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Physical Vacuum Problems for the Full Compressible Euler Equations: Low-regularity Hadamard-style Local Well-posedness

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Pith's one-line read The full compressible Euler equations with a physical vacuum are locally well-posed in low-regularity weighted Sobolev spaces, in every space dimension, even when the gas-vacuum interface has unbounded curvature.

desk verdict Serious extension of Ifrim-Tataru to non-isentropic Euler, but the rough-solution bootstrap in §6.2 has a gap that excludes data with small B0 and large H2κ norm. read the letter →

arxiv 2411.17153 v4 pith:4OR23BGS submitted 2024-11-26 math.AP math-phmath.MP

Sicheng Liu , Tao Luo This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35Q3535Q3176N1035L60
keywords compressibleEulerequationsphysicalvacuumfreeboundaryproblemslocalwell-posednessweightedSobolevspacesnon-isentropicgasdynamicscontinuationcriteriaEuleriancoordinates
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

This paper claims that the full non-isentropic compressible Euler equations with a physical vacuum are locally well-posed in low-regularity weighted Sobolev spaces, in any space dimension, for initial data so rough that the gas-vacuum interface may have unbounded curvature. In other words, existence, uniqueness, and continuous dependence on the data all hold for such rough states, a question that was previously open for high-dimensional non-isentropic flows. The paper also proves sharp a priori energy estimates and a continuation criterion that guarantees the flow can be extended as long as the boundary stays non-degenerate and certain control parameters remain bounded. The proof works in Eulerian coordinates, avoiding the regularity loss of the flow map that would come with Lagrangian coordinates.

What carries the argument

The argument is carried by a reformulation and by adapted 'good unknowns'. The change of variables q = \frac{1+\$\beta$}{\$\beta$} $p^{{\beta/(1+\beta)}}$ and \$\sigma$ = $e^{{S/(1+\beta)}}$ turns the pressure gradient into a bilinear term, giving the system D_t q + \$\beta$ q \nabla\cdot v = 0, D_t v + \$\sigma$ \nabla q = 0, D_t \$\sigma$ = 0. Local well-posedness is then proved inside weighted Sobolev spaces built from powers of q, using self-adjoint degenerate-elliptic operators such as L_1 u = \$\beta$ q \$\Delta$ u + \nabla q\cdot\nabla u and L_2 w = \$\beta$ \nabla(q\nabla\cdot w) + (\nabla w)^*\nabla q to construct coercive energy functionals. The good unknowns s_{2k} = $D_t^{{2k}}$q - \nabla q\cdot $D_t^{{2k-1}}$v and w_{2k} = $D_t^{{2k}}$v satisfy linearized systems with admissible error terms, which keeps the energy propagation estimates free of derivative loss. Existence for rough data is obtained by Euler's polygonal iteration with carefully bi-scale regularized data, and the passage to fractional regularity uses frequency envelopes and interpolation.

What would settle it

Construct a non-isentropic initial state in d=2 with \$\beta$=1, an interface with unbounded curvature, and \kappa just above 1 + \frac12 + \frac12, then run the one-step polygonal iteration at decreasing time steps: if the discrete energy increments ever exceed (1+C\varepsilon) times the initial energy, the a priori estimate underlying the existence proof fails. Alternatively, exhibit two distinct $H^{{2\kappa}}$ limiting solutions from the same initial data, which would refute the uniqueness claim.

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Extended reading notes

Core claim

The central claim is Theorem 1.3: the full compressible Euler system written in the new variables (q, v, $\sigma$) is locally well-posed in the state space $H^{{2\kappa}}$ for every real \kappa > \kappa_0 + \frac12 = 1 + \frac d2 + \frac{1}{2\$\beta$}, where d is the space dimension and \$\beta$ is the polytropic exponent. In these spaces, q, v, and $\sigma$ are controlled by weighted Sobolev norms with weights $q^{{\kappa+(\alpha-1)/2}}$, $q^{{\kappa+\alpha/2}}$, and $q^{{\kappa+\alpha/2}}$, where q is a renormalized effective pressure and \$\alpha$ = \$beta^{{-1}}$. The Sobolev embeddings then imply that the free boundary is at least $C^{{1.5+}}$ and may have unbounded curvature, so the result covers interfaces outside the classical smooth-boundary theory. The paper establishes solutions in C([0,T]; $H^{{2\kappa}}$), uniqueness in a larger Lipschitz-type solution class, weak Lipschitz dependence on the data in an $L^{2}$-type distance, continuous dependence in the $H^{{2\kappa}}$ topology, sharp a priori energy estimates, and a continuation criterion.

Load-bearing premise

The load-bearing assumption is the physical-vacuum boundary scaling: the sound speed squared must vanish like the distance to the gas-vacuum interface, with the renormalized entropy bounded above and below, so that q has nonzero gradient at the boundary; if the sound speed decayed with a different power of distance, the weighted Sobolev spaces would not be adapted and the problem is expected to be ill-posed.

Editorial extensions

If this is right

  • Free gas-vacuum interfaces with unbounded curvature evolve deterministically for a short time, since unique local solutions now exist for such rough initial data.
  • The a priori energy estimate gives a Gronwall bound with no loss of derivatives, so the lifespan of a solution depends only on the size of the initial state and the control parameters.
  • Solutions can be continued as long as the boundary remains non-degenerate, \inf_{\Gamma_t}|\nabla q| \geq c_0 > 0, and the control parameters A_* and B remain bounded as required; in particular, splash-type singularities are excluded while these conditions hold.
  • Continuous dependence on initial data holds in the H^{2\kappa} topology, so nearby initial states produce nearby solutions in a concrete, checkable sense.
  • The same Eulerian framework works in all space dimensions d \geq 1 and for every polytropic exponent \beta > 0.

Reading between the lines

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

  • Because the entropy variable \sigma simply transports along particle paths, a similar Eulerian treatment may extend to other systems with transport coefficients, such as Euler-Poisson equations or multi-species gas models, once a renormalized conserved quantity with a uniformly bounded inverse is identified; this is beyond what the paper proves.
  • The threshold \kappa > 1 + \frac d2 + \frac{1}{2\beta} suggests that the entropy adds roughly half a derivative of difficulty: the most singular new terms involve \nabla\sigma weighted by powers of q, so pushing the regularity lower would likely require a different treatment of entropy waves.
  • The bi-scale regularization construction described in the paper could be read as a blueprint for numerical schemes: low-scale regularized data combined with high-scale corrections reproduces the energy without accumulating derivative loss, making the polygonal iteration a plausible computational strategy.
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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

1 major / 5 minor

Summary. The manuscript develops an Eulerian-coordinate theory for the non-isentropic compressible Euler equations with a physical vacuum, using the variables (q, v, σ) introduced in (1.5)-(1.6). Its central result, Theorem 1.3, asserts Hadamard-style local well-posedness in weighted Sobolev spaces H^{2κ} for any κ > κ0 + 1/2 = 1 + d/2 + 1/(2β), allowing gas-vacuum interfaces with unbounded curvature. The paper also proves a uniqueness theorem in a Lipschitz-type regularity class (Theorem 1.1 and the quantitative Theorem 3.1), sharp a priori energy estimates (Theorem 1.5), and a continuation criterion (Theorem 1.4). The method follows the Eulerian scheme of Ifrim-Tataru: linearized energy estimates, weighted interpolation and coercivity arguments, Euler-polygonal construction of high-regularity solutions via a two-scale regularization, and finally frequency-envelope/interpolation arguments for rough solutions.

Significance. If the main theorem is correct, this is a substantial advance: it extends the isentropic Eulerian physical-vacuum theory of Ifrim-Tataru [20] to variable entropy, while providing continuous dependence, a priori estimates, and continuation criteria in a low-regularity setting. The reformulation that reduces the fully nonlinear entropy coupling to bilinear terms is a genuine step, and the a priori estimates are derived from the equations rather than fitted to data. The uniqueness theorem covers almost all classical solutions. The main reservation is a bootstrap gap in §6.2 that affects the proof of existence in Theorem 1.3 for a non-negligible part of the stated data class; this is a load-bearing issue rather than a presentation concern.

major comments (1)
  1. [§6.2 (after Eq. (6.13))] The bootstrap closure for the lifespan of the regularized solutions is not valid for the full data class stated in Theorem 1.3. The text says 'if B0 is chosen so that B0 ≫ A∗0 and B0 ≫ M0', but B0 is not a free parameter: it is the control parameter (1.17) of the given initial datum, and Definition 1.2 imposes no relation between B0, A∗0, and M0 = ∥(q0,v0,σ0)∥_{H^{2κ}}. Data with small B0 and order-one M0 are easy to construct: fix a smooth φ supported away from Γ0 and add a_N φ(Nx) to v0 or σ0 with a_N = N^{-2κ}; then M0 remains of order one while ∥∇(·)∥_{L∞} + ∥·∥_{C^{1/2+ε}} tends to zero because κ > κ0 + 1/2 > 1/2. For such data the bootstrap assumptions (6.7), the estimate (6.13), and the claimed improvement do not close, so the uniform lifespan T independent of h is not established and Theorem 1.3 is not proved as stated. The proof needs either a bootstrap whose constants are functions of M0 (with T0 also allowed to depend on M0 and c0), or an explicit smallness/relation condition on the data; as written, the existence theorem is incomplete.
minor comments (5)
  1. [§1.3, Eq. (1.9)] The displayed physical energy Ephy lacks the volume element dx, and the following sentence 'one can first infer from (1.9) that ∫ ... dx' is grammatically incomplete; the intended conclusion 'is conserved' should be written out.
  2. [§3.1, Eq. (3.6)] The notation κ := σ1 + σ2 in (3.6) collides with the Sobolev regularity index κ used throughout the paper; renaming the sum (for example Σ) would remove a source of confusion.
  3. [§5.5.2, Eq. (5.25)] The last displayed condition in (5.25) mixes ε and ϵ in the term '(ϵ2t)2χ2ε', and the left-hand side '(εt)χε · (1 − χε)' would be clearer with a consistent product notation.
  4. [§6.1, Proposition 6.2] The statement introduces δ as an arbitrarily small constant in the envelope estimate and later uses δ and δ′ for a boundary-layer separation parameter; these uses should be distinguished notationally.
  5. [§4–§6, imported results] Several load-bearing technical tools (Propositions 4.1–4.4, 5.4, and 6.2) are quoted from [20] without proof. This is acceptable as a citation practice, but the manuscript should explicitly flag at each first use that these are imported results and state the exact hypotheses under which they are being invoked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the well-posedness proof is self-contained from the PDE system, with all load-bearing external references drawn from prior authors and no fitted parameters or definitional reductions.

full rationale

The central claim, Theorem 1.3, is established by an explicit derivation chain: the reformulation (1.6), the weighted Sobolev state spaces, the linearized estimates (Proposition 2.1), the a priori energy estimates (Theorem 4.8), the high-regularity construction via Euler's polygonal method and two-scale regularization, and finally the rough-solution limit via frequency envelopes and interpolation. No parameter is fitted to any data set; the scaling analysis in Section 1.3 is used only to motivate function spaces, not to predetermine the estimates. The energy coercivity and propagation estimates are proved from the equations themselves using elliptic estimates and interpolation lemmas; where results are imported, they are external results of Ifrim-Tataru, Tao, and Lunardi, none of which assume the target theorem. The paper contains self-citations (e.g., [39]-[41]), but these concern related physical-vacuum problems and are not load-bearing for the present well-posedness theorem; the uniqueness result is proved in Section 3 rather than imported. The skeptical observation about the bootstrap condition 'if B0 is chosen so that B0 ≫ A∗0 and B0 ≫ M0' in Section 6.2 is a possible correctness concern about the choices of control parameters, but it is not a circularity: it does not identify an output that is equivalent by construction to an input, nor a fitted parameter renamed as a prediction. Accordingly, no circular step can be exhibited from the text.

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

The paper introduces no new physical entities or fitted parameters. The axioms listed are the physical and structural assumptions that define the physical-vacuum problem and the technical tools imported from the cited literature. The renormalized variables q and σ are changes of variables, not invented entities.

assumptions (5)
  • domain assumption Physical vacuum scaling: c_s^2 ≃ dist(x, Γ_t), equivalently |∇q| ~ const > 0 on Γ_t (equations (1.3)-(1.4), (1.7)).
    Defines the problem class; if λ ≠ 1/2 the weighted spaces are not adapted and ill-posedness is expected (Jang-Masmoudi [24]).
  • domain assumption Polytropic gas law p = ρ^{1+β} e^S with constant β > 0 (equation (1.2)).
    The state equation fixes the algebraic structure used in the reparametrization (1.5)-(1.6).
  • domain assumption Uniform entropy bounds: ∥σ∥_{L∞} + ∥σ^{-1}∥_{L∞} < ∞, and σ transported (Dt σ = 0).
    Used to pass from (1.2) to (1.6) and to control A*; without it the physical vacuum condition is not stable.
  • domain assumption Boundary of initial domain is a finite union of disjoint C^{1+} hypersurfaces; q ∈ C^{1+}, non-degenerate |∇q| > 0 on Γ (Definition 1.2).
    Needed for the weighted Sobolev calculus and elliptic estimates (Lemmas 4.10-4.13).
  • standard math Weighted interpolation inequalities (Propositions 4.1-4.4), regularization operator estimates (Proposition 5.4), and frequency-envelope decomposition (Proposition 6.2) hold as stated.
    Taken from Ifrim-Tataru [20] and Tao [49]; used throughout to close the energy estimates and to construct and limit regular solutions.

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Pith. "Pith review of Physical Vacuum Problems for the Full Compressible Euler Equations: Low-regularity Hadamard-style Local Well-posedness." pith.science (2026). https://pith.science/paper/4OR23BGS

@misc{pith2026241117153,
  author       = {Pith},
  title        = {Pith review of: Physical Vacuum Problems for the Full Compressible Euler Equations: Low-regularity Hadamard-style Local Well-posedness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OR23BGS}},
  note         = {Machine review of arXiv:2411.17153}
}
read the original abstract

This manuscript concerns the dynamics of non-isentropic compressible Euler equations in a physical vacuum. We establish the Hadamard-style local well-posedness in low-regularity weighted Sobolev spaces, where the gas-vacuum interface is allowed to have unbounded curvature, demonstrating existence, uniqueness, and continuous dependence on initial data. Additionally, we prove sharp a priori energy estimates and continuation criteria. The approach is based on the framework of Eulerian coordinates, avoiding the regularity issues of the flow map and the high nonlinearity induced by the Lagrangian transformation.

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Works this paper leans on

61 extracted references · 59 canonical work pages

  1. [20]

    Mihaela Ifrim and Daniel Tataru, The compressible Euler equations in a physical vacuum: a compre- hensive Eulerian approach, Ann. Inst. H. Poincar´e C Anal. Non Lin´eaire 41 (2024), 405–495

    work page 2024

  2. [1]

    Alazard, N

    T. Alazard, N. Burq, and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math. 198 (2014), 71–163

    work page 2014

  3. [2]

    Lars Andersson and Huali Zhang, Well-posedness for rough solutions of the 3D compressible euler equations, arXiv preprint (2023), arXiv:2208.10132

    work page arXiv 2023

  4. [3]

    Jean Bourgain and Dong Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math. 201 (2015), 97–157

    work page 2015

  5. [4]

    Chemin, Dynamique des gaz `a masse totale finie, Asymptotic Anal

    J.-Y. Chemin, Dynamique des gaz `a masse totale finie, Asymptotic Anal. 3 (1990), 215–220

    work page 1990

  6. [5]

    Chorin and Jerrold E

    Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, third ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993

    work page 1993

  7. [6]

    Daniel Coutand, Hans Lindblad, and Steve Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys. 296 (2010), 559–587

    work page 2010

  8. [7]

    Pure Appl

    Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum , Comm. Pure Appl. Math. 64 (2011), 328– 366

    work page 2011

Show all 61 references
  1. [8]

    Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for the moving- boundary three-dimensional compressible Euler equations in physical vacuum , Arch. Ration. Mech. Anal. 206 (2012), 515–616

    2012

  2. [9]

    Marcelo M. Disconzi, Mihaela Ifrim, and Daniel Tataru, The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation crite- rion, Arch. Ration. Mech. Anal. 245 (2022), 127–182

    2022

  3. [10]

    Disconzi, Chenyun Luo, Giusy Mazzone, and Jared Speck, Rough sound waves in 3D compressible Euler flow with vorticity, Selecta Math

    Marcelo M. Disconzi, Chenyun Luo, Giusy Mazzone, and Jared Speck, Rough sound waves in 3D compressible Euler flow with vorticity, Selecta Math. (N.S.) 28 (2022), Paper No. 41, 153

    2022

  4. [11]

    21 (2019), 231–266

    Yongcai Geng, Yachun Li, Dehua Wang, and Runzhang Xu,Well-posedness of non-isentropic Euler equations with physical vacuum, Interfaces Free Bound. 21 (2019), 231–266

    2019

  5. [12]

    Daniel Ginsberg, Hans Lindblad, and Chenyun Luo, Local well-posedness for the motion of a com- pressible, self-gravitating liquid with free surface boundary, Arch. Ration. Mech. Anal. 236 (2020), 603–733. 72 Nonisentropic Ideal Gases in A Physical V acuum REFERENCES

    2020

  6. [13]

    Mahir Had ˇzi´c and Juhi Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math. 214 (2018), 1205–1266

    2018

  7. [14]

    Pure Appl

    Mahir Had ˇzi´c and Juhi Jang,Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system, Comm. Pure Appl. Math. 71 (2018), 827–891

    2018

  8. [15]

    Mahir Had ˇzi´c, Juhi Jang, and King Ming Lam, Nonradial stability of self-similarly expanding goldreich-weber stars, arXiv preprint (2022), arXiv:2212.11420

    2022 arXiv

  9. [16]

    III , Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition

    Lars H ¨ormander, The analysis of linear partial differential operators. III , Classics in Mathematics, Springer, Berlin, 2007, Pseudo-differential operators, Reprint of the 1994 edition

    2007

  10. [17]

    Mihaela Ifrim, Ben Pineau, Daniel Tataru, and Mitchell A Taylor, Sharp well-posedness for the free boundary mhd equations, arXiv preprint (2024), arXiv:2412.15625

    2024 arXiv

  11. [18]

    PDE 11 (2025), Paper No

    Mihaela Ifrim, Ben Pineau, Daniel Tataru, and Mitchell A Taylor, Sharp Hadamard local well- posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free bound- ary Euler equations, Ann. PDE 11 (2025), Paper No. 16, 154 pp

    2025

  12. [19]

    Mihaela Ifrim and Daniel Tataru, Local well-posedness for quasi-linear problems: a primer , Bull. Amer. Math. Soc. (N.S.) 60 (2023), 167–194

    2023

  13. [21]

    Juhi Jang, Jiaqi Liu, and Nader Masmoudi, Waiting time solutions in gas dynamics, arXiv preprint (2025), arXiv:2501.07831

    2025 arXiv

  14. [22]

    Pure Appl

    Juhi Jang and Nader Masmoudi, Well-posedness for compressible Euler equations with physical vac- uum singularity, Comm. Pure Appl. Math. 62 (2009), 1327–1385

    2009

  15. [23]

    Juhi Jang and Nader Masmoudi, Vacuum in gas and fluid dynamics, Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 315–329

    2011

  16. [24]

    Juhi Jang and Nader Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012), 115625, 11

    2012

  17. [25]

    Pure Appl

    Juhi Jang and Nader Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math. 68 (2015), 61–111

    2015

  18. [26]

    Rational Mech

    Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems , Arch. Rational Mech. Anal. 58 (1975), 181–205

    1975

  19. [27]

    Lannes, Modeling shallow water waves, Nonlinearity 33 (2020), R1–R57

    D. Lannes, Modeling shallow water waves, Nonlinearity 33 (2020), R1–R57

    2020

  20. [28]

    Lax, Hyperbolic partial differential equations , Courant Lecture Notes in Mathematics, vol

    Peter D. Lax, Hyperbolic partial differential equations , Courant Lecture Notes in Mathematics, vol. 14, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006, With an appendix by Cathleen S. Morawetz

    2006

  21. [29]

    Hans Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary , Comm. Math. Phys. 260 (2005), 319–392

    2005

  22. [30]

    Pure Appl

    Hans Lindblad and Chenyun Luo, A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit, Comm. Pure Appl. Math.71 (2018), 1273– 1333

    2018

  23. [31]

    T.-P Liu and J.A Smoller, On the vacuum state for the isentropic gas dynamics equations, Advances in Applied Mathematics 1 (1980), 345–359

    1980

  24. [32]

    Tai-Ping Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math.13 (1996), 25–32. 73 REFERENCES Sicheng LIU and Tao LUO

    1996

  25. [33]

    Differential Equations 140 (1997), 223–237

    Tai-Ping Liu and Tong Yang, Compressible Euler equations with vacuum, J. Differential Equations 140 (1997), 223–237

    1997

  26. [34]

    Tai-Ping Liu and Tong Yang, Compressible flow with vacuum and physical singularity , Methods Appl. Anal. 7 (2000), 495–509, Cathleen Morawetz: a great mathematician

    2000

  27. [35]

    Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes

    Alessandra Lunardi, Interpolation theory, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 16, Edizioni della Normale, Pisa, 2018, Third edition [of MR2523200]

    2018

  28. [36]

    PDE 4 (2018), Paper No

    Chenyun Luo, On the motion of a compressible gravity water wave with vorticity , Ann. PDE 4 (2018), Paper No. 20, 71

    2018

  29. [37]

    Differential Equations 332 (2022), 333–403

    Chenyun Luo and Junyan Zhang, Local well-posedness for the motion of a compressible gravity water wave with vorticity, J. Differential Equations 332 (2022), 333–403

    2022

  30. [38]

    Tao Luo and Joel Smoller, On the Euler-Poisson equations of self-gravitating compressible fluids , Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 415–431

    2011

  31. [39]

    Tao Luo, Zhouping Xin, and Huihui Zeng, Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation, Arch. Ration. Mech. Anal. 213 (2014), 763–831

    2014

  32. [40]

    Pure Appl

    Tao Luo and Huihui Zeng, Global existence of smooth solutions and convergence to Barenblatt so- lutions for the physical vacuum free boundary problem of compressible Euler equations with damping , Comm. Pure Appl. Math. 69 (2016), 1354–1396

    2016

  33. [41]

    Press, Boston, MA, [2020] ©2020, pp

    Tao Luo and Huihui Zeng, Some results on fluid free boundary problems, Proceedings of the Inter- national Consortium of Chinese Mathematicians 2017, Int. Press, Boston, MA, [2020] ©2020, pp. 453–464

    2017

  34. [42]

    Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol

    A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984

    1984

  35. [43]

    Tetu Makino, Seiji Ukai, and Shuichi Kawashima, Sur la solution `a support compact de l’´equations d’Euler compressible, Japan J. Appl. Math. 3 (1986), 249–257

    1986

  36. [44]

    Calum Rickard, Mahir Had ˇzi´c, and Juhi Jang, Global existence of the nonisentropic compressible Euler equations with vacuum boundary surrounding a variable entropy state, Nonlinearity 34 (2021), 33–91

    2021

  37. [45]

    Denis Serre, Expansion of a compressible gas in vacuum , Bull. Inst. Math. Acad. Sin. (N.S.) 10 (2015), 695–716

    2015

  38. [46]

    Jalal Shatah and Chongchun Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, 61 (2008), 698–744

    2008

  39. [47]

    Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch

    Steve Shkoller and Thomas C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal. 234 (2019), 115–180

    2019

  40. [48]

    Smith and Daniel Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann

    Hart F. Smith and Daniel Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), 291–366

    2005

  41. [49]

    Terence Tao, Global regularity of wave maps. II. Small energy in two dimensions , Comm. Math. Phys. 224 (2001), 443–544

    2001

  42. [50]

    Pure Appl

    Yuri Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic com- pressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math. 62 (2009), 1551–1594. 74 Nonisentropic Ideal Gases in A Physical V acuum REFERENCES

    2009

  43. [51]

    Chao Wang, Zhifei Zhang, Weiren Zhao, and Yunrui Zheng,Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary , Mem. Amer. Math. Soc. 270 (2021), v + 119

    2021

  44. [52]

    PDE 3 (2017), Paper No

    Qian Wang, A geometric approach for sharp local well-posedness of quasilinear wave equations, Ann. PDE 3 (2017), Paper No. 12, 108

    2017

  45. [53]

    Qian Wang, Rough solutions of the 3-D compressible Euler equations, Ann. of Math. (2)195 (2022), 509–654

    2022

  46. [54]

    Qian Wang, On global dynamics of 3-D irrotational compressible fluids , arXiv preprint (2024), arXiv:2407.13649

    2024 arXiv

  47. [55]

    Wei Wang, Zhifei Zhang, and Wenbin Zhao, Well-posedness of the free boundary problem for the compressible Euler equations and the incompressible limit , Commun. Math. Anal. Appl. 1 (2022), 410–456

    2022

  48. [56]

    Sijue Wu, Wellposedness of the 2D full water wave equation in a regime that allows for non-C1 inter- faces, Invent. Math. 217 (2019), 241–375

    2019

  49. [57]

    Differential Equations 210 (2005), 217–231

    Chao-Jiang Xu and Tong Yang, Local existence with physical vacuum boundary condition to Euler equations with damping, J. Differential Equations 210 (2005), 217–231

    2005

  50. [58]

    Tong Yang, Singular behavior of vacuum states for compressible fluids, J. Comput. Appl. Math. 190 (2006), 211–231

    2006

  51. [59]

    Huihui Zeng, Global resolution of the physical vacuum singularity for three-dimensional isentropic in- viscid flows with damping in spherically symmetric motions, Arch. Ration. Mech. Anal.226 (2017), 33–82

    2017

  52. [60]

    Huihui Zeng, Time-asymptotics of physical vacuum free boundaries for compressible inviscid flows with damping, Calc. Var. Partial Differential Equations 61 (2022), Paper No. 59, 32

    2022

  53. [61]

    Huihui Zeng, Global solution to the physical vacuum problem of compressible Euler equations with damping and gravity, SIAM J. Math. Anal. 55 (2023), 6375–6424. 75

    2023

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Reviewed August 12, 2026 · model on record in the stance chip above.