Wave maps into the sphere
DOI:
https://doi.org/10.33044/revuma.3159Abstract
In this note we discuss some geometric analogs of the classical harmonic functions on $\mathbb{R}^n$ and their associated evolutions.
Downloads
References
R. Côte, On the soliton resolution for equivariant wave maps to the sphere, Comm. Pure Appl. Math. 68 (2015), no. 11, 1946–2004. MR 3403756.
R. Côte, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I, Amer. J. Math. 137 (2015), no. 1, 139–207. MR 3318089.
R. Côte, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II, Amer. J. Math. 137 (2015), no. 1, 209–250. MR3318090.
R. Côte, C. Kenig, A. Lawrie and W. Schlag, Profiles for the radial focusing $4d$ energy-critical wave equation, Comm. Math. Phys. 357 (2018), no. 3, 943–1008. MR 3769743.
R. Côte, C. E. Kenig and W. Schlag, Energy partition for the linear radial wave equation, Math. Ann. 358 (2014), no. 3-4, 573–607. MR 3175135.
T. Duyckaerts, H. Jia, C. Kenig and F. Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal. 27 (2017), no. 4, 798–862. MR 3678502.
T. Duyckaerts, H. Jia, C. Kenig and F. Merle, Universality of blow up profile for small blow up solutions to the energy critical wave map equation, Int. Math. Res. Not. 2018, no. 22, 6961–7025. MR 3878592.
T. Duyckaerts, C. Kenig, Y. Martel and F. Merle, Soliton resolution for critical co-rotational wave maps and radial cubic wave equation, Commun. Math. Phys. 391 (2022), no. 2, 779–871. MR 4397184.
T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 533–599. MR 2781926.
T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1389–1454. MR 2966655.
T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math. 1 (2013), no. 1, 75–144. MR 3272053.
T. Duyckaerts, C. Kenig and F. Merle, Soliton resolution for the radial critical wave equation in all odd space dimensions. https://arxiv.org/abs/1912.07664 [math.AP], 2019. To appear in Acta Math.
T. Duyckaerts, C. Kenig and F. Merle, Exterior energy bounds for the critical wave equation close to the ground state, Comm. Math. Phys. 379 (2020), no. 3, 1113–1175. MR 4163362.
T. Duyckaerts, C. Kenig and F. Merle, Decay estimates for nonradiative solutions of the energy-critical focusing wave equation, J. Geom. Anal. 31 (2021), no. 7, 7036–7074. MR 4289254.
J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 0164306.
J. Eells, Jr. and J. C. Wood, Restrictions on harmonic maps of surfaces, Topology 15 (1976), no. 3, 263–266. MR 0420708.
M. Engelstein and D. Mendelson, Non-uniqueness of bubbling for wave maps. https://arxiv.org/abs/2005.14128 [math.AP], 2020.
E. Fermi, J. Pasta and S. Ulam, Studies of Nonlinear Problems. I, Los Alamos Scientific Laboratory, Report LA-1940, 1955. https://doi.org/doi:10.2172/4376203.
R. Grinis, Quantization of time-like energy for wave maps into spheres, Comm. Math. Phys. 352 (2017), no. 2, 641–702. MR 3627409.
R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 0664497.
J. Jendrej and A, Lawrie, Continuous time soliton resolution for two-bubble equivariant wave maps, https://arxiv.org/abs/2010.12506 [math.AP], 2020. To appear in Math. Res. Lett.
J. Jendrej and A. Lawrie, Soliton resolution for energy-critical wave maps in the equivariant case, https://arxiv.org/abs/2106.10738 [math.AP], 2021. To appear in J. Amer. Math. Soc.
H. Jia and C. Kenig, Asymptotic decomposition for semilinear wave and equivariant wave map equations, Amer. J. Math. 139 (2017), no. 6, 1521–1603. MR 3730929.
C. Kenig, A. Lawrie, B. Liu and W. Schlag, Channels of energy for the linear radial wave equation, Adv. Math. 285 (2015), 877–936. MR 3406517.
C. E. Kenig, A. Lawrie and W. Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data, Geom. Funct. Anal. 24 (2014), no. 2, 610–647. MR 3192036.
J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. MR 2372807.
M. Kruskal, The birth of the soliton, in Nonlinear Evolution Equations Solvable by the Spectral Transform (Internat. Sympos., Accad. Lincei, Rome, 1977), 1–8, Res. Notes in Math., 26, Pitman, Boston, MA, 1978. MR 0521004.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, https://arxiv.org/abs/math/0211159 [math.DG], 2002.
P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122. MR 2929728.
I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical ${rm O}(3)$ $sigma$-model, Ann. of Math. (2) 172 (2010), no. 1, 187–242. MR 2680419.
C. Rodriguez, Profiles for the radial focusing energy-critical wave equation in odd dimensions, Adv. Differential Equations 21 (2016), no. 5-6, 505–570. MR 3473583.
J. Shatah and A. S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. MR 1278351.
J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys. 298 (2010), no. 1, 139–230. MR 2657817.
J. Sterbenz and D. Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys. 298 (2010), no. 1, 231–264. MR 2657818.
M. Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math. 56 (2003), no. 7, 815–823. MR 1990477.
T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), no. 2, 443–544. MR 1869874.
D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. MR 1827277.
P. M. Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 (1997), no. 3, 593–610. MR 1472890.
N. J. Zabusky and M. D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), no. 6, 240–243. https://doi.org/10.1103/PhysRevLett.15.240.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Carlos Kenig
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.
