{"id":"17448974-2d97-4e89-9948-7f74b6efae57","arxiv_id":"2412.00390","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global well-posedness is proven for the 3D inhomogeneous Navier-Stokes system with discontinuous density and small velocity in critical Besov spaces, yielding the first forward self-similar solutions.","lead":"This paper proves that the 3D inhomogeneous Navier-Stokes equations, which model fluids with non-uniform density, have global solutions for a wide class of discontinuous initial densities and small critical-space velocities. The result includes the first forward self-similar solutions for this system, a step toward understanding singularity formation in variable-density fluids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 depends on an unproved ρ*-uniformity claim; the critical pressure identity in Lemma 4.3 is imported from [25],[26] without constant tracking as the density lower bound tends to zero.","rationale":"The paper's central claim is Theorem 1.4: global well-posedness for initial velocities merely small in ˙B^{1/2}_{2,∞} with densities only bounded above, including vacuum. This strengthens previous critical Besov results and is the basis for the claimed first forward self-similar solution for (INS). I read Section 4 in detail to see whether the proof supports this. The a priori estimates mostly have constants depending only on ‖ρ0‖_{L∞}: for example, (4.4) uses √ρ0 ≤ ‖ρ0‖_{L∞}^{1/2}, and the Stokes estimates use ‖ρ ˙u‖_{L2} ≤ ‖ρ‖_{L∞}^{1/2}‖√ρ ˙u‖_{L2}. This suggests that the core estimates may indeed be uniform as the lower bound ρ* tends to zero, so the reader's concern is appropriately focused. However, the crucial Lemma 4.3 is not self-contained: its pressure identity and the bounds (4.14)–(4.15) are explicitly imported from two unpublished preprints, [25] and [26]. Those preprints are also used for the uniqueness method and for the weak-strong uniqueness argument of Theorem 1.3. No constant tracking with respect to the density lower bound is provided for this imported step, and the present text gives no way to verify that the constants remain finite as ρ* → 0. This is an honest gap: if the imported pressure estimates were to degrade as the density develops vacuum, the approximation argument in Section 4.2 would not yield a solution for merely bounded-above densities, and Theorem 1.4—and hence the headline self-similar claim—would not follow. The proposed check directly addresses this by re-deriving the pressure identity with explicit constants and with ρ allowed to vanish. Because the reader already identified the same weakest assumption and rendered a CONDITIONAL verdict, my stress-test does not move the verdict; it sharpens the specific step that must be verified: the ρ*-uniformity of Lemma 4.3 and its imported proofs.","tokens_in":38792,"tokens_out":35679,"duration_ms":317326,"concrete_test":"Independently derive (4.14)–(4.15) from the momentum equation and the pressure equation −ΔP = ∂_i∂_j(ρu_i u_j), allowing ρ to vanish, and track every constant against inf ρ; if any constant diverges as inf ρ → 0, or if any integration by parts requires uniform control of ρ^{-1}, the passage to vacuum in Theorem 1.4 fails. A direct check is to take the proof of Lemma 3.5 in arXiv:2406.19907, specialize it to the setting of Section 4, and verify that the expressions for J, Ψ, and the Gronwall inequality remain valid with constants depending only on ‖ρ0‖_{L∞} as the regularization ρ^ε = max(ρ0, ε) is sent to ε → 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.4 claims a unique global weak solution for u0 ∈ ˙B^{1/2}_{2,∞} small when the density is only bounded above, hence may have vacuum. The proof approximates smooth positive-density solutions and passes to the limit, which is legitimate only if every a priori estimate in Section 4 is uniform as the lower bound ρ* tends to zero. Footnote 3 asserts this uniformity but does not demonstrate it. The routine estimates in Lemmas 4.1 and 4.2 use only upper bounds on ρ: the Stokes estimates bound Δu by √ρ ˙u via ‖ρ‖_{L∞}^{1/2}, never by 1/ρ*. But Lemma 4.3, which supplies the crucial L1(T1,2T1;L∞) control of ∇u and the bounds on √ρ ˙u and ∇˙u used in Corollaries 4.1–4.2 and in the uniqueness proof, relies on a nontrivial pressure identity: J(t) = d/dt∫P∂_i u_j∂_j u_i − 3∫P∂_i u_j∂_j ˙u_i + 2∫P∂_i u_k∂_k u_j∂_j u_i, 'along the same lines as [25, Lemma 3.2] and [26, Lemma 3.5]'. The present paper does not prove this identity or the resulting inequalities (4.14)–(4.15); if their constants depended on ρ* in any way, the approximation argument for vacuum would collapse. The same imported estimates underpin weak-strong uniqueness (Theorem 1.3) via (3.27)–(3.29). Thus the load-bearing condition is that the pressure analysis of [25],[26] remains quantitatively independent of the lower density bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies global well-posedness for the three-dimensional inhomogeneous incompressible Navier-Stokes system in critical Besov spaces. The first main result (Theorem 1.2) extends the Cannone-Meyer-Planchon theory to data with density close to 1 in L∞ and momentum small in Ḃ_{p,∞}^{-1+3/p}, with uniqueness for 3<p<6. The second theorem (Theorem 1.3) establishes weak-strong uniqueness between this solution and Lions weak solutions under an additional L2 condition on the initial velocity, using a Besov-space decomposition of Barker. The third theorem (Theorem 1.4) claims global well-posedness with merely bounded above density, allowing vacuum, for initial velocity small in Ḃ_{2,∞}^{1/2}, and the abstract promotes this as the first existence result for forward self-similar solutions of the inhomogeneous Navier-Stokes system. The proofs are based on a series of a priori estimates, maximal regularity for the Stokes system, approximation by smooth positive-density solutions, and a final compactness passage.","tokens_in":39117,"tokens_out":10589,"duration_ms":100942,"significance":"If the results are correct, the paper makes a substantial contribution: it pushes the critical-space well-posedness theory for the inhomogeneous Navier-Stokes system to the larger space Ḃ_{2,∞}^{1/2}, permits discontinuous densities with vacuum, establishes a weak-strong uniqueness theorem, and provides a route to forward self-similar solutions. The proof strategy is largely explicit: the smallness conditions are stated quantitatively, the a priori estimates are derivable in a standard maximal-regularity framework, and the compactness arguments are indicated in detail. No fitted parameters or post hoc assumptions appear, and most technical ingredients are either proved in the paper or drawn from well-established references. The main caveat is the paper's reliance on an imported pressure identity in Lemma 4.3, whose uniformity with respect to the lower density bound is asserted but not demonstrated; this is exactly the point on which the vacuum case of Theorem 1.4 rests.","major_comments":[{"comment":"The proof of Theorem 1.4 passes to the limit as the lower density bound ρ* tends to zero, in order to allow initial densities that are only bounded above and may vanish. The estimates that make this passage legitimate are those of Lemma 4.3, in particular (4.12)–(4.15). The key identity for J(t) and the resulting inequalities (4.14)–(4.15) are imported 'along the same lines as [25, Lemma 3.2] and [26, Lemma 3.5]', with no proof in the present paper and no explicit tracking of the dependence of constants on inf ρ. Footnote 3 asserts that all constants are independent of ρ*, but this assertion is exactly the load-bearing fact for both the existence part (approximation by positive-density solutions) and the uniqueness part (via Corollaries 4.1–4.2 and Proposition 4.1). The authors should either include a self-contained proof of the J(t) identity and of (4.14)–(4.15) with constants depending only on ‖ρ0‖_{L∞}, or quote a precise lemma from [26] that covers the case of merely bounded density. As written, the manuscript does not demonstrate the asserted uniformity.","section":"§4.1, Lemma 4.3 and Footnote 3"},{"comment":"The advertised claim that Theorem 1.4 gives 'the first existence result of the forward self-similar solution for (INS)' is not actually derived. Remark 1.3 verifies only that the model velocity u0(x)=ε0|x|^{-1}(-x2/|x|,x1/|x|,0) belongs to Ḃ_{2,∞}^{1/2} and not to Ḣ^{1/2}; it does not show that the solution produced by Theorem 1.4 is self-similar. Since Theorem 1.4 provides uniqueness in a scaling-invariant class, the conclusion would follow by taking the initial density also invariant under the scaling and applying the uniqueness statement, but this argument is absent. The authors should add this argument or soften the claim to match what is proved.","section":"Remark 1.3 and Abstract"}],"minor_comments":[{"comment":"The notation ~L^1 in the statement of Theorem 1.1 is not defined in the notation section; if this denotes a Chemin-Lerner type time-space Besov space, it should be defined in Appendix A or replaced by the standard L^1(0,∞;·) notation used elsewhere.","section":"Theorem 1.1 and §1.3"},{"comment":"The compactness passage in the proof of the existence part of Theorem 1.4 is described as following by the Ascoli-Arzelà theorem after uniform bounds on ∂_t u_ε and on u_ε in L^2(δ0,t;̇H^1∩̇H^2). A more precise justification would cite the Aubin-Lions lemma or state the compact embedding used, since Ascoli-Arzelà normally requires continuity in space, which is not established here.","section":"§4.2"},{"comment":"The remark that ρ* is required only for smoothness and decay, while all constants are independent of ρ*, is central to the vacuum case; this statement should be moved into the body of §4 and proved or accompanied by a precise reference to the corresponding lemma in [26].","section":"Footnote 3"},{"comment":"In Proposition 4.1, the first two assumptions are essentially the statement B(t)≤Ct^{1/4} for the quantity B(t) defined in (4.22); the proof would be easier to follow if this equivalence were stated explicitly after the definition of B.","section":"§4.3, Proposition 4.1"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the reliance on the authors' unpublished preprints [25] and [26] for the pressure identity in Lemma 4.3, together with the asserted but unproved independence of the constants from the lower density bound. This is a load-bearing point for the advertised vacuum case of Theorem 1.4. If the authors can supply the missing proof or a precise quotation from [26] with the required uniformity, the main theorems are plausible and likely correct. I would not reject on this basis, but I would not accept the manuscript in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is a genuine step forward for the critical-space theory of the inhomogeneous Navier-Stokes system. It extends the Cannone-Meyer-Planchon result to p in (3,∞) with density close to a constant, proves weak-strong uniqueness against Lions weak solutions, and—the headline item—gets global well-posedness for u0 in Ḃ^{1/2}_{2,∞} small with the density only bounded above. That last part is new and, if correct, gives the first forward self-similar solutions for INS. The example u0(x)=ε0/|x|(-x2/|x|, x1/|x|,0) is in that space but not Ḣ^{1/2}, so the improvement over [26] is real.\n\nThe proof in Sections 2 and 3 is detailed and looks sound to me. The high-low frequency argument in Section 4 is genuinely clever: Lemma 4.1 shows how to get the L^{3,∞} bound on √ρu directly from Littlewood-Paley pieces without summability in the third index. The use of Barker's decomposition for the weak-strong uniqueness is also well executed.\n\nThe soft spot is exactly what the stress-test note flags. Theorem 1.4 allows vacuum, so every estimate has to be uniform as the density lower bound ρ* tends to zero. The footnote claims this uniformity, but the key pressure identity in Lemma 4.3 is simply imported from the authors' unpublished preprints [25] and [26]—with no derivation and no constant tracking. If the constants in (4.14)–(4.15) depend on ρ* in any way, the approximation argument collapses. This is load-bearing, not cosmetic. I don't think it's wrong: the identity is algebraic and the ρ* dependence usually cancels, but the paper does not give a referee the material to check it.\n\nThere is also a smaller issue: the uniqueness in Theorem 1.2 is restricted to p<6, which looks like a technical limitation rather than a genuine boundary. That's fine, but it should be stated more clearly.\n\nBottom line: this is a serious paper by serious people, and it deserves a real referee. My recommendation is to send it to review, but with the condition that the authors provide the missing proofs or make [25] and [26] available, and that the ρ*-uniformity argument be written out explicitly. If that comes through, the theorem stands. I would cite it if I worked in this area.","headline":"Genuine critical-space advance for inhomogeneous Navier-Stokes with a load-bearing but likely fixable gap in the vacuum-allowance proof.","tokens_in":39736,"tokens_out":6106,"would_cite":true,"duration_ms":54925,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D03","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system for initial velocities small in the critical Besov space $\\dot B^{1/2}_{2,\\infty}$ and densities merely bounded above, with the first…","keywords":["inhomogeneous Navier-Stokes equations","global well-posedness","critical Besov spaces","self-similar solutions","vacuum states","density patches","weak-strong uniqueness","Lorentz spaces"],"falsifier":"Check the constant dependence in Lemma 4.1 explicitly: for a fixed small $u_0$ in $\\dot B^{1/2}_{2,\\infty}$ and a family of smooth densities $\\rho_\\varepsilon$ with $\\inf\\rho_\\varepsilon=\\varepsilon$, if the claimed bound on $\\|\\sqrt{\\rho}u\\|_{L^\\infty L^{3,\\infty}}$ or $\\|t^{1/4}\\nabla u\\|_{L^\\infty L^2}$ grows like a power of $\\varepsilon^{-1}$, the vacuum passage in Theorem 1.4 collapses; likewise, if two different positive-density approximation sequences converge to different limits for the same data, uniqueness would fail.","tokens_in":38573,"feed_emoji":"🌊","tokens_out":8429,"duration_ms":68360,"temperature":0.7,"pith_summary":"Global well-posedness is proved for the 3-D inhomogeneous incompressible Navier-Stokes system when the initial velocity is small in the critical Besov space $\\dot B^{1/2}_{2,\\infty}$ and the initial density is merely nonnegative and bounded above, so discontinuous densities and vacuum regions are allowed. This yields the first existence result for forward self-similar solutions of the inhomogeneous system. The paper also establishes a Cannone-Meyer-Planchon analogue for variable density: small momentum $\\rho_0 u_0$ in $\\dot B^{-1+3/p}_{p,\\infty}$ with $\\rho_0$ close to 1 in $L^\\infty$ gives global strong solutions, unique for $3<p<6$, and these are weak-strong unique among Lions weak solutions when $u_0\\in L^2$. A sympathetic reader should care because these results remove regularity and lower-bound restrictions on the density that earlier critical-space theory required.","feed_headline":"First self-similar solutions for variable-density Navier-Stokes","feed_subtitle":"Small critical-space velocities force global uniqueness, even for densities with vacuum regions.","key_machinery":"The central mechanism is to rewrite the momentum equation for the projected variable $v=\\mathcal P(\\rho u)$, where $\\mathcal P$ is the Leray projector; its equation is $\\partial_t v-\\Delta v=\\Delta(u-v)-\\mathcal P\\operatorname{div}(\\rho u\\otimes u)$, and smallness of $\\|\\rho_0-1\\|_{L^\\infty}$ and $\\|m_0\\|_{\\dot B^{-1+3/p}_{p,\\infty}}$ lets the $\\Delta(u-v)$ term be absorbed, yielding the critical $L^{q,\\infty}_tL^p_x$ estimate through O'Neil's convolution inequality and Stokes maximal regularity. For the $\\dot B^{1/2}_{2,\\infty}$ result, the machinery is a dyadic high-low frequency decomposition of the velocity, estimating each block $u_j$ and summing to obtain $\\|\\sqrt{\\rho}u\\|_{L^\\infty L^{3,\\infty}}$ and $\\|t^{1/4}\\nabla u\\|_{L^\\infty L^2}$; uniqueness uses a relative-energy inequality in which the difference $(\\delta\\rho,\\delta u)$ is controlled by $t^{1/4}$ bounds and a Gr\\\"onwall step that lets the initial time $\\delta\\to 0$.","core_discovery":"The central claim is that the borderline regularity class for global well-posedness of the inhomogeneous Navier-Stokes system can be pushed to $\\dot B^{1/2}_{2,\\infty}$: for divergence-free $u_0$ with $\\|u_0\\|_{\\dot B^{1/2}_{2,\\infty}}$ small and $0\\le \\rho_0\\le \\|\\rho_0\\|_{L^\\infty}$ with $\\rho_0\\not\\equiv 0$, there is a unique global weak solution, and the solution obeys bounds such as $t^{1/4}\\nabla u\\in L^\\infty(\\mathbb R_+;L^2)$ and $\\sqrt{\\rho}u\\in C([0,\\infty);L^{3,\\infty})$. This is the first existence of forward self-similar solutions for the inhomogeneous system, because an example such as $u_0(x)=\\varepsilon_0 |x|^{-1}(-x_2/|x|,x_1/|x|,0)$ lies in $\\dot B^{1/2}_{2,\\infty}$ but not in $\\dot H^{1/2}$. Alongside this, the paper proves a Cannone-Meyer-Planchon type global strong solution for variable density under smallness of $\\rho_0u_0$ in $\\dot B^{-1+3/p}_{p,\\infty}$ and closeness of $\\rho_0$ to 1 in $L^\\infty$, with uniqueness for $3<p<6$ and weak-strong uniqueness against Lions weak solutions when $u_0\\in L^2$.","pith_inferences":["An implicit corollary is that the forward self-similar profiles can carry density discontinuities and vacuum at every positive time, so the result speaks directly to patch and bubble configurations in a critical scaling.","The use of a third Besov index $+\\infty$ plus a high-low frequency split suggests the borderline velocity class is not tied to summability; a different low-frequency regularization might reach spaces such as $BMO^{-1}$, though the methods here stop at $p<+\\infty$.","The asserted independence of the density lower bound could be tested numerically or analytically: take densities with $\\inf\\rho=\\varepsilon$, keep $u_0$ fixed and small in $\\dot B^{1/2}_{2,\\infty}$, and verify that all constants in Lemma 4.1 stay independent of $\\varepsilon$ as $\\varepsilon\\to 0$."],"forward_implications":["Forward self-similar solutions of the inhomogeneous Navier-Stokes system exist for the first time; the example $u_0(x)=\\varepsilon_0 |x|^{-1}(-x_2/|x|, x_1/|x|, 0)$ is admissible because it lies in $\\dot B^{1/2}_{2,\\infty}$ but not in $\\dot H^{1/2}$.","Unique global weak solutions exist for densities that are only bounded above, including vacuum regions and density discontinuities, provided the critical Besov norm of $u_0$ is small.","Cannone-Meyer-Planchon critical-space theory carries over to variable density: global strong solutions exist for $3<p<\\infty$ and are unique for $3<p<6$.","Any Lions weak solution with the same data coincides with the constructed strong solution when $u_0\\in L^2$, giving weak-strong uniqueness.","Highly oscillatory initial velocities with critical scaling are admissible, widening the class of initial data beyond $\\dot H^{1/2}$ and $L^3$."],"supporting_citations":[{"why":"Provides the Cannone-Meyer-Planchon theorem for classical Navier-Stokes that Theorem 1.2 extends to variable density.","marker":"[10]"},{"why":"Supplies the bounded-density critical framework and uniqueness and weak-strong uniqueness techniques used throughout.","marker":"[20]"},{"why":"Establishes global well-posedness with $u_0\\in\\dot H^{1/2}$ and bounded density, the result Theorem 1.4 sharpens to $\\dot B^{1/2}_{2,\\infty}$, and supplies the treatment of $\\delta\\rho$.","marker":"[26]"},{"why":"Gives the first critical-space global existence for the inhomogeneous system with bounded density, the baseline for allowing only upper-bounded density.","marker":"[37]"},{"why":"Defines the Leray weak-solution class that the weak-strong uniqueness result of Theorem 1.3 is measured against.","marker":"[30]"},{"why":"Provides the Lions weak solutions for the inhomogeneous system that Theorem 1.3 identifies with the constructed solution.","marker":"[31]"},{"why":"Supplies the Besov-space decomposition used to prove weak-strong uniqueness in Theorem 1.3.","marker":"[9]"},{"why":"Provides the transport-equation estimates that control the density difference in the uniqueness proofs.","marker":"[7]"},{"why":"Supplies maximal regularity estimates for the Stokes and heat systems in Lorentz spaces used in the critical estimates.","marker":"[18]"},{"why":"Gives the O'Neil convolution inequality in Lorentz spaces that underlies the critical $L^{q,\\infty}_tL^p_x$ estimate.","marker":"[32]"}],"fun_headline_variants":["First self-similar solutions for variable-density Navier-Stokes","Self-similar flows emerge at borderline regularity for Navier-Stokes","Global uniqueness for variable-density flows at critical regularity","New existence for inhomogeneous Navier-Stokes at borderline space","Self-similar solutions open up for variable-density Navier-Stokes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that all the a priori estimates in Section 4 remain valid with constants independent of the positive lower bound on the density, so that solutions with vacuum can be obtained as limits of positive-density solutions.","fun_headline_variants_meta":{"raw":{"variants":["First self-similar solutions for variable-density Navier-Stokes","Self-similar flows emerge at borderline regularity for Navier-Stokes","Global uniqueness for variable-density flows at critical regularity","New existence for inhomogeneous Navier-Stokes at borderline space","Self-similar solutions open up for variable-density Navier-Stokes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000557,"raw_usage":{"total_tokens":2743,"prompt_tokens":1128,"completion_tokens":1615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":1530}},"tokens_in":744,"tokens_out":1615,"duration_ms":12805,"temperature":1.0,"reasoning_tokens":1530,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:25:47.270179+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the constant dependence in Lemma 4.1 explicitly: for a fixed small $u_0$ in $\\dot B^{1/2}_{2,\\infty}$ and a family of smooth densities $\\rho_\\varepsilon$ with $\\inf\\rho_\\varepsilon=\\varepsilon$, if the claimed bound on $\\|\\sqrt{\\rho}u\\|_{L^\\infty L^{3,\\infty}}$ or $\\|t^{1/4}\\nabla u\\|_{L^\\infty L^2}$ grows like a power of $\\varepsilon^{-1}$, the vacuum passage in Theorem 1.4 collapses; likewise, if two different positive-density approximation sequences converge to different limits for the same data, uniqueness would fail.","supporting_citations":[{"cited_title":"Cannone, Y","cited_arxiv_id":null,"evidence_quote":"Provides the Cannone-Meyer-Planchon theorem for classical Navier-Stokes that Theorem 1.2 extends to variable density."},{"cited_title":"Danchin and S","cited_arxiv_id":null,"evidence_quote":"Supplies the bounded-density critical framework and uniqueness and weak-strong uniqueness techniques used throughout."},{"cited_title":"Zhang, Global Fujita-Kato solution of 3-D inhomogen eous incompressible Navier-Stokes system","cited_arxiv_id":null,"evidence_quote":"Gives the first critical-space global existence for the inhomogeneous system with bounded density, the baseline for allowing only upper-bounded density."},{"cited_title":"Leray, Sur le mouvement d’un liquide visqueux emplis sant l’espace","cited_arxiv_id":null,"evidence_quote":"Defines the Leray weak-solution class that the weak-strong uniqueness result of Theorem 1.3 is measured against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Lions weak solutions for the inhomogeneous system that Theorem 1.3 identifies with the constructed solution."},{"cited_title":"Barker, Uniqueness results for weak Leray-Hopf solut ions of the Navier-Stokes system with initial values in critical spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the Besov-space decomposition used to prove weak-strong uniqueness in Theorem 1.3."},{"cited_title":"Bahouri, J","cited_arxiv_id":null,"evidence_quote":"Provides the transport-equation estimates that control the density difference in the uniqueness proofs."},{"cited_title":"Danchin, P","cited_arxiv_id":null,"evidence_quote":"Supplies maximal regularity estimates for the Stokes and heat systems in Lorentz spaces used in the critical estimates."},{"cited_title":"O’Neil, Convolution operators and L(p, q ) spaces","cited_arxiv_id":null,"evidence_quote":"Gives the O'Neil convolution inequality in Lorentz spaces that underlies the critical $L^{q,\\infty}_tL^p_x$ estimate."}],"review_version":1}