Globally stable blowup profile for supercritical wave maps in all dimensions (2207.06952v2)
Abstract: We consider wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere. It is known from the work of Bizo\'n and Biernat \cite{BizBie15} that in the energy-supercritical case, i.e., for $d \geq 3$, this model admits a closed-form corotational self-similar blowup solution. We show that this blowup profile is globally nonlinearly stable for all $d \geq 3$, thereby verifying a perturbative version of the conjecture posed in \cite{BizBie15} about the generic large data blowup behavior for this model. To accomplish this, we develop a novel stability analysis approach based on similarity variables posed on the whole space $\mathbb{R}d$. As a result, we draw a general road map for studying spatially global stability of self-similar blowup profiles for nonlinear wave equations in the radial case for arbitrary dimension $d \geq 3$.
- Threshold for blowup for equivariant wave maps in higher dimensions. Nonlinearity, 30(4):1513–1522, 2017.
- Hyperboloidal similarity coordinates and a globally stable blowup profile for supercritical wave maps. Int. Math. Res. Not. IMRN, (21):16530–16591, 2021.
- Piotr Bizoń. Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere. Comm. Math. Phys., 215(1):45–56, 2000.
- Piotr Bizoń. Threshold behavior for nonlinear wave equations. volume 8, pages 35–41. 2001. Nonlinear evolution equations and dynamical systems (Kolimbary, 1999).
- Piotr Bizoń. An unusual eigenvalue problem. Acta Phys. Polon. B, 36(1):5–15, 2005.
- Generic self-similar blowup for equivariant wave maps and Yang-Mills fields in higher dimensions. Comm. Math. Phys., 338(3):1443–1450, 2015.
- Dispersion and collapse of wave maps. Nonlinearity, 13(4):1411–1423, 2000.
- Formation of singularities for equivariant (2+1)21(2+1)( 2 + 1 )-dimensional wave maps into the 2-sphere. Nonlinearity, 14(5):1041–1053, 2001.
- On the division problem for the wave maps equation. Ann. PDE, 4(2):Paper No. 17, 61, 2018.
- An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998.
- On blowup of co-rotational wave maps in odd space dimensions. J. Differential Equations, 263(8):5090–5119, 2017.
- Mode stability of self-similar wave maps in higher dimensions. Comm. Math. Phys., 351(3):959–972, 2017.
- A proof for the mode stability of a self-similar wave map. Nonlinearity, 29(8):2451–2473, 2016.
- On blowup for the supercritical quadratic wave equation. to appear in Analysis & PDE, page arXiv:2109.11931.
- Roland Donninger. Asymptotics and analytic modes for the wave equation in similarity coordinates. J. Evol. Equ., 9(3):511–523, 2009.
- Roland Donninger. Nonlinear stability of self-similar solutions for semilinear wave equations. Comm. Partial Differential Equations, 35(4):669–684, 2010.
- Roland Donninger. The radial wave operator in similarity coordinates. J. Math. Phys., 51(2):023527, 10, 2010.
- Roland Donninger. On stable self-similar blowup for equivariant wave maps. Comm. Pure Appl. Math., 64(8):1095–1147, 2011.
- On the existence and stability of blowup for wave maps into a negatively curved target. Anal. PDE, 12(2):389–416, 2019.
- A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory. arXiv e-prints, page arXiv:2108.13668, August 2021.
- On stable self-similar blow up for equivariant wave maps: the linearized problem. Ann. Henri Poincaré, 13(1):103–144, 2012.
- Optimal blowup stability for supercritical wave maps. arXiv e-prints, page arXiv:2201.11419, January 2022.
- One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
- Optimal polynomial blow up range for critical wave maps. Commun. Pure Appl. Anal., 14(5):1705–1741, 2015.
- Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps. J. Differential Equations, 265(7):2968–3047, 2018.
- The Cauchy problem for the O(N),𝐂P(N−1),O𝑁𝐂P𝑁1{\rm O}(N),\,{\bf C}{\rm P}(N-1),roman_O ( italic_N ) , bold_C roman_P ( italic_N - 1 ) , and G𝐂(N,p)subscript𝐺𝐂𝑁𝑝G_{{\bf C}}(N,\,p)italic_G start_POSTSUBSCRIPT bold_C end_POSTSUBSCRIPT ( italic_N , italic_p ) models. Ann. Physics, 142(2):393–415, 1982.
- Irfan Glogić. On the Existence and Stability of Self-Similar Blowup in Nonlinear Wave Equations. ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–The Ohio State University.
- Threshold for blowup for the supercritical cubic wave equation. Nonlinearity, 33(5):2143–2158, mar 2020.
- Co-dimension one stable blowup for the supercritical cubic wave equation. Adv. Math., 390:Paper No. 107930, 79, 2021.
- Irfan Glogić. Stable blowup for the supercritical hyperbolic Yang-Mills equations. Adv. Math., 408:Paper No. 108633, 52 pp., 2022.
- Loukas Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, second edition, 2008.
- Chao Hao Gu. On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space. Comm. Pure Appl. Math., 33(6):727–737, 1980.
- The Kolmogorov-Riesz compactness theorem. Expo. Math., 28(4):385–394, 2010.
- Local and global well-posedness of wave maps on ℝ1+1superscriptℝ11\mathbb{R}^{1+1}blackboard_R start_POSTSUPERSCRIPT 1 + 1 end_POSTSUPERSCRIPT for rough data. Internat. Math. Res. Notices, (21):1117–1156, 1998.
- Sergiu Klainerman. On the regularity of classical field theories in Minkowski space-time 𝐑3+1superscript𝐑31{\bf R}^{3+1}bold_R start_POSTSUPERSCRIPT 3 + 1 end_POSTSUPERSCRIPT. In Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995), volume 29 of Progr. Nonlinear Differential Equations Appl., pages 29–69. Birkhäuser, Basel, 1997.
- On the optimal local regularity for gauge field theories. Differential Integral Equations, 10(6):1019–1030, 1997.
- On the global regularity of wave maps in the critical Sobolev norm. Internat. Math. Res. Notices, (13):655–677, 2001.
- Remark on the optimal regularity for equations of wave maps type. Comm. Partial Differential Equations, 22(5-6):901–918, 1997.
- Joachim Krieger. Null-form estimates and nonlinear waves. Adv. Differential Equations, 8(10):1193–1236, 2003.
- On the stability of blowup solutions for the critical corotational wave-map problem. Duke Math. J., 169(3):435–532, 2020.
- Concentration compactness for critical wave maps. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2012.
- Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math., 171(3):543–615, 2008.
- Charles W. Misner. Harmonic maps as models for physical theories. Phys. Rev. D (3), 18(12):4510–4524, 1978.
- Charles W. Misner. Nonlinear model field theories based on harmonic mappings, pages x+189. University of Texas Press, Austin, Tex., 1982. The Alfred Schild Lectures.
- Classical and multilinear harmonic analysis. Vol. I, volume 137 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2013.
- On the well-posedness of the wave map problem in high dimensions. Comm. Anal. Geom., 11(1):49–83, 2003.
- NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX).
- Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems. Publ. Math. Inst. Hautes Études Sci., pages 1–122, 2012.
- An introduction to partial differential equations, volume 13 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2004.
- On the formation of singularities in the critical O(3)O3{\rm O}(3)roman_O ( 3 ) σ𝜎\sigmaitalic_σ-model. Ann. of Math. (2), 172(1):187–242, 2010.
- Walter Rudin. Principles of mathematical analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976.
- Jalal Shatah. Weak solutions and development of singularities of the SU(2)SU2{\rm SU}(2)roman_SU ( 2 ) σ𝜎\sigmaitalic_σ-model. Comm. Pure Appl. Math., 41(4):459–469, 1988.
- Geometric wave equations, volume 2 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998.
- The Cauchy problem for wave maps. Int. Math. Res. Not., (11):555–571, 2002.
- Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds. Comm. Pure Appl. Math., 45(8):947–971, 1992.
- On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math., 47(5):719–754, 1994.
- Energy dispersed large data wave maps in 2+1212+12 + 1 dimensions. Comm. Math. Phys., 298(1):139–230, 2010.
- Regularity of wave-maps in dimension 2+1212+12 + 1. Comm. Math. Phys., 298(1):231–264, 2010.
- Terence Tao. Ill-posedness for one-dimensional wave maps at the critical regularity. Amer. J. Math., 122(3):451–463, 2000.
- Terence Tao. Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices, (6):299–328, 2001.
- Terence Tao. Global regularity of wave maps. II. Small energy in two dimensions. Comm. Math. Phys., 224(2):443–544, 2001.
- Terence Tao. Global regularity of wave maps III-VII. arXiv preprints, 2008-2009.
- Daniel Tataru. Local and global results for wave maps. I. Comm. Partial Differential Equations, 23(9-10):1781–1793, 1998.
- Daniel Tataru. On global existence and scattering for the wave maps equation. Amer. J. Math., 123(1):37–77, 2001.
- Daniel Tataru. The wave maps equation. Bull. Amer. Math. Soc. (N.S.), 41(2):185–204, 2004.
- Global texture and the microwave background. Phys. Rev. Lett., 64:2736–2739, Jun 1990.