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arxiv: 2601.19515 · v2 · submitted 2026-01-27 · 🧮 math.AP · math-ph· math.MP· math.SP

Mode stability of self-similar wave maps without symmetry in higher dimensions

Roland Donninger , Frederick Moscatelli This is my paper

Pith reviewed 2026-05-16 10:45 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPmath.SP
keywords modestabilitycorotationalfirstmapsself-similarsolutionsymmetry
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The pith

The explicit self-similar wave map solution is mode stable without symmetry assumptions for every dimension d at least 4.

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

This paper proves mode stability of the known self-similar wave map solution from (1+d)-dimensional Minkowski space to the d-sphere without assuming corotational symmetry. Previous work established this only for d=3, and the authors extend it to all d greater than or equal to 4. They do so by successfully applying the quasi-solution method with two extra parameters that compensate for the lack of symmetry. If this holds, the finite-time blowup behavior appears robust to asymmetric perturbations. Readers might care because it broadens the conditions under which such blowups are expected to occur.

Core claim

The authors prove that the explicit self-similar solution to the wave map equation is mode stable in the full space of functions, without corotational symmetry, for all dimensions d greater than or equal to 4. This extends the d=3 case by using the quasi-solution method adapted to include two additional parameters.

What carries the argument

The quasi-solution method implemented with two additional parameters to remove symmetry assumptions.

Load-bearing premise

The quasi-solution method can be successfully implemented and closed when two additional parameters are introduced to remove symmetry assumptions.

What would settle it

Detection of a growing eigenmode for the linearized operator around the self-similar solution in the non-symmetric sector for some d >= 4 would disprove the claim.

read the original abstract

We consider wave maps from $(1+d)$-dimensional Minkowski space into the $d$-sphere. For every $d \geq 3$, there exists an explicit self-similar solution that exhibits finite time blowup. This solution is corotational and its mode stability in the class of corotational functions is known. Recently, Weissenbacher, Koch, and the first author proved mode stability without symmetry assumptions in $d =3$. In this paper we extend this result to all $d \geq 4$. On a technical level, this is the first successful implementation of the quasi-solution method where two additional parameters are present.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper extends the known mode stability of the explicit self-similar wave-map solution from (1+d)-dimensional Minkowski space into the d-sphere, previously established without symmetry only for d=3, to all d≥4. The extension is achieved by implementing the quasi-solution method with two additional parameters that enforce orthogonality conditions against the unstable modes of the linearized operator, yielding uniform error bounds in the relevant function spaces.

Significance. If the result holds, the work is significant because it supplies the first successful implementation of the quasi-solution method in the presence of two free parameters, thereby removing symmetry assumptions in higher dimensions. The explicit constructions and closed estimates without residual terms strengthen the analytic understanding of mode stability for self-similar blow-up solutions in wave maps.

minor comments (2)
  1. [Introduction] In the introduction, the two additional parameters are introduced without an immediate comparison to the single-parameter implementation used for d=3; a short sentence clarifying the new orthogonality conditions would improve readability.
  2. [Section 4] Section 4 (or the main estimates section) tracks dimension-dependent constants; adding an explicit remark that the bounds remain uniform for all d≥4 would help readers verify the claimed uniformity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful reading, and recommendation to accept the manuscript. We are pleased that the significance of extending mode stability without symmetry assumptions to all dimensions d ≥ 4, via the first implementation of the quasi-solution method with two free parameters, has been recognized.

Circularity Check

0 steps flagged

No significant circularity; extension uses independent quasi-solution implementation

full rationale

The derivation extends the d=3 mode stability result (cited to Weissenbacher-Koch-Donninger) to d>=4 via an explicit quasi-solution method with two new parameters enforcing orthogonality. The central estimates and constructions close uniformly without reducing to fitted inputs, self-definitions, or load-bearing self-citations; the prior result supplies only the base case while the higher-d argument supplies independent technical content. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard PDE analysis tools and the successful closure of the quasi-solution method with two extra parameters; no explicit free parameters or invented entities are named in the abstract.

free parameters (1)
  • two additional parameters in quasi-solution method
    Abstract states these parameters are present in the first successful implementation of the method.
axioms (1)
  • standard math Standard functional-analytic setup for wave maps and mode stability (Sobolev spaces, spectral theory)
    Typical background assumptions for papers in math.AP on nonlinear wave equations.

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [1]

    Kendall Atkinson and Weimin Han.Spherical harmonics and approximations on the unit sphere: an introduction. Vol. 2044. Lecture Notes in Mathematics. Springer, Heidelberg, 2012, pp. x+244.isbn: 978-3-642-25982-1.doi:10 . 1007 / 978 - 3 - 642 - 25983 - 8.url:https : //doi.org/10.1007/978-3-642-25983-8

    work page doi:10.1007/978-3-642-25983-8 2044

  2. [2]

    An unusual eigenvalue problem

    Piotr Bizo´ n. “An unusual eigenvalue problem”. In:Acta Phys. Polon. B36.1 (2005), pp. 5–15. issn: 0587-4254,1509-5770

    work page 2005

  3. [3]

    Generic self-similar blowup for equivariant wave maps and Yang-Mills fields in higher dimensions

    Piotr Bizo´ n and Pawe l Biernat. “Generic self-similar blowup for equivariant wave maps and Yang-Mills fields in higher dimensions”. In:Comm. Math. Phys.338.3 (2015), pp. 1443–1450. issn: 0010-3616,1432-0916.doi:10.1007/s00220-015-2404-y.url:https://doi.org/10. 1007/s00220-015-2404-y

    work page doi:10.1007/s00220-015-2404-y.url:https://doi.org/10 2015

  4. [4]

    Large Deviations Estimates for Dy- namical Systems without the Specification Property. Application to the β-shifts,

    Piotr Bizo´ n, Tadeusz Chmaj, and Zbis law Tabor. “Dispersion and collapse of wave maps”. In:Nonlinearity13.4 (2000), pp. 1411–1423.issn: 0951-7715,1361-6544.doi:10.1088/0951- 7715/13/4/323.url:https://doi.org/10.1088/0951-7715/13/4/323

    work page doi:10.1088/0951- 2000

  5. [5]

    Formation of singularities for equivariant (2 + 1)-dimensional wave maps into the 2-sphere

    Piotr Bizo´ n, Tadeusz Chmaj, and Zbis law Tabor. “Formation of singularities for equivariant (2 + 1)-dimensional wave maps into the 2-sphere”. In:Nonlinearity14.5 (2001), pp. 1041– 1053.issn: 0951-7715,1361-6544.doi:10.1088/0951-7715/14/5/308.url:https://doi. org/10.1088/0951-7715/14/5/308. 29

    work page doi:10.1088/0951-7715/14/5/308.url:https://doi 2001

  6. [6]

    On blowup of co-rotational wave maps in odd space dimensions

    Athanasios Chatzikaleas, Roland Donninger, and Irfan Glogi´ c. “On blowup of co-rotational wave maps in odd space dimensions”. In:J. Differential Equations263.8 (2017), pp. 5090– 5119.issn: 0022-0396,1090-2732.doi:10.1016/j.jde.2017.06.011.url:https://doi. org/10.1016/j.jde.2017.06.011

    work page doi:10.1016/j.jde.2017.06.011.url:https://doi 2017

  7. [7]

    A proof for the mode stability of a self-similar wave map

    O. Costin, R. Donninger, and X. Xia. “A proof for the mode stability of a self-similar wave map”. In:Nonlinearity29.8 (2016), pp. 2451–2473.issn: 0951-7715,1361-6544.doi:10.1088/ 0951-7715/29/8/2451.url:https://doi.org/10.1088/0951-7715/29/8/2451

    work page doi:10.1088/0951-7715/29/8/2451 2016

  8. [8]

    Conformal structures of static vacuum d ata,

    Ovidiu Costin, Roland Donninger, and Irfan Glogi´ c. “Mode stability of self-similar wave maps in higher dimensions”. In:Comm. Math. Phys.351.3 (2017), pp. 959–972.issn: 0010- 3616,1432-0916.doi:10.1007/s00220- 016- 2776- 7.url:https://doi.org/10.1007/ s00220-016-2776-7

    work page doi:10.1007/s00220- 2017

  9. [9]

    On the stability of self-similar solutions to nonlinear wave equations

    Ovidiu Costin et al. “On the stability of self-similar solutions to nonlinear wave equations”. In:Comm. Math. Phys.343.1 (2016), pp. 299–310.issn: 0010-3616,1432-0916.doi:10.1007/ s00220-016-2588-9.url:https://doi.org/10.1007/s00220-016-2588-9

    work page doi:10.1007/s00220-016-2588-9 2016

  10. [10]

    On blowup for the supercritical quadratic wave equation

    Elek Csobo, Irfan Glogi´ c, and Birgit Sch¨ orkhuber. “On blowup for the supercritical quadratic wave equation”. In:Anal. PDE17.2 (2024), pp. 617–680.issn: 2157-5045,1948-206X.doi: 10.2140/apde.2024.17.617.url:https://doi.org/10.2140/apde.2024.17.617. [11]NIST Digital Library of Mathematical Functions.https://dlmf.nist.gov/, Release 1.2.3 of 2024-12-15. F...

    work page doi:10.2140/apde.2024.17.617.url:https://doi.org/10.2140/apde.2024.17.617 2024

  11. [11]

    Singer and H.-T

    Roland Donninger. “On stable self-similar blowup for equivariant wave maps”. In:Comm. Pure Appl. Math.64.8 (2011), pp. 1095–1147.issn: 0010-3640,1097-0312.doi:10.1002/cpa. 20366.url:https://doi.org/10.1002/cpa.20366

    work page doi:10.1002/cpa 2011

  12. [12]

    Spectral theory and self-similar blowup in wave equations

    Roland Donninger. “Spectral theory and self-similar blowup in wave equations”. In:Bull. Amer. Math. Soc. (N.S.)61.4 (2024), pp. 659–685.issn: 0273-0979,1088-9485.doi:10.1090/ bull/1845.url:https://doi.org/10.1090/bull/1845

    work page doi:10.1090/bull/1845 2024

  13. [13]

    Preprint

    Roland Donninger and Frederick Moscatelli.Blowup stability of wave maps without symmetry. Preprint. 2026

    work page 2026

  14. [14]

    Roland Donninger and Matthias Ostermann.On stable self-similar blowup for corotational wave maps and equivariant Yang-Mills connections. 2024. arXiv:2409 . 14733 [math.AP]. url:https://arxiv.org/abs/2409.14733

    work page arXiv 2024

  15. [15]

    Quantum Optimal Transport with Quantum Channels

    Roland Donninger, Birgit Sch¨ orkhuber, and Peter C. Aichelburg. “On stable self-similar blow up for equivariant wave maps: the linearized problem”. In:Ann. Henri Poincar´ e13.1 (2012), pp. 103–144.issn: 1424-0637,1424-0661.doi:10.1007/s00023- 011- 0125- 0.url:https: //doi.org/10.1007/s00023-011-0125-0

    work page doi:10.1007/s00023- 2012

  16. [16]

    Optimal blowup stability for supercritical wave maps

    Roland Donninger and David Wallauch. “Optimal blowup stability for supercritical wave maps”. In:Adv. Math.433 (2023), Paper No. 109291, 86.issn: 0001-8708,1090-2082.doi: 10.1016/j.aim.2023.109291.url:https://doi.org/10.1016/j.aim.2023.109291

    work page doi:10.1016/j.aim.2023.109291.url:https://doi.org/10.1016/j.aim.2023.109291 2023

  17. [17]

    Optimal blowup stability for three-dimensional wave maps

    Roland Donninger and David Wallauch. “Optimal blowup stability for three-dimensional wave maps”. In:Anal. PDE18.4 (2025), pp. 895–962.issn: 2157-5045,1948-206X.doi:10.2140/ apde.2025.18.895.url:https://doi.org/10.2140/apde.2025.18.895

    work page doi:10.2140/apde.2025.18.895 2025

  18. [18]

    Saber Elaydi.An introduction to difference equations. Third. Undergraduate Texts in Math- ematics. Springer, New York, 2005, pp. xxii+539.isbn: 0-387-23059-9

    work page 2005

  19. [19]

    William Fulton and Joe Harris.Representation theory. Vol. 129. Graduate Texts in Mathemat- ics. A first course, Readings in Mathematics. Springer-Verlag, New York, 1991, pp. xvi+551. isbn: 0-387-97527-6.doi:10.1007/978-1-4612-0979-9.url:https://doi.org/10.1007/ 978-1-4612-0979-9. 30

    work page doi:10.1007/978-1-4612-0979-9.url:https://doi.org/10.1007/ 1991

  20. [20]

    Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

    T. Ghoul, S. Ibrahim, and V. T. Nguyen. “Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps”. In:J. Differential Equations265.7 (2018), pp. 2968–3047.issn: 0022-0396,1090-2732.doi:10.1016/j.jde.2018.04.058.url:https: //doi.org/10.1016/j.jde.2018.04.058

    work page doi:10.1016/j.jde.2018.04.058.url:https: 2018

  21. [21]

    Globally stable blowup profile for supercritical wave maps in all dimensions

    Irfan Glogi´ c. “Globally stable blowup profile for supercritical wave maps in all dimensions”. In: Calc. Var. Partial Differential Equations64.2 (2025), Paper No. 46, 34.issn: 0944-2669,1432- 0835.doi:10.1007/s00526-024-02901-7.url:https://doi.org/10.1007/s00526-024- 02901-7

    work page doi:10.1007/s00526-024-02901-7.url:https://doi.org/10.1007/s00526-024- 2025

  22. [22]

    Thesis (Ph.D.)–The Ohio State University

    Irfan Glogi´ c.On the Existence and Stability of Self-Similar Blowup in Nonlinear Wave Equa- tions. Thesis (Ph.D.)–The Ohio State University. ProQuest LLC, Ann Arbor, MI, 2018, p. 108. isbn: 978-0438-58934-6.url:http://gateway.proquest.com/openurl?url_ver=Z39.88- 2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat= xri:pqdiss:11005215

    work page 2018

  23. [23]

    Co-dimension one stable blowup for the supercritical cubic wave equation

    Irfan Glogi´ c and Birgit Sch¨ orkhuber. “Co-dimension one stable blowup for the supercritical cubic wave equation”. In:Adv. Math.390 (2021), Paper No. 107930, 79.issn: 0001-8708,1090- 2082.doi:10.1016/j.aim.2021.107930.url:https://doi.org/10.1016/j.aim.2021. 107930

    work page doi:10.1016/j.aim.2021.107930.url:https://doi.org/10.1016/j.aim.2021 2021

  24. [24]

    Hall.Quantum theory for mathematicians

    Brian C. Hall.Quantum theory for mathematicians. Vol. 267. Graduate Texts in Mathematics. Springer, New York, 2013, pp. xvi+554.isbn: 978-1-4614-7115-8.doi:10.1007/978-1-4614- 7116-5.url:https://doi.org/10.1007/978-1-4614-7116-5

    work page doi:10.1007/978-1-4614- 2013

  25. [25]

    Humphreys.Introduction to Lie algebras and representation theory

    James E. Humphreys.Introduction to Lie algebras and representation theory. Vol. 9. Gradu- ate Texts in Mathematics. Second printing, revised. Springer-Verlag, New York-Berlin, 1978, pp. xii+171.isbn: 0-387-90053-5

    work page 1978

  26. [26]

    Renormalization and blow up for charge one equivari- ant critical wave maps

    J. Krieger, W. Schlag, and D. Tataru. “Renormalization and blow up for charge one equivari- ant critical wave maps”. In:Invent. Math.171.3 (2008), pp. 543–615.issn: 0020-9910,1432- 1297.doi:10.1007/s00222-007-0089-3.url:https://doi.org/10.1007/s00222-007- 0089-3

    work page doi:10.1007/s00222-007-0089-3.url:https://doi.org/10.1007/s00222-007- 2008

  27. [27]

    Duke Math

    Joachim Krieger and Shuang Miao. “On the stability of blowup solutions for the critical corotational wave-map problem”. In:Duke Math. J.169.3 (2020), pp. 435–532.issn: 0012- 7094,1547-7398.doi:10.1215/00127094- 2019- 0053.url:https://doi.org/10.1215/ 00127094-2019-0053

    work page doi:10.1215/00127094- 2020

  28. [28]

    Joachim Krieger, Shuang Miao, and Wilhelm Schlag.A stability theory beyond the co-rotational setting for critical Wave Maps blow up. 2024. arXiv:2009.08843 [math.AP].url:https: //arxiv.org/abs/2009.08843

    work page arXiv 2024

  29. [29]

    Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems

    Pierre Rapha¨ el and Igor Rodnianski. “Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems”. In:Publ. Math. Inst. Hautes ´Etudes Sci. 115 (2012), pp. 1–122.issn: 0073-8301,1618-1913.doi:10.1007/s10240-011-0037-z.url: https://doi.org/10.1007/s10240-011-0037-z

    work page doi:10.1007/s10240-011-0037-z.url: 2012

  30. [30]

    On the formation of singularities in the critical O(3) σ-model

    Igor Rodnianski and Jacob Sterbenz. “On the formation of singularities in the critical O(3) σ-model”. In:Ann. of Math. (2)172.1 (2010), pp. 187–242.issn: 0003-486X,1939-8980.doi: 10.4007/annals.2010.172.187.url:https://doi.org/10.4007/annals.2010.172.187

    work page doi:10.4007/annals.2010.172.187.url:https://doi.org/10.4007/annals.2010.172.187 2010

  31. [31]

    Weak solutions and development of singularities of the SU(2)σ-model

    Jalal Shatah. “Weak solutions and development of singularities of the SU(2)σ-model”. In: Comm. Pure Appl. Math.41.4 (1988), pp. 459–469.issn: 0010-3640,1097-0312.doi:10.1002/ cpa.3160410405.url:https://doi.org/10.1002/cpa.3160410405

    work page doi:10.1002/cpa.3160410405 1988

  32. [32]

    Equivariant wave maps in two space dimensions

    Michael Struwe. “Equivariant wave maps in two space dimensions”. In: vol. 56. 7. Dedicated to the memory of J¨ urgen K. Moser. 2003, pp. 815–823.doi:10 . 1002 / cpa . 10074.url: https://doi.org/10.1002/cpa.10074. 31

    work page doi:10.1002/cpa.10074 2003

  33. [33]

    Gerald Teschl.Ordinary differential equations and dynamical systems. Vol. 140. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012, pp. xii+356. isbn: 978-0-8218-8328-0.doi:10.1090/gsm/140.url:https://doi.org/10.1090/gsm/140

    work page doi:10.1090/gsm/140.url:https://doi.org/10.1090/gsm/140 2012

  34. [34]

    Global texture and the microwave background

    Neil Turok and David Spergel. “Global texture and the microwave background”. In:Phys. Rev. Lett.64 (23 1990), pp. 2736–2739.doi:10.1103/PhysRevLett.64.2736.url:https: //link.aps.org/doi/10.1103/PhysRevLett.64.2736

    work page doi:10.1103/physrevlett.64.2736.url:https: 1990

  35. [35]

    Polynomials Whose Zeros Have Negative Real Parts

    H. S. Wall. “Polynomials Whose Zeros Have Negative Real Parts”. In:The American Mathe- matical Monthly52.6 (1945), pp. 308–322.doi:10.1080/00029890.1945.11991574. eprint: https://doi.org/10.1080/00029890.1945.11991574.url:https://doi.org/10.1080/ 00029890.1945.11991574

    work page doi:10.1080/00029890.1945.11991574 1945

  36. [36]

    Max Weissenbacher, Herbert Koch, and Roland Donninger.Mode stability of blow-up for wave maps in the absence of symmetry. 2025. arXiv:2503.02632 [math.AP].url:https: //arxiv.org/abs/2503.02632. Email address:roland.donninger@univie.ac.at Universit¨at Wien, F akult¨at f ¨ur Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Email address:frederick...

    work page arXiv 2025