Exponential decay for the 3D Boussinesq equations with Navier boundary conditions

Weinan Wang; Wen Feng; Xiang Xu

Exponential decay for the 3D Boussinesq equations with Navier boundary conditions

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that for the 3D Boussinesq equations with Navier boundary conditions, any global weak solution's total energy decays exponentially whenever the boundary friction is nonnegative and positive on a set of positive surface mea

desk verdict Solid, honest extension of the Navier-Stokes decay theory to the 3D Boussinesq system; the weighted Korn-Poincare inequality is a real new tool and the main theorem holds up. read the letter →

arxiv 2608.00972 v1 pith:45XHRTFY submitted 2026-08-02 math.AP

Wen Feng , Weinan Wang , Xiang Xu This is my paper · ORCID

classification math.AP MSC 35Q3035Q3535B4076D0576R50

keywords BoussinesqequationsNavierboundaryconditionsexponentialdecayweaksolutionsLeray–HopfenergyinequalityKorn–PoincaréGalerkinapproximationslipfriction

topics Navier-Stokes Regularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the three-dimensional Boussinesq equations—the coupled system for a viscous fluid and a diffusing scalar such as temperature—in a bounded container whose boundary allows slip with friction. The authors construct global weak solutions for any square-integrable initial data and prove that the total energy decays exponentially whenever the friction coefficient is nonnegative and positive on some boundary patch of positive surface area. The key mechanism is a weighted Korn–Poincaré inequality that controls the velocity's L2 norm by its symmetric-gradient dissipation plus the frictional boundary dissipation, together with a two-time Gronwall inequality that handles the buoyancy forcing as an exponentially decaying term. In the frictionless case, the scalar and the non-rigid part of the velocity decay, and if there are no tangential rigid motions, the whole velocity decays. A corollary gives exponential decay for the Navier–Stokes equations with slip friction on solids of revolution, settling a case left open in [10].

What carries the argument

The central objects are (i) the weighted Korn–Poincaré inequality (Proposition 4.3), which asserts that ‖u‖² ≤ C(‖S(u)‖² + ∫∂Ω α|u|² dS) whenever α≥0 has positive support on the boundary, so frictional dissipation controls even rigid motions; and (ii) a two-time Gronwall-type inequality (Proposition A.1) that turns the scalar's exponential decay into a forcing term for the velocity without requiring a sign condition on the energy. The Leray–Hopf energy inequalities (Lemma 4.1), obtained by a one-sided time-mollification passage from the Galerkin approximation, feed both.

What would settle it

Take a concrete domain (e.g., unit ball) and α = 1 on an open cap, 0 elsewhere; compute the Galerkin sequence's boundary traces and check whether they converge strongly in L²(∂Ω). If a subsequence loses trace compactness, the energy inequality (4.2) might fail and exponential decay would not follow from this proof; alternatively, if a numerical simulation of the full Boussinesq system with such localized friction shows energy persisting indefinitely, the theorem's conclusion is false.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 5.1: for any global Leray–Hopf weak solution of the 3D Boussinesq system with Navier boundary conditions, if the friction coefficient α ∈ L∞(∂Ω) is nonnegative and positive on a boundary subset of positive surface measure, then the total L2 energy decays exponentially: ‖u(t)‖² + ‖ρ(t)‖² ≤ D(‖u0‖² + ‖ρ0‖²)$e^{{-Kt}}$ for all t≥0, with K,D depending only on ν, κ, Ω, and α. The proof does not require the rigid-motion kernel to be trivial nor any geometric restriction on the domain; the weighted boundary term in the Korn–Poincaré inequality accounts for the kernel component. When α≡0, the scalar ρ and the velocity component orthogonal to the rigid-motion kernel K

Load-bearing premise

The whole decay proof starts from an energy inequality for the limiting weak solution that is obtained by a mollification-and-limit argument tailored to continuous friction; extending it to essentially bounded friction requires the Galerkin approximations' boundary traces to converge strongly, a step that is only sketched.

Editorial extensions

If this is right

Every Leray–Hopf weak solution of the 3D Boussinesq system with localized positive wall friction decays to zero in L²; there is no need for small initial data or for the friction to be positive everywhere.
The decay rate is explicit in terms of the weighted Korn–Poincaré constant and the Poincaré constant: K = min{2ν/C(KP,α) − ε, 2κ/CP}, so the result gives quantitative rates, not just qualitative stability.
For the Navier–Stokes equations (ρ0≡0), any nonnegative L∞ friction that is positive on a boundary set of positive surface measure yields exponential decay of the velocity, including on solids of revolution—resolving the open case in [10].
In the frictionless case, the buoyancy force breaks conservation of the rigid-motion projection, but the non-rigid part decays; if the domain admits no tangential rigid motions, the full velocity decays.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The weighted Korn–Poincaré inequality is likely to transfer to other slip-boundary problems where damping acts only on part of the boundary, since it shows that a control set of positive surface measure is enough to tame the rigid-motion kernel.
The two-time Gronwall inequality without a sign condition may have independent use in coupled parabolic systems where one component decays exponentially and drives another; the proof's shift trick avoids nonnegativity assumptions.
One testable extension: the same machinery should yield exponential decay for the partially dissipative Boussinesq models (e.g., zero diffusivity) only if the scalar equation itself provides decay; when κ=0 the scalar does not decay, so the present argument stops, suggesting a threshold condition on κ.
The result also suggests that the friction coefficient could be replaced by a measure supported on a positive-measure set, since only the support matters in the weighted Korn–Poincaré proof.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it over the weekend. The paper does what it says: it extends Kelliher et al.'s 3D Navier-Stokes decay results to the Boussinesq system, and it genuinely weakens the hypothesis from continuous positive friction everywhere to L∞ friction positive on a boundary patch. The weighted Korn-Poincare inequality (Prop. 4.3) is the right new tool, and the proof via compactness and rigid-motion classification is clean. The two-time Gronwall lemma is also solid and useful on its own, and the paper is upfront about borrowing the limiting energy-inequality argument from [10].

The main decay theorem is proved without hidden assumptions: the scalar field decays because rho=0 on the boundary, the buoyancy term becomes an exponentially decaying forcing, and the weighted Korn-Poincare inequality closes the velocity estimate. The frictionless section is a sensible counterpart: the kernel projection is tracked, the orthogonal component decays, and they recover the Navier-Stokes rigid-motion limit. The Navier-Stokes corollary for L∞ alpha with a positive patch does resolve the open case in [10].

Soft spots are minor. Lemma 4.1 does rely on the energy-inequality limiting passage from [10, Thm. 4.4]; for L∞ alpha, the one-sided mollifier and trace-limit argument is sketched rather than fully written out. I checked the trace convergence in (3.6), and it is enough—the sign-changing (Lambda-alpha) term is controlled by strong trace convergence—so I do not see a real gap, but a referee should ask for the details to be made explicit. Also, in the proof of Theorem 5.1, the displayed inequality appears to use an implicit doubling of (4.2); the algebra works if you multiply by 2 first, but as written it is easy to trip over. Both are presentation fixes.

The citation pattern is appropriate: [10] is the direct antecedent and is cited with precision; self-citations are to relevant prior work, not padding. No fitted parameters, no invented entities.

This is a paper for people working on long-time behavior of fluids with slip boundary conditions. It deserves a serious referee and likely publication after minor revision. I would bring it to reading group and cite it.

Referee Report

0 major / 5 minor

Summary. The paper studies the 3D incompressible Boussinesq system on a bounded smooth domain with Navier slip boundary conditions and an L∞ nonnegative friction coefficient α. It constructs global weak solutions by a Galerkin approximation and proves Leray–Hopf energy inequalities for the constructed solution (Lemma 4.1). The main decay result, Theorem 5.1, states that if α ≥ 0 and H²({α>0}) > 0 then the total L² energy of every global Leray–Hopf weak solution decays exponentially, with constants depending only on ν, κ, Ω, α. The proof combines a new weighted Korn–Poincaré inequality (Prop. 4.3), which controls the full velocity whenever the friction is positive on a set of positive boundary measure, with a new two-time Gronwall inequality (Prop. A.1) that does not require a sign assumption. In the frictionless case (Theorem 6.1) the scalar field and the velocity component orthogonal to the rigid-motion kernel Ker S decay exponentially, and the kernel component converges exponentially to a (generally nonzero) rigid motion; when Ker S is trivial this gives full energy decay. Corollary 5.2 gives the corresponding Navier–Stokes decay result for nonnegative L∞ friction, including the case left open by [10].

Significance. If correct, the paper is a meaningful extension of [10]: it replaces continuity of α by L∞, replaces positivity everywhere by positivity on a positive-measure boundary subset, and passes from Navier–Stokes to the full Boussinesq system. The two auxiliary lemmas are independently useful: the weighted Korn–Poincaré inequality is a clean compactness/geometry statement, and the sign-free two-time Gronwall lemma removes an unnecessary hypothesis in the corresponding lemma of [10]. The Galerkin existence proof is essentially complete, and the crucial step of passing the energy inequality from the Galerkin sequence to the weak limit, including the boundary term for merely L∞ α, is supplied rather than merely quoted. The decay mechanism—the scalar acting as an exponentially decaying forcing on the velocity—is natural and the argument is internally consistent. I consider the central claims sound.

minor comments (5)

[Lemma 4.1, Eq. (4.8)] The passage from the mollified Galerkin inequality to the limit is compressed. The strong trace convergence of the boundary term, proved in (3.6), is used but not restated; a sentence making explicit the convergence of ∫∫(Λ−α)|u_m|² and ∫∫α|u_m|² would improve readability.
[Theorem 5.1, Eq. (5.5)] The displayed inequality is obtained after dropping the nonnegative term 2ν∫∥S(u)∥² from a combined estimate. This should be indicated, as otherwise the coefficient (2ν/C−ε) appears to follow directly from (4.2) without an extra step.
[Proposition A.1, around (A.4)] The calculation leading to (A.4) is terse. Expanding the algebra for z_s would help, especially because the nonnegativity of y is deliberately not assumed.
[Section 2, Proposition 2.2] The notation for the shifted operator, typeset as 'eBβ', is unusual and not defined clearly; please use a standard symbol such as \tilde B_β.
[Throughout] There are several places where inline derivations are slightly too compressed (e.g., the shape-operator bound following (4.4) and the trace interpolation estimates in (3.6)). Expanding these by one or two lines would make the paper easier to check. Also, the running header on page 1 contains an accidental line break in 'DECAY'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the decay theorems follow from independently proven energy inequalities, a weighted Korn–Poincaré inequality proved by compactness, and a two-time Gronwall lemma proved from scratch.

full rationale

The central claim (Theorem 5.1) rests on three ingredients, none of which assumes the conclusion. Lemma 4.1 derives the Leray–Hopf energy inequalities by a liminf passage from Galerkin approximations; it borrows the one-sided mollifier technique from the external reference [10, Theorem 4.4], but [10] shares no authors with the present paper, and the extension to L^∞ friction is supplied here via the strong boundary-trace convergence (3.6), which handles the sign-indefinite boundary term, while dissipation is lower semicontinuous and the buoyancy term converges strongly. Proposition 4.3 (weighted Korn–Poincaré) is proved by contradiction using compactness, trace interpolation, and the standard rigid-motion classification of Ker S cited to [10, Section 7]; the contradiction sequence forces a limit rigid motion vanishing on a positive-measure boundary set, which is impossible unless it is zero, so the inequality is an independent functional-analytic fact rather than a restatement of decay. Appendix A proves the two-time Gronwall inequality from scratch using the shifted variable z_s, with Corollary A.2 following as the q≡0 case. In Theorem 5.1, the scalar decay is obtained first from the scalar energy inequality and Poincaré, and then used only as an exponentially decaying forcing term in the velocity estimate; the velocity decay rate is controlled by the weighted Korn–Poincaré constant, not by any assumed decay. The frictionless Theorem 6.1 similarly uses the external kernel-orthogonal Korn–Poincaré inequality (4.16) from [10]. The only self-citations ([12],[14],[21],[22]) appear as literature context and are not load-bearing. No parameter is fitted to the quantity being predicted, and no derivation step reduces by definition to its input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. Its new content is a weighted Korn-Poincare inequality and a Gronwall-type lemma, both proved from standard functional analysis and the geometric setup for Navier slip inherited from [10].

assumptions (7)

standard math Standard Sobolev embedding, trace interpolation, Rellich-Kondrachov and Aubin-Lions compactness theorems
Used throughout Sections 3-4 for compactness, strong trace convergence and the Galerkin limit.
standard math Lax-Milgram and spectral theorem for compact self-adjoint operators provide a Hilbert basis of eigenfunctions for the shifted velocity form a_beta
Invoked in the Galerkin construction in Theorem 3.2.
standard math Poincare inequality on H^1_0(Omega) and on zero-mean H^1 functions
Used for the scalar decay estimate (5.1) and the H^1-norm equivalence on H^1_sigma_tan.
domain assumption Classification of the kernel of the symmetric gradient: KerS = {a + b x x : a,b in R^3}
Quoted from [10, Section 7]; used in Proposition 4.3 and Theorem 6.1 to rule out nontrivial limits vanishing on a positive-measure boundary set.
domain assumption Geometric boundary identity (Lemma 2.1): for fields satisfying the Navier condition, v dot (grad(u) n) = v dot [d_n(u) - alpha u] on the boundary
Quoted from [10, Lemma 2.1]; it is the bridge between the strong and weak forms of the Navier boundary condition used in the bilinear form.
standard math Viscous identity Proposition 2.3 (extension of [10, Proposition 4.3])
Proved in the paper for smooth fields via the cited identity and extended by density; used to derive the symmetric-gradient energy inequality (4.2).
standard math Zero set of a + b x x is empty or a line when b != 0
Linear algebra fact used in Proposition 4.3 to show a rigid motion vanishing on a positive-measure surface patch is zero.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Exponential decay for the 3D Boussinesq equations with Navier boundary conditions." pith.science (2026). https://pith.science/paper/45XHRTFY

@misc{pith2026260800972,
  author       = {Pith},
  title        = {Pith review of: Exponential decay for the 3D Boussinesq equations with Navier boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45XHRTFY}},
  note         = {Machine review of arXiv:2608.00972}
}

abstract

We study the three-dimensional incompressible Boussinesq equations on a bounded domain with smooth boundary and Navier boundary conditions. We construct global weak solutions by a Galerkin approximation and establish the associated Leray--Hopf energy inequalities. For nonnegative boundary friction, the total energy decays exponentially when the friction coefficient is positive on a boundary subset of positive surface measure. In the frictionless case, the scalar field and the velocity component orthogonal to the rigid-motion kernel decay exponentially; when the kernel is trivial, this is exponential decay of the total energy. When the scalar initial datum vanishes, this also proves exponential decay for the Navier--Stokes system on solids of revolution for every friction coefficient $\alpha\in L^\infty(\partial\Omega)$ such that $\alpha\ge0$ almost everywhere and $\alpha\not\equiv0$, resolving the corresponding case left open in \cite{Kelliher2025}. The proof uses a weighted Korn--Poincar\'e inequality and a two-time Gronwall-type inequality with an exponentially decaying forcing term.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[10]

Kelliher, Christophe Lacave, Milton C

James P. Kelliher, Christophe Lacave, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, and Edriss S. Titi. Large time behavior for the 3D Navier–Stokes with Navier boundary conditions.Journal of Math- ematical Fluid Mechanics, 27(3):48, 2025. Article 48, 23 pp

work page 2025
[1]

Large-time behavior of the 2D thermally non- diffusive Boussinesq equations with Navier-slip boundary conditions.Zeitschrift f¨ ur Angewandte Math- ematik und Physik, 76:58, 2025

Fabian Bleitner, Elizabeth Carlson, and Camilla Nobili. Large-time behavior of the 2D thermally non- diffusive Boussinesq equations with Navier-slip boundary conditions.Zeitschrift f¨ ur Angewandte Math- ematik und Physik, 76:58, 2025. Article 58

work page 2025
[2]

Stability and enhanced decay rate for 3D anisotropic Boussinesq equations near the hydrostatic balance.Journal of Differential Equations, 425:300–341, 2025

Jianfeng Cheng, Ruihong Ji, Ling Tian, and Jiahong Wu. Stability and enhanced decay rate for 3D anisotropic Boussinesq equations near the hydrostatic balance.Journal of Differential Equations, 425:300–341, 2025

work page 2025
[3]

Global existence and Rayleigh-Taylor instability for the semi-dissipative Boussinesq system with Naiver boundary conditions

Huafei Di, Liang Li, Xiaoming Peng, and Quan Wang. Global existence and Rayleigh–Taylor insta- bility for the semi-dissipative Boussinesq system with Navier boundary conditions. arXiv preprint arXiv:2405.16074, 2024

work page Pith review arXiv 2024
[4]

The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term inR 3.Nonlinear Analysis, 255:113775, 2025

Lihua Dong. The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term inR 3.Nonlinear Analysis, 255:113775, 2025. Article 113775, 17 pp

work page 2025
[5]

On asymptotic stability of the 3D Boussinesq equations without heat conduction.Science China Mathematics, 67(2):253–280, 2024

Lihua Dong and Yongzhong Sun. On asymptotic stability of the 3D Boussinesq equations without heat conduction.Science China Mathematics, 67(2):253–280, 2024

work page 2024
[6]

The 2D Boussinesq–Navier–Stokes equations with logarithmically supercritical dissipation.Journal of Mathematical Study, 57(1):101–132, 2024

Durga Jang K.C., Dipendra Regmi, Lizheng Tao, and Jiahong Wu. The 2D Boussinesq–Navier–Stokes equations with logarithmically supercritical dissipation.Journal of Mathematical Study, 57(1):101–132, 2024

work page 2024
[7]

Nonlinear Rayleigh–Taylor instability of inho- mogeneous incompressible geophysical fluids with partial dissipation.Nonlinearity, 38(5):055016, 2025

Yanlong Fan, Daozhi Han, Quan Wang, and Chao Xing. Nonlinear Rayleigh–Taylor instability of inho- mogeneous incompressible geophysical fluids with partial dissipation.Nonlinearity, 38(5):055016, 2025. Article 055016, 41 pp

work page 2025

Show all 25 references

[8]

Partially dissipative 2D Boussinesq equations with Navier type boundary conditions.Physica D: Nonlinear Phenomena, 376:39–48, 2018

Weiwei Hu, Yanzhen Wang, Jiahong Wu, Bei Xiao, and Jia Yuan. Partially dissipative 2D Boussinesq equations with Navier type boundary conditions.Physica D: Nonlinear Phenomena, 376:39–48, 2018. Combined volumes 376–377

2018
[9]

Uniqueness of weak solutions to the Boussinesq equations with fractional dissipation.Communications in Mathematical Sciences, 21(6):1531–1548, 2023

Ruihong Ji, Dan Li, and Jiahong Wu. Uniqueness of weak solutions to the Boussinesq equations with fractional dissipation.Communications in Mathematical Sciences, 21(6):1531–1548, 2023. 22 WEN FENG, WEINAN W ANG AND XIANG XU

2023
[11]

Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity.Pure and Applied Functional Analysis, 5(1):27–45, 2020

Igor Kukavica and Weinan Wang. Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity.Pure and Applied Functional Analysis, 5(1):27–45, 2020

2020
[12]

Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity.Journal of Dynamics and Differential Equations, 32(4):2061–2077, 2020

Igor Kukavica and Weinan Wang. Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity.Journal of Dynamics and Differential Equations, 32(4):2061–2077, 2020
[13]

Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation.Journal of Differential Equations, 271:764–796, 2021

Suhua Lai, Jiahong Wu, and Yueyuan Zhong. Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation.Journal of Differential Equations, 271:764–796, 2021

2021
[14]

Norm inflation for the Boussinesq system.Discrete and Continuous Dynamical Systems – Series B, 26(10):5449–5463, 2021

Zongyuan Li and Weinan Wang. Norm inflation for the Boussinesq system.Discrete and Continuous Dynamical Systems – Series B, 26(10):5449–5463, 2021

2021
[15]

The stability and decay of 2D in- compressible Boussinesq equation with partial vertical dissipation.Communications in Analysis and Mechanics, 17(1):100–127, 2025

Hongxia Lin, Sabana, Qing Sun, Ruiqi You, and Xiaochuan Guo. The stability and decay of 2D in- compressible Boussinesq equation with partial vertical dissipation.Communications in Analysis and Mechanics, 17(1):100–127, 2025

2025
[16]

Stability of 3D perturbations near a special 2D solution to the rotating Boussinesq equations.Studies in Applied Mathematics, 148(4):1624–1655, 2022

Liangliang Ma, Jiahong Wu, and Qian Zhang. Stability of 3D perturbations near a special 2D solution to the rotating Boussinesq equations.Studies in Applied Mathematics, 148(4):1624–1655, 2022

2022
[17]

M´ emoire sur les lois du mouvement des fluides.M´ emoires de l’Acad´ emie Royale des Sciences de l’Institut de France, 6:389–440, 1827

Claude-Louis-Marie-Henri Navier. M´ emoire sur les lois du mouvement des fluides.M´ emoires de l’Acad´ emie Royale des Sciences de l’Institut de France, 6:389–440, 1827. Read to the Acad´ emie Royale des Sciences on 18 March 1822; volume for the year 1823
[18]

Maria Alessandra Ragusa and Fan Wu. Global regularity and stability of solutions to the 3D double- diffusive convection system with Navier boundary conditions.Advances in Differential Equations, 26(7– 8):281–304, 2021

2021
[19]

A global regularity result for the 2D Boussinesq equations with critical dissipation.Journal d’Analyse Math´ ematique, 137(1):269–290, 2019

Atanas Stefanov and Jiahong Wu. A global regularity result for the 2D Boussinesq equations with critical dissipation.Journal d’Analyse Math´ ematique, 137(1):269–290, 2019

2019
[20]

Global regularity results of the 2D fractional Boussinesq equations.Mathematische Annalen, 391(4):5965–6012, 2025

Atanas Stefanov, Jiahong Wu, Xiaojing Xu, and Zhuan Ye. Global regularity results of the 2D fractional Boussinesq equations.Mathematische Annalen, 391(4):5965–6012, 2025

2025
[21]

On the global regularity for a 3D Boussinesq model without thermal diffusion.Zeitschrift f¨ ur Angewandte Mathematik und Physik, 70(6):174, 2019

Weinan Wang. On the global regularity for a 3D Boussinesq model without thermal diffusion.Zeitschrift f¨ ur Angewandte Mathematik und Physik, 70(6):174, 2019. Article 174, 6 pp

2019
[22]

On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions.Archiv der Mathematik, 116(4):445–456, 2021

Weinan Wang. On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions.Archiv der Mathematik, 116(4):445–456, 2021

2021
[23]

On 2D incompressible Boussinesq systems: global stabilization under dynamic boundary conditions.Journal of Differential Equations, 367:246–289, 2023

Jiahong Wu and Kun Zhao. On 2D incompressible Boussinesq systems: global stabilization under dynamic boundary conditions.Journal of Differential Equations, 367:246–289, 2023

2023
[24]

Well-posedness and exponential stability for Boussinesq sys- tems on real hyperbolic manifolds and application.Asymptotic Analysis, 148(3):803–824, 2026

Pham Truong Xuan and Tran Thi Ngoc. Well-posedness and exponential stability for Boussinesq sys- tems on real hyperbolic manifolds and application.Asymptotic Analysis, 148(3):803–824, 2026. First published online 16 October 2025

2026
[25]

On certain classes of mild solutions for Boussinesq systems: a framework of curved spaces.Zeitschrift f¨ ur Angewandte Mathematik und Physik, 76(5):214, 2025

Pham Truong Xuan, Nguyen Thi Van, and Tran Thi Ngoc. On certain classes of mild solutions for Boussinesq systems: a framework of curved spaces.Zeitschrift f¨ ur Angewandte Mathematik und Physik, 76(5):214, 2025. Article 214, 32 pp. 1 Department of Mathematics and Statistics, S...

2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [10]

Kelliher, Christophe Lacave, Milton C

James P. Kelliher, Christophe Lacave, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, and Edriss S. Titi. Large time behavior for the 3D Navier–Stokes with Navier boundary conditions.Journal of Math- ematical Fluid Mechanics, 27(3):48, 2025. Article 48, 23 pp

work page 2025

[2] [1]

Large-time behavior of the 2D thermally non- diffusive Boussinesq equations with Navier-slip boundary conditions.Zeitschrift f¨ ur Angewandte Math- ematik und Physik, 76:58, 2025

Fabian Bleitner, Elizabeth Carlson, and Camilla Nobili. Large-time behavior of the 2D thermally non- diffusive Boussinesq equations with Navier-slip boundary conditions.Zeitschrift f¨ ur Angewandte Math- ematik und Physik, 76:58, 2025. Article 58

work page 2025

[3] [2]

Stability and enhanced decay rate for 3D anisotropic Boussinesq equations near the hydrostatic balance.Journal of Differential Equations, 425:300–341, 2025

Jianfeng Cheng, Ruihong Ji, Ling Tian, and Jiahong Wu. Stability and enhanced decay rate for 3D anisotropic Boussinesq equations near the hydrostatic balance.Journal of Differential Equations, 425:300–341, 2025

work page 2025

[4] [3]

Global existence and Rayleigh-Taylor instability for the semi-dissipative Boussinesq system with Naiver boundary conditions

Huafei Di, Liang Li, Xiaoming Peng, and Quan Wang. Global existence and Rayleigh–Taylor insta- bility for the semi-dissipative Boussinesq system with Navier boundary conditions. arXiv preprint arXiv:2405.16074, 2024

work page Pith review arXiv 2024

[5] [4]

The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term inR 3.Nonlinear Analysis, 255:113775, 2025

Lihua Dong. The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term inR 3.Nonlinear Analysis, 255:113775, 2025. Article 113775, 17 pp

work page 2025

[6] [5]

On asymptotic stability of the 3D Boussinesq equations without heat conduction.Science China Mathematics, 67(2):253–280, 2024

Lihua Dong and Yongzhong Sun. On asymptotic stability of the 3D Boussinesq equations without heat conduction.Science China Mathematics, 67(2):253–280, 2024

work page 2024

[7] [6]

The 2D Boussinesq–Navier–Stokes equations with logarithmically supercritical dissipation.Journal of Mathematical Study, 57(1):101–132, 2024

Durga Jang K.C., Dipendra Regmi, Lizheng Tao, and Jiahong Wu. The 2D Boussinesq–Navier–Stokes equations with logarithmically supercritical dissipation.Journal of Mathematical Study, 57(1):101–132, 2024

work page 2024

[8] [7]

Nonlinear Rayleigh–Taylor instability of inho- mogeneous incompressible geophysical fluids with partial dissipation.Nonlinearity, 38(5):055016, 2025

Yanlong Fan, Daozhi Han, Quan Wang, and Chao Xing. Nonlinear Rayleigh–Taylor instability of inho- mogeneous incompressible geophysical fluids with partial dissipation.Nonlinearity, 38(5):055016, 2025. Article 055016, 41 pp

work page 2025