Pith. sign in

REVIEW 3 major objections 5 minor 154 references

Prandtl Equations and Related Boundary Layer Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A book-length survey claims five new well-posedness theorems for boundary-layer equations, with the proofs said to live in unpublished chapters.

desk verdict The advertised new theorems are not in this arXiv file; what remains is a useful but unpolished survey, so the book's main claim is currently unassessable. read the letter →

arxiv 2411.14081 v1 pith:QAHLM4KV submitted 2024-11-21 math.AP

Yuming Qin , Xiaolei Dong , Xiuqing Wang This is my paper · ORCID
classification math.AP MSC 35Q3576D1035A0135A0235Q30
keywords PrandtlequationsboundarylayerMHDwell-posednessanalyticregularityweightedSobolevspacesshearflowPrandtl-Hartmannregime
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 book aims to establish new well-posedness results for five boundary-layer models, and its survey chapter organizes the state of the art up to 2020. The authors state in the preface that Chapters 2 through 6 contain previously unpublished proofs: global well-posedness for the two-dimensional Prandtl-Hartmann equations in an analytic framework, local existence for the two-dimensional Prandtl equations in a weighted Sobolev space, local well-posedness for the two-dimensional mixed Prandtl equations without monotonicity and lower bound, local existence for the two-dimensional magnetic Prandtl model in the Prandtl-Hartmann regime, and local existence for three-dimensional Prandtl equations with a special structure. If those proofs are correct, the book delivers new existence, uniqueness, and stability results for equations that describe thin viscous layers near solid boundaries. The arXiv text visible here contains only the survey chapter and front matter; none of the chapter proofs appears in this posting.

What carries the argument

The mechanisms named in the chapter openings are analytic a priori estimates for the Prandtl-Hartmann system, a shear-flow decomposition that separates a one-dimensional heat-equation profile from the perturbation, weighted Sobolev spaces with polynomial weights in the normal variable, regularized systems with uniform estimates, and a structural assumption on the 3D velocity field that prevents the loss of tangential derivative. Each chapter is organized around uniform estimates, then existence, then uniqueness.

What would settle it

Obtain the full book text and check the key a priori estimate in Chapter 2 that controls the analytic norm globally in time; alternatively, construct an initial datum satisfying the stated hypotheses of Chapter 4 whose solution loses regularity before the claimed existence time or violates the claimed uniqueness, which would directly contradict the theorem.

Watch

Extended reading notes

Core claim

On its own terms, the book's original contribution is a set of existence and uniqueness theorems whose proofs are claimed to occupy Chapters 2 through 6. For the 2D Prandtl-Hartmann equations, it claims global well-posedness in an analytic class; for the 2D Prandtl equations, local existence in a weighted Sobolev space; for the 2D mixed Prandtl equations, local well-posedness in a Sobolev space with neither the usual monotonicity assumption nor a lower bound on the tangential velocity; for the 2D magnetic Prandtl model, local existence in the Prandtl-Hartmann regime; and for the 3D Prandtl equations, local existence under a special structural condition. The survey chapter that is actually present documents the prior landscape—steady and unsteady problems, local versus global existence, Gevrey and analytic classes, ill-posedness examples, and viscosity limits—against which these new results are positioned.

Load-bearing premise

The load-bearing premise is that the proofs advertised for Chapters 2 through 6 are correct, even though this arXiv posting does not show them; if any central a priori estimate or fixed-point argument in those chapters fails, the corresponding well-posedness theorem collapses.

Editorial extensions

If this is right

  • If Chapter 2 is right, the 2D Prandtl-Hartmann equations are known to be globally well-posed in a class with analytic regularity.
  • If Chapter 3 is right, local existence for the 2D Prandtl equations holds in a weighted Sobolev space, extending the range of admissible data beyond the analytic setting.
  • If Chapter 4 is right, monotonicity and the lower-bound condition are not necessary for local well-posedness of the 2D mixed Prandtl system.
  • If Chapter 5 is right, the 2D magnetic Prandtl model gains a local existence theory in the Prandtl-Hartmann regime.
  • If Chapter 6 is right, a special structural class of 3D Prandtl equations is locally well-posed even though generic 3D boundary layers are known to be ill-posed in Sobolev classes.

Reading between the lines

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

  • Because the proofs are not visible in the arXiv posting, readers cannot yet verify the advertised theorems from this text; the book's contribution becomes checkable only once the full chapters are available.
  • If the Chapter 4 claim against monotonicity holds, it likely relies on the magnetic-field coupling in the mixed Prandtl system, and the same strategy may transfer to other regularized boundary-layer models.
  • The weighted Sobolev technique of Chapter 3, if valid, could be applied to non-monotone shear flows, where existing energy methods typically require a monotonicity condition.
  • A natural test is to compare the claimed global existence time in Chapter 2 with numerical or asymptotic behavior of Prandtl-Hartmann solutions for shear flows with large data.
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

3 major / 5 minor

Summary. This manuscript presents the front matter and Chapter 1 of a planned monograph on Prandtl and MHD boundary-layer equations. The Abstract and Preface announce that Chapters 2–6 contain previously unpublished proofs of well-posedness for five models: the 2D Prandtl–Hartmann equations in an analytic framework (Chapter 2), the 2D Prandtl equations in a weighted Sobolev space (Chapter 3), the 2D mixed Prandtl equations without monotonicity or lower bound (Chapter 4), the 2D magnetic Prandtl model in the Prandtl–Hartmann regime (Chapter 5), and the 3D Prandtl equations with special structure (Chapter 6). The submitted text, however, contains only the table of contents, the Preface, and Chapter 1, which is a survey of results up to 2020. None of the announced proofs or theorem statements for Chapters 2–6 is present. The Abstract and the Table of Contents also disagree on whether Chapter 5 concerns global or local existence. As submitted, the manuscript’s original contribution cannot be checked.

Significance. If the announced theorems and their proofs were supplied and correct, the book would provide a useful unified reference and several new well-posedness results for boundary-layer models. The survey chapter alone collects a broad range of classical and recent results and may be of bibliographic value. However, the central contribution of the work—the new proofs in Chapters 2–6—cannot be evaluated from the submitted text. There are no machine-checked proofs, reproducible code, or parameter-free derivations visible to offset this omission. In its current form the manuscript supports only the survey part of its stated program, and the claimed new results remain unsupported.

major comments (3)
  1. [Preface, p. vii; Abstract; Table of Contents] The central claim that Chapters 2–6 contain new, previously unpublished proofs is unsupported: those chapters are absent from the submission. The text stops after Chapter 1, and the proofs, or even the statements of the theorems, for e.g. §5.3 (“Local-in-time existence and uniqueness”) and §6.3 (“Local-in-time existence and uniqueness”) are not included. A referee cannot check the a priori estimates, fixed-point arguments, or uniqueness arguments announced there. Since these chapters are the book’s original contribution, this omission is load-bearing and cannot be repaired by local editing.
  2. [Abstract vs. Table of Contents, Chapter 5] The Abstract states that Chapter 5 concerns “global existence of solutions to the 2D magnetic Prandtl equations,” while the Table of Contents and Preface both state “Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime.” This conflict concerns the main result attributed to Chapter 5, and because the chapter is absent the reader cannot determine which claim is intended. The manuscript must be internally consistent on this point before the announced results can be assessed.
  3. [Preface, p. vii] The Preface explicitly states that the results in Chapters 2–6 “have been obtained recently by the authors and have never been published before.” Since none of those chapters appears in the submission, the claimed novelty is entirely unverifiable. This is not a matter of an arguable step within a visible proof; the proofs are missing altogether, so the manuscript does not currently satisfy the standards of a self-contained mathematical work.
minor comments (5)
  1. [Title page and front matter] The author names contain spacing artifacts such as “Y uming Qin” and “Y uming’s father,” and the dedication page contains the typo “Xiuqiung’s husband.” These should be cleaned before publication.
  2. [§1.1.1, Theorem 1.1.3] The hypothesis “u0(y) for y > 0” is incomplete, and “/nequal0” appears in place of a mathematical relation; as printed the theorem statement cannot be read precisely.
  3. [§1.1.1, Theorem 1.1.2] The condition “uy > 0 on y = 0 if x > 0” is repeated, and the following clause “uy > 0 on y = 0 if x > 0, and uy < 0 on y = 0 if x < 0” appears garbled; the intended statement should be reformulated.
  4. [§1.1.2, Theorem 1.1.39] The stability estimate contains the expression “max{√β − Cη}, √δ,” which is not defined in the text and makes the displayed inequality ambiguous.
  5. [Chapter 1 generally] The survey cites many works by bracketed numbers, but the submitted excerpt contains no bibliography, so the reader cannot verify the cited literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning detectable in the reviewed excerpt; the new results are announced, not derived, so there is no derivation chain to reduce.

full rationale

The reviewed material consists of front matter, a table of contents, and Chapter 1, which is a survey of previously published results. The original contribution is announced in the Preface: 'The results in Chapters 2-6 have been obtained recently by the authors and have never been published before.' None of the proofs, estimates, or existence arguments for Chapters 2-6 is present in the text provided, so there is no visible derivation chain whose premises can be compared with its conclusions. No parameter is fitted and then renamed as a prediction; no uniqueness theorem or ansatz is imported from the authors' prior work to force the announced conclusions; and no stated result is defined in terms of another stated result. The only internal discrepancy noted is that the Abstract describes Chapter 5 as 'global existence' while the Preface and Table of Contents say 'Local well-posedness,' but this is an inconsistency in claims, not a circular argument. A missing proof is a verification and completeness concern, not evidence of circularity. Therefore the appropriate finding is no significant circularity, score 0.

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

The reviewed excerpt does not contain the derivations of the claimed new results, so no free parameters, axioms beyond standard PDE background, or invented entities can be identified from the available material. The survey chapter quotes many theorems from the literature but contributes no new mathematical entities.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Prandtl Equations and Related Boundary Layer Equations." pith.science (2026). https://pith.science/paper/QAHLM4KV

@misc{pith2026241114081,
  author       = {Pith},
  title        = {Pith review of: Prandtl Equations and Related Boundary Layer Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QAHLM4KV}},
  note         = {Machine review of arXiv:2411.14081}
}
read the original abstract

This book aims to present some recent results on Prandtl equations and MHD boundary layer equations. This book is essentially divided into two parts. Chapter 1 as the first part systematically surveys the results till 2020 on Prandtl equations and MHD boundary layer equations. Chapter 2 to 6 are the main part of the book, which presents the local and the global well-posedness of solutions to the Prandtl equations and MHD boundary layer equations. In detail, Chapter 2 is concerned with global well-posedness of solutions to the 2D Prandtl-Hartmann equations in an analytic framework. Chapter 3 investigates the local existence of solutions to the 2D Prandtl equations in a weighted Sobolev space. Chapter 4 studies the local well-posedness of solutions to the 2D mixed Prandtl equations in a Sobolev space without monotonicity and lower bound. Chapter 5 is concerned with global existence of solutions to the 2D magnetic Prandtl equations in the Prandtl-Hartmann regime. Chapter 6 proves the local existence of solutions to the 3D Prandtl equations with a special structure. Mathematicians and physicists who are interested in fluid dynamics will find this book helpful.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

154 extracted references · 78 canonical work pages

  1. [1]

    Acrivos, Solution of the laminar boundary layer energ y equation at high Prandtl numbers, The Physics of Fluids, 3(4)(1960), 10.1063 /1.1706102

    A. Acrivos, Solution of the laminar boundary layer energ y equation at high Prandtl numbers, The Physics of Fluids, 3(4)(1960), 10.1063 /1.1706102

    1960

  2. [2]

    R. A. Adams and J. J. F. Fournier, Sobolev Spaces, V ol. 140 , 2003

    2003

  3. [3]

    Alexander, Y

    R. Alexander, Y . Wang, C. Xu and T. Y ang, Well posedness of the Prandtl eqauation in Sobolev spaces, J. Amer. Math. Soc., 28(3)(2015), 745-784

    2015

  4. [4]

    J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equa- tions, Math. Comput., 37(1981), 243-259

    1981

  5. [5]

    R. E. Caflisch and M. Sammartino, Existence and singulari ties for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80(11-12)(2000), 733-74 4

    2000

  6. [6]

    Cannone, M

    M. Cannone, M. Lombardo and M. Sammartino, Existence and uniqueness for the Prandtl equa- tions, C. R. Acad. Sci. Paris., 332(2001), 277-282

    2001

  7. [7]

    Cannone, M

    M. Cannone, M. Lombardo and M. Sammartino, Well-posedne ss of Prandtl equations with non- compatible data, Nonlinearity, 26(2013), 3077-3100

    2013

  8. [8]

    Cannone, M

    M. Cannone, M. Lombardo and M. Sammartino, On the Prandtl boundary layer equations in pres- ence of corner singularities, Acta. Appl. Math., 132(2014) , 139-149

    2014

Show all 154 references
  1. [9]

    D. Chen, S. Ren, Y . Wang and Z. Zhang, Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime, Asymptotic Anal., 120(3-4 )(2020), 1-21

    2020

  2. [10]

    D. Chen, S. Ren, Y . Wang and Z. Zhang, Long time well-pose dness of the MHD boundary layer equation in Sobolev space, Anal. Theory Appl., 36(1)(2020) , 1-18

    2020

  3. [11]

    D. Chen, Y . Wang and Z. Zhang, Well-posedness of the line arized Prandtl equation around a non- monotonic shear flow, Ann. H. Poincar´ e, 35(2018), 1119-1142

    2018

  4. [12]

    D. Chen, Y . Wang and Z. Zhang, Well-posedness of the Pran dtl equation with monotonicity in Sobolev spaces, J. Di fferential. Equations, 264(2018), 5870-5893

    2018

  5. [13]

    W. F. Cope and D. R. Hartree, The laminar boundary layer i n compressible flow, Philos. Trans. R. Soc. A., 241(827)(1951), 1-69

    1951

  6. [14]

    T. G. Cowling, Magnetohydrodynamics, Interscience Tr acts on Physics and Astronomy, 4. Inter- science, New Y ork; Interscience, London, 1957

    1957

  7. [15]

    A. L. Dalibard, H. Dietert, D. G´ erard-V aret and F. Marbach, High frequency analysis of the un - steady interactive boundary layer model, SIAM J. Math. Anal ., (2017), arXiv:1710.04510v1. 257 258 BIBLIOGRAPHY

    2017 arXiv

  8. [16]

    A. L. Dalibard and N. Masmoudi, Separation for the stati onary Prandtl equation, Publications Math´ematiques de l’IH´eS, 130(2019), 187-297

    2019

  9. [17]

    A. L. Dalibard, M. Paddick, An existence result for the s teady rotating Prandtl equation, J. Math. Fluid Mech., 23(2021), Article number: 13

    2021

  10. [18]

    P . A. Davidson, An Introduction to Magnetohydrodynami cs, Cambridge Texts in Applied Mathe- matics, Cambridge University Press, Cambridge, 2001

    2001

  11. [19]

    Dietert and D

    H. Dietert and D. G´erard-V aret, Well-posedness of the Prandtl equations without any structural as- sumption, Ann. Partial Differential Equations, (2019), https://doi.org/10.1007/s40818-019-0063-6

    2019 doi

  12. [20]

    Ding, Remarks on the Prandtl boundary layer, J

    Y . Ding, Remarks on the Prandtl boundary layer, J. Math. Anal. Appl., 389(2012), 412-419

    2012

  13. [21]

    Ding and S

    M. Ding and S. Gong, Global existence of weak solution to the compressible Prandtl equations, J. Math. Fluid. Mech., 19(2017), 239-254

    2017

  14. [22]

    S. Ding, Z. Ji and Z. Lin, V alidity of Prandtl layer theor y for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows, J. Di fferential. Equations, 278(2021), 220- 293

    2021

  15. [23]

    Ding and Q

    S. Ding and Q. Li, Symmetrical Prandtl boundary layer ex pansions of steady Navier-Stokes equa- tions on bounded domain, J. Di fferential Equations, 268(2020), 1771-1819

    2020

  16. [24]

    Dong and Y

    X. Dong and Y . Qin, Global well-posedness of solutions t o 2D Prandtl-Hartmann equations in analytic framework, J. Part. Di ff. Eq., 3(35)(2022), 289-306

    2022

  17. [25]

    Drasin, Stability of parallel flow in a parallel magne tic field at small magnetic Reynolds number, J

    P . Drasin, Stability of parallel flow in a parallel magne tic field at small magnetic Reynolds number, J. Fluid Mech., 8 (1960), 130-142

    1960

  18. [26]

    E, Boundary layer theory and the zero-viscosity limi t of the Navier-Stokes equation, Acta Math

    W. E, Boundary layer theory and the zero-viscosity limi t of the Navier-Stokes equation, Acta Math. Sin., 16(2000), 207-218

    2000

  19. [27]

    W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Commun. Pur. Appl. Math., 50(12)(1997), 1287-1293

    1997

  20. [28]

    L. C. Evans, Partial di fferential equations, American Mathematical Society, V ol 19, 1997

    1997

  21. [29]

    L. Fan, L. Ruan and A. Y ang, Local well-posedness of solu tions to the boundary layer equations for 2D compressible flow, J. Math. Anal. Appl., 493(2021), 12 4565

    2021

  22. [30]

    M. Fei, T. Tao and Z. Zhang, On the zero-viscosity limit o f the Navier-Stokes equations in R3 + without analyticity, J. Math. Pure. Appl., 112(9)(2018), 1 70-229

    2018

  23. [31]

    P . C. Fife, Toward the validity of Prandtl’s approximation in a boundary layer, Arch. Ration. Mech. Anal., 18(1)(1965), 1-13

    1965

  24. [32]

    P . C. Fife, Considerations regarding the mathematical basis for Prandtl’s boundary layer theory, Arch. Ration. Mech. An., 28(1968), 184-216

    1968

  25. [33]

    Filho, Boundary layers amd the vanishing viscosity l imit for incompressible 2D flow, Mathematics-Basel, (2007), arXiv:0712.0875

    M. Filho, Boundary layers amd the vanishing viscosity l imit for incompressible 2D flow, Mathematics-Basel, (2007), arXiv:0712.0875

    2007 arXiv

  26. [34]

    J. Gao, B. Guo and D. Huang, Local-in-time well-posedne ss of boundary layer system for the full incompressible MHD equations by energy methods, (2017), ar Xiv:1704.06766v1. BIBLIOGRAPHY 259

    2017 arXiv

  27. [35]

    J. Gao, D. Huang and Z. Y ao, Boundary layer problems for t he two-dimensional inhomogeneous incompressible magnetohydrodynamics equations, (2018), arXiv:1810.11258v2

    2018 arXiv

  28. [36]

    J. Gao, D. Huang and Z. Y ao, A note on the well-posedness o f Prandtl equations in dimension two, (2018), arXiv:1811.11931v1

    2018 arXiv

  29. [37]

    J. Gao, M. Li and Z. Y ao, Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime, math.AP ., arXiv:2111.04016v1

    arXiv

  30. [38]

    Gao, and L

    C. Gao, and L. Zhang, On the steady Prandtl boundary laye r expansions, (2020), arXiv:2001.10700v1

    2020 arXiv

  31. [39]

    Gargano, M

    F. Gargano, M. Sammartino and V . Sciacca, Singularity f ormation for Prandtl’s equations, Phys. D., 238(19)(2009), 1975-1991

    2009

  32. [40]

    G´erard-V aret and E

    D. G´erard-V aret and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23(2)(2010), 591-609

    2010

  33. [41]

    G´erard-V aret and Y

    D. G´erard-V aret and Y . Maekawa, Sobolev stability of Prandtl expansions for the steady Navier- Stokes equations, Arch. Ration. Mech. Anal., 233(2019), 13 19-1382

    2019

  34. [42]

    G´erard-V aret, Y

    D. G´erard-V aret, Y . Maekawa and N. Masmoudi, Gevrey stability of Prandtl expansions for 2D Navier- Stokes flows, Duke Math. J., 167(13) (2018), 2531-26 31

    2018

  35. [43]

    G´erard-V aret and N

    D. G´erard-V aret and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. ´Ecole. Norm. S., 48(4)(2015), 1273-1325

    2015

  36. [44]

    G´erard-V aret and T

    D. G´erard-V aret and T. Nguyen, Remarks on the ill-posedness of th e Prandtl equation, Asymptot. Anal., 77(1-2)(2012), 71-88

    2012

  37. [45]

    G´ erard-V aret and M

    D. G´ erard-V aret and M. Prestipino, Formal derivation and stabil ity analysis of boundary layer models in MHD, Z. Angew. Math. Phys., 68(3)(2017), Art., 76

    2017

  38. [46]

    S. Gong, Y . Guo and Y . Wang, Boundary layer problems for t he two-dimensional compressible Navier-Stokes equations, Anal. Appl., 14(1)(2016), 1-37

    2016

  39. [47]

    Gong and X

    S. Gong and X. Wang, On a global weak solution and back flow of the mixed Prandtl-Hartmann boundary layer problem, J. Math. Fluid Mech., 23(2021), Art . 11

    2021

  40. [48]

    Greniner, On the nonlinear instability of Euler and P randtl equations, Comm

    E. Greniner, On the nonlinear instability of Euler and P randtl equations, Comm. Pure Appl. Math., 53(9)(2000), 1067-1091

    2000

  41. [49]

    Greniner, Y

    E. Greniner, Y . Guo and T. Nguyen, Spectral stability of Prandtl boundary layers: an overview, Analysis, 35(4)(2015), 343-355

    2015

  42. [50]

    Greniner, Y

    E. Greniner, Y . Guo and T. Nguyen, Spectral instability of characteristic boundary layer flows, Duke Math. J., 16(2016), 3085-3146

    2016

  43. [51]

    Guo and S

    Y . Guo and S. Lyer, V alidity of steady prandtl layer expa nsions with external forcing, (2018), arXiv:1810.06662v1

    2018 arXiv

  44. [52]

    Guo and T

    Y . Guo and T. Nguyen, A note on Prandtl boundary layers, C omm. Pure Appl. Math., 64(10)(2011), 1416-1438. 260 BIBLIOGRAPHY

    2011

  45. [53]

    Guo and T

    Y . Guo and T. Nguyen, Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plane, Ann. Partial Di fferential Equations, 3(10)(2017), DOI: 10.1007 /s40818-016-0020- 6

    2017

  46. [54]

    Guo, and I

    Y . Guo, and I. Tice, Compressible, inviscid Rayleigh-T aylor instability, Indiana Univ. Math. J., 2(2009), 677-712

    2009

  47. [55]

    Hong and J

    L. Hong and J. Hunter, Singularity formation and instab ility in the unsteady inviscid and viscous Prandtl equations, Comm. Math. Sci, 1(2)(2003), 293-316

    2003

  48. [56]

    H ¨ormander, The Analysis of Linear Partial Di fferential Equations III, V ol.257, Springer, Berlin, New Y ork, 1983

    L. H ¨ormander, The Analysis of Linear Partial Di fferential Equations III, V ol.257, Springer, Berlin, New Y ork, 1983

    1983

  49. [57]

    Q. Hou, C. Liu, Y . Wang and Z. Wang, Stability of boundarylayers for a viscous hyperbolic system arising from chemotaxys: one dimensional case, SIAM J. Math . Anal., 50(3)(2018), 3058-3091

    2018

  50. [58]

    Huang, C

    Y . Huang, C. Liu and T. Y ang, Local-in-time well-posedn ess for compressible MHD boundary layer, J. Differential. Equations, 266(2019), 2978-3013

    2019

  51. [59]

    Ignatova and V

    M. Ignatova and V . Vicol, Almost global existence for th e Prandtl boundary layer equations, Arch. Ration. Mech. Anal., 220(2)(2016), 809-848

    2016

  52. [60]

    Iyer and N

    S. Iyer and N. Masmoudi, Global-in- x stability of steady Prandtl expansions for 2D Navier-Stoke s flows, (2020), arXiv:2008.12347v1

    2020 arXiv

  53. [61]

    Jiang and J

    S. Jiang and J. Zhang, On the non-resistive limit and the magnetic boundary-layer for one- dimensional compressible magnetohydrodynamics, Nonline arity, 30(2017), 3587-3612

    2017

  54. [62]

    K. T. Joseph and P . G. Lefloch, Boundary layers in weak sol utions of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147(1999), 47-88

    1999

  55. [63]

    T. V . K´arm´an and H. S. Tsien, Boundary layer in compressible fluids, J. Ae ronautical Science, 5(1938), 227-232

    1938

  56. [64]

    S. A. Korpela, On the viscous dissipation in the boundar y layer of a high Prandtl number fluid in laminar flow over a flat plate, Z. Angew. Math. Phys., 36(1985) , 624-628

    1985

  57. [65]

    Kukavica, N

    I. Kukavica, N. Masmoudi, V . Vicol and T. K. Wong, On the l ocal well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotoni city regions, SIAM J. Math. Anal., 46(2014), 3865-3890

    2014

  58. [66]

    Kukavica and V

    I. Kukavica and V . Vicol, On the local existence of analy tic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., 11(2013), 269-292

    2013

  59. [67]

    Kukavica, V

    I. Kukavica, V . Vicol and F. Wang, The van dommelen and sh en singularity in the Prandtl equa- tions, Adv. Math., 307(2017), 288-311

    2017

  60. [68]

    V . V . Kuznetsov, On solvability conditions for a boundary value problem for the prandtl boundary layer, Siberian Math. J., 35(1994), 558-563

    1994

  61. [69]

    Li and S

    Q. Li and S. Ding, Symmetrical Prandtl boundary layer ex pansions of steady Navier-Stokes equa- tions on bounded domain, J. Di fferential. Equations, 268(2020), 1771-1819

    2020

  62. [70]

    W. Li, N. Masmoudi and T. Y ang, Well-posedness in Gevrey function space for 3D Prandtl equa- tions without structural assumption, (2020), arXiv:2001. 10222v2. BIBLIOGRAPHY 261

    2020

  63. [71]

    Li and S

    W. Li and S. Wang, Boundary layer e ffects for the nonlinear evolution equations with the vanishing diffusion limit, Discrete Dyn. Nat. Soc., (2018), 1-11

    2018

  64. [72]

    W. Li, D. Wu and C. Xu, Gevrey class smoothing e ffect for the Prandtl equation, SIAM J. Math. Anal., 48(3)(2016), 1672-1726

    2016

  65. [73]

    Li and R

    W. Li and R. Xu, Gevrey well-posedness of the hyperbolic Prandtl equations, Commun. Math. Res., 38()(2022), 605-624

    2022

  66. [74]

    W. Li, R. Xu and T. Y ang, Global well-posedness of a Prandtl model from MHD in Gevrey function spaces, Acta Math. Sci., 42B(6)(2022), 2343-2366

    2022

  67. [75]

    Li and T

    W. Li and T. Y ang, Well-posedness in Gevrey function spa ce for the three-dimensional Prandtl equations, (2019), arXiv:1708.08217v4

    2019 arXiv

  68. [76]

    Li and T

    W. Li and T. Y ang, Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc., 22(3)(2020), 717-775

    2020

  69. [77]

    Li and T

    W. Li and T. Y ang, Well-posedness of the MHD boundary lay er system in Gevrey function space without structural assumption, (2020), arXiv:2009.06513 v1

    2020 arXiv

  70. [80]

    C. Liu, Y . Wan, A simple exact solution of the Prandtl bou ndary layer equations containing a point of separation, Arch. Ration. Mech. Anal., 89(1985), 177-18 5

    1985

  71. [81]

    Liu and Y

    C. Liu and Y . Wang, Derivation of Prandtl boundary layer equations for the incompressible Navier- Stokes equations in a curved domain, Appl. Math. Lett., 34(2 014), 81-85

  72. [82]

    Liu and Y

    C. Liu and Y . Wang, Stability of boundary layers for the n onisentropic compressible circularly symmetric 2D flow, SIAM J. Math. Anal., 46(1)(2014), 256-309

    2014

  73. [83]

    C. Liu, D. Wang, F. Xie and T. Y ang, Magnetic e ffects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. F unct. Anal., 279(2020), 108637

    2020

  74. [84]

    C. Liu, Y . Wang and T. Y ang, On the ill-posedness of the Pr andtl equations in three-dimensional space, Arch. Ration. Mech. Anal., 220 (2016), 83-108

    2016

  75. [85]

    C. Liu, Y . Wang and T. Y ang, Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure, Discrete Cont. Dyn. Sys t., 9(6) (2016), 2011-2029

    2016

  76. [86]

    C. Liu, Y . Wang and T. Y ang, A well-posedness theory for t he Prandtl equations in three space variables, Adv. Math., 308 (2017), 1074-1126

    2017

  77. [87]

    C. Liu, F. Xie and T. Y ang, A note on the ill-posedness of s hear flow for the MHD boundary layer equations, Sci. China Math., 61 (2018), 2065-2078

    2018

  78. [88]

    C. Liu, F. Xie and T. Y ang, Justification of Prandtl ansat z for MHD boundary layer, SIAM J. Math. Anal., 51(3)(2019), 2748-2791. 262 BIBLIOGRAPHY

    2019

  79. [89]

    C. Liu, F. Xie and T. Y ang, MHD boundary layers theory in S obolev spaces without monotonicity I: well-posedness theory, Commun. Pure Appl. Math., 1 (2019 ), 63-121

    2019

  80. [90]

    C. Liu, F. Xie and T. Y ang, MHD boundary layers in Sobolev spaces without monotonicity, II. convergence theory, Analysis of PDEs, (2018), arXiv:1704. 00523v4

    2018

  81. [91]

    Liu and T

    C. Liu and T. Y ang, Ill-posedness of the Prandtl equatio ns in Sobolev spaces around a shear flow with general decay, J. Math. Pures Appl., 108(2)(2017), 150 -162

    2017

  82. [92]

    Lin and T

    X. Lin and T. Zhang, Local well-posedness for 2D incompr essible magneto-micropolar boundary layer system, Appl. Anal., (2019), 1-22

    2019

  83. [93]

    Lin and T

    X. Lin and T. Zhang, Almost global existence for the 3D Pr andtl boundary layer equations, Acta. Appl. Math., 169(2020), 383-410

    2020

  84. [94]

    Liu and P

    N. Liu and P . Zhang, Global small analytic solutions of M HD boundary layer equations, J. Di ffer. Equations, 281(2021), 199-257

    2021

  85. [95]

    M. C. Lombardo, M. Cannone and M. Sammartino, Well-pose dness of the boundary layer equa- tions, SIAM J. Math. Anal., 35(4)(2003), 987-1004

    2003

  86. [96]

    Luo and Z

    T. Luo and Z. Xin, H ¨ older continuous solutions to the three-dimensional Prandt l system, (2018), arXiv:1804.04285

    2018 arXiv

  87. [97]

    Maekawa, On the inviscid limit problem of the vortici ty equations for viscous incompressible flows in the half-plane, Comm

    Y . Maekawa, On the inviscid limit problem of the vortici ty equations for viscous incompressible flows in the half-plane, Comm. Pure Appl. Math., 67(2014), 10 45-1128

    2014

  88. [98]

    Masmoudi and T

    N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68(10) (2015), 1683-1741

    2015

  89. [99]

    Medve ˇd, A new approach to an analysis of Henry type integral inequal ities and their Bihari type versions, J

    M. Medve ˇd, A new approach to an analysis of Henry type integral inequal ities and their Bihari type versions, J. Math. Anal. Appl., 214(2)(1997), 349-366

    1997

  90. [100]

    D. S. Mitrinovi´c, J. E. Pˇecari´c and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications, V ol 63, 1991

    1991

  91. [101]

    Moore, Three-dimensional boundary layer theory

    F.K. Moore, Three-dimensional boundary layer theory . Adv. Appl. Mech., 4(1956), 159-228

    1956

  92. [102]

    Narasimha, S

    R. Narasimha, S. V asantha, Laminar boundary layer on a flat plate at high Prandtl number, Z. Angew. Math. Phys., 17(1966), 585-592

    1966

  93. [103]

    Nickel, Prandtl’s boundary layer theory from the vi ew-point of a mathematician, Annu

    K. Nickel, Prandtl’s boundary layer theory from the vi ew-point of a mathematician, Annu. Rev. Fluid Mech., 5(1973), 405-428

    1973

  94. [104]

    O. A. Oleinik, On the system of Prandtl equations in bou ndary-layer theory, Dokl. Akad. Nauk SSSR, 150(1)(1963), 28-31

    1963

  95. [105]

    O. A. Oleinik, On the mathematical theory of boundary l ayer for unsteady flow of incompressible fluid, PMM-J. Appl. Math. Mec., 30(5)(1966), 951-974

    1966

  96. [106]

    O. A. Oleinik and V . N. Samokhin, Mathematical Models i n Boundary Layer Theory, Applied Mathematics and Mathematical Computation, V ol. 15, Chapman and Hall /CRC, Boca Raton, FL, 1999. BIBLIOGRAPHY 263

    1999

  97. [107]

    Paicu and P

    M. Paicu and P . Zhang, Global well-posdness of Prandtl system with small analytic data, (2019), arXiv: 1911.03690v1

    2019 arXiv

  98. [108]

    Pan and C

    X. Pan and C. Xu, Global tangentially analytical solut ions of the 3D axially symmetric Prandtl equations, https://doi.org/10.48550/arXiv.2202.05969

    doi arXiv

  99. [109]

    Pan and C

    X. Pan and C. Xu, Long-time existence of Gevrey-2 solut ions to the 3D Prandtl boundary layer equations, https://doi.org/10.48550/arXiv.2212.02113

    doi arXiv

  100. [110]

    V . E. Petrov and T.A. Suslina, Regularity of the soluti on of the Prandtl equation, (2020), arXiv: 2008.06715v2

    2020 arXiv

  101. [111]

    Prandtl, Uber Flussigkeitsbewegung bei sehr klein er Reibung, in: V erhandlung des III Intern

    L. Prandtl, Uber Flussigkeitsbewegung bei sehr klein er Reibung, in: V erhandlung des III Intern. Math. Kongresses, Heidelberg, 1904, 484-491

    1904

  102. [112]

    Qin, Integral and Discrete Inequalities and Their A pplications: V olume I: Nonlinear Inequali- ties, Birkh¨auser, 2016

    Y . Qin, Integral and Discrete Inequalities and Their A pplications: V olume I: Nonlinear Inequali- ties, Birkh¨auser, 2016

    2016

  103. [113]

    Qin, Integral and Discrete Inequalities and Their A pplications: V olume II: Nonlinear Inequali- ties, Birkh¨auser, 2016

    Y . Qin, Integral and Discrete Inequalities and Their A pplications: V olume II: Nonlinear Inequali- ties, Birkh¨auser, 2016

    2016

  104. [114]

    Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017

    Y . Qin, Analytic Inequalities and Their Applications in PDEs, Springer International Publishing, 2017

    2017

  105. [115]

    Qin and X

    Y . Qin and X. Dong, Local existence of solutions to 2D Pr andtl equations in a weighted Sobolev space, Anal. Math. Phy., (2022) 12:16, https //doi.org/10.1007/s13324-021-00615-z

    2022 doi

  106. [116]

    Qin and X

    Y . Qin and X. Dong, Local well-posedness of solutions t o 2D mixed Prandtl equations in Sobolev space without monotonicity and a lower bound, Nonlinear Ana l. RW A, 80 (2024) 104140

    2024

  107. [117]

    Qin and X

    Y . Qin and X. Wang, Global existence of solutions to 3D Prandtl equations with a special structure, submitted

  108. [118]

    Qin and X

    Y . Qin and X. Wang, Local well-posedness of solutions t o 2D magnetic Prandtl model in the Prandtl-Hartmann regime, submitted

  109. [119]

    Renardy, Well-posedness of the Prandtl boundary la yer equations for the upper convected maxwell fluid, J

    M. Renardy, Well-posedness of the Prandtl boundary la yer equations for the upper convected maxwell fluid, J. Dyn. Di ffer. Equ., 27(2015), 981-988

    2015

  110. [120]

    Renardy and X

    M. Renardy and X. Wang, Well-posedness of boundary lay er equations for time-dependent flow of non-newtonian fluids, J. Math. Fluid Mech., 16(2014), 179 -191

    2014

  111. [121]

    Sammartino, R

    M. Sammartino, R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space, I. Existence for Euler and Prandtl Equations Comm. Math. Phys., 192(1998), 433-461

    1998

  112. [122]

    Sammartino, R

    M. Sammartino, R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space, II. Construction of the Navier-St okes solution, Comm. Math. Phys., 192(1998), 463-491

    1998

  113. [123]

    Schlichting and K

    H. Schlichting and K. Gersten, Boundary-Layer Theory , Ninth Edition, Springer-V erlag Berlin Heidelberg, 2017

    2017

  114. [124]

    Sermange and R

    M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Com- puter Compacts, 1(4)(1983), 635-664. 264 BIBLIOGRAPHY

    1983

  115. [125]

    Serrin, Asymptotic behaviour of velocity profiles i n the Prandtl boundary layer theory, Proc

    J. Serrin, Asymptotic behaviour of velocity profiles i n the Prandtl boundary layer theory, Proc. Roy. Soc. London Ser. A., 299(1967), 491-507

    1967

  116. [126]

    Serrin, On the mathematical basis for Prandtl’s bou ndary layer theory: an example, Arch

    J. Serrin, On the mathematical basis for Prandtl’s bou ndary layer theory: an example, Arch. Ra- tion. Mech. An., 28(1968), 217-225

    1968

  117. [127]

    W. Shen, Y . Wang and Z. Zhang, Boundary layer separatio n and local behavior for the steady Prandtl equation, (2019), arXiv:1904.08055v1

    2019 arXiv

  118. [128]

    Sun, A blow-up criterion for classical solutions to the Prandtl equations, J

    Y . Sun, A blow-up criterion for classical solutions to the Prandtl equations, J. Math. Phys., 60(11)(2019), 111502

    2019

  119. [129]

    Temam, Boundary layers associated with incompress ible Navier-Stokes equations: The non- characteristic boundary case, J

    R. Temam, Boundary layers associated with incompress ible Navier-Stokes equations: The non- characteristic boundary case, J. Di fferential. Equations, 179(2002), 647-686

    2002

  120. [130]

    Triebel, Theory of Function Spaces, Modern Birkh¨ auser classics, Birkh¨ auser, 1983

    H. Triebel, Theory of Function Spaces, Modern Birkh¨ auser classics, Birkh¨ auser, 1983

    1983

  121. [131]

    Walter, On the asymptotic behavior of solutions of t he Prandtl boundary layer equations, Indi- ana Univ

    W. Walter, On the asymptotic behavior of solutions of t he Prandtl boundary layer equations, Indi- ana Univ. Math. J., 20(1970), 829-841

    1970

  122. [132]

    S. Wang, L. Jiang and C. Liu, Quasi-neutral limit and th e boundary layer problem of Planck- Nernst-Poisson-Navier-Stokes equations for electro-hyd rodynamics, J. Di fferential Equations, 267(2019), 3475-3523

    2019

  123. [133]

    Wang and C

    S. Wang and C. Liu, Boundary layer problem and quasineu tral limit of compressible Euler-Poisson system, Comm. Pure Appl. Anal., 16(6)(2017), 2177-2199

    2017

  124. [134]

    Wang and N

    S. Wang and N. Wang, Boundary layer problem of MHD syste m with non-characteristic perfect conducting wall, Appl. Anal., (2017), 1-21

    2017

  125. [135]

    Wang and G

    S. Wang and G. Wang, Well-posedness and blowup of the ge ophysical boundary layer problem, J. Math. Fluid Mech., 22(2020), Art., 52

    2020

  126. [136]

    S. Wang, B. Wang, C. Liu and N. Wang, Boundary layer prob lem and zero viscosity-di ffusion limit of the incompressible magnetohydrodynamic system wi th no-slip boundary conditions, J. Differential. Equations, 263(2017), 4723-4749

    2017

  127. [137]

    X. Wang, Y . Wang and Z. Xin, Boundary layers in incompre ssible Navier-Stokes equations with Navier friction boundary condition for vanishing viscosit y limit, Comm. Math. Sci., 8(4)(2010), 965-998

    2010

  128. [138]

    C. Wang, Y . Wang and Z. Zhang, Gevrey stability of hydrostatic approximate for the Navier-Stokes equations in a thin domain, arXiv:2004.05537v1

    2004 arXiv

  129. [139]

    C. Wang, Y . Wang and P . Zhang, On the global small soluti on of 2-D Prandtl system with initial data in the optimal Gevrey class, (2021), arXiv:2103.00681 v1

    2021 arXiv

  130. [140]

    Wang and F

    Y . Wang and F. Xie and T. Y ang, Local well-posedness of Prandtl equations for compressible flow in two space variables, SIAM J. Math. Anal., 47(1)(2015), 32 1-346

    2015

  131. [141]

    Wang and X

    S. Wang and X. Xin, Boundary layer problems in the visco sity-diffusion vanishing limits for the incompressible MHD systems, (2017), arXiv:1706.07898v1

    2017 arXiv

  132. [142]

    Wang and Z

    Y . Wang and Z. Zhang, Global C∞ regularity of the steady prandtl equation, (2019), arXiv:1912.08402v1. BIBLIOGRAPHY 265

    2019 arXiv

  133. [143]

    Wang and S

    Y . Wang and S. Zhu, Back flow of the two-dimensional unst eady Prandtl boundary layers under adverse pressure gradient, SIAM J. Math. Anal., 52(1)(2020 ), 954-966

    2020

  134. [144]

    Wang and S

    Y . Wang and S. Zhu, Blow-up criterion for the 2-D Prandtl equation, (2020), arXiv:2002.06327v1

    2020 arXiv

  135. [145]

    Wu, Well-posedness of the nonlinear unsteady Prandtl equations with Robin boundary condition in weighted Sobolev spaces, (2015), arXiv:1503.00682v3

    F. Wu, Well-posedness of the nonlinear unsteady Prandtl equations with Robin boundary condition in weighted Sobolev spaces, (2015), arXiv:1503.00682v3

    2015 arXiv

  136. [146]

    Wu, Full regularity and well-posedness of the nonli near unsteady Prandtl equations with Robin or Dirichlet boundary condition, (2016), arXiv:1603.0772 5v1

    F. Wu, Full regularity and well-posedness of the nonli near unsteady Prandtl equations with Robin or Dirichlet boundary condition, (2016), arXiv:1603.0772 5v1

    2016

  137. [147]

    Xie and T

    F. Xie and T. Y ang, Global-in-time stability of 2D MHD b oundary layer in the Prandtl-Hartmann regime, SIAM. J. Math. Anal. 50(6)(2018), 5749-5760

    2018

  138. [148]

    Xie and T

    F. Xie and T. Y ang, Lifespan of solutions to MHD Boundar y layer equations with analytic pertur- bation of general shear flow, Acta Math. Appl. Sin-E, 35(1)(2 019), 209-229

  139. [149]

    Xin and L

    Z. Xin and L. Zhang, On the global existence of solution s to the Prandtl’s system, Adv. Math., 181(2004), 88-133

    2004

  140. [150]

    Xu and X

    C. Xu and X. Zhang, Well-posedness of the Prandtl equat ion in Sobolev space without monotonic- ity, (2015), arXiv: 1511.04850v3

    2015 arXiv

  141. [151]

    Xu and X

    C. Xu and X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Di ffer- ential. Equations, 263(2017), 8749-8803

    2017

  142. [152]

    Xu and J

    X. Xu and J. Zhao, On the global existence and uniquenes s of solutions to Prandtl’s system, Acta Math. Sin., 25(1)(2009), 109-132

    2009

  143. [153]

    Xue, Well-posedness and zero microrotation viscos ity limit of the 2D micropolar fluid equa- tions, Math

    L. Xue, Well-posedness and zero microrotation viscos ity limit of the 2D micropolar fluid equa- tions, Math. Meth. Appl. Sci., 34(2011), 1760-1777

    2011

  144. [154]

    Y ang and W

    T. Y ang and W. Li, Well-posedness in Gevrey function sp aces for the Prandtl equations with non- degenerate critical points, J. Eur. Math. Soc., (2019), 717 -775

    2019

  145. [155]

    Y e and J

    X. Y e and J. Zhang, The behavior of boundary layer for th e compressible planar MHD equations at small shear viscosity, J. Dyn. Di ffer. Equ., 32(2020), 281-311

    2020

  146. [156]

    Zhang and Z

    P . Zhang and Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., 270(7)(2016), 2591-2615

    2016

Pith tools

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