{"id":"db59a14d-0165-494b-ad22-28991fa55535","arxiv_id":"2607.13694","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Norm inflation is proven for 2D inviscid and fully dissipative Boussinesq systems in almost all supercritical Besov spaces, with the density, not the velocity, carrying the blow-up.","lead":"This paper proves that the 2D Boussinesq equations are strongly ill-posed in almost all supercritical Besov spaces: arbitrarily small smooth initial data can develop a density component whose Besov norm is arbitrarily large in arbitrarily short time. The result covers inviscid and fully dissipative versions, with the inflation always carried by the density while the velocity stays small.","discovery_kind":"new_application","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proves norm inflation (strong ill-posedness) for the 2D Boussinesq system in supercritical homogeneous Besov spaces. For the inviscid system, Theorem 1.1 asserts that for β≠0, 1<p≤∞, 1≤q,r≤∞, and −2<β−2/p<1, for every ε>0 there are C_c^∞ initial data with sum of the B^β_{p,1} norms <ε and a time t*<ε such that the density's B^β_{p,∞} norm exceeds 1/ε, while the velocity remains bounded. For the fully dissipative system, Theorem 1.2 asserts the same conclusion in the range −2<β−2/p<−1. The construction is explicit: a stationary radial vorticity creates an angular shear, and the density is transported by the associated velocity. For β>0 the shear creates rapid radial oscillations by time t*; for β<0 the initial density is premixed and then unwound. A perturbation (stability) argument transfers the growth from the approximate solution to the exact solution. The main λ-exponent bookkeeping appears consistent, with the admissible ranges exactly where the Gronwall factors and the final error terms decay.","tokens_in":25237,"tokens_out":25074,"duration_ms":298375,"significance":"If correct, the paper establishes a strong ill-posedness result in almost all supercritical locally integrable Besov spaces for the inviscid Boussinesq system and in a substantial supercritical range for the fully dissipative system, including negative regularity. This is a significant advance over the existing critical-space ill-posedness results and is directly in line with the recent Euler/Navier–Stokes norm-inflation program of Luo and others. The construction is explicit and largely parameter-free: the approximate solution is given in closed form, the smallness of the premixed negative-β data is proved by duality rather than assumed, and the lower bound at t* is computed directly. These are genuine strengths. The main weaknesses are two load-bearing technical points that are not fully written out: the endpoint case of the Besov duality statement used for p=∞, and the compressed stability proof in the fully dissipative case. Both are fixable, but they need to be addressed before the claims are fully verified.","major_comments":[{"comment":"The stated Besov duality “for 1≤p,r≤∞” is not standard at p=∞ or r=∞. In the usual form, ∙B^s_{p,r}∙ is the dual of ∙B^{-s}_{p',r'}∙ only in the reflexive range; at p=∞ or r=∞ one must work with the appropriate “small Besov” completion. This proposition is used one-directionally at endpoints in Lemma 3.1 (q=∞) and Lemma 3.5 (p=∞), and those endpoint cases are part of Theorems 1.1 and 1.2. Please state the precise predual characterization, give a proof or a precise reference, and confirm that the estimates (3.10), (3.23)–(3.24) indeed yield the claimed Besov bounds when p=∞ or q=∞.","section":"§2.1, Proposition 2.3"},{"comment":"The stability transfer in the fully dissipative case is the most compressed step of the paper. The proof asserts that “using the standard commutator estimate gives inductively” the bound (4.15), but it does not display the commutator expansions, the pressure contributions, or the precise role of the bootstrap assumption (4.13). Since (4.12) is exactly what forces the admissible range β−2/p<−1, this is load-bearing. A complete induction with all product, commutator, pressure, and diffusion error terms should be supplied, at least for the first nontrivial derivative order.","section":"§4.1, Corollary 4.1"},{"comment":"In the inviscid stability lemma, the high-order L^2 induction and its closure are also only sketched. In particular, (3.35) is asserted rather than derived, and the final closure of the L^∞ bootstrap (3.31) relies on the Gagliardo–Nirenberg interpolation in Step 4 without displaying the interpolation parameters for all needed ranges of k and q. Since this lemma is the mechanism that transfers the approximate norm inflation to the exact solution, the missing details should be written out. The argument is plausible, but as written it is not fully verifiable.","section":"§3.2, Lemma 3.6"}],"minor_comments":[{"comment":"The construction of f and g is correct, but the verification that f^{(-2)} is compactly supported uses the convention f^{(-1)}(r)=∫_{-∞}^r f(s)ds; since f vanishes for r<1/2, this is fine but should be stated explicitly once for clarity.","section":"§2.2, Lemma 2.6 / Appendix A"},{"comment":"The notation “Eλ≤exp(CL^2)=λ^{o(1)}” is used without tracking the constants C. It would help to state that C may depend on k and q but not on λ, and that all λ-powers are uniform.","section":"§3.2, Lemma 3.6"},{"comment":"At the end of the proof, the phrase “standard commutator estimates” should be accompanied by a precise reference to a textbook inequality (e.g., Kato–Ponce or the commutator estimates in [1]) for the Besov/Leibniz products used in both the inviscid and dissipative stability proofs.","section":"§4.1, Corollary 4.1"},{"comment":"The claim that the argument also applies to the fractionally dissipative system and gives ill-posedness in H^s(R^2) for −1<s<2−max{α,γ}, s≠0, is not proved or even sketched. This is a minor issue if the remark is intended only as an outlook, but it should be labeled as such.","section":"§1, Remark 1.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in substance and is a strong contribution to the ill-posedness literature for fluid equations. The two main gaps — the endpoint Besov duality and the compressed dissipative stability proof — are fixable but genuinely load-bearing, and the current text does not allow a referee to verify them without substantial additional work. I would support publication after the authors supply the missing details and clarify the endpoint statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a genuinely useful paper, and I think the reader's conditional verdict is about right. Chen and Xie construct an explicit family of initial data showing norm inflation for the 2D Boussinesq system in supercritical Besov spaces, for both the inviscid and fully dissipative cases, in ranges that nearly exhaust the locally integrable supercritical regime. The new ingredient compared with the Boussinesq ill-posedness results at critical scale (Elgindi–Masmoudi, Bianchini–Hientzsch–Iandoli, Li–Wang) is that the construction is a norm-inflation one: a stationary radial vorticity shears a density profile, and the density component grows while the velocity stays bounded. For negative β they premix and let the shear unwind, which is a neat twist.\n\nThe paper is honest and carefully written. The approximate solution is explicit, the profile lemma is proven in the appendix, and the λ-exponent bookkeeping I checked is consistent. The smallness of the initial data, including the β<0 premixed case, is proven by direct duality estimates rather than assumed. The ranges β−2/p<1 and β−2/p<−1 are exactly where the error terms (3.46)/(4.19) decay, so the thresholds match the known local well-posedness scales. I also appreciate Remark 1.3 admitting the fractional-dissipation gap and the abstract's explanation of why ρ can inflate in a space that is subcritical for ρ itself.\n\nThe soft spots are exactly the two the reader flagged. Corollary 4.1, the fully dissipative stability transfer, is compressed: the induction is said to close via 'standard commutator estimates' at precisely the step that produces the more restrictive range β−2/p<−1. A referee should ask to see the induction written out, including the bootstrap norm (4.13) and why the viscous forcing terms don't change the structure. The second is Proposition 2.3, the Besov duality characterization, stated at the endpoint p=∞/r=∞ without comment; it's used one-directionally in Lemmas 3.1 and 3.5, where it is standard, so this is a footnote-level fix rather than a substantive gap. The stability argument also relies on classical local well-posedness/continuation for smooth solutions; that is standard for smooth data, but at p=∞ a one-sentence reference would help.\n\nNone of these look load-bearing. The central construction is sound, the citation pattern is appropriate, and the result is new. This deserves a serious referee: I would recommend accepting it for peer review, with the expectation of a revision that expands Corollary 4.1 and qualifies the duality statement. It's not a paper for everyone, but anyone working on ill-posedness for fluid equations or using norm-inflation machinery will want it.","headline":"Solid, likely correct norm-inflation construction for 2D Boussinesq in supercritical Besov spaces, with sharp thresholds; the dissipative stability step needs more detail before publication.","tokens_in":25826,"tokens_out":2883,"would_cite":true,"duration_ms":577796,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35R25","35A01"],"pacs":[],"model":"deepseek-v4-flash","headline":"The 2D Boussinesq system is strongly ill-posed in nearly every supercritical Besov space: arbitrarily small smooth data can grow the density's norm past 1/ε in time <ε, while velocity stays bounded.","keywords":["Boussinesq system","norm inflation","strong ill-posedness","Besov spaces","supercritical regularity","inviscid","fully dissipative","density transport"],"falsifier":"Run a high-resolution numerical simulation of the inviscid Boussinesq system with the paper's compactly supported initial data and measure $\\|\\rho(t_*)\\|_{\\dot B^{\\beta}_{p,\\infty}}$ at $t_* = \\lambda^{\\beta - 2/p - 1} \\log\\log\\lambda$: if the density norm fails to exceed $1/\\epsilon$ for arbitrarily small $\\epsilon$ while the velocity stays bounded, the stability transfer is wrong. A targeted analytic check is whether the bootstrap bound (3.31) closes for $\\beta - 2/p$ just below $1$; if the Gronwall factor no longer tends to $0$, the transfer step is the point of failure.","tokens_in":25111,"feed_emoji":"📈","tokens_out":9531,"duration_ms":79834,"temperature":0.7,"texified_at":"2026-08-05T21:21:24.141561+00:00","pith_summary":"This paper proves that the two-dimensional Boussinesq system is strongly ill-posed in almost every supercritical Besov space, for both the inviscid and the fully dissipative versions. Concretely, for arbitrarily small $\\epsilon$, smooth compactly supported initial data with Besov norm below $\\epsilon$ can evolve so that the density's Besov norm exceeds $1/\\epsilon$ before time $\\epsilon$, while the velocity stays bounded. This establishes norm inflation, the classic signature of strong ill-posedness, and shows that the density transport mechanism, not the velocity equation, drives the instability. The result holds in $\\dot B^{\\beta}_{p,q}(\\mathbb{R}^2)\\times \\dot B^{\\beta}_{p,r}(\\mathbb{R}^2)$ with $\\beta\\neq 0$, $1<p\\leq\\infty$, and $-2<\\beta-2/p<1$ in the inviscid case, $-2<\\beta-2/p<-1$ in the dissipative case. Because the inflation time tends to $0$ and the initial data are smooth, the solution map is discontinuous at the origin in these spaces.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8606,"prompt_tokens":1017,"completion_tokens":7589,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":1017,"completion_tokens_details":{"reasoning_tokens":6503}},"feed_headline":"Density norm blows up for Boussinesq systems in supercritical spaces","feed_subtitle":"Arbitrarily small smooth initial data push density's Besov norm above 1/ε in time <ε; velocity stays bounded.","key_machinery":"Central object: the approximate solution is a stationary radial vorticity $\\bar w = \\lambda^{2/p+1-\\beta}(\\log\\log\\lambda)^{-|\\beta|/(2+|\\beta|)} f(\\lambda r)$, whose Biot–Savart velocity $\\bar u$ is time-independent, purely angular, with angular frequency $\\Omega_\\lambda(r) = \\lambda^{2/p-\\beta}(\\log\\log\\lambda)^{-|\\beta|/(2+|\\beta|)} u_\\theta[f](\\lambda r)/r$. The density is the initial profile advected by this flow; the profiles $f,g$ have disjoint supports and nondegenerate shear $|h'|\\geq c_0$ on supp $g$, so at $t_* = \\lambda^{-2/p-1+\\beta}\\log\\log\\lambda$ the shear creates or unwinds rapid radial oscillations and yields the Besov lower bound. A stability estimate transfers the growth to the exact solution; in the dissipative case the Laplacians are perturbative errors, shifting the range by two derivativ","core_discovery":"Inviscid case ($\\mu=\\nu=0$): norm inflation in $\\dot B^{\\beta}_{p,q}\\times \\dot B^{\\beta}_{p,r}$ for $\\beta\\neq 0$, $1<p\\leq\\infty$, $-2<\\beta-2/p<1$. Fully dissipative case ($\\mu=\\nu=1$): same conclusion for $-2<\\beta-2/p<-1$. In both, for every $\\epsilon>0$ there is a $C_c^\\infty$ datum $(u_0,\\rho_0)$ with $\\|u_0\\|_{\\dot B^{\\beta}_{p,1}}+\\|\\rho_0\\|_{\\dot B^{\\beta}_{p,1}}<\\epsilon$ and a time $0<t_*<\\epsilon$ with $\\|\\rho(t_*)\\|_{\\dot B^{\\beta}_{p,\\infty}}>1/\\epsilon$. Embedding $\\dot B^{\\beta}_{p,1}\\hookrightarrow \\dot B^{\\beta}_{p,q}$ and $\\|\\cdot\\|_{\\dot B^{\\beta}_{p,r}}\\geq \\|\\cdot\\|_{\\dot B^{\\beta}_{p,\\infty}}$ give the result for every $q,r$. The velocity stays bounded in $\\dot B^{\\beta}_{p,\\infty}$; only the density inflates.","pith_inferences":["The paper's mechanism suggests a general criterion: any equation in which a passive scalar is advected by a flow with supercritical Besov regularity can exhibit density norm inflation, even when the scalar's own scaling looks subcritical; testing this on other stratified-flow models (anelastic, primitive) would show whether the phenomenon is generic.","A quantitative next step is to determine whether the rate (log log λ)^{|β|/(2+|β|)} is optimal for these spaces; if a faster growth rate can be constructed, the current bound is not sharp.","Because t*→0 as λ→∞, the construction implies instantaneous loss of regularity at t=0 in the supercritical regime; this could be made explicit by extracting a sequence of solutions with blow-up times tending to zero.","The condition β−2/p>−2 confines the result to locally integrable data; the complementary distributional range is untouched by this construction and may require a genuinely different mechanism."],"forward_implications":["The solution map of the Boussinesq system is discontinuous at the origin in every Besov space covered: strong ill-posedness holds throughout the supercritical locally integrable range.","For the inviscid system the upper threshold β−2/p<1 is sharp against the known local well-posedness at one derivative above, and for the dissipative system β−2/p<−1 matches the velocity-critical parabolic scaling.","Norm inflation is driven purely by the density transport: even when the density's own scaling is subcritical (dissipative case), a velocity in the supercritical regime amplifies ρ above 1/ε while u stays small.","The same construction yields norm inflation in H^s for the fractionally dissipative system in the range −1<s<2−max{α,γ}, s≠0."],"fun_headline_variants":["Boussinesq density blows up in supercritical Besov spaces","Norm inflation for Boussinesq: density explodes, velocity stays bounded","Small data, huge density: Boussinesq norm inflation in supercritical spaces","Inviscid and dissipative Boussinesq: density norm inflates, velocity bounded"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the exact Boussinesq solution with the paper's initial data stays within $o(1)$ of the explicit shear-transport approximate solution in the inflation norm up to the inflation time $t_*$; the Gronwall/bootstrap estimates that enforce this close only under the stated regime conditions, and the transfer also assumes smooth solutions exist and extend through $[0,t_*]$.","fun_headline_variants_meta":{"raw":{"variants":["Boussinesq density blows up in supercritical Besov spaces","Norm inflation for Boussinesq: density explodes, velocity stays bounded","Small data, huge density: Boussinesq norm inflation in supercritical spaces","Inviscid and dissipative Boussinesq: density norm inflates, velocity bounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1375,"prompt_tokens":857,"completion_tokens":518,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":601,"tokens_out":518,"duration_ms":5611,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T04:02:33.049325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a high-resolution numerical simulation of the inviscid Boussinesq system with the paper's compactly supported initial data and measure $\\|\\rho(t_*)\\|_{\\dot B^{\\beta}_{p,\\infty}}$ at $t_* = \\lambda^{\\beta - 2/p - 1} \\log\\log\\lambda$: if the density norm fails to exceed $1/\\epsilon$ for arbitrarily small $\\epsilon$ while the velocity stays bounded, the stability transfer is wrong. A targeted analytic check is whether the bootstrap bound (3.31) closes for $\\beta - 2/p$ just below $1$; if the Gronwall factor no longer tends to $0$, the transfer step is the point of failure.","supporting_citations":[],"review_version":1}