{"id":"63e10676-a238-404b-b1f1-41f6970b7835","arxiv_id":"2412.10875","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global weak martingale solutions exist for the stochastic 1D quantum Navier-Stokes equations with density-dependent viscosity and vacuum, for viscosity exponents alpha in (1/2, 1].","lead":"This paper proves that the one-dimensional quantum Navier-Stokes equations, a fluid model with capillarity and density-dependent viscosity, have global weak solutions even when driven by random noise and starting from random data, and with vacuum regions allowed. It is the first existence result for the stochastic version of this model in the weak solution framework.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Existence proof rests on unproved Theorem 4.3 imported from companion preprint arXiv:2401.10064; the sketched verification does not establish the ε-independent estimates, so the main theorem's foundation is currently unverified.","rationale":"The paper's stated goal is to prove global existence of weak dissipative martingale solutions. The proof strategy is standard: build a strong approximating sequence, derive uniform-in-ε estimates, use stochastic compactness, and pass to the limit. The weakest point is the source of those uniform estimates: Theorem 4.3 is quoted from a companion preprint and is not derived in this manuscript. The text itself flags this in the proof sketch of Section 4 ('we refer the reader to [26] for the detailed computations') and in Remark 4.4, where the lower density bound is said to degenerate in ε. This external dependency is load-bearing because every later proposition assumes the approximating sequence exists with the specified ε-independent bounds. The reader's verdict of CONDITIONAL is appropriate: the result is plausible and the arguments align with the deterministic literature, but the main theorem cannot be accepted as proven until Theorem 4.3 is independently verified. I considered whether an internal gap in Proposition 5.3 (the claimed tightness on L^2(0,T;H^1)) should be the primary concern; it is a real flaw in the written proof, but it appears repairable by choosing a weaker path space such as L^2(0,T;L^2), because the subsequent limit passages use strong convergence of ρ_ε in L^p and of ∂_xρ_ε^{α/2} in L^2, not strong H^1 convergence of ρ_ε. The unverified imported theorem is therefore the more fundamental obstruction to the central claim.","tokens_in":33162,"tokens_out":10472,"duration_ms":98555,"concrete_test":"Inspect arXiv:2401.10064, Theorem 2.4, and verify that its hypotheses cover the exact system (4.1) with the ε∂_xxu term and the chosen noise coeffcients, and that estimates (4.6)-(4.9) hold with constants independent of ε as claimed. In particular, re-derive the Gronwall step leading to (4.7) to confirm that no ε-dependent lower bound on ρ enters; if any constant in (4.6)-(4.8) degenerates as ε→0, the tightness in Section 5.3 and the limit identification fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 3.3 is established by passing to the limit in the approximating system (4.1). Every uniform bound used for tightness (Propositions 5.3-5.9) and for the limit passages in Section 5.4 is taken from Theorem 4.3, which asserts global well-posedness of (4.1) with regularity estimates (4.6)-(4.9). This theorem is not proved here: the authors state it is 'obtained in [26], Theorem 2.4' and provide only a sketch that refers to [26] for the detailed computations. The sketch itself derives (4.6) and indicates (4.7), but the lower bound (4.9) is explicitly not uniform in ε (Remark 4.4), and the proof of the energy/BD estimates for the ε-dependent extra viscosity term is not carried out. If Theorem 4.3 is false, or if the constants in (4.6)-(4.8) depend on ε in an uncontrolled way, then the family (ρ_ε,u_ε) may not exist with the regularity needed for the Skorokhod compactness argument, and the whole convergence proof has no starting point. Since the paper gives no self-contained verification of this black box, the main theorem is conditionally supported by an unreviewed companion preprint. A related internal issue is that Proposition 5.3 claims tightness on L^2(0,T;H^1) via a compactness argument that appears invalid (a bounded set in H^1 is not compact in H^1), though that gap looks repairable by working with a weaker path space; the import of Theorem 4.3 is the more load-bearing dependency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims global existence of weak dissipative martingale solutions for the one-dimensional stochastic Quantum-Navier-Stokes equations with density-dependent viscosity and multiplicative noise, allowing vacuum states. The proof introduces an epsilon-level artificial viscosity, invokes a global strong well-posedness result with epsilon-independent estimates for the approximating system, derives uniform energy and BD-entropy bounds, and then passes to the limit via a velocity truncation and a Jakubowski-Skorokhod compactness argument. The main theorem is Theorem 3.3, with the novel content being the first stochastic weak-solution existence result for this QNS model. The overall strategy is recognizable from the deterministic literature ([8], [33]) and appears plausible, but the manuscript as submitted contains several load-bearing gaps, including an unproved foundational theorem imported from a companion preprint and an invalid compactness step.","tokens_in":33488,"tokens_out":16797,"duration_ms":149164,"significance":"If completed, the result would be a meaningful extension of the deterministic weak-solution theory for Quantum-Navier-Stokes systems to the stochastic setting, covering the viscosity range alpha in (1/2,1] with possible vacuum and multiplicative noise of rather low regularity. The combination of BD entropy, velocity truncation, and stochastic compactness is a natural and potentially valuable framework. The manuscript does not ship machine-checked proofs or reproducible code; its value rests entirely on the analytic verification. The proof is not self-contained: the existence of the approximating sequence and all uniform bounds come from Theorem 4.3, which is quoted from the authors' companion preprint, and one of the tightness arguments used later is invalid as written. These issues do not necessarily destroy the central claim, but they make the present version unsuitable for publication without substantial revision.","major_comments":[{"comment":"The main theorem depends on Theorem 4.3, which asserts global strong well-posedness of the approximating system and the epsilon-independent estimates (4.6)-(4.9). This theorem is not proved in the present manuscript; the proof sketch derives only the energy identity (4.11) and says that (4.7) follows by the same argument, while the local and maximal existence theory is referred to [26]. All tightness bounds in Section 5.2 are taken from (4.6)-(4.8), so if Theorem 4.3 is not available with the claimed uniformity, the compactness argument has no starting sequence. The paper must either give a complete proof of Theorem 4.3, or state it as a clearly delineated external result with precise epsilon-uniform constants and a verification that the estimates used here are indeed those proved in [26]. Remark 4.4 also makes item (3) ambiguous: (4.9) is displayed without an epsilon factor and then said to degenerate as epsilon tends to 0.","section":"§4, Theorem 4.3"},{"comment":"The proof of tightness on X_rho = L^2(0,T;H^1) is invalid. The Aubin-Lions embedding cited in (5.11) gives compactness only in L^2(0,T;L^2), not in L^2(0,T;H^1). The set K = B_L ∩ {||partial_x rho||_{L^2_t,x} <= L~} is bounded in L^2(0,T;H^1), but a bounded set in H^1 is not precompact in H^1 because H^1 does not compactly embed into itself. Therefore the conclusion that K is compact in X_rho is false, and the convergence rho_epsilon -> rho in L^2(0,T;H^1) asserted in Proposition 5.11 and used in Lemma 5.12 is not justified. This gap appears repairable by working on the weaker path space L^2(0,T;L^2) for rho and proving strong convergence of partial_x rho^{alpha/2} separately through the interpolation argument in (5.12), but the current text does not do that.","section":"§5.3, Proposition 5.3"},{"comment":"The passage to the limit in the pressure term is not justified. Equation (5.36) claims convergence of the term involving 2 rho_epsilon^{gamma/2} partial_x rho_epsilon^{gamma/2} beta'_delta(u_epsilon) by appealing to (5.24) and (5.1). However, Lemma 5.12(3) is stated and proved for partial_x rho^{alpha/2}, not for partial_x rho^{gamma/2}. No uniform bound for partial_x rho^{gamma/2} appears in the estimates (5.4), and gamma is arbitrary in (1,infty) while alpha is in (1/2,1]. The manuscript needs an explicit argument showing that the pressure term converges in this topology, for example by proving the necessary strong convergence of rho^gamma or of the product rho^{gamma/2} partial_x rho^{gamma/2}.","section":"§5.4, Step 1, equation (5.36)"},{"comment":"The 'dissipative' part of the solution definition is not fully specified for the limit object. The energy inequality (3.6) contains the term integral mu(rho) |partial_x u_epsilon|^2, but the solution variables in Definition 3.1 are (rho, Lambda, zeta) and no velocity field u is defined; the subscript epsilon makes the term ill-posed. Moreover, in the limit passage the manuscript does not identify the limit of the dissipation term: after passing to the limit in (5.57), the text simply says that the uniform bounds allow the limit to be taken in the Ito correction term, without explaining which object in Definition 3.1 represents liminf integral mu(rho_epsilon)|partial_x u_epsilon|^2. Similarly, the limiting momentum equation (5.59) contains integral rho^{alpha/2} M partial_x psi, but the paper never proves that this equals the combination integral rho^{alpha/2} zeta partial_x psi plus the two capillarity terms appearing in (3.4). The later proof of (3.5) is a separate identity and does not by itself identify M with zeta.","section":"Definition 3.1 and §5.4, Step 1"}],"minor_comments":[{"comment":"The regularity statement rho in L^infty(0,T;H^1) cap L^2(0,T;H^2) does not follow from the estimates proved in the paper; the available bounds concern partial_xx rho^{alpha/2}, not rho in H^2. This should be corrected to match the actual regularity used.","section":"Definition 3.1, item (5)"},{"comment":"The condition 'u_epsilon(· wedge tau) > 0' appears to be a typo; positivity is expected for the density rho_epsilon, not for the velocity u_epsilon. Please correct this.","section":"Definition 4.1, item (3)"},{"comment":"The statements say the sets {L[mu^{d,delta}], delta in (0,1)} are tight, but the parameter indexing the approximating family is epsilon. As written this is confusing; the intended quantifier is over epsilon in (0,1) for each fixed delta.","section":"§5.3, Propositions 5.8 and 5.9"},{"comment":"The statements of these stochastic-convergence tools contain several notation slips: the symbol L^R is used without definition, the convergence 'W_n -> W in C([0,T];U_0) in probability' is not aligned with the Hilbert-space setting of Lemma 2.6, and the displayed equality in Theorem 2.5 seems to repeat the test function psi inconsistently. These should be cleaned up because the arguments in Section 5.4 rely on these tools.","section":"Theorem 2.5 and Lemma 2.6"},{"comment":"The statement of the no-vacuum estimates is ambiguous: if (4.9) is meant to hold uniformly in epsilon, it contradicts Remark 4.4; if it is not uniform, the dependence on epsilon should be displayed as in the proof, which yields epsilon ||1/sqrt(rho_epsilon)|| in L^p bounded uniformly.","section":"§4, equations (4.8)-(4.9)"}],"recommendation":"major_revision","confidential_remarks":"The main concern for the editor is the manuscript's dependence on the companion preprint [26] for Theorem 4.3. This is not a fatal objection if the companion result is available and correct, but the present paper should either contain the full proof or state the external theorem with explicit hypotheses and uniform constants. The invalid tightness on L^2(0,T;H^1) is a local gap that is likely repairable, but it affects a stated convergence in Proposition 5.11 and should be fixed carefully. The unresolved identification of the dissipation term and of the limit M in the momentum equation is the most serious mathematical issue after the external dependency; the authors should address it explicitly before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the headline: this is the first existence result for weak dissipative martingale solutions of the stochastic 1D quantum Navier-Stokes equations, and the main theorem is believable. The paper adapts the Lacroix-Violet-Vasseur velocity truncation and the Breit-Feireisl-Hofmanova stochastic compactness machinery to the QNS structure, with BD entropy handling the density-dependent viscosity. That is real work, not a copy-and-paste: the Korteweg term forces new identifications (Λ, ζ) and the vacuum issue needs the truncated formulation. The deterministic byproduct (G=0) is already known, but the stochastic extension with multiplicative noise and random initial data is new.\n\nThe paper is clearly written and the plan is coherent. My main reservations are the two the stress-test flagged, and I think they are both real.\n\nFirst, Theorem 4.3, the global well-posedness of the approximating system, is not proved in the paper. It is quoted from the authors' companion arXiv:2401.10064, with only a sketch. A sketch that says 'same lines of argument' for the BD estimates might be acceptable for a small lemma, but here it is the entire foundation: every uniform bound used later comes from that theorem. A referee cannot verify the chain without checking an unreviewed preprint. This is a fixable dependency, but the paper as submitted is not self-contained, and the authors should either include a full proof or wait for [26] to appear.\n\nSecond, Proposition 5.3 claims tightness of the laws of ρ_ε on L²(0,T;H¹). The set K built from an Aubin-Lions compact set in L²(L²) plus a uniform bound on ∂xρ is not compact in H¹; bounded sets in H¹ are not compact. The argument as written does not work. The gap looks repairable: working in a weaker path space for ρ (e.g., L²(L²) strong plus weak topologies for the derivative) would likely suffice for the limit passages, since the momentum equation only needs distributional convergence. But it is a genuine hole in the current version.\n\nThere are also minor slips, like a stray ε in the energy inequality (3.6). Nothing fatal.\n\nThe citation pattern is fair; the reliance on [26] is the only heavy self-citation, and it's a load-bearing one. Given that the result is new, plausible, and the gaps are technical rather than conceptual, I'd send this to a competent referee. The referee should demand the missing details, but the paper doesn't deserve a desk reject.\n\nSummary: useful for anyone working on stochastic compressible fluids or QNS; the main theorem is worth having, but only after the approximating-system proof and the strong-density compactness are nailed down.","headline":"A genuinely new stochastic extension of the 1D QNS existence theory, but the main theorem currently leans on an unproved companion preprint and a shaky tightness argument.","tokens_in":34053,"tokens_out":4634,"would_cite":true,"duration_ms":43767,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","60H15","76M35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes global existence of weak dissipative martingale solutions for the stochastic 1D Quantum-Navier-Stokes system, allowing vacuum regions, for viscosity exponent $\\alpha \\in (1/2,1]$ and pressure exponent $\\gamma>1$.","keywords":["stochastic compressible fluids","Quantum-Navier-Stokes equations","Navier-Stokes-Korteweg equations","weak martingale solutions","vacuum regions","BD entropy","stochastic compactness","one-dimensional torus"],"falsifier":"Check the companion preprint [26] for Theorem 4.3: if one can exhibit initial data satisfying (2.10) and noise satisfying (2.3)-(2.5) for which the approximating system (4.1) does not have a global strong pathwise solution, or for which the constants in (4.6)-(4.9) depend on $\\epsilon$, then the compactness argument has no limiting sequence and Theorem 3.3 collapses.","tokens_in":32938,"feed_emoji":"🌊","tokens_out":6474,"duration_ms":53168,"temperature":0.7,"pith_summary":"The paper claims that the one-dimensional stochastic Quantum-Navier-Stokes equations, a compressible fluid model with a Bohm-potential capillarity term and density-dependent viscosity, admit global weak dissipative martingale solutions for viscosity exponent $\\alpha \\in (1/2,1]$ and pressure exponent $\\gamma>1$. \"Weak\" means weak in the PDE sense and weak in the probabilistic sense: the probability space and the driving Wiener process are part of the solution, and the density may develop vacuum regions. The proof constructs an approximating system with added dissipation, derives uniform energy and BD-entropy estimates, introduces a truncation of the velocity in the momentum equation, and passes to the limit by a stochastic compactness argument. If correct, this is the first global existence result for the stochastic Quantum-Navier-Stokes system in the weak-solution framework, and it also yields global weak solutions in the deterministic case.","feed_headline":"Stochastic quantum fluid equations admit global weak solutions","feed_subtitle":"Existence proof allows vacuum regions for viscosity exponents in (1/2, 1] and all pressure exponents γ>1.","key_machinery":"The machinery has four interacting parts. First, the approximating system (4.1), which adds an extra dissipation term $\\epsilon\\partial_{xx}u$ to the momentum equation; Theorem 4.3, quoted from [26], gives global strong pathwise solutions with uniform energy estimate (4.6), BD-entropy estimate (4.7), an upper bound on $\\sqrt{\\rho}$, and a lower bound on $\\sqrt{\\rho}$ that degenerates as $\\epsilon\\to0$. Second, the BD entropy, which yields the extra density regularity $\\partial_x\\rho^{\\alpha-1/2}\\in L^\\infty_t L^2_x$ and, for $\\alpha\\in(1/2,1]$, permits vacuum. Third, the velocity truncation $\\beta_\\delta(u)$, a smooth compactly supported approximation of the identity on $\\mathbb{R}$, applied inside the momentum equation to control nonlinear terms where the velocity is not defined on $\\{\\rho=0\\}$; the identities $\\sqrt{\\rho}\\Lambda=m$ and $\\rho^{\\alpha/2}\\zeta=\\partial_x(\\rho^{\\alpha-1/2}\\Lambda)-2\\rho^{(\\alpha-1)/2}\\Lambda\\,\\partial_x\\rho^{\\alpha/2}$ let the weak formulation avoid dividing by $\\rho$. Fourth, a Jakubowski-Skorokhod compactness argument on a sub-Polish path space, together with It\\^o calculus identities and continuity-of-solution arguments, identifies the limit as a weak dissipative martingale solution.","core_discovery":"On the paper's own terms, the central discovery is Theorem 3.3: under smoothness and integrability assumptions on the random initial data and on the noise coefficients, there exists a weak dissipative martingale solution $((\\Omega,\\mathcal{F},(\\mathcal{F}_t),P),\\rho,\\Lambda,\\zeta)$ to system (1.1)-(1.3) in the sense of Definition 3.1. The velocity is not directly part of the solution; instead the momentum is encoded by $\\sqrt{\\rho}\\Lambda=m$, and the dissipation variable $\\zeta$ is defined by $\\rho^{\\alpha/2}\\zeta=\\partial_x(\\rho^{\\alpha-1/2}\\Lambda)-2\\rho^{(\\alpha-1)/2}\\Lambda\\,\\partial_x\\rho^{\\alpha/2}$, which is what allows vacuum regions. The solution satisfies the continuity and momentum equations in distribution form, an energy inequality, and a BD-entropy inequality, with enough density regularity to control $\\partial_{xx}\\rho^{\\alpha/2}$, $\\partial_x\\rho^{(\\gamma+\\alpha-1)/2}$, and related quantities. The proof relies on Theorem 4.3, quoted from the companion preprint [26], which asserts global strong pathwise well-posedness of the approximating system with estimates uniform in the approximation parameter; the present paper sketches but does not prove that theorem.","pith_inferences":["The main theorem is conditional on Theorem 4.3 being fully proved in the companion paper [26]; the present text only sketches its proof, so a reader should treat the quoted uniform estimates (4.6)-(4.9) as load-bearing and verify them first.","A testable extension would be to push the method toward the boundary $\\alpha=1/2$, where the BD entropy starts to exclude vacuum; the paper's choice $\\alpha\\in(1/2,1]$ appears driven by the technique rather than by the model itself.","The same weak formulation, with $\\Lambda$ and $\\zeta$ replacing $u$, could serve as a template for stochastic versions of other degenerate-viscosity compressible systems, such as shallow-water or degenerate Navier-Stokes equations, where similar vacuum issues arise.","Because the result holds for arbitrary $\\gamma>1$, the pressure does not need to dominate the viscosity; this suggests the existence mechanism is carried by the viscosity-plus-capillarity dissipation rather than by the pressure law."],"forward_implications":["If Theorem 3.3 is correct, global weak dissipative martingale solutions exist for all viscosity exponents $\\alpha\\in(1/2,1]$ and all pressure exponents $\\gamma>1$, complementing the range $\\alpha\\in[0,1/2]$ from [26] and covering the full exponent range accessible to the BD-entropy method.","The deterministic Quantum-Navier-Stokes system with $G=0$ inherits global weak solutions, since the theorem is stated for general noise satisfying (2.3)-(2.5), including zero noise.","The existence framework tolerates additive noise $\\sigma(x)dW$ as a special case, which opens the door to invariant-measure questions for stochastic compressible fluids with capillarity.","Because the solution concept uses $(\\rho,\\Lambda,\\zeta)$ rather than a pointwise velocity, the theorem provides a workable notion of solution precisely in the regime where vacuum regions make $u$ undefined on $\\{\\rho=0\\}$."],"supporting_citations":[{"why":"Supplies Theorem 4.3, the global strong pathwise well-posedness of the approximating system with $\\epsilon$-uniform estimates that the compactness argument needs.","marker":"[26]"},{"why":"Introduces the velocity-truncation technique for the deterministic quantum Navier-Stokes equations that the paper adapts to the stochastic setting.","marker":"[33]"},{"why":"Extends the weak-solution approach to Navier-Stokes-Korteweg systems and is used as a template for the convergence argument.","marker":"[8]"},{"why":"Provides the stochastic compactness tools, It\\^o calculus identities, and the weak-martingale-solution framework for compressible flows.","marker":"[12]"},{"why":"Introduces the BD entropy, the a priori estimate giving extra density regularity and controlling vacuum behaviour.","marker":"[13]"},{"why":"Gives the tame-capillarity condition that selects the viscosity range $\\alpha\\in(1/2,1]$ used here.","marker":"[29]"},{"why":"Provides the fact that $\\partial_x\\rho=0$ almost everywhere on $\\{\\rho=0\\}$, used to pass limits on the vacuum set.","marker":"[27]"},{"why":"Underpins the infinite-dimensional stochastic integration background used throughout the paper.","marker":"[24]"}],"fun_headline_variants":["Global weak martingale solutions for stochastic 1D quantum Navier-Stokes","Stochastic 1D quantum fluids: existence of martingale solutions with vacuum","Weak solutions to stochastic quantum Navier-Stokes with vacuum regions","Randomly forced 1D quantum fluid admits global weak solutions","Global dissipative martingale solutions for 1D stochastic quantum fluids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the quoted Theorem 4.3 from [26]: global strong pathwise well-posedness of the approximating system with estimates uniform in $\\epsilon$; the present paper does not prove it, and if it fails the main theorem has no starting sequence.","fun_headline_variants_meta":{"raw":{"variants":["Global weak martingale solutions for stochastic 1D quantum Navier-Stokes","Stochastic 1D quantum fluids: existence of martingale solutions with vacuum","Weak solutions to stochastic quantum Navier-Stokes with vacuum regions","Randomly forced 1D quantum fluid admits global weak solutions","Global dissipative martingale solutions for 1D stochastic quantum fluids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000866,"raw_usage":{"total_tokens":3727,"prompt_tokens":895,"completion_tokens":2832,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":2739}},"tokens_in":511,"tokens_out":2832,"duration_ms":17125,"temperature":1.0,"reasoning_tokens":2739,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:31:32.247406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the companion preprint [26] for Theorem 4.3: if one can exhibit initial data satisfying (2.10) and noise satisfying (2.3)-(2.5) for which the approximating system (4.1) does not have a global strong pathwise solution, or for which the constants in (4.6)-(4.9) depend on $\\epsilon$, then the compactness argument has no limiting sequence and Theorem 3.3 collapses.","supporting_citations":[{"cited_title":"Lacroix-Violet and A","cited_arxiv_id":null,"evidence_quote":"Introduces the velocity-truncation technique for the deterministic quantum Navier-Stokes equations that the paper adapts to the stochastic setting."},{"cited_title":"Antonelli and S","cited_arxiv_id":null,"evidence_quote":"Extends the weak-solution approach to Navier-Stokes-Korteweg systems and is used as a template for the convergence argument."},{"cited_title":"Breit, E","cited_arxiv_id":null,"evidence_quote":"Provides the stochastic compactness tools, It\\^o calculus identities, and the weak-martingale-solution framework for compressible flows."},{"cited_title":"Bresch and B","cited_arxiv_id":null,"evidence_quote":"Introduces the BD entropy, the a priori estimate giving extra density regularity and controlling vacuum behaviour."},{"cited_title":"Germain and P","cited_arxiv_id":null,"evidence_quote":"Gives the tame-capillarity condition that selects the viscosity range $\\alpha\\in(1/2,1]$ used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fact that $\\partial_x\\rho=0$ almost everywhere on $\\{\\rho=0\\}$, used to pass limits on the vacuum set."},{"cited_title":"Da Prato and J","cited_arxiv_id":null,"evidence_quote":"Underpins the infinite-dimensional stochastic integration background used throughout the paper."}],"review_version":1}