Pith. sign in

REVIEW 3 major objections 5 minor 37 references

Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbb{R}^2$

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The hydrostatic balance of the 2D Boussinesq equations is globally stable under horizontal-only dissipation, with explicit decay rates.

desk verdict A plausible solution to a notable open problem in anisotropic Boussinesq stability, but the linear dispersive estimate has a real gap around the Hilbert transform and the bootstrap margins are razor-thin. read the letter →

arxiv 2607.19071 v1 pith:SQEP75SS submitted 2026-07-21 math.AP

Jiahong Wu , Mengxin Yan , Ning Zhu This is my paper · ORCID
classification math.AP MSC 35Q3535B3535B4076D0376D50
keywords Boussinesqequationshydrostaticequilibriumhorizontaldissipationstabilitylarge-timebehaviordispersiveestimatesinternalgravitywavesanisotropicdecay
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 is trying to prove that the hydrostatic equilibrium $(U,\Theta)=(0,x_2)$ of the two-dimensional Boussinesq equations remains stable even when the only dissipation is horizontal. The claim is that for initial perturbations small in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge 14$, there is a unique global classical solution whose $H^k$ norm stays bounded, whose horizontal derivatives are square-integrable in time, and whose velocity and temperature decay at explicit componentwise rates. The reason this matters is that horizontal dissipation alone cannot control the nonlinearity by standard energy estimates; the paper argues that the coupling between velocity and temperature creates internal gravity waves whose dispersive decay supplies the missing stabilization. If the claim is right, it identifies wave propagation as a stabilizing mechanism that can compensate for incomplete dissipation in stratified fluids.

What carries the argument

The load-bearing object is the coupled oscillatory-dissipative semigroup $S_\sigma(t)=e^{t\partial_1^2}e^{\sigma t R_1}Q_\sigma$, $\sigma\in\{+, -\}$, where $R_1$ is the Fourier multiplier $i|\xi_1|/|\xi|$ and $Q_\sigma$ projects onto the eigenspace of the skew-adjoint coupling operator. The phase $\varphi(\xi)=\xi_1/|\xi|$ is anisotropic: its Hessian degenerates along $\xi_2=0$ and the phase itself is not smooth at $\xi_1=0$, so the proof decomposes frequency space and uses two mechanisms at once, stationary phase for the non-degenerate part and horizontal heat-kernel decay on Littlewood-Paley blocks for the degenerate part. The resulting kernel bound $2^{2j}(1+t)^{-1}+2^{2j}e^{-c2^{2j}t}(1+t)^{-1/2}$ is interpolated to the $L^4$ estimate, and this estimate is what makes the Duhamel nonlinear terms time-integrable in the bootstrap.

What would settle it

Evaluate numerically, for $j=0$, the oscillatory integral $I(t)=\int e^{ix\cdot\xi}e^{i\xi_1 t/|\xi|}\psi_2(\xi_1)\tilde\varphi(\xi)\,d\xi$ in the sector $|\xi_1|\le 1/3$, $|\xi_2|\approx 1$. The proof needs $|I(t)|\lesssim (1+t)^{-1}$ uniformly in $x$; if the true decay is only $(1+t)^{-1/2}$ or worse, the kernel bound (3.7) is too optimistic and the bootstrap closing in Sections 4 and 5 collapses.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the perturbation system around the hydrostatic balance has a hidden oscillatory structure that standard energy estimates miss. After writing the perturbation $(u_1,u_2,\theta)$ as a vector and applying an extended Helmholtz projection, the linearized operator has eigenvalues $\pm i|\xi_1|/|\xi|$ and $0$; the zero mode is killed by divergence freeness, so the linear flow is a sum of two semigroups $e^{t\partial_1^2}e^{\pm t R_1}$ acting on the corresponding eigenprojections, where $R_1$ is the Fourier multiplier with symbol $i|\xi_1|/|\xi|$. The paper establishes an $L^4$ dispersive estimate of order $(1+t)^{-1/2}$ for these semigroups on $W^{1+\gamma,4/3}$ data, splitting frequency space: in the region where the phase $\xi_1/|\xi|$ is non-degenerate, stationary phase gives the decay, while in the degenerate region the one-dimensional horizontal heat kernel $e^{-\xi_1^2 t}$ provides it. Combining this linear decay with a bootstrap of $H^k$ energy estimates yields the global existence, uniqueness, and componentwise decay rates of Theorem 1.1, including the faster decay of the vertical velocity.

Load-bearing premise

The proof stands on the linear estimate that the oscillatory-dissipative semigroup decays like $(1+t)^{-1/2}$ in $L^4$; if the stationary-phase argument does not survive in the degenerate frequency region $|\xi_1|\le |\xi_2|$, where the phase $|\xi_1|/|\xi|$ is not smooth at $\xi_1=0$, the bootstrap exponents fail and the decay rates would not close.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, the hydrostatic balance is globally nonlinearly stable on the whole space $\mathbb{R}^2$ with only horizontal dissipation, with no periodicity or Poincaré inequality needed: the dispersive decay replaces that mechanism.
  • The vertical velocity $u_2$ decays faster than the horizontal velocity and the temperature componentwise, for example $\|u_2(t)\|_{L^2}\le C_0\varepsilon(1+t)^{-1/4}$ versus $\|(u_1|\theta)(t)\|_{L^2}\le C_0\varepsilon(1+t)^{-1/8-\eta}$, and horizontal derivatives decay faster still.
  • The proof yields uniform time-integrability of the horizontal dissipation: $\int_0^t (\|\partial_1 u(\tau)\|_{H^k}^2 + \|\partial_1\theta(\tau)\|_{H^k}^2)\,d\tau$ is bounded independently of $t$.
  • The same wave-dispersion mechanism separates the Boussinesq system from the anisotropically dissipative Navier–Stokes equations, for which horizontal-only dissipation has no known robust stabilizing mechanism.

Reading between the lines

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

  • One extension the authors do not pursue: the same semigroup mechanism should likely work for fractional horizontal dissipation $(-\partial_1^2)^\alpha$ with $\alpha$ below 1, provided the degenerate-frequency kernel estimate is re-derived with the fractional heat kernel; the present paper treats the endpoint case $\alpha=1$ in the horizontal direction.
  • The requirement $k\ge 14$ likely reflects the interpolation-heavy bootstrap rather than a threshold set by the physics; sharper product estimates might lower the needed regularity, though the smallness condition would remain.
  • A direct numerical evaluation of the oscillatory integral at the degenerate sector would provide a clean independent test of the mechanism, since the linear decay rate is what carries the whole nonlinear argument.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the two-dimensional Boussinesq equations with only horizontal dissipation near the hydrostatic equilibrium (U,Θ)=(0,x_2). The perturbation system is rewritten as a coupled dispersive-dissipative system, symmetrized in Section 2, and the linearized operator is diagonalized through internal gravity wave modes. Section 3 develops linear estimates, including an L^4 dispersive estimate for the semigroup; Sections 4 and 5 then propose a bootstrap argument to prove global existence and componentwise decay rates for sufficiently small data in H^k(R^2)∩W^{3,1}(R^2), k≥14. The claimed decay rates include ||(u_1|θ)||_{L^2}≤C_0ε(1+t)^{-1/8-η}, ||u_2||_{L^2}≤C_0ε(1+t)^{-1/4}, and ||u_2||_{L^4}≤C_0ε(1+t)^{-7/8-η+δ} with η=1/120 and δ=10^{-5}.

Significance. If the proof were complete, the paper would be a significant contribution to the anisotropic Boussinesq stability problem: it identifies a concrete stabilizing mechanism (dispersive decay of internal gravity waves combined with horizontal dissipation) that compensates for the absence of vertical dissipation, and it provides explicit anisotropic and componentwise rates. The organization is clear, the bootstrap assumptions and improved estimates are stated explicitly, and the linear semigroup representation is worked out in detail. However, the central linear estimate on which the whole bootstrap rests is not proved as stated, and several exponent checks that are needed to close the convolution estimates are only verified numerically. The result is plausible, but the manuscript currently does not provide a complete proof.

major comments (3)
  1. [§3.3, Lemma 3.7 and Proposition 3.9] Lemma 3.7 states (3.7) as an L^1(R^2)→L^8(R^2) estimate, but the proof bounds the kernel in L^∞ and uses Young's inequality with the L^1 norm of the data, which yields an L^1→L^∞ estimate. The subsequent Riesz–Thorin interpolation in Proposition 3.9 is algebraically the one associated with the L^∞ endpoint: interpolating (3.6) with an L^1→L^∞ bound at θ=1/2 gives L^{4/3}→L^4 with the dyadic factor 2^j and decay (1+t)^{-1/2} displayed in (3.12). Interpolating the stated L^1→L^8 bound would give L^{6/5}→L^4 and decay (1+t)^{-2/3}. Thus, as written, Proposition 3.9 is not proved. This is load-bearing because the Duhamel estimates in Sections 5.1–5.4 invoke (3.11) to close the bootstrap. The lemma and proposition must be reconciled: either correct (3.7) to an L^∞ estimate or provide a genuine L^8 estimate that still yields (3.12).
  2. [§3.3, proof of Proposition 3.9] The passage from (3.12) to the W^{1+γ,4/3} bound contains an unshown dyadic summation. The estimate (3.12) has a factor 2^j, while the next display introduces 2^{j/2}+2^j without explanation, and the reduction of ∑_j 2^j‖Δ_j f_0‖_{L^{4/3}} to ‖f_0‖_{L^{4/3}}+‖Λ^{1+γ}f_0‖_{L^{4/3}} requires a separate treatment of the positive and negative frequency sums. Please provide the complete summation argument, or state the estimate with the appropriate Besov norm on the right-hand side.
  3. [§5.1 and §5.5, exponent verifications] The text verifies that s_1, s_2, m_1, m_2, m_3 are larger than 1 only by approximate decimal evaluations for the chosen parameters (for example, m_2≈1.000030553 for η=1/120, k=14, δ=10^{-5}). No derivation or exact algebraic inequality is shown, and the preceding estimates have unspecified multiplicative constants. Since Lemma 3.3 requires the exponent β>1 to avoid a logarithmic factor, the closing of the bootstrap depends on these inequalities holding exactly. The numerical checks should be replaced by a rigorous verification, e.g., exact rational arithmetic or a clear monotonicity argument for the stated parameter values.
minor comments (5)
  1. [§3.3, equation (3.4)] The uniform L^p bound for R_jΔ_j is correct for all 1≤p≤∞ because the symbol is smooth on the dyadic annulus and the rescaled kernel has a uniformly integrable profile. The text should say this explicitly rather than invoking global Riesz-transform boundedness for p=1,∞, where the latter statement would be false.
  2. [§3.3, equation (3.12) and following display] The factor 2^{j/2} appears in the dyadic sum after (3.12) but not in (3.12) itself. Please remove the typographical discrepancy or justify where the 2^{j/2} comes from.
  3. [§5.2] The exponent (1+(1−δ)α)/2 in the nonlinear Duhamel estimate is stated without derivation, whereas the corresponding linear exponent is (1+α)/2. Please show explicitly how the horizontal heat semigroup and Proposition 3.9 combine to produce the δ-dependent exponent.
  4. [§5.4, equation (5.1)] In the estimates for G_11 and G_12, the replacement of B_1Λ^{-1}Δ_j by a bounded dyadic multiplier is not stated explicitly. Because ξ_1/|ξ| is uniformly bounded on each dyadic annulus, one has B_1Λ^{-1}Δ_j∼Δ_j in L^{4/3}; this step should be written out.
  5. [Global] The symbol L^8 appears in Lemma 3.7 and its proof; if it is a rendering of L^∞, it should be corrected throughout. The reader should not have to infer the intended exponent from the interpolation argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a self-contained bootstrap built on standard linear dispersive estimates; self-citations are contextual and not load-bearing.

full rationale

The central claim is not equivalent to its inputs. Section 3 derives the linear L^4 decay (Prop. 3.9) from stationary phase (Lemma 3.6), heat-semigroup bounds (Lemmas 3.4, 3.5) and Riesz-Thorin interpolation (Lemma 3.8), all external standard results. The nonlinear bootstrap in Sections 4 and 5 assumes bounds of the same form as the conclusions with constant C0 and then proves the same bounds with constant C0/2 using the Duhamel formulas (2.3)-(2.5) and the linear estimates; this is a standard continuity argument, not a circular prediction. The cited prior works by the same authors (e.g., [11], [17], [24], [25], [35], [36]) are contextual and are not used as the load-bearing argument. One possible concern is technical: Prop. 3.9's interpolation from Lemma 3.7 appears to require an L1-to-L∞ estimate, whereas Lemma 3.7 is stated as L1-to-L8, and the Hilbert transform is not bounded on L1. This is a correctness gap, not a circularity, and does not make the derivation equivalent to its inputs. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The papers contributions rest on standard PDE tools plus two hand-tuned small parameters (η, δ) and a high Sobolev index k. The main free parameters are the exponents that make the bootstrap close; they are not fitted to data but are chosen to make inequalities work, which is a form of parameter fitting to the proof. The linear wave structure is derived, not postulated.

free parameters (3)
  • η = 1/120
    A small power added to decay exponents. It is chosen at the end of Section 5.1 (s1≈1.1088) and Section 5.4 to make certain convolution integrals converge. The value is not derived from any physical or mathematical principle; it is a free parameter tuned to close the bootstrap.
  • δ = 10^{-5}
    A second small power used in several decay rates and exponent estimates (e.g., s2≈1.01028, m2≈1.000030553). It is introduced ad hoc to ensure the exponents exceed 1 by a small margin.
  • k = 14
    The order of the Sobolev space H^k. The theorem requires k≥14, and the energy estimates in Section 4 use k=14 in the interpolation formulas. The threshold is not derived from a general principle but is the smallest integer for which the displayed interpolation exponents stay positive.
assumptions (4)
  • standard math Standard Littlewood-Paley decomposition and Bernstein inequalities (Lemma 3.1), Kato-Ponce inequalities (Lemma 3.2), heat semigroup estimates (Lemmas 3.4, 3.5), and stationary phase estimate (Lemma 3.6) are valid as stated.
    These are classical tools cited to references [26], [27], [28], [32], and [2]. The paper does not prove them but relies on them for the linear and nonlinear estimates.
  • domain assumption The operator -P~J~P has eigenvalues ±i|ξ1|/|ξ| and 0, with the given eigenvectors forming an orthonormal basis, and the divergence-free condition exactly kills the zero mode.
    This is verified by direct computation in Section 2 but not fully proved in the text. The entire semigroup representation depends on this spectral decomposition.
  • standard math The Riesz transform is bounded on L^p for 1<p<∞, including the Hilbert transform on L^4 and L^{4/3}.
    Invoked repeatedly, e.g., equation (3.4), and is standard, but it is used for the projected operators Qσ, whose boundedness is not proved in detail.
  • standard math The bootstrap method is valid: if all a priori estimates (4.1)-(4.6) hold on [0,T) and imply the improved estimates (4.7)-(4.12) with constants divided by 2, then the solution can be extended to infinity.
    This is a standard continuity argument in PDE theory, but its application here requires that the local well-posedness be established for the given data class, which the paper states is standard via Friedrichs method.
invented entities (1)
  • Internal gravity wave modes arising from the symmetrized linear system
    purpose: The two eigenmodes with eigenvalues ±i|ξ1|/|ξ| are used to represent the solution as a superposition of waves that disperse and decay.
    These modes are not physically new entities; they are a mathematical consequence of the linearized Boussinesq system. They are introduced through the spectral decomposition in Section 2 and have no independent evidence outside the equations. However, they are not invented postulates but derived mathematical objects. This entry is included for completeness, but the entity is a mathematical construct, not a new physical entity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbb{R}^2$." pith.science (2026). https://pith.science/paper/SQEP75SS

@misc{pith2026260719071,
  author       = {Pith},
  title        = {Pith review of: Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbbR^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQEP75SS}},
  note         = {Machine review of arXiv:2607.19071}
}
abstract

The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier--Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on $\mathbb{R}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,\Theta)=(0,x_2)$. We show that the velocity--temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 35 canonical work pages

  1. [1]

    Adhikari, O

    D. Adhikari, O. Ben Said, U. R. Pandey and J. Wu, Stability and large-time behav- ior for the 2D Boussinesq system with horizontal dissipation and vertical thermal diffusion,Nonlinear Differ. Equ. Appl.29(2022), Art. 42

    work page 2022

  2. [2]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Vol. 343, Springer, Heidelberg, 2011

    work page 2011

  3. [3]

    Ben Said, U

    O. Ben Said, U. R. Pandey and J. Wu, The stabilizing effect of the temperature on buoyancy-driven fluids,Indiana Univ. Math. J.71(2022), 2605–2645

    work page 2022

  4. [4]

    Brandolese and M

    L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system,Trans. Amer. Math. Soc.364(2012), no. 10, 5057–5090

    work page 2012

  5. [5]

    Castro, D

    ´A. Castro, D. C´ ordoba and D. Lear, On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term,Math. Models Methods Appl. Sci.29(2019), 1227–1277

    work page 2019

  6. [6]

    Chen and T

    J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data, arXiv:2210.07191v3

    arXiv

  7. [7]

    Chen and T

    J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous Numerics,Multiscale Model. Simul. 23(2025), 25–130

    work page 2025

  8. [8]

    Choi and I.-J

    K. Choi and I.-J. Jeong, Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole,Nonlinear Anal. Real World Appl.65(2022), Paper No. 103470, 20 pp

    work page 2022

Show all 37 references
  1. [9]

    C´ ordoba, L

    D. C´ ordoba, L. Mart´ ınez-Zoroa and F. Zheng, Finite time singularities to the 3D incompressible Euler equations for solutions inC 8pR3zt0uqXC 1,αXL 2,Ann. PDE 11(2025), Art. 14

    2025

  2. [10]

    Doering, J

    C. Doering, J. Wu, K. Zhao and X. Zheng, Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion,Physica D376/377(2018), 144– 159

    2018

  3. [11]

    B. Dong, J. Wu, X. Xu and N. Zhu, Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation,Calc. Var. Partial Dif- ferential Equations60(2021), Art. 116, 21 pp. 38

    2021

  4. [12]

    Dong and Y

    L. Dong and Y. Sun, Asymptotic stability of the 2D Boussinesq equations without thermal conduction,J. Differential Equations337(2022), 507–540

    2022

  5. [13]

    Dong and Y

    L. Dong and Y. Sun, Asymptotic stability of the 3D Boussinesq equations without thermal conduction,Sci. China Math.67(2024), 253–280

    2024

  6. [14]

    T. M. Elgindi, Finite-time singularity formation forC 1,α solutions to the incompress- ible Euler equations onR 3,Ann. of Math.194(2021), 647–727

    2021

  7. [15]

    G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, 1999

    1999

  8. [16]

    Fujii and Y

    M. Fujii and Y. Li, Linear and nonlinear stability for the three dimensional stratified Boussinesq equations with horizontal viscosity and diffusivity,SIAM J. Math. Anal. 58(2026), 411–449

    2026

  9. [17]

    R. Ji, L. Yan and J. Wu, Optimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balance,Calc. Var. Partial Differential Equations61(2022), no. 4, Art. 136

    2022

  10. [18]

    Jeong, Y

    I.-J. Jeong, Y. Yao and T. Zhou, Superlinear gradient growth for 2D Euler equation without boundary, arXiv:2507.15739 (2025)

    2025 arXiv

  11. [19]

    K. Kang, J. Lee and D. Nguyen, Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium, arXiv:2306.08286 (2023)

    2023 arXiv

  12. [20]

    Kim and J

    J. Kim and J. Lee, Stratified Boussinesq equations with a velocity damping term, Nonlinearity35(2022), 3059–3094

    2022

  13. [21]

    Kiselev, J

    A. Kiselev, J. Park and Y. Yao, Small scale formation for the 2-dimensional Boussi- nesq equation,Anal. PDE18(2025), no. 1, 171–198

    2025

  14. [22]

    Kiselev and V

    A. Kiselev and V. ˇSver´ ak, Small scale creation for solutions of the incompressible two dimensional Euler equation,Ann. of Math.180(2014), 1205–1220

    2014

  15. [23]

    Kukavica and W

    I. Kukavica and W. Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity,J. Dynam. Differential Equations32(2020), no. 4, 2061–2077

    2020

  16. [24]

    S. Lai, J. Wu and Y. Zhong, Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation,J. Differential Equations271(2021), 764–796. 39

    2021

  17. [25]

    S. Lai, J. Wu, X. Xu, J. Zhang and Y. Zhong, Optimal decay estimates for 2D Boussinesq equations with partial dissipation,J. Nonlinear Sci.31(2021), Art. 16

    2021

  18. [26]

    Li, On Kato–Ponce and fractional Leibniz,Rev

    D. Li, On Kato–Ponce and fractional Leibniz,Rev. Mat. Iberoam.35(2019), no. 1, 23–100

    2019

  19. [27]

    H. Lin, J. Wu and X. Zheng, Global stability and decay for the 2D Boussinesq system with only velocity damping, submitted for publication, 2026

    2026

  20. [28]

    C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations,Nonlinear Anal.68(2008), no. 3, 461–484

    2008

  21. [29]

    M. E. Schonbek,L 2 decay for weak solutions of the Navier–Stokes equations,Arch. Rational Mech. Anal.88(1985), 209–222

    1985

  22. [30]

    Shang and L

    H. Shang and L. Xu, Stability near hydrostatic equilibrium to the three-dimensional Boussinesq equations with partial dissipation,Z. Angew. Math. Phys.72(2021), no. 2, Art. 80, 1–14

    2021

  23. [31]

    F. Shao, D. Wei, P. Zhang and Z. Zhang, Self-similar blow-up solutions of incom- pressible Euler equations inR d,dě3, withC 1,1´ 2 d´ velocity, arXiv:2605.19716

    arXiv

  24. [32]

    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Os- cillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, NJ, 1993

    1993

  25. [33]

    L. Tao, J. Wu, K. Zhao and X. Zheng, Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion,Arch. Ration. Mech. Anal.237 (2020), 585–630

    2020

  26. [34]

    Wan, Global well-posedness for the 2D Boussinesq equations with a velocity damp- ing term,Discrete Contin

    R. Wan, Global well-posedness for the 2D Boussinesq equations with a velocity damp- ing term,Discrete Contin. Dyn. Syst.39(2019), no. 5, 2709–2730

    2019

  27. [35]

    Z. Wang, J. Wu and N. Zhu, Stability and decay for the 2D anisotropic Navier–Stokes equations with fractional horizontal dissipation onR 2, arXiv:2601.16110v2 (2026)

    2026 arXiv

  28. [36]

    Wu and Q

    J. Wu and Q. Zhang, Stability and optimal decay for a system of 3D anisotropic Boussinesq equations,Nonlinearity34(2021), no. 8, 5456–5484

    2021

  29. [37]

    Zlatoˇ s, Exponential growth of the vorticity gradient for the Euler equation on the torus,Adv

    A. Zlatoˇ s, Exponential growth of the vorticity gradient for the Euler equation on the torus,Adv. Math.268(2015), 396–403. 40

    2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.