{"id":"39a6522e-1cc2-45cc-a47d-513713a763a7","arxiv_id":"2501.00690","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A proof that Couette flow in the stably stratified Boussinesq system on R^2 is asymptotically stable for Richardson number R>1/4, with explicit enhanced dissipation, Taylor dispersion, and inviscid damping rates.","lead":"This mathematics paper proves that the stably stratified Couette flow (a simple shear flow with a linear density profile) is stable on the whole plane R^2, provided the initial perturbation is small enough and viscosity and density dissipation are comparable. It is the first stability result of this type on the fully unbounded domain, and it quantifies the decay of disturbances at explicit rates.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The nonlinear bootstrap depends on interpolation Lemmas 6.4, 6.6, and 6.7, which are stated without proof, and on the mixed terms T_gamma,m1 and T_gamma,m2, which are asserted to follow identically; if any of these bounds fails, Lemma 4.1 and hence Theorem 1.1 collapse.","rationale":"The reader's weakest assumption identifies the missing proofs of the interpolation lemmas and the uncomputed mixed nonlinear terms as the main reason for a conditional verdict. My independent stress-test confirms that these are indeed the most load-bearing gaps: Lemma 4.1 is the only nonlinear estimate used to close the bootstrap, and the specific estimates (4.14) and (4.20) rely on Lemmas 6.6 and 6.4, both explicitly marked as omitted. The mixed terms are not merely cosmetic, since they involve cross-products of the nonlinearities with the opposite symmetrized variable and carry different multipliers. The paper contains substantial original work: the symmetrization, the linear pointwise energy estimates in Section 3, and the successful reduction to Lemma 4.1 are plausible and largely consistent. I found no internal contradiction in the computed linear estimates, and the bootstrap argument in Remark 2.1 is logically sound assuming Lemma 2.3. The concern is therefore not that the theorem is false, but that the proof as written is incomplete at a crucial point. This is exactly what a CONDITIONAL verdict should reflect, so I do not change the reader's verdict. I also note a smaller technical defect: A(k) and B(k) are used before being defined, and the statement of Theorem 1.1 uses a slightly different m-range from Theorem 2.1; these are minor and fixable compared to the unproved lemmas. If the author supplies the missing proofs and the endpoint checks succeed, I would regard the central claim as supported.","tokens_in":1395,"tokens_out":1121,"duration_ms":59455,"concrete_test":"Provide complete proofs of Lemmas 6.4, 6.6, and 6.7 in an appendix, and write out the bounds for T_gamma,m1 and T_gamma,m2 using the explicit convolution expressions (4.9)-(4.10). Then verify the arithmetic in the two most delicate applications: the chain of inequalities leading to (4.20) for T^y_{alpha,Z1,LH}, and the chain leading to (4.14) for T^y_{gamma,Z1,HL,(.,H)}. Specifically, test Lemma 6.4 on the sharp endpoint cases |k| about mu and |eta-kt| about 0, where the claimed bound E^{1/2} + mu^{-delta*} E^{1/2} must be saturated without losing more than mu^{-delta*}; if the bound fails there, the bootstrap estimate (4.8) is false and Theorem 1.1 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 is proved via the bootstrap Lemma 2.3, which follows from Lemma 4.1 by the bound |T_gamma|+|T_alpha|+|T_beta| <= C mu^{-1/2-delta*} D E^{1/2}. That bound is the load-bearing nonlinear estimate. The proof of Lemma 4.1 repeatedly invokes interpolation lemmas that are not proved: Lemma 6.4 is used in the bound of T^y_{alpha,Z1,LH} at (4.20), Lemma 6.6 is used for T^y_{gamma,Z1,HL,(.,H)} at (4.14), and Lemma 6.7 is used for T^x_{alpha,Z2,LH} and for T^x_{beta,Q2,HL}. Each of these estimates is needed to close one of the T_* terms; if any one fails, the bootstrap inequality (2.17) may fail and the threshold mu^{1/2+delta*} is not obtained. Section 4.3 also explicitly declines to bound T_gamma,m1 and T_gamma,m2, asserting only that they follow 'schematically identically' from T_gamma,Z and T_gamma,Q; but those mixed terms contain Re(NL^{(Z)}_k Q_k-bar) and Re(Z_k-bar NL^{(Q)}_k), where the nonlinearities in Z and Q interact with the opposite symmetrized variable. The claimed schematic identity is not automatically valid because the powers of p and the multiplier (partial_t p)/(|k| p^{1/2}) differ in the two cases. Additionally, the paper uses symbols A(k) and B(k) in Sections 4.4-4.7 without defining them, presumably A(k)^2 = alpha_k and B(k)^2 = beta_k; this is fixable but adds to the difficulty of verifying the missing estimates. Since these gaps sit directly between the linear estimates and the nonlinear theorem, they are the weakest point of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the 2D Boussinesq system on R^2 near the stably stratified Couette flow. Its main result, Theorem 1.1, asserts that if R > 1/4, ν and κ satisfy (1.4), and the initial perturbation has size ζ ≤ δ µ^{1/2+δ*} in a low-order anisotropic Sobolev norm V(0), then the solution obeys the quantitative estimates in (1.9): growth of (ω, ∇θ) at the rate ⟨t⟩^{1/2}, decay of ∂_x u_1 and ∂_x θ at the rate ⟨t⟩^{-1/2}, and decay of ∂_x |∂_x, ∂_y+t∂_x|^{-1} ∂_x u_2 at the rate ⟨t⟩^{-3/2}, together with an integrated dissipation bound. The proof introduces symmetrized variables in a moving frame, proves a pointwise-in-frequency linear energy estimate (Proposition 2.2), and then uses a bootstrap argument based on the nonlinear estimate Lemma 4.1. A second theorem (Theorem 1.2) is formulated with an additional L∞_k component in the norm and is treated in Section 5.","tokens_in":82016,"tokens_out":9256,"duration_ms":81974,"significance":"If the proof is completed, the result is a significant advance: it is the first quantitative nonlinear stability result for the Boussinesq system near Couette flow on the fully unbounded domain R^2, and it simultaneously provides explicit enhanced dissipation, Taylor dispersion, and inviscid damping rates in a low-order anisotropic norm. The linear part of the paper is careful: the pointwise energy (2.7), the derivation of Proposition 2.2 in Section 3, and the explicit choice of constants in Remark 3.1 are presented in detail, and the rates in (1.9) are concrete and testable. However, the nonlinear estimate Lemma 4.1, which is the load-bearing step for Theorem 1.1, is not proven in the written manuscript: several interpolation lemmas it relies on are stated without proof, and several mixed nonlinear terms are not estimated. These gaps can in principle be filled within the same framework, so I do not regard the central idea as unsound, but the proof as written is incomplete.","major_comments":[{"comment":"The proof of Lemma 4.1 uses interpolation lemmas that are not proved. Lemma 6.4 is introduced with the note that its proof is not presented, Lemma 6.6 with the note that its proof is omitted for the sake of space, and Lemma 6.7 with the note that its proof has no new ideas. These are not peripheral: Lemma 6.4 is used in the bound of T^y_{α,Z1,LH} at (4.20), Lemma 6.6 in the bound of T^y_{γ,Z1,HL,(·,H)} at (4.14), and Lemma 6.7 in the bounds for T^x_{α,Z2,LH} and T^x_{β,Q2,HL}. Without these estimates, the bound (4.8) on |T_γ|+|T_α|+|T_β| is not established, and consequently the bootstrap inequality (2.17) that yields Theorem 2.1 and then Theorem 1.1 is not justified.","section":"Appendix (Lemmas 6.4, 6.6, 6.7); Section 4 estimates (4.14), (4.20)"},{"comment":"The decomposition (4.7) defines mixed terms T_{γ,m1}, T_{γ,m2}, T_{α,m1}, T_{α,m2}, T_{β,m1}, and T_{β,m2}, yet no estimates are given for these quantities. Section 4.3 states only that T_{γ,m1} and T_{γ,m2} follow schematically from the Z and Q arguments, citing boundedness of ∂_t p/(|k|p^{1/2}); Sections 4.4–4.7 then estimate only the T_{*,Z} and T_{*,Q} components. This is not an automatic reduction: the mixed integrands contain Re(NL^{(Z)}_k \\overline{Q_k}) and Re(\\overline{Z_k} NL^{(Q)}_k), where the nonlinearities (4.9)–(4.10) carry different powers of p and different k-sign factors, so an explicit estimate is required. Moreover the mixed terms for the α and β components are not mentioned after (4.7). Since Lemma 4.1 requires bounds for all of T_γ, T_α, and T_β, the proof of the main nonlinear estimate is incomplete.","section":"Section 4.3 and (4.7)"},{"comment":"The alternate theorem has the same structural gap. Lemma 5.3 bounds T_∞ together with time-integrated versions of T_γ, T_α, and T_β, but the proof in Sections 5.3–5.4 estimates only T_∞,Z and T_∞,Q. The mixed terms T_∞,m1 and T_∞,m2, defined just before (5.7), are never estimated. The text says that the bounds on T_γ, T_α, and T_β are proven in a similar manner to Sections 4.1–4.7, but since those sections already omit the mixed terms and rely on unproved lemmas, Theorem 1.2 is not established as written.","section":"Section 5, Lemma 5.3 and (5.6)–(5.7)"}],"minor_comments":[{"comment":"The symbols A(k) and B(k) are used throughout the nonlinear estimates without being defined; the notation strongly suggests A(k)^2 = α_k and B(k)^2 = β_k from (2.8), but this identification is never stated. The estimates systematically use identities such as B(k)^2 |k| ≲ A(k), so the definition should be given explicitly at first use.","section":"Sections 4.4–4.7"},{"comment":"The displayed u_2 estimate has the positive power ⟨t⟩^{(1+d)/2}, which is inconsistent with the inviscid damping statements of Corollary 3.4 and with the decay of ∂_x u_2 in (1.9); the exponent should presumably be negative.","section":"Corollary 3.5"},{"comment":"In the displayed final estimate, the two integral terms after e^{2cλ_kt} E_k are not separated by a plus sign; the inequality appears to have a missing '+'.","section":"Proposition 2.2"},{"comment":"The statement refers to 'the corresponding solution (Z,Q) to (1.3)' in the line following (2.5); the symmetrized system is (2.4), not (1.3).","section":"Theorem 2.1"},{"comment":"There are numerous typos in headings and formulas, for example 'Dam ping' in the title, 'cmpletes' at the end of Section 4, and inconsistent spellings of Riesz; these should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is long and several of the most important nonlinear estimates are delegated either to schematic reductions or to appendix lemmas without proofs. Because the nonlinear estimate Lemma 4.1 is the bridge between the linear theory and the main theorem, the editor may wish to require that the missing proofs be supplied in full or in a clearly identified supplement before the paper is accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [name],\n\nThe one thing you should know: this is the first nonlinear stability result for the stably stratified Couette flow in the Boussinesq system on the fully unbounded R^2, and the low-frequency Taylor dispersion estimates are genuinely new. The theorem is plausible and the overall argument follows a standard bootstrap architecture. But as written, the proof has real unfinished spots that sit between the linear machinery and the main theorem.\n\nWhat is actually new: the R^2 domain, which forces a treatment of the low x-frequency that the periodic T×R papers discard; the symmetrized-variable energy with the inviscid damping operator is adapted to the unbounded setting; and the paper gives explicit enhanced dissipation, Taylor dispersion, and inviscid damping rates. It also translates known linear results from T×R honestly, and the citation pattern is appropriate—[1] really does supply the norm framework.\n\nThe soft spots are all in Section 4. Lemma 6.4 is stated without proof, Lemma 6.6 and 6.7 are omitted 'for the sake of space,' and each is used in a load-bearing estimate (4.14), (4.20), and the alpha x-terms. These are not one-line interpolation facts; they carry the μ-power bookkeeping that produces the final μ^{-1/2-δ*} coefficient. Section 4.3 also declines to bound T_{γ,m1} and T_{γ,m2}, saying they follow 'schematically identically.' That is not automatically convincing: those mixed terms contain Re(NL^{(Z)} \\bar{Q}) and Re(\\bar{Z} NL^{(Q)}), where the Z and Q nonlinearities interact with the opposite symmetrized variable, and the powers of p and the multiplier ∂_t p/(|k|p^{1/2}) are different. Finally, A(k) and B(k) are used in Sections 4.4–4.7 without being defined; presumably A^2=α_k and B^2=β_k, but that should be stated.\n\nNone of this looks like a counterexample, and I don't see circularity. The gaps are technical and probably fillable by a careful reader. But the main nonlinear estimate Lemma 4.1 is not currently verifiable from the text, so the paper needs heavy revision, not light polishing.\n\nWho it's for: people working on quantitative stability of shear flows, enhanced dissipation, or stratified fluids. It deserves a serious referee. I'd send it out, with the expectation that the referee asks for the missing interpolation proofs and the mixed-term estimates before acceptance.","headline":"A genuine first nonlinear stability result for Boussinesq–Couette on R^2, but the written proof has load-bearing gaps that need real work before the theorem is verifiable.","tokens_in":82550,"tokens_out":5070,"would_cite":true,"duration_ms":48665,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35B35","76D05","76E05","76D50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stratified Couette flow on the whole plane is quantitatively stable, with explicit enhanced dissipation, Taylor dispersion, and inviscid damping rates.","keywords":["Boussinesq system","Couette flow","enhanced dissipation","Taylor dispersion","inviscid damping","stratified shear flow","asymptotic stability","anisotropic Sobolev spaces"],"falsifier":"A direct calculation of the mixed nonlinear terms $T_{\\gamma,m1}$ and $T_{\\gamma,m2}$ in Section 4.3 that violates the claimed $\\mu^{-1/2-\\delta^*} D E^{1/2}$ bound, or a counterexample to any of Lemmas 6.4, 6.6, or 6.7, would invalidate the bootstrap and hence the main theorem.","tokens_in":81363,"feed_emoji":"🌊","tokens_out":8457,"duration_ms":76131,"temperature":0.7,"pith_summary":"This paper proves that the stably stratified Couette flow $(y,0)$, $\\rho=1-by$, on the whole plane $\\mathbb{R}^2$ is quantitatively asymptotically stable under the fully dissipative 2D Boussinesq equations, provided the Richardson number $R>1/4$ and the viscosity and density dissipations are comparable as in (1.4). For initial perturbations of size $\\mu^{1/2+\\delta^*}$ in a low-regularity anisotropic Sobolev norm, with $\\mu$ roughly the smaller of $\\nu$ and $\\kappa$ corrected by a factor depending on $R$, the paper establishes explicit decay rates for enhanced dissipation, Taylor dispersion, and inviscid damping, together with transient growth of vorticity and density gradient. This is the first nonlinear result of this type for the Boussinesq system on the fully unbounded domain $\\mathbb{R}^2$, removing the periodicity-in-$x$ restriction of previous work on $\\mathbb{T}\\times\\mathbb{R}$.","feed_headline":"Proved: stratified Couette flow on R^2 is asymptotically stable","feed_subtitle":"Small perturbations decay at explicit enhanced-dissipation, Taylor-dispersion, and inviscid-damping rates.","key_machinery":"The argument is carried by the frequency-by-frequency energy $E_k[Z_k,Q_k]$ in the moving reference frame $X=x-yt$, built on symmetrized variables $Z = \\langle\\partial_X\\rangle^{1/2}p^{-1/4}\\Omega$ and $Q = \\sqrt{R}\\,\\partial_X|\\partial_X|^{-1}\\langle\\partial_X\\rangle^{1/2}p^{1/4}\\Theta$, where $p=k^2+(\\eta-kt)^2$. The energy contains cross-terms between $Z$ and $Q$ that turn the coupling into dissipation, an exponential multiplier $N_k$ tied to the stratification, and the inviscid damping operator $J_k(t,\\eta)=\\frac12\\arctan(\\eta/k-t)$. The linearized system satisfies the pointwise differential inequality $\\frac{d}{dt}E_k \\leq -c_1 D_k - c_0\\lambda_k E_k$, with $\\lambda_k = \\mu^{1/3}|k|^{2/3}$ for $|k|\\geq\\mu$ and $|k|^2/\\nu$ for $|k|\\leq\\mu$; this single multiplier encodes both enhanced dissipation at high $x$-frequencies and Taylor dispersion at low $x$-frequencies. The nonlinear problem is closed by a bootstrap in the energy $E[Z,Q]=\\int\\!\\int \\langle c\\lambda_k t\\rangle^{2J} M_k(t)\\langle k,\\eta\\rangle^{2n}\\langle k\\rangle^{2m} E_k\\, d\\eta\\, dk$, where $M_k$ solves a designed ODE that absorbs time-derivatives of the decay weight, and the nonlinear terms are bounded by $\\mu^{-1/2-\\delta^*} D E^{1/2}$ through a sequence of interpolation lemmas in frequency space.","core_discovery":"The central discovery is a set of explicit quantitative estimates for the nonlinear problem. Theorem 1.1 states that if $\\|\\omega_{in}\\|_{V(0)} + \\sqrt{R}\\|\\nabla\\theta_{in}\\|_{V(0)} = \\zeta \\leq \\delta \\mu^{1/2+\\delta^*}$, then for all $t\\geq 0$, $\\|\\omega\\|_{V(t)} + \\sqrt{R}\\|\\nabla\\theta\\|_{V(t)} \\leq 2\\langle t\\rangle^{1/2}\\zeta$, $\\|\\partial_x u_1\\|_{V(t)} + \\sqrt{R}\\|\\partial_x\\theta\\|_{V(t)} \\leq 2\\langle t\\rangle^{-1/2}\\zeta$, and $\\|\\partial_x|\\partial_x,\\partial_y+t\\partial_x|^{-1}\\partial_x u_2\\|_{V(t)} \\leq 2\\langle t\\rangle^{-3/2}\\zeta$, with an integral control on $u_2$. Here $V(t)$ is a time-dependent anisotropic Sobolev norm adapted to the sheared coordinates. In the symmetrized variables the proof reduces to a bootstrap showing that the energy $E[Z,Q]$ and dissipation $D[Z,Q]$ satisfy $\\frac{d}{dt}E \\leq -4cD + \\mu^{-1/2-\\delta^*} C^{1/2} E^{1/2} D$, which closes once the initial energy is below $O(\\mu^{1+2\\delta^*})$.","pith_inferences":["Editorial inference: the same frequency-by-frequency energy construction should extend to other strictly monotone shear profiles on unbounded domains, such as Poiseuille-type flows, whenever the analogous pointwise dissipation terms dominate the nonlinear convolution.","Editorial inference: because $\\delta^*\\in(0,1/12)$ is arbitrary, the method leaves open whether the sharp stability threshold is exactly $\\mu^{1/2}$; testing the bootstrap at $\\delta^*=0$ would decide whether the $\\mu^{1/2+\\delta^*}$ loss is an artifact of the interpolation scheme.","Editorial inference: the three interpolation lemmas stated without proof (Lemmas 6.4, 6.6, and 6.7) are the natural place to look first; a direct verification or numerical check of those frequency-space estimates would either certify or refute the nonlinear bounds in Lemma 4.1."],"forward_implications":["Any perturbation of Couette flow in the stably stratified Boussinesq system on $\\mathbb{R}^2$ with initial size $\\zeta \\leq \\delta\\mu^{1/2+\\delta^*}$ in the $V(0)$ norm remains globally controlled, with the bounds (1.9) holding for all time.","Nonzero $x$-frequencies decay at the enhanced dissipation rate $\\exp(-c\\lambda_k t)$, which is faster than heat diffusion when $|k|\\geq\\mu$, with rate $\\mu^{1/3}|k|^{2/3}$.","Low $x$-frequencies are controlled by Taylor dispersion with the rate $|k|^2/\\nu$, the shear-accelerated diffusion rate.","The velocity and density perturbations exhibit inviscid damping: $u_1$ and $\\theta$ decay like $\\langle t\\rangle^{-1/2}$, the second velocity component decays like $\\langle t\\rangle^{-3/2}$, while vorticity and density gradient grow like $\\langle t\\rangle^{1/2}$ as in the linear theory.","The result on the fully unbounded plane $\\mathbb{R}^2$ removes the periodicity-in-$x$ restriction of earlier nonlinear results on $\\mathbb{T}\\times\\mathbb{R}$, and the alternate theorem gives an $L^\\infty_k$ version of the bounds that controls an additional half-derivative at low frequencies."],"supporting_citations":[{"why":"Supplies the linear inviscid damping estimates on $\\mathbb{T}\\times\\mathbb{R}$ for stratified Couette flow that the paper translates and extends to the unbounded plane.","marker":"[10]"},{"why":"Introduces the dissipation condition (1.4) and the parameter $\\mu$ in (1.5), together with the symmetrized-variable framework used throughout the nonlinear proof.","marker":"[11]"},{"why":"Provides the hypocoercive scheme that simultaneously proves enhanced dissipation and Taylor dispersion for parallel shear flows, which the paper adapts to the Boussinesq setting.","marker":"[13]"},{"why":"Supplies the bootstrap energy/dissipation machinery on unbounded domains, including the role of the $J_k$ damping operator in the nonlinear estimates.","marker":"[1]"},{"why":"Introduces the inviscid damping operator $J_k$ used in the pointwise energy functional.","marker":"[6]"},{"why":"Gives the prior nonlinear stability threshold for the Boussinesq-Couette problem on $\\mathbb{T}\\times\\mathbb{R}$, which the present work improves in regularity and extends to $\\mathbb{R}^2$.","marker":"[33]"},{"why":"Establishes linear enhanced dissipation for Boussinesq-Couette on $\\mathbb{T}\\times\\mathbb{R}$, serving as a starting point for the frequency-by-frequency analysis.","marker":"[36]"},{"why":"First uses the comparability condition and the definition of $\\mu$, and proves linear enhanced dissipation for the stratified Couette flow in three dimensions.","marker":"[32]"}],"fun_headline_variants":["Stratified Couette flow on R^2 is asymptotically stable","First stability proof for Couette flow on unbounded plane","Explicit decay rates for stratified Couette flow on R^2","New damping estimates for Couette flow on the full plane"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole nonlinear proof rests on the frequency-space interpolation estimates collected in Lemmas 6.1-6.8 and on the assertion that the mixed nonlinear terms, which the paper does not compute, obey the same bounds as the principal terms; several of those lemmas are stated without proof, and if any one of them fails, the bootstrap argument proving Theorem 2.1 and hence Theorem 1.1 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Stratified Couette flow on R^2 is asymptotically stable","First stability proof for Couette flow on unbounded plane","Explicit decay rates for stratified Couette flow on R^2","New damping estimates for Couette flow on the full plane"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1530,"prompt_tokens":1072,"completion_tokens":458,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":386}},"tokens_in":688,"tokens_out":458,"duration_ms":5016,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:44:35.502641+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct calculation of the mixed nonlinear terms $T_{\\gamma,m1}$ and $T_{\\gamma,m2}$ in Section 4.3 that violates the claimed $\\mu^{-1/2-\\delta^*} D E^{1/2}$ bound, or a counterexample to any of Lemmas 6.4, 6.6, or 6.7, would invalidate the bootstrap and hence the main theorem.","supporting_citations":[{"cited_title":"Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows","cited_arxiv_id":"2309.12738","evidence_quote":"Supplies the linear inviscid damping estimates on $\\mathbb{T}\\times\\mathbb{R}$ for stratified Couette flow that the paper translates and extends to the unbounded plane."},{"cited_title":"Coti Zelati and T","cited_arxiv_id":null,"evidence_quote":"Provides the hypocoercive scheme that simultaneously proves enhanced dissipation and Taylor dispersion for parallel shear flows, which the paper adapts to the Boussinesq setting."},{"cited_title":"Zhai and W","cited_arxiv_id":null,"evidence_quote":"Gives the prior nonlinear stability threshold for the Boussinesq-Couette problem on $\\mathbb{T}\\times\\mathbb{R}$, which the present work improves in regularity and extends to $\\mathbb{R}^2$."},{"cited_title":"Zillinger, On enhanced dissipation for the boussinesq equations , Journal of Differential Equations 282 (2021), 507–445","cited_arxiv_id":null,"evidence_quote":"Establishes linear enhanced dissipation for Boussinesq-Couette on $\\mathbb{T}\\times\\mathbb{R}$, serving as a starting point for the frequency-by-frequency analysis."},{"cited_title":"Coti Zelati and A","cited_arxiv_id":null,"evidence_quote":"First uses the comparability condition and the definition of $\\mu$, and proves linear enhanced dissipation for the stratified Couette flow in three dimensions."}],"review_version":1}