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Dissipative measure-valued solutions and weak--strong uniqueness for a viscous Baer--Nunziato system with pressure relaxation

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Pith's one-line read For any finite-energy initial data, a viscous two-fluid model with pressure relaxation has a global dissipative measure-valued solution, and any such solution must collapse to a strong solution when one exists.

desk verdict A serious first global existence and weak-strong uniqueness theory for the pressure-relaxation Baer-Nunziato system, but the uniqueness theorem leans on a strong-solution regularity class whose existence is deferred to future work. read the letter →

arxiv 2608.13348 v1 pith:RCLY7S7E submitted 2026-08-13 math.AP

Nilasis Chaudhuri , Milan Pokorný , Ewelina Zatorska This is my paper · ORCID
classification math.AP MSC 35Q3576N10
keywords one-velocityBaer-Nunziatosystempressurerelaxationdissipativemeasure-valuedsolutionweak-stronguniquenessrelativeenergyYoungmeasuresvolumefractioncompressibletwo-phaseflow
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 develops a global existence and stability theory for a viscous one-velocity two-phase flow model in which the volume fraction $\alpha$ evolves by a differential pressure-relaxation law rather than being transported or fixed by an algebraic equilibrium constraint. For arbitrary finite-energy initial data, encoded as a Young measure, and for any adiabatic exponents $\gamma^\pm>1$, it constructs a dissipative measure-valued solution on any time interval $(0,T)$. Its second main claim is weak-strong uniqueness: if a sufficiently regular strong solution with the same initial data exists, then every such measure-valued solution is that strong solution pointwise, and the dissipation defect is zero. The reason to care is that this supplies the first global solution framework for this pressure-relaxation closure and turns the measure-valued formulation into a convergence criterion for approximations, contingent on the existence of a suitably strong comparison solution.

What carries the argument

The central object is the augmented relative energy $E_{\mathrm{mv}}(V|\beta,\widetilde{R},\widetilde{Q},w)$: the kinetic and thermodynamic relative-entropy terms for the two-fluid system plus the extra quadratic control $\frac12|\alpha-\beta|^2$ of the volume fraction. The argument is carried by the identity (5.18) for the evolution of $\langle V_{t,x};(s-\beta)^2\rangle$, derived from the renormalized volume-fraction equation, together with an endpoint cutoff: the relaxation source $\omega(s,r,q)$ is dominated by $\mathcal{R}(s,r,q)=\frac{s(1-s)}{\lambda+2\mu}(p_+(r/s)-p_-(q/(1-s)))^2$, whose integral is already part of the dissipation, so the singular behavior at $s=0$ and $s=1$ is controlled without forcing the volume fraction away from the endpoints. A measure-valued conformal Korn-Poincare inequality then absorbs the velocity and symmetric-gradient deviations into the shear dissipation and closes the Gronwall argument.

What would settle it

One concrete test is to take data satisfying the higher-regularity assumption of Remark 3.5 and run two different consistent approximation schemes; if their limiting Young measures differ, or if the dissipation defect is positive, while a strong solution of that regularity exists, the collapse assertion fails. A more targeted calculation is to check whether the constant in (3.15) remains uniform as $\beta\nearrow 2$; if it degenerates, the absorption step closing the Gronwall argument would break down.

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Extended reading notes

Core claim

On its own terms the paper establishes two results. Theorem 3.2 states that for any initial Young measure with finite energy and any $\gamma^\pm>1$, a dissipative measure-valued solution $(V,\mathcal{D})$ exists globally in time on $(0,T)$ for arbitrary $T$; the Young measure records oscillations of the volume fraction, the partial masses, the velocity, and its symmetric gradient, while the dissipation defect absorbs concentrations. Theorem 6.1 states that if a strong solution $(\beta,\widetilde{R},\widetilde{Q},w)$ with the regularity described in Remark 3.5 exists from the same initial data, then the measure-valued relative energy satisfies $E_{\mathrm{mv}}(\tau)+\mathcal{D}(\tau)+\text{dissipation}\le C_\kappa E_{\mathrm{mv}}(0)$; consequently, when the initial data coincide as measures, $V_{\tau,x}$ is the Dirac mass concentrated on $(\beta(\tau,x),\widetilde{R}(\tau,x),\widetilde{Q}(\tau,x),w(\tau,x),Dw(\tau,x))$ and $\mathcal{D}(\tau)=0$ for almost every $\tau$. The collapse to a single trajectory is the load-bearing conclusion.

Load-bearing premise

The paper's uniqueness claim presupposes that a strong solution one derivative smoother than the one Theorem 3.3 actually proves exists; Remark 3.5 only says this stronger existence can be anticipated, so for data where it fails the theorem has no comparison solution.

Editorial extensions

If this is right

  • If a strong solution with the higher regularity of Remark 3.5 exists, every dissipative measure-valued solution from the same data is a single Dirac mass at that solution, and the dissipation defect vanishes identically.
  • The global existence result covers the full range $\gamma^\pm>1$ and arbitrary finite-energy initial Young measures, so the framework does not restrict the adiabatic exponents.
  • The collapse result supplies a convergence criterion: any consistent energy-stable approximation whose limits generate dissipative measure-valued solutions must converge to the strong solution on its lifespan.
  • The extra $\frac12|\alpha-\beta|^2$ term gives direct control of the volume fraction, so the stability estimate controls $\alpha-\beta$ together with the phase densities and the velocity.
  • The separated limit procedure, parabolic regularization first and Galerkin projection second, provides a construction that can be adapted to related differential-closure multiphase models.

Reading between the lines

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

  • If the stronger local existence anticipated in Remark 3.5 fails for some admissible data, the weak-strong uniqueness statement would have an empty hypothesis for those data; proving or disproving that regularity class is the natural next step.
  • The same augmented-relative-energy structure should transfer to differential-closure two-phase models with heat conduction or phase transitions, since the endpoint difficulty is generic to volume fractions reaching $0$ or $1$.
  • The measure-valued solution space is a plausible setting for low-Mach-number limits with pressure relaxation, where the new volume-fraction control may prevent oscillations from escaping into uncontrolled concentration defects.
  • A concrete testable extension is to compute the constant in (3.15) as $\beta\nearrow 2$; if the constant is not uniform, the absorption step that closes the Gronwall argument would need a different smallness mechanism.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper develops a dissipative measure-valued (DMV) theory for a viscous one-velocity Baer–Nunziato system with pressure relaxation in a bounded three-dimensional domain. Theorem 3.2 constructs global-in-time DMV solutions for arbitrary finite-energy initial Young measures and for the full range of adiabatic exponents gamma^±>1, using a parabolic regularization, a Faedo–Galerkin approximation, two separated approximation limits, endpoint cutoffs, and a lower-semicontinuous defect analysis. Theorem 6.1 (and its condensed Theorem 3.6) proves DMV–strong uniqueness: if a sufficiently regular strong solution exists on (0,T), then every DMV solution emanating from the same Dirac initial data collapses to the strong solution and the dissipation defect vanishes. The relative-entropy argument augments the thermodynamic relative energy with an (alpha-beta)^2 term and controls the pressure-relaxation terms through the renormalized volume-fraction equation. The paper is carefully structured and the existence construction is presented in detail, but the uniqueness theorem relies on a strong-solution regularity class whose existence is explicitly deferred rather than proved.

Significance. If the existence theory and the weak–strong uniqueness result are accepted, this is a substantial contribution: it appears to be the first global DMV existence theory for the viscous barotropic one-velocity Baer–Nunziato system with differential pressure-relaxation closure, and it covers the full range gamma^±>1. The handling of the endpoint singularities at alpha=0,1 via lower-semicontinuous extensions and defect inequalities is technically serious, and the momentum defect control (4.30) is a clean way to close the formulation. The paper is also honest about its approximation limits and does not tune parameters against benchmarks. The principal concern is formal rather than computational: the comparison class for the uniqueness theorem is not proved to be non-empty, because the higher-regularity strong-solution existence is left to a remark. If repaired by a proof or by an explicitly conditional statement, the stability framework is a valuable contribution.

major comments (2)
  1. [§3.2/Remark 3.5; Theorems 3.6 and 6.1; Lemma 5.1] The collapse statement in Theorems 3.6 and 6.1 is not fully supported because the strong solution with the regularity used in Sections 5–6 is never proved to exist. Theorem 3.3 supplies only u in W^{1,p}(0,T;L^q) cap L^p(0,T;W^{2,q}) and (R,Q,alpha) in W^{1,p}(0,T;W^{1,q}), and Remark 3.5 explicitly states that these bounds are insufficient for the relative-entropy estimates and that the W^{2,q}-based maximal-regularity theorem "can be anticipated" but is not proved, requiring Omega in C^3, (R0,Q0,alpha0) in (W^{2,q})^3, u0 in B^{3-2/p}_{q,p}, and compatibility conditions. Since Lemma 5.1 and the estimates of J1–J12 in Section 6.1 use exactly the missing bounds (partial_t w + w·nabla w in L^1(0,T;L^infty) and nabla beta, nabla tilde-rho^± in L^infty), the hypothesis of Theorem 6.1 may be empty for data covered by Theorem 3.2. Please either prove the differentiated existence theorem, cite a proved version, or reformulate Theorems 3.6 and 6.1 as conditional statements with the strong-solution existence as an explicit assumption; the current wording in Section 3.3 even promises that the regular solution "emanating from the same initial data exists", which is not established.
  2. [§4.4.3, Eq. (3.15)] The proof of the Korn–Poincaré compatibility condition (3.15) passes beta pointing to 2 in the family of L^beta Young-measure inequalities. The text does not justify that the constants C_beta from the cited L^beta conformal Korn estimates are uniformly bounded as beta tends to 2; if they blow up, the passage is invalid and (3.15) is not established. Since (3.15) is used in Section 6.3 to absorb the kappa-terms in (6.8), this is a load-bearing step; please provide the uniform-constant argument or state the explicit p-dependence of the constant in [46].
minor comments (6)
  1. [§3.1 heading] The heading "Dissipative measure-valued solutions solutions" contains a duplicated word and should be corrected.
  2. [§5.1, Eqs. (5.4) and (5.8)] In equations (5.4) and the first line of (5.8), the time integrals contain left angle bracket V_{tau,x}; ... right angle bracket, but the integration variable is t; these should read V_{t,x}.
  3. [Eq. (4.26)] The displayed definition of omega(s,r,q) has an unbalanced parenthesis and writes the pressure difference in a confusing way; it should be s(1-s)/(lambda+2mu) ((r/s)^{gamma+} - (q/(1-s))^{gamma-}).
  4. [§4.4.2] Shortly after the definition of the relative entropy, the text writes P(r,p,q) where the arguments should be (s,r,q); please correct this typo.
  5. [§4.4] The phrase "bu (4.9)" should read "by (4.9)".
  6. [Theorem 3.2, display (3.18)] The condition alpha0 := int_Omega <V0,x;s> dx in [0,1] is automatic because s in [0,1] V0,x-a.s.; it can be deleted or marked as a consistency observation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation chain: the collapse of the measure-valued solution in Theorem 6.1 follows from a Gronwall estimate on an augmented relative energy, while the main caveat is a deferred strong-solution regularity hypothesis in Remark 3.5 that is missing support rather than circular reasoning.

full rationale

The paper's derivation chain is not circular. Theorem 3.2 constructs dissipative measure-valued solutions through parabolic regularization, a Faedo-Galerkin approximation, separated limits, lower-semicontinuous extension of the pressure, and concentration-defect control; no parameter is fitted to the target conclusion and no 'prediction' is reintroduced as an input. The weak-strong uniqueness conclusion of Theorem 6.1 is obtained by applying Gronwall's inequality to the augmented relative energy (3.27)/(5.1); the extra volume-fraction term |α-β|^2 is not assumed small but is controlled through the renormalized volume-fraction equation, with the identity (5.18) in Lemma 5.2 supplying the needed evolution. The estimates of the remainder terms J1-J12 in Sections 6.1-6.2 are standard relative-entropy estimates; the only author-overlapping citations, such as [43] for the 'similar strategy' and [52] as the starting point for Theorem 3.3, are used as methodological precedents, not as imports of the target theorem. Thus there is no step in which a quantity that should be derived is instead used as an input by construction. The genuine caveat is conditional, not circular: Remark 3.5 explicitly states that the W^{1,q}-regularity of Theorem 3.3 'does not, by itself, imply' the bounds ∂t w + w·∇w ∈ L^1(0,T;L∞) and ∇β, ∇ϱ± ∈ L∞ needed in the relative-entropy argument, and then says that the stronger existence result 'can be anticipated' only under C^3 domains, W^{2,q} initial data, and extra compatibility conditions, without being proved. Consequently Theorem 6.1 is a conditional weak-strong uniqueness statement relative to any sufficiently regular solution in the class of Remark 3.5; if no such solution exists for the stated data, the collapse conclusion has an empty hypothesis. This is a gap in support, not a reduction of the theorem to its own inputs, so the circularity score remains low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data: the viscosity coefficients µ, ξ, the adiabatic exponents γ±, the domain Ω, and the relaxation coefficient are model inputs, while ε, n, δ, η, κ, ϵ are technical parameters internal to the proofs. The dissipation defect D and the concentration measures σ∞, R∞, r_M are standard components of the dissipative measure-valued framework ([1, 16, 28]), not new physical entities. The list above records the mathematical and modeling premises the central claims rest on.

assumptions (5)
  • standard math Standard theory of Young measures, parametrized measures, and the lower-semicontinuity defect lemma (Lemma 4.4, after [28, Lemma 2.1], [17, Lemma 4.2], [32]).
    Invoked in Section 4.4 to pass to the measure-valued limit and to define nonnegative defects E∞, σ∞, R∞.
  • standard math Conformal (trace-free) Korn-Poincare inequality in W^{1,p}_0, including its Young-measure version (3.15).
    Used in Definition 1, condition (3.15), and verified in Section 4.4.3 via the L^β estimate [6, Lemma 2.2] and [20, 46]; the passage β↗2 requires a bounded constant C_β, which the paper does not quantify.
  • standard math Lp-Lq maximal regularity theory for the weighted Lame operator (after [24, 53]), as the basis of Theorem 3.3.
    Remark 3.4 sketches the Lagrangian fixed-point argument; Theorem 3.3 states existence of local strong solutions for data in W^{1,q} × B^{2-2/p}_{q,p}.
  • domain assumption Existence of a strong solution with the higher regularity of Remark 3.5: (R₀,Q₀,α₀) ∈ (W^{2,q}(Ω))³, u₀ ∈ B^{3-2/p}_{q,p}(Ω), Ω ∈ C³, with compatibility conditions.
    Flagged by the authors as 'can be anticipated' (Remark 3.5) but not proved. Theorems 3.6 and 6.1 require exactly this comparison class; Theorem 3.3 provides only the weaker W^{1,q}-based regularity (3.21)-(3.22). This is the load-bearing premise of the weak-strong uniqueness result.
  • domain assumption Physical closure of the Baer-Nunziato model: one velocity, identical Newtonian stress (2.4) with µ>0, ξ≥0, barotropic pressures p±(z)=z^{γ±}, and relaxation source ω = α(1-α)(λ+2µ)^{-1}(p+ - p-).
    These are the model equations, motivated by the derivations cited in [7, 9, 35]; they are inputs, not consequences of the proof.

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Pith. "Pith review of Dissipative measure-valued solutions and weak--strong uniqueness for a viscous Baer--Nunziato system with pressure relaxation." pith.science (2026). https://pith.science/paper/RCLY7S7E

@misc{pith2026260813348,
  author       = {Pith},
  title        = {Pith review of: Dissipative measure-valued solutions and weak--strong uniqueness for a viscous Baer--Nunziato system with pressure relaxation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCLY7S7E}},
  note         = {Machine review of arXiv:2608.13348}
}
abstract

We study a viscous one-velocity Baer--Nunziato system for two barotropic compressible fluids in a bounded three-dimensional domain. In contrast with models in which the volume fraction is merely transported or determined by an algebraic equilibrium constraint, it satisfies a differential closure driven by the pressure gap between the phases. For arbitrary finite-energy initial data and adiabatic exponents $\gamma^\pm>1$, we construct global-in-time dissipative measure-valued solutions and prove dissipative measure-valued--strong uniqueness relative to any sufficiently regular solution with the same initial data. The principal difficulties are the singular and non-continuous behavior of the pressure-relaxation source at vanishing volume fractions, the associated concentration defects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These are resolved by an endpoint cutoff argument and an augmented relative energy coupled to the renormalized volume-fraction equation.

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Works this paper leans on

58 extracted references · 55 canonical work pages

  1. [46]

    Lewintan, P., Neff, P.:L p-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions. Z. Angew. Math. Phys.72, Paper No. 127 (2021)

    work page 2021

  2. [1]

    Abbatiello, E

    A. Abbatiello, E. Feireisl, and A. Novotn´ y. Generalized solutions to models of com- pressible viscous fluids.Discrete & Continuous Dynamical Systems, 41(1): 1–28, 2021

    work page 2021

  3. [2]

    Brenier, C

    Y. Brenier, C. De Lellis, and L. Sz´ ekelyhidi Jr.: Weak–strong uniqueness for measure- valued solutions. Comm. Math. Phys.305, 351–361 (2011)

    work page 2011

  4. [3]

    Baer, M., Nunziato, J.: A two-phase mixture theory for the deflagration-to- detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12, 861–889 (1986)

    work page 1986

  5. [4]

    Bouchut, F., Brenier, Y., Cortes, J. et al. A Hierarchy of Models for Two-Phase Flows. J. Nonlinear Sci.10, 639–660 (2000)

    work page 2000

  6. [5]

    Bresch, D., Jabin, P.-E.: Global existence of weak solutions for compressible Navier– Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Ann. of Math. (2)188, 577–684 (2018)

    work page 2018

  7. [6]

    Bˇ rezina, J., Feireisl, E., Novotn´ y, A.: Stability of strong solutions to the Navier– Stokes–Fourier system. SIAM J. Math. Anal.52, 1761–1785 (2020)

    work page 2020

  8. [7]

    Bresch, D., Hillairet, M.: A compressible multifluid system with new physical relax- ation terms. Ann. Sci. ´Ec. Norm. Sup´ er. (4)52, 255–295 (2019)

    work page 2019

Show all 58 references
  1. [8]

    Bresch, D., Mucha, P.B., Zatorska, E.: Finite-energy solutions for compressible two- fluid Stokes system. Arch. Rational Mech. Anal.232, 987–1029 (2019)

    2019

  2. [9]

    Bresch, D., Burtea, C., Lagouti` ere, F.: Mathematical justification of a compressible bifluid system with different pressure laws: a continuous approach. Appl. Anal.101, 4235–4266 (2022)

    2022

  3. [10]

    Burtea, T

    C. Burtea, T. Crin-Barat, and P. Gonin–Joubert: Pressure and temperature relax- ation limit for a one-velocity Baer–Nunziato model. arXiv:2605.23693 (2026)

    2026 arXiv

  4. [11]

    arXiv:2507.21850 (2025)

    Burtea, C., G´ erard-Varet, D.: Weak solutions of a viscous model for fluid–bubbles interaction. arXiv:2507.21850 (2025)

    2025 arXiv

  5. [12]

    arXiv:2605.11938 (2026)

    Burtea, C., G´ erard-Varet, D.: A variational approach to the derivation of reduced models for bubbly flows. arXiv:2605.11938 (2026)

    2026 arXiv

  6. [13]

    Stud Appl Math.153, Paper No

    Bresch, D., Narbona-Reina, G., Burgisser, A., and Collombet, M.: Mathematical topics in compressible flows from single-phase systems to two-phase averaged sys- tems. Stud Appl Math.153, Paper No. e12739 (2024)

    2024

  7. [14]

    Chen, G.-Q., Levermore, C.D., Liu, T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math.47, 787–830 (1994)

    1994

  8. [15]

    Burtea, C., Crin-Barat, T., Tan, J.: Relaxation limit for a damped one-velocity Baer- Nunziato model to a Kapila model. Math. Models Methods Appl. Sci.33, 687–753 (2023)

    2023

  9. [16]

    Chaudhuri, N.: On weak (measure-valued)-strong uniqueness for compressible Navier-Stokes system with non-monotone pressure law. J. Math. Fluid Mech.22, Paper No. 17 (2020)

    2020

  10. [17]

    Chaudhuri, N.: Limit of a consistent approximation to the complete compressible Euler system. J. Math. Fluid Mech.23, Paper No. 97 (2021)

    2021

  11. [18]

    NoDEA Nonlinear Differential Equations Appl.31, Paper No

    Chaudhuri, N.: On weak (measure valued)-strong uniqueness for Navier-Stokes- Fourier system with Dirichlet boundary condition. NoDEA Nonlinear Differential Equations Appl.31, Paper No. 5 (2024)

    2024

  12. [19]

    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal.70, 167–179 (1979). 38

    1979

  13. [20]

    Dain, S.: Generalized Korn’s inequality and conformal Killing vectors. Calc. Var. Partial Differential Equations25, 535–540 (2006)

    2006

  14. [21]

    DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal.88, 223–270 (1985)

    1985

  15. [22]

    Demoulini, S., Stuart, D.M.A., Tzavaras, A.E.: Weak–strong uniqueness of dissipa- tive measure-valued solutions for polyconvex elastodynamics. Arch. Rational Mech. Anal.205, 927–961 (2012)

    2012

  16. [23]

    Sci- ences, vol

    Drew, D., Passman, S.L.: Theory of Multicomponent Fluids, Applied Math. Sci- ences, vol. 135, Springer Verlag, (1999)

    1999

  17. [24]

    Funkcial

    Enomoto, Y., Shibata, Y.: On theR-sectoriality and the initial boundary value problem for the viscous compressible fluid flow. Funkcial. Ekvac.56, 441–505 (2013)

    2013

  18. [25]

    Evje, S., Karlsen, K.H.: Global existence of weak solutions for a viscous two-phase model. J. Differential Equations245, 2660–2703 (2008)

    2008

  19. [26]

    Evje, S., Wen, H., Zhu, C.: On global solutions to the viscous liquid–gas model with unconstrained transition to single-phase flow. Math. Models Methods Appl. Sci.27, 323–346 (2017)

    2017

  20. [27]

    Feireisl, E.: Compressible Navier-Stokes equations with a non-monotone pressure law. J. Differential Equations.184, 97–108 (2002)

    2002

  21. [28]

    Feireisl, E., Gwiazda, P., ´Swierczewska-Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Par- tial Differential Equations55, Paper No. 141 (2016)

    2016

  22. [29]

    Feireisl, E., Klein, R., Novotn´ y, A., Zatorska, E.: On singular limits arising in the scale analysis of stratified fluid flows. Math. Models Methods Appl. Sci.26, 419–443 (2016)

    2016

  23. [30]

    Feireisl, E., Luk´ aˇ cov´ a-Medvid’ov´ a, M.: Convergence of a mixed finite element–finite volume scheme for the isentropic Navier–Stokes system via dissipative measure- valued solutions. Found. Comput. Math.18, 703–730 (2018)

    2018

  24. [31]

    ESAIM Math

    Feireisl, E., Luk´ aˇ cov´ a-Medvid’ov´ a, M., Mizerov´ a, H., She, B.: Convergence of a finite volume scheme for the compressible Navier–Stokes system. ESAIM Math. Model. Numer. Anal.53, 1957–1979 (2019)

    2019

  25. [32]

    Feireisl, E., Luk´ aˇ cov´ a-Medvid’ov´ a, M., Mizerov´ a, H., She, B.: Numerical analy- sis of compressible fluid flows. MS&A. Modeling, Simulation and Applications, 20. Springer, Cham (2021)

    2021

  26. [33]

    S., Mishra, S., and Tadmor, E

    Fjordholm, U. S., Mishra, S., and Tadmor, E.. On the computation of measure- valued solutions.Acta Numer., 25:567–679, 2016

    2016

  27. [34]

    Springer (2006)

    Ishii, M., Hibiki, T.: Thermo-fluid dynamics of two-phase flow. Springer (2006)

    2006

  28. [35]

    Part II: Derivation of a macroscopic model

    Hillairet, M., Mathis, H., Seguin, N.: Analysis of compressible bubbly flows. Part II: Derivation of a macroscopic model. ESAIM Math. Model. Numer. Anal.57, 2865– 2906 (2023)

    2023

  29. [36]

    and Novotn´ y, A.: Existence and stability of dissipative turbulent solutions to a simple bi-fluid model of compressible fluids

    Jin, B., Kwon, Y.S., Neˇ casov´ a, ˇS. and Novotn´ y, A.: Existence and stability of dissipative turbulent solutions to a simple bi-fluid model of compressible fluids. J Elliptic Parabolic Equations.7, 537–570 (2021)

    2021

  30. [37]

    Jin, B.J., Novotn´ y, A.: Weak–strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids. J. Differential Equations.268, 204–238 (2019)

    2019

  31. [38]

    Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., Stewart, D.S.: Two-phase model- ing of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids13, 3002–3024 (2001)

    2001

  32. [39]

    Kraˇ cmar, S., Kwon, Y-S., Neˇ casov´ a,ˇS., Novotn´ y, A.: Weak solutions for a bifluid model for a mixture of two compressible noninteracting fluids with general boundary data. SIAM J. Math Anal.54, 818–871 (2022). 39

    2022

  33. [40]

    arXiv:2406.08383 (2024)

    Kalousek, M., Neˇ casov´ a,ˇS.: On existence of weak solutions to a Baer–Nunziato type system. arXiv:2406.08383 (2024)

    2024

  34. [41]

    arXiv:2512.16286 (2025)

    Lebot, C.: Low-Mach-number limit for multiphase flows. arXiv:2512.16286 (2025)

    2025 arXiv

  35. [42]

    arXiv:2602.23189 (2026)

    Lebot, C.: Low-Mach-number limit of a compressible two-phase flow system with algebraic closure. arXiv:2602.23189 (2026)

    2026 arXiv

  36. [43]

    Li, Y., Luk´ aˇ cov´ a-Medvid’ov´ a, M., Pokorn´ y, M., Zatorska, E.: Weak-strong unique- ness for bi-fluid compressible system with algebraic closure, arXiv:2602.15970 (2026)

    2026

  37. [44]

    arXiv:2603.07720 (2026)

    Li, Y., Luk´ aˇ cov´ a-Medvid’ov´ a, M., Zatorska, E.: Low Mach number limit and con- vergence rates for a compressible two-fluid model with algebraic pressure closure. arXiv:2603.07720 (2026)

    2026

  38. [45]

    Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Science Publications. Oxford Lecture Series in Mathematics and its Appli- cations 10, (1998)

    1998

  39. [47]

    Stability of Strong Solutions to the Full Compressible Magnetohydro- dynamic System with Non-Conservative Boundary Conditions

    Mizerov´ a, H. Stability of Strong Solutions to the Full Compressible Magnetohydro- dynamic System with Non-Conservative Boundary Conditions. J. Math. Fluid Mech. 27, 61 (2025)

    2025

  40. [48]

    Murrone, A., Guillard, H.: A five equation reduced model for compressible two phase flow problems. J. Comput. Phys.202, 664–698 (2005)

    2005

  41. [49]

    Sci China Math.63, 2399–2414 (2020)

    Novotn´ y, A.: Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids. Sci China Math.63, 2399–2414 (2020)

    2020

  42. [50]

    Novotn´ y, A., Pokorn´ y, M.: Weak solutions for some compressible multicomponent fluid models. Arch. Rational Mech. Anal.235, 355-403 (2020)

    2020

  43. [51]

    Oxford Lecture Series in Mathematics and Its Applications, p

    Novotn´ y, A., Straˇ skraba, I.: Introduction to the Mathematical Theory of Compress- ible Flow. Oxford Lecture Series in Mathematics and Its Applications, p. 27. Oxford University Press, Oxford (2004)

    2004

  44. [52]

    Piasecki, T., Zatorska, E.: Maximal regularity for compressible two-fluid system. J. Math. Fluid Mech.24, Paper No. 39, 23 pp (2022)

    2022

  45. [53]

    Piasecki, T., Shibata, Y., Zatorska, E.: On the maximalL p–Lq regularity of solutions to a general linear parabolic system. J. Differential Equations268, 3332–3369 (2020)

    2020

  46. [54]

    Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.150, 425–467 (1999)

    1999

  47. [55]

    Tzavaras, A.E.: Relative entropy in hyperbolic relaxation. Commun. Math. Sci.3, 119–132 (2005)

    2005

  48. [56]

    Wen, H., Yu, C.: Global weak solution to the viscous two-phase model with finite energy

    Vasseur, A. Wen, H., Yu, C.: Global weak solution to the viscous two-phase model with finite energy. J. Math. Pure. Appl.125, 247–282 (2019)

    2019

  49. [57]

    Nonlinear Anal.163, 34–59 (2017)

    Vauchelet, N., Zatorska, E.: Incompressible limit of the Navier–Stokes model with a growth term. Nonlinear Anal.163, 34–59 (2017)

    2017

  50. [58]

    Wen, H.: On global solutions to a viscous compressible two-fluid model with un- constrained transition to single-phase flow in three dimensions. Calc. Var. PDE.60, Paper No. 158 (2021). 40

    2021

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Reviewed August 14, 2026 · model on record in the stance chip above.