alone are sufficient to determine the corresponding temperature trajectories for the given boundary conditions.

What would settle it

Exhibiting two distinct temperature evolutions that arise from the same velocity trajectory under the Rayleigh-Bénard equations would falsify the determining property.

read the original abstract

We construct a determining form for the 2D Rayleigh-B\'enard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.

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

They build a determining form for 2D Rayleigh-Bénard that lives only on velocity trajectories, with temperature recovered from them.

read the letter

The paper constructs a determining form for the 2D Rayleigh-Bénard system in a strip, for both no-slip and stress-free boundaries. The ODE is written in a space of velocity trajectories, and the claim is that these trajectories determine the temperature trajectories on the global attractor. Steady states of the ODE recover the long-time dynamics, and solutions on the attractor satisfy a scalar equation obtained by reduction. This is the main new step beyond earlier determining-form work on Navier-Stokes: closing the system without keeping temperature as an independent variable in the trajectory space. The construction itself looks direct and follows the pattern from the NSE case, with the velocity-to-temperature map used to eliminate the temperature equation. That map is the part that needs to hold for the reduction to work. The abstract states it as a fact for the given boundary conditions, so the paper presumably proves uniqueness for the temperature equation when velocity is fixed along a trajectory on the attractor. If the estimates close without extra restrictions on the Rayleigh number or domain aspect ratio, the result stands. The soft spot is exactly there: the bidirectional coupling means the uniqueness step is not automatic, and any gap in the fixed-point or energy argument for temperature would break the velocity-only reduction. No circularity appears in the setup. The work is aimed at people who already follow determining forms and global attractors for dissipative PDEs. It is a straightforward but useful extension, and the proofs are the part worth checking. I would send it to a referee who knows the prior determining-form papers; the central claim is specific enough to be worth a careful read.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a determining form for the 2D Rayleigh-Bénard system in a strip with solid horizontal boundaries, for both no-slip and stress-free cases. The determining form is an ODE in a Banach space of velocity trajectories whose steady states comprise the long-time dynamics of the full RB system. Solutions on the global attractor are further identified as zeros of a scalar equation obtained by reducing the ODE for each initial trajectory. The key feature is that velocity trajectories alone determine the corresponding temperature trajectories.

Significance. If the velocity-to-temperature determination property holds rigorously, the result supplies a new reduction of the bidirectionally coupled RB system to an ODE whose equilibria recover the global attractor, extending prior determining-form constructions to this setting. This could facilitate analysis of long-time behavior and attractor structure without directly evolving the temperature equation.

major comments (2)

[§3] §3 (Determination property): The uniqueness argument that a velocity trajectory determines a unique temperature trajectory (used to close the determining form) relies on an energy estimate for the temperature equation with fixed velocity. For the stress-free case the boundary integrals arising from integration by parts on the advection term are not shown to vanish under the stated regularity of the velocity trajectory space; this step is load-bearing for the reduction to a velocity-only ODE.
[Theorem 4.1] Theorem 4.1 (Main existence result): The proof that steady states of the determining form coincide with the RB global attractor invokes the velocity-to-temperature map to project the buoyancy term, but the a-priori estimates closing the fixed-point argument for the temperature do not explicitly control the coupling back into the momentum equation when the velocity is taken only from the trajectory space; a gap here would invalidate the claim that the ODE equilibria recover the full attractor.

minor comments (2)

[p. 5] Notation for the trajectory space X (p. 5) is introduced without an explicit norm; adding the precise definition would clarify subsequent estimates.
[§5] The scalar equation whose zeros identify attractor solutions is stated in §5 but its derivation from the ODE is only sketched; a short appendix deriving the reduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments. Below we respond point-by-point to the major remarks. We agree that additional explicit verification is needed in both places identified and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (Determination property): The uniqueness argument that a velocity trajectory determines a unique temperature trajectory (used to close the determining form) relies on an energy estimate for the temperature equation with fixed velocity. For the stress-free case the boundary integrals arising from integration by parts on the advection term are not shown to vanish under the stated regularity of the velocity trajectory space; this step is load-bearing for the reduction to a velocity-only ODE.

Authors: We agree that the vanishing of the boundary integrals must be verified explicitly for the stress-free case. The trajectory space is equipped with sufficient regularity (H^1 in space, continuous in time) and the stress-free conditions imply that the normal velocity and the relevant tangential derivatives vanish on the horizontal boundaries. Consequently the boundary terms arising from integration by parts of the advection term are identically zero. We will insert a short lemma in §3 that records this calculation under the precise regularity stated for the velocity trajectories, thereby closing the uniqueness argument rigorously. revision: yes
Referee: [Theorem 4.1] Theorem 4.1 (Main existence result): The proof that steady states of the determining form coincide with the RB global attractor invokes the velocity-to-temperature map to project the buoyancy term, but the a-priori estimates closing the fixed-point argument for the temperature do not explicitly control the coupling back into the momentum equation when the velocity is taken only from the trajectory space; a gap here would invalidate the claim that the ODE equilibria recover the full attractor.

Authors: The observation is correct: the current write-up of the fixed-point argument in the proof of Theorem 4.1 does not spell out the a-priori control of the buoyancy term’s feedback into the momentum equation when the velocity is taken from the trajectory space. The trajectory space is defined so that each element satisfies the momentum equation for some temperature; the determining map then supplies the unique temperature that is consistent with that velocity. To make the closure explicit we will augment the estimates in the proof by deriving uniform bounds on the buoyancy forcing that remain controlled by the determining-form norm, thereby confirming that the fixed point lies on the global attractor of the full system. This revision will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct construction

full rationale

The paper constructs a determining form as an ODE on velocity trajectories whose steady states recover the RB attractor dynamics, with temperature trajectories claimed to be uniquely determined by velocity on the attractor. This is presented as a mathematical construction from the coupled PDE system and boundary conditions rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. No quoted step reduces the claimed result to its inputs by construction. The uniqueness property for temperature is a separate analytic claim (energy estimates or fixed-point), not a tautology. This matches the default case of a self-contained construction in PDE dynamical systems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the standard existence of a global attractor for the RB system under the stated boundary conditions; no free parameters or new invented entities are introduced.

axioms (1)

domain assumption The 2D Rayleigh-Bénard system with the given boundary conditions possesses a global attractor whose long-time dynamics can be captured by a determining form.
The determining form is defined to have steady states that comprise exactly those long-time dynamics.

pith-pipeline@v0.9.0 · 5632 in / 1340 out tokens · 28348 ms · 2026-05-25T12:31:47.715397+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 determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system... trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

dv/ds = −∥v − IhW(v)∥²_X (v − Ihu*)

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

28 extracted references · 28 canonical work pages

[1]

Azouani, E

A. Azouani, E. Olson, and E. S. Titi. Continuous data assi milation using general interpolant observables. J. Nonlinear Sci. , 24(2):277–304, 2014

work page 2014
[2]

Azouani and E

A. Azouani and E. S. Titi. Feedback control of nonlinear d issipative systems by ﬁnite determining parameters—a reaction-diﬀusion paradigm. Evol. Equ. Control Theory , 3(4):579–594, 2014

work page 2014
[3]

Bai and M

L. Bai and M. Yang. A determining form for a nonlocal syste m. Adv. Nonlinear Stud. , 17(4):705–713, 2017

work page 2017
[4]

Biswas, C

A. Biswas, C. Foias, C. F. Mondaini, and E. S. Titi. Downsc aling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 36(2):295–326, 2019

work page 2019
[5]

Brézis and T

H. Brézis and T. Gallouet. Nonlinear Schrödinger evolut ion equations. Nonlinear Anal., 4(4):677–681, 1980

work page 1980
[6]

Y. Cao. Determining form and data assimilation algorithm for the 2D Rayleigh-Bénard problem . PhD thesis, Indiana University, 2019

work page 2019
[7]

Y. Cao, M. S. Jolly, E. S. Titi, and J. P. Whitehead. Algebr aic bounds on the Rayleigh-Bénard attractor. arXiv:1905.01399 [math.AP], 2019

work page arXiv 1905
[8]

Celik, E

E. Celik, E. Olson, and E. S. Titi. Spectral Filtering of I nterpolant Observables for a Discrete-in-Time Down- scaling Data Assimilation Algorithm. SIAM J. Appl. Dyn. Syst. , 18(2):1118–1142, 2019

work page 2019
[9]

Cockburn, D

B. Cockburn, D. A. Jones, and E. S. Titi. Determining degr ees of freedom for nonlinear dissipative equations. C. R. Acad. Sci. Paris Sér. I Math. , 321(5):563–568, 1995

work page 1995
[10]

Constantin and C

P. Constantin and C. Foias. Navier-Stokes equations . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988

work page 1988
[11]

Constantin, C

P. Constantin, C. Foias, B. Nicolaenko, and R. Temam. Integral manifolds and inertial manifolds for dissipative partial diﬀerential equations , volume 70. Springer Science & Business Media, 2012

work page 2012
[12]

Farhat, M

A. Farhat, M. S. Jolly, and E. S. Titi. Continuous data as similation for the 2D Bénard convection through velocity measurements alone. Phys. D , 303:59–66, 2015

work page 2015
[13]

Farhat, E

A. Farhat, E. Lunasin, and E. S. Titi. Continuous data as similation for a 2D Bénard convection system through horizontal velocity measurements alone. J. Nonlinear Sci. , 27(3):1065–1087, 2017

work page 2017
[14]

Foiaş and G

C. Foiaş and G. Prodi. Sur le comportement global des sol utions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova , 39:1–34, 1967

work page 1967
[15]

Foias, M

C. Foias, M. S. Jolly, R. Kravchenko, and E. S. Titi. A det ermining form for the two-dimensional Navier-Stokes equations: the Fourier modes case. J. Math. Phys. , 53(11):115623, 30, 2012. 32 YU CAO 1, MICHAEL S. JOLLY 1,† , AND EDRISS S. TITI 2

work page 2012
[16]

Foias, M

C. Foias, M. S. Jolly, R. Kravchenko, and E. S. Titi. A uni ﬁed approach to determining forms for the 2D Navier-Stokes equations—the general interpolants case. Russian Mathematical Surveys , 69(2):359, 2014

work page 2014
[17]

Foias, M

C. Foias, M. S. Jolly, D. Lithio, and E. S. Titi. One-dime nsional parametric determining form for the two- dimensional Navier-Stokes equations. J. Nonlinear Sci. , 27(5):1513–1529, 2017

work page 2017
[18]

Foias, O

C. Foias, O. Manley, R. Rosa, and R. Temam. Navier-Stokes equations and turbulence , volume 83 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2001

work page 2001
[19]

Foias, O

C. Foias, O. Manley, and R. Temam. Attractors for the Bén ard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal., 11(8):939–967, 1987

work page 1987
[20]

Foias, B

C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam. Inerti al manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. J. Math. Pures Appl. , 67(3):197–226, 1988

work page 1988
[21]

Foias, G

C. Foias, G. R. Sell, and R. Temam. Inertial manifolds fo r nonlinear evolutionary equations. J. Diﬀerential Equations, 73(2):309–353, 1988

work page 1988
[22]

M. S. Jolly, V. R. Martinez, T. Sadigov, and E. S. Titi. A d etermining form for the subcritical surface quasi- geostrophic equation. J Dyn Diﬀ Equat , 2018. https://doi.org/10.1007/s10884-018-9652-4

work page doi:10.1007/s10884-018-9652-4 2018
[23]

M. S. Jolly, T. Sadigov, and E. S. Titi. A determining for m for the damped driven nonlinear Schrödinger equation—Fourier modes case. J. Diﬀerential Equations , 258(8):2711–2744, 2015

work page 2015
[24]

M. S. Jolly, T. Sadigov, and E. S. Titi. Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case. Nonlinear Anal. Real World Appl. , 36:287–317, 2017

work page 2017
[25]

D. A. Jones and E. S. Titi. Upper bounds on the number of de termining modes, nodes, and volume elements for the Navier-Stokes equations. Indiana Univ. Math. J. , 42(3):875–887, 1993

work page 1993
[26]

Mallet-Paret and G

J. Mallet-Paret and G. R. Sell. Inertial manifolds for r eaction diﬀusion equations in higher space dimensions. J. Amer. Math. Soc. , 1(4):805–866, 1988

work page 1988
[27]

R. Temam. Inﬁnite-dimensional dynamical systems in mechanics and ph ysics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997

work page 1997
[28]

E. S. Titi. On a criterion for locating stable stationar y solutions to the Navier-Stokes equations. Nonlinear Anal., 11(9):1085–1102, 1987. † corresponding author 1Depar tment of Ma thema tics, Indiana University, Bloomington, IN 47405 2Depar tment of Ma thema tics, Texas A&M University, 3368 TAMU , College Sta tion, TX 77843- 3368, USA. Depar tment of C...

work page 1987

[1] [1]

Azouani, E

A. Azouani, E. Olson, and E. S. Titi. Continuous data assi milation using general interpolant observables. J. Nonlinear Sci. , 24(2):277–304, 2014

work page 2014

[2] [2]

Azouani and E

A. Azouani and E. S. Titi. Feedback control of nonlinear d issipative systems by ﬁnite determining parameters—a reaction-diﬀusion paradigm. Evol. Equ. Control Theory , 3(4):579–594, 2014

work page 2014

[3] [3]

Bai and M

L. Bai and M. Yang. A determining form for a nonlocal syste m. Adv. Nonlinear Stud. , 17(4):705–713, 2017

work page 2017

[4] [4]

Biswas, C

A. Biswas, C. Foias, C. F. Mondaini, and E. S. Titi. Downsc aling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 36(2):295–326, 2019

work page 2019

[5] [5]

Brézis and T

H. Brézis and T. Gallouet. Nonlinear Schrödinger evolut ion equations. Nonlinear Anal., 4(4):677–681, 1980

work page 1980

[6] [6]

Y. Cao. Determining form and data assimilation algorithm for the 2D Rayleigh-Bénard problem . PhD thesis, Indiana University, 2019

work page 2019

[7] [7]

Y. Cao, M. S. Jolly, E. S. Titi, and J. P. Whitehead. Algebr aic bounds on the Rayleigh-Bénard attractor. arXiv:1905.01399 [math.AP], 2019

work page arXiv 1905

[8] [8]

Celik, E

E. Celik, E. Olson, and E. S. Titi. Spectral Filtering of I nterpolant Observables for a Discrete-in-Time Down- scaling Data Assimilation Algorithm. SIAM J. Appl. Dyn. Syst. , 18(2):1118–1142, 2019

work page 2019

[9] [9]

Cockburn, D

B. Cockburn, D. A. Jones, and E. S. Titi. Determining degr ees of freedom for nonlinear dissipative equations. C. R. Acad. Sci. Paris Sér. I Math. , 321(5):563–568, 1995

work page 1995

[10] [10]

Constantin and C

P. Constantin and C. Foias. Navier-Stokes equations . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988

work page 1988

[11] [11]

Constantin, C

P. Constantin, C. Foias, B. Nicolaenko, and R. Temam. Integral manifolds and inertial manifolds for dissipative partial diﬀerential equations , volume 70. Springer Science & Business Media, 2012

work page 2012

[12] [12]

Farhat, M

A. Farhat, M. S. Jolly, and E. S. Titi. Continuous data as similation for the 2D Bénard convection through velocity measurements alone. Phys. D , 303:59–66, 2015

work page 2015

[13] [13]

Farhat, E

A. Farhat, E. Lunasin, and E. S. Titi. Continuous data as similation for a 2D Bénard convection system through horizontal velocity measurements alone. J. Nonlinear Sci. , 27(3):1065–1087, 2017

work page 2017

[14] [14]

Foiaş and G

C. Foiaş and G. Prodi. Sur le comportement global des sol utions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova , 39:1–34, 1967

work page 1967

[15] [15]

Foias, M

C. Foias, M. S. Jolly, R. Kravchenko, and E. S. Titi. A det ermining form for the two-dimensional Navier-Stokes equations: the Fourier modes case. J. Math. Phys. , 53(11):115623, 30, 2012. 32 YU CAO 1, MICHAEL S. JOLLY 1,† , AND EDRISS S. TITI 2

work page 2012

[16] [16]

Foias, M

C. Foias, M. S. Jolly, R. Kravchenko, and E. S. Titi. A uni ﬁed approach to determining forms for the 2D Navier-Stokes equations—the general interpolants case. Russian Mathematical Surveys , 69(2):359, 2014

work page 2014

[17] [17]

Foias, M

C. Foias, M. S. Jolly, D. Lithio, and E. S. Titi. One-dime nsional parametric determining form for the two- dimensional Navier-Stokes equations. J. Nonlinear Sci. , 27(5):1513–1529, 2017

work page 2017

[18] [18]

Foias, O

C. Foias, O. Manley, R. Rosa, and R. Temam. Navier-Stokes equations and turbulence , volume 83 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2001

work page 2001

[19] [19]

Foias, O

C. Foias, O. Manley, and R. Temam. Attractors for the Bén ard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal., 11(8):939–967, 1987

work page 1987

[20] [20]

Foias, B

C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam. Inerti al manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. J. Math. Pures Appl. , 67(3):197–226, 1988

work page 1988

[21] [21]

Foias, G

C. Foias, G. R. Sell, and R. Temam. Inertial manifolds fo r nonlinear evolutionary equations. J. Diﬀerential Equations, 73(2):309–353, 1988

work page 1988

[22] [22]

M. S. Jolly, V. R. Martinez, T. Sadigov, and E. S. Titi. A d etermining form for the subcritical surface quasi- geostrophic equation. J Dyn Diﬀ Equat , 2018. https://doi.org/10.1007/s10884-018-9652-4

work page doi:10.1007/s10884-018-9652-4 2018

[23] [23]

M. S. Jolly, T. Sadigov, and E. S. Titi. A determining for m for the damped driven nonlinear Schrödinger equation—Fourier modes case. J. Diﬀerential Equations , 258(8):2711–2744, 2015

work page 2015

[24] [24]

M. S. Jolly, T. Sadigov, and E. S. Titi. Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case. Nonlinear Anal. Real World Appl. , 36:287–317, 2017

work page 2017

[25] [25]

D. A. Jones and E. S. Titi. Upper bounds on the number of de termining modes, nodes, and volume elements for the Navier-Stokes equations. Indiana Univ. Math. J. , 42(3):875–887, 1993

work page 1993

[26] [26]

Mallet-Paret and G

J. Mallet-Paret and G. R. Sell. Inertial manifolds for r eaction diﬀusion equations in higher space dimensions. J. Amer. Math. Soc. , 1(4):805–866, 1988

work page 1988

[27] [27]

R. Temam. Inﬁnite-dimensional dynamical systems in mechanics and ph ysics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997

work page 1997

[28] [28]

E. S. Titi. On a criterion for locating stable stationar y solutions to the Navier-Stokes equations. Nonlinear Anal., 11(9):1085–1102, 1987. † corresponding author 1Depar tment of Ma thema tics, Indiana University, Bloomington, IN 47405 2Depar tment of Ma thema tics, Texas A&M University, 3368 TAMU , College Sta tion, TX 77843- 3368, USA. Depar tment of C...

work page 1987