Abstract
We consider co-rotational wave maps from Minkowski space in d + 1 dimensions to the d-sphere. Recently, Bizoń and Biernat found explicit self-similar solutions for each dimension \({d\geq 4}\). We give a rigorous proof for the mode stability of these self-similar wave maps.
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Bizoń P., Biernat P.: Generic self-similar blowup for equivariant wave maps and Yang–Mills fields in higher dimensions. Commun. Math. Phys. 338(3), 1443–1450 (2015)
Bizoń Piotr: An unusual eigenvalue problem. Acta Phys. Polon. B. 36(1), 5–15 (2005)
Bizoń Piotr, Chmaj Tadeusz, Tabor Zbisław: Dispersion and collapse of wave maps. Nonlinearity 13(4), 1411–1423 (2000)
Buslaev V.I., Buslaeva S.F.: Poincaré’s theorem on difference equations. Mat. Zametki 78(6), 943–947 (2005)
Cazenave Thierry, Shatah Jalal, Tahvildar-Zadeh A. Shadi: Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang–Mills fields. Ann. Inst. H. Poincaré Phys. Théor. 68(3), 315–349 (1998)
Costin O., Costin R.D., Lebowitz J.L., Rokhlenko A.: Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Commun. Math. Phys. 221(1), 1–26 (2001)
Costin O., Huang M., Schlag W.: On the spectral properties of \({L_\pm}\) in three dimensions. Nonlinearity 25(1), 125–164 (2012)
Costin O., Donninger R., Glogić I., Huang M.: On the stability of self-similar solutions to nonlinear wave equations. Commun. Math. Phys. 343(1), 299–310 (2016)
Costin O., Donninger R., Xia X.: A proof for the mode stability of a self-similar wave map. Nonlinearity 29(8), 2451 (2016)
Costin O., Huang M., Tanveer S.: Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of P I . Duke Math. J. 163(4), 665–704 (2014)
Costin O., Kim T.E., Tanveer S.: A quasi-solution approach to nonlinear problems—the case of the Blasius similarity solution. Fluid Dyn. Res. 46(3), 031419,19 (2014)
Côte R., Kenig C.E., Lawrie A., Schlag W.: Characterization of large energy solutions of the equivariant wave map problem: I. Amer. J. Math. 137(1), 139–207 (2015)
Côte R., Kenig C.E., Lawrie A., Schlag W.: Characterization of large energy solutions of the equivariant wave map problem: II. Amer. J. Math. 137(1), 209–250 (2015)
Donninger Roland: On stable self-similar blowup for equivariant wave maps. Comm. Pure Appl. Math. 64(8), 1095–1147 (2011)
Donninger R.: Stable self-similar blowup in energy supercritical Yang-Mills theory. Math. Z. 278(3-4), 1005–1032 (2014)
Donninger Roland: Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation. arXiv:1509.02041 (2015) (Preprint)
Donninger R., Schörkhuber Birgit: Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Differ. Equ. 9(1), 63–87 (2012)
Donninger R., Schörkhuber B.: Stable blow up dynamics for energy supercritical wave equations. Trans. Amer. Math. Soc. 366(4), 2167–2189 (2014)
Donninger R., Schörkhuber B.: On blowup in supercritical wave equations. Comm. Math. Phys. 346(3), 907–943 (2016)
Donninger R., Schörkhuber B., Aichelburg P.C.: On stable self-similar blow up for equivariant wave maps: the linearized problem. Ann. Henri Poincaré 13(1), 103–144 (2012)
Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2005)
Jaffé, G.: Zur theorie des wasserstoffmolekülions. Zeitschrift für Physik 87(7), 535– (1934)
Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008)
Krieger J., Schlag W.: Concentration Compactness for Critical Wave Maps. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2012)
Leaver E.W.: An analytic representation for the quasi-normal modes of kerr black holes. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 402(1823), 285–298 (1985)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, Charles W. (eds): NIST Handbook of Mathematical Functions. US Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010)
Phillips G.M.: Interpolation and Approximation by Polynomials. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 14. Springer, New York (2003)
Raphaël, P., Rodnianski, I.: Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci., pp. 1–122 (2012)
Rodnianski I., Sterbenz J.: On the formation of singularities in the critical \({{\rm O(3)}}\) \({\sigma}\)-model. Ann. Math. (2) 172(1), 187–242 (2010)
Shatah Jalal: Weak solutions and development of singularities of the \({{\rm SU(2)}}\) \({\sigma}\)-model. Comm. Pure Appl. Math. 41(4), 459–469 (1988)
Sterbenz J., Tataru D.: Regularity of wave-maps in dimension \({{\rm 2+1}}\). Comm. Math. Phys. 298(1), 231–264 (2010)
Struwe, M.: Equivariant wave maps in two space dimensions. Comm. Pure Appl. Math., 56(7), 815–823 (2003) (Dedicated to the memory of Jürgen K. Moser)
Titchmarsh, E.C.: The theory of functions. Oxford University Press, Oxford, (1958). Reprint of the second (1939) edition.
Turok N., Spergel D.: Global texture and the microwave background. Phys. Rev. Lett. 64, 2736–2739 (1990)
Wall H.S.: Polynomials whose zeros have negative real parts. Amer. Math. Mon. 52, 308–322 (1945)
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Communicated by W. Schlag
O. C. and I. G. were partially supported by the NSF DMS grant 1515755. R. D. is supported by a Sofja Kovalevskaja Award Granted by the Alexander von Humboldt Foundation and the German Federal Ministry of Education and Research; partial support by the DFG, CRC 1060, is also gratefully acknowledged.
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Costin, O., Donninger, R. & Glogić, I. Mode Stability of Self-Similar Wave Maps in Higher Dimensions. Commun. Math. Phys. 351, 959–972 (2017). https://doi.org/10.1007/s00220-016-2776-7
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DOI: https://doi.org/10.1007/s00220-016-2776-7