Pith. sign in

REVIEW 4 major objections 5 minor 37 references

Weak martingale solutions to the stochastic 1D Quantum-Navier-Stokes equations

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read 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 $\gamma>1$.

desk verdict 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. read the letter →

arxiv 2412.10875 v1 pith:PZ2Y3K5N submitted 2024-12-14 math.AP

Donatella Donatelli , Lorenzo Pescatore , Stefano Spirito This is my paper · ORCID
classification math.AP MSC 35Q3560H1576M35
keywords stochasticcompressiblefluidsQuantum-Navier-StokesequationsNavier-Stokes-KortewegweakmartingalesolutionsvacuumregionsBDentropycompactnessone-dimensionaltorus
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\{\rho=0\}$.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [§4, Theorem 4.3] 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.
  2. [§5.3, Proposition 5.3] 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.
  3. [§5.4, Step 1, equation (5.36)] 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}.
  4. [Definition 3.1 and §5.4, Step 1] 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.
minor comments (5)
  1. [Definition 3.1, item (5)] 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.
  2. [Definition 4.1, item (3)] 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.
  3. [§5.3, Propositions 5.8 and 5.9] 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.
  4. [Theorem 2.5 and Lemma 2.6] 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.
  5. [§4, equations (4.8)-(4.9)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the compactness limit from the epsilon-approximating system is a genuine derivation, and the unproved companion theorem is a verification dependency rather than a circular step.

full rationale

The claimed derivation chain is: Theorem 4.3 provides strong pathwise solutions to the epsilon-system (4.1) with epsilon-uniform estimates (4.6)-(4.8); Section 5.2 collects these into the uniform bounds (5.4)-(5.7); Propositions 5.3-5.11 establish tightness and Jakubowski-Skorokhod convergence; and Section 5.4 passes to the limit in the truncated momentum equation and in the energy/entropy inequalities. None of these steps defines a quantity in terms of the target conclusion, fits a parameter, or renames a fitted quantity as a prediction. The fields Lambda and zeta are introduced algebraically through sqrt(rho)Lambda = m and the dissipation identity (3.5), which is a reformulation designed for vacuum regions, not a circular definition. The energy and entropy inequalities (3.6)-(3.7) are limits of the approximating balance laws obtained by Ito's formula and the BD entropy, with no undischarged assumption of the desired weak solution. The main caveat is Theorem 4.3: it is quoted from the authors' companion preprint [26] and its proof is only sketched in Section 4, while Remark 4.4 states that the lower vacuum bound (4.9) degenerates as epsilon goes to zero. If Theorem 4.3 were false, or if the constants in (4.6)-(4.8) depended on epsilon in an uncontrolled way, the compactness argument would have no sequence to start from; this is a load-bearing verification risk. It is not, however, circular, because [26] is a separate stated result whose hypotheses (2.3)-(2.5) and (2.10) do not include the existence of a weak dissipative martingale solution and whose conclusion concerns the epsilon-regularized strong system. There is also a technical gap in Proposition 5.3, where tightness on L2(0,T;H1) is inferred from a compactness argument that more directly gives compactness in L2(0,T;L2); this is a correctness issue, not a circularity. No step reduces to its own input by construction, so the circularity score is set to zero.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new physical entities or fitted parameters. The auxiliary fields Lambda and zeta are part of the solution concept used to handle vacuum, and they are defined through the momentum and density relations. No numerical constants are fitted; the only free exponents gamma > 1 and alpha in (1/2,1] are parameters of the model, not chosen to make the proof work.

assumptions (4)
  • domain assumption Theorem 4.3 from [26]: the approximating system (4.1) has a unique global strong pathwise solution satisfying estimates (4.6)-(4.9), including epsilon-independent energy and BD entropy bounds.
    The proof of the main theorem starts from these approximate solutions and their uniform bounds. The companion preprint [26] is by the same authors, is not included in this submission, and is not machine-checked, so this is a load-bearing external assumption.
  • standard math Standard stochastic calculus tools as stated in Section 2: Ito's formula (Theorems 2.1 and 2.2), Burkholder-Davis-Gundy inequality, Jakubowski-Skorokhod representation theorem, and Lemma 2.6 on convergence of stochastic integrals.
    These are standard tools from the stochastic PDE literature, quoted from Breit-Feireisl-Hofmanova [12] and related references.
  • domain assumption Structural assumptions on the noise coefficients Gk in (2.3)-(2.5) and on the initial data in (2.10), with the optional relaxation in Remark 3.5.
    These conditions define the class of problems for which Theorem 3.3 is stated. They are hypotheses of the theorem rather than ad hoc additions.
  • standard math The BD entropy (Bresch-Desjardins) provides control on rho^{alpha-1/2} in L-infinity_t H1_x, which is a key tool for the estimates in Section 4.
    The BD entropy is a known a priori estimate for density-dependent viscosity; the present paper invokes it via [26] and the classical literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weak martingale solutions to the stochastic 1D Quantum-Navier-Stokes equations." pith.science (2026). https://pith.science/paper/PZ2Y3K5N

@misc{pith2026241210875,
  author       = {Pith},
  title        = {Pith review of: Weak martingale solutions to the stochastic 1D Quantum-Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZ2Y3K5N}},
  note         = {Machine review of arXiv:2412.10875}
}
read the original abstract

In this paper we prove the existence of global weak dissipative martingale solutions for a one-dimensional compressible fluid model with capillarity and density dependent viscosity, driven by random initial data and a stochastic forcing term. These solutions are weak in both PDEs and Probability sense and may have vacuum regions. The proof relies on the construction of an approximating system which provides extra dissipation properties and the convergence is based on an appropriate truncation of the velocity field in the momentum equation and a stochastic compactness argument

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 35 canonical work pages

  1. [26]

    Donatelli, L

    D. Donatelli, L. Pescatore and S. Spirito : Global regularity for the one-dimensional stochastic Quan tum- Navier-Stokes equations (2024), Preprint, available at arXiv:2401.10064. 28 D. DONATELLI, L. PESCATORE, S. SPIRITO

    arXiv 2024

  2. [8]

    Antonelli and S

    P. Antonelli and S. Spirito : Global existence of weak solutions to the Navier-Stokes-Ko rteweg equations , Ann. Inst. H. Poincar` e C. Anal. Non Lin` eaire 39 (2022), no. 1, pp 171-200

    work page 2022

  3. [33]

    Lacroix-Violet and A

    I. Lacroix-Violet and A. V asseur: Global weak solutions to the compressible quantum Navier-S tokes equation and its semi-classical limit , J. Math. Pures Appl., 114 (2017), 191–210

    work page 2017

  4. [1]

    Antonelli, G

    P. Antonelli, G. Cianfarani Carnevale, C. Lattanzio, and S. Spirito: Relaxation limit from the quantum Navier-Stokes equations to the quantum drift-diffusion equa tion, J. Nonlinear Sci. 31 (2021), no. 5, Paper No. 71, 32 pp. 1

    work page 2021

  5. [2]

    Antonelli and P

    P. Antonelli and P. Marcati : On the finite energy weak solutions to a system in quantum fluid dynamics, Comm.Math. Phys., 287, (2009), no.2, 657-686

    work page 2009

  6. [3]

    Antonelli and P

    P. Antonelli and P. Marcati : The Quantum Hydrodynamics system in two space dimensions , Arch. Ration. Mech. Anal., 203, (2012), no.2, 499-527

    work page 2012

  7. [4]

    Antonelli, P

    P. Antonelli, P. Marcati, and H. Zheng : An intrinsically hydrodynamic approach to multidimension al QHD systems, Arch. Ration. Mech. Anal., 247, (2023), Paper No. 24, 58

    work page 2023

  8. [5]

    Antonelli, P

    P. Antonelli, P. Marcati, and H. Zheng : Genuine hydrodynamic analysis to the 1-D QHD system:existe nce, dispersion and stability , Comm. Math. Phys., 383, (2021), no.3, 2113–2161

    work page 2021

Show all 37 references
  1. [6]

    Antonelli, L

    P. Antonelli, L. E. Hientzsch, and P. Marcati : On the low Mach number limit for quantum Navier-Stokes equations, SIAM J. Math. Anal., 52, (2020), no.6, 6105-6139

    2020

  2. [7]

    Antonelli, L.E

    P. Antonelli, L.E. Hientzsch, and S. Spirito : Global existence of finite energy weak solutions to the quant um Navier-Stokes equations with non-trivial far-field behavi or,J. Differential Equations, 290, (2021), 147-177

    2021

  3. [9]

    Antonelli and S

    P. Antonelli and S. Spirito : On the compactness of finite energy weak solutions to the quan tum Navier-Stokes equations, J. Hyperbolic Differ. Equ., 15, (2018), no.1, pp. 133-147

    2018

  4. [10]

    Audiard and B

    C. Audiard and B. Haspot : Global well-posedness of the Euler-Korteweg system for sma ll irrotational data , Comm. Math. Phys., 351, (2017), no. 1, 201-247

    2017

  5. [11]

    Benzoni-Gavage, R

    S. Benzoni-Gavage, R. Danchin, and S. Descombes : On the well-posedness for the Euler-Korteweg model in several space dimensions , Indiana Univ. Math. J., 56, (2007), no.4, 1499-1579

    2007

  6. [12]

    Breit, E

    D. Breit, E. Feireisl, and M. Hofmanova : Stochastically Forced Compressible Fluid Flows , De Gruyter Series in Applied and Numerical Mathematics. De Gruyter (20 18)

  7. [13]

    Bresch and B

    D. Bresch and B. Desjardins : Sur un mod` e` oe de Saint-Venant visqueux et sa limite quasi g ´ eostrophique, [ On viscous shallow-water equations (Saint-Venant model) and the quasi-geostrophic limit] , C.R. Math. Acad. Sci. Paris, 335, (2002), no.12, 1079-1084

    2002

  8. [14]

    Bresch, A

    D. Bresch, A. V asseur, and C. Yu : Global existence of entropy-weak solutions to the compress ible Navier- Stokes equations with non-linear density dependent viscos ities, J. Eur. Math. Soc., 24, (2022), no.5, 1791–1837

    2022

  9. [15]

    Dhariw al and E

    Z.Brze´zniak, G. Dhariw al and E. Zatorska : Sequential stability of weak martingale solutions to stoch as- tic compressible Navier-Stokes equations with viscosity v anishing on vacuum ,Journal of Differential Equations, Volume 416, Part 2,2025,Pages 1285-1346

    2025

  10. [16]

    Burtea and B

    C. Burtea and B. Haspot : Existence of global strong solution for the Navier–Stokes– Korteweg system in one dimension for strongly degenerate viscosity coefficients , Pure Appl. Anal., 4, (2022), no. 3, 449-485

    2022

  11. [17]

    Burtea and B

    C. Burtea and B. Haspot : Vanishing capillarity limit of the Navier-Stokes-Kortewe g system in one dimension with degenerate viscosity coefficient and discontinuous ini tial density , SIAM J. Math. Anal., 54, (2022), no.2, 1428–1469

    2022

  12. [18]

    Caggio and D

    M. Caggio and D. Donatelli : High Mach number limit for Korteweg fluids with density depen dent viscosity , J. Differential Equations, 277, (2021), 1-37

    2021

  13. [19]

    Charve and B

    F. Charve and B. Haspot : Existence of global strong solution and vanishing capillar ity-viscosity limit in one dimension for the Korteweg system , SIAM J. Math. Anal. 45, (2013), no.2, 469-494

    2013

  14. [20]

    Z. Chen, X. Chai, B. Dong, and H. Zhao : Global classical solutions to the one-dimensional compres sible fluid models of Korteweg type for large initial data , J. Differential Equations, 259, (2015), no.8, 4376-4411

    2015

  15. [21]

    Constantin, T.D

    P. Constantin, T.D. Drivas, H.Q. Nguyen, and F. P asqualotto : Compressible fluids and active potentials , Ann. Inst. H. Poincar` e C. Anal. Non Lin` eaire, 37, (2020), n o.1, pp. 145-180

    2020

  16. [22]

    Constantin, T.D

    P. Constantin, T.D. Drivas, and R. Shvydkoy : Entropy hierarchies for equations of compressible fluids an d self-organized dynamics, SIAM J. Math. Anal., 52, (2020), no. 3, 3073-3092

    2020

  17. [23]

    Coti-Zelati, N

    M. Coti-Zelati, N. Glatt-Holtz, and K. Trivisa : Invariant measure for the stochastic compressible Navier- Stokes equations , Appl. Math. Optim., 83, (2021), no. 3, 1487-1522

    2021

  18. [24]

    Da Prato and J

    G. Da Prato and J. Zabczyk : Stochastic Equations in Infinite Dimensions , volume 44 of Encyclopedia of Mathematics and its Applications , Cambridge University Press, Cambridge, 1992

    1992

  19. [25]

    Donatelli, E

    D. Donatelli, E. Feireisl, and P. Marcati : Well/ill Posedness for the Euler-Korteweg-Poisson System and Related Problems, Comm. Partial Differential Equations, 40, (2015), no.7, pp . 1314-1335

    2015

  20. [27]

    L. C. Evans and R. F. Gariepy ; Measure theory and fine properties of functions . Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992

    1992

  21. [28]

    Dunn and J

    J.E. Dunn and J. Serrin : On the thermomechanics of interstitial working , Arch. Ration. Mech. Anal., 88, no.2 (1985), 95-133

    1985

  22. [29]

    Germain and P

    P. Germain and P. LeFloch : Finite Energy Method for Compressible Fluids: The Navier-S tokes-Korteweg Model, Comm. Pure Appl. Math., 69, (2015), no.1, 3-61

    2015

  23. [30]

    Haspot : Existence of global strong solution for the compressible Na vier-Stokes equations with degenerate viscosity coefficients in 1D , Math

    B. Haspot : Existence of global strong solution for the compressible Na vier-Stokes equations with degenerate viscosity coefficients in 1D , Math. Nachr. 291 (2018), no. 14-15, 2188–2203

    2018

  24. [31]

    Jiu and Z

    Q. Jiu and Z. Xin , The Cauchy problem for 1D compressible flows with density-de pendent viscosity coefficients . Kinet. Relat. Models 1 (2008), no. 2, 313–330

    2008

  25. [32]

    J ¨ungel: Global weak solutions to compressible Navier–Stokes equat ions for quantum fluids , SIAM J

    A. J ¨ungel: Global weak solutions to compressible Navier–Stokes equat ions for quantum fluids , SIAM J. Math. Anal. 42 (2010), 1025–1045

    2010

  26. [34]

    Landau and E

    L. Landau and E. Lifschitz : Quantum Mechanics: Non-relativistic Theory. Pergamon Press, New York, 1977

    1977

  27. [35]

    Li and Z

    J. Li and Z. Xin : Global existence of weak solutions to the barotropic compre ssible Navier-Stokes flows with degenerate viscosities. arXiv:1504.06826

    arXiv

  28. [36]

    Mellet and A

    A. Mellet and A. V asseur : Existence and uniqueness of global strong solutions for one -dimensional com- pressible Navier-Stokes equations , SIAM J. Math. Anal., 39, (2008), no. 4, pp. 1344–1365

    2008

  29. [37]

    V asseur and C

    A.F. V asseur and C. Yu : Existence of global weak solutions for 3D degenerate compre ssible Navier-Stokes equations, Invent. Math. 206 (2016), no. 3, 935–974. (D.Donatelli) DISIM - Dipartimento di Ingegneria e Scienze dell’Informaz ione e Matematica, Univer- sit`a degli Studi...

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.