A class of large global solutions for the wave-map equation

Chiodaroli, Elisabetta; Krieger, Joachim

doi:10.1090/tran/6805

A class of large global solutions for the wave-map equation
HTML articles powered by AMS MathViewer

by Elisabetta Chiodaroli and Joachim Krieger PDF

Trans. Amer. Math. Soc. 369 (2017), 2747-2773 Request permission

Abstract:

In this paper we consider the equation for equivariant wave maps from $\mathbb {R}^{3+1}$ to $\mathbb {S}^3$ and we prove global in forward time existence of certain $C^\infty$-smooth solutions which have infinite critical Sobolev norm $\dot {H}^{\frac {3}{2}}(\mathbb {R}^3)\times \dot {H}^{\frac {1}{2}}(\mathbb {R}^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $\|u(0, \cdot )\|_{L^\infty (|x|\geq 1)}>M$ for arbitrarily chosen $M>0$. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self–similar solutions.

References

Piotr Bizoń, Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere, Comm. Math. Phys. 215 (2000), no. 1, 45–56. MR 1799875, DOI 10.1007/s002200000291
Thierry Cazenave, Jalal Shatah, and A. Shadi Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 3, 315–349 (English, with English and French summaries). MR 1622539
Roland Donninger, On stable self-similar blowup for equivariant wave maps, Comm. Pure Appl. Math. 64 (2011), no. 8, 1095–1147. MR 2839272, DOI 10.1002/cpa.20366
Roland Donninger and Peter C. Aichelburg, Spectral properties and linear stability of self-similar wave maps, J. Hyperbolic Differ. Equ. 6 (2009), no. 2, 359–370. MR 2543325, DOI 10.1142/S0219891609001812
Roland Donninger and Peter C. Aichelburg, A note on the eigenvalues for equivariant maps of the $\rm SU(2)$ sigma-model, Appl. Math. Comput. Sci. 1 (2010), no. 1, 73–82. MR 2978622
Roland Donninger, Birgit Schörkhuber, and Peter C. Aichelburg, On stable self-similar blow up for equivariant wave maps: the linearized problem, Ann. Henri Poincaré 13 (2012), no. 1, 103–144. MR 2881965, DOI 10.1007/s00023-011-0125-0
Pierre Germain, Besov spaces and self-similar solutions for the wave-map equation, Comm. Partial Differential Equations 33 (2008), no. 7-9, 1571–1596. MR 2450171, DOI 10.1080/03605300802031630
Joachim Krieger, Global regularity of wave maps from $\textbf {R}^{3+1}$ to surfaces, Comm. Math. Phys. 238 (2003), no. 1-2, 333–366. MR 1990880, DOI 10.1007/s00220-003-0836-2
Joachim Krieger and Wilhelm Schlag, Large global solutions for energy supercritical nonlinear wave equations on $\textbf {R}^{3+1}$, J. Anal. Math. (accepted).
Richard Schoen and Karen Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Invent. Math. 78 (1984), no. 1, 89–100. MR 762354, DOI 10.1007/BF01388715
Jalal Shatah, Weak solutions and development of singularities of the $\textrm {SU}(2)$ $\sigma$-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459–469. MR 933231, DOI 10.1002/cpa.3160410405
Jalal Shatah and A. Shadi Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. MR 1278351, DOI 10.1002/cpa.3160470507
Thomas C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math. 42 (1989), no. 1, 1–13. MR 973742, DOI 10.1002/cpa.3160420102
Atsushi Tachikawa, Rotationally symmetric harmonic maps from a ball into a warped product manifold, Manuscripta Math. 53 (1985), no. 3, 235–254. MR 807098, DOI 10.1007/BF01626399
Neil Turok and David Spergel, Global texture and the microwave background, Phys. Rev. Lett. 64 (1990), no. 23, 2736–2739.