Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Elisabetta Chiodaroli and Joachim Krieger
Journal: Trans. Amer. Math. Soc. 369 (2017), 2747-2773
MSC (2010): Primary 35L05
DOI: https://doi.org/10.1090/tran/6805
Published electronically: June 20, 2016
MathSciNet review: 3592527
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ \Vert u(0, \cdot )\Vert _{L^\infty (\vert x\vert\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 [Enhancements On Off] (What's this?)

  • [1] Piotr Bizoń, Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere, Comm. Math. Phys. 215 (2000), no. 1, 45-56. MR 1799875 (2001k:58055), https://doi.org/10.1007/s002200000291
  • [2] 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 (2000g:58042)
  • [3] Roland Donninger, On stable self-similar blowup for equivariant wave maps, Comm. Pure Appl. Math. 64 (2011), no. 8, 1095-1147. MR 2839272 (2012f:58034), https://doi.org/10.1002/cpa.20366
  • [4] 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 (2010h:35268), https://doi.org/10.1142/S0219891609001812
  • [5] 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
  • [6] 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, https://doi.org/10.1007/s00023-011-0125-0
  • [7] 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 (2009m:35332), https://doi.org/10.1080/03605300802031630
  • [8] Joachim Krieger, Global regularity of wave maps from $ {\bf R}^{3+1}$ to surfaces, Comm. Math. Phys. 238 (2003), no. 1-2, 333-366. MR 1990880 (2004e:58050), https://doi.org/10.1007/s00220-003-0836-2
  • [9] Joachim Krieger and Wilhelm Schlag, Large global solutions for energy supercritical nonlinear wave equations on $ {\bf R}^{3+1}$, J. Anal. Math. (accepted).
  • [10] Richard Schoen and Karen Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Invent. Math. 78 (1984), no. 1, 89-100. MR 762354 (86a:58024), https://doi.org/10.1007/BF01388715
  • [11] Jalal Shatah, Weak solutions and development of singularities of the $ {\rm SU}(2)$ $ \sigma $-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459-469. MR 933231 (89f:58044), https://doi.org/10.1002/cpa.3160410405
  • [12] 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 (96c:58049), https://doi.org/10.1002/cpa.3160470507
  • [13] Thomas C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math. 42 (1989), no. 1, 1-13. MR 973742 (89k:58069), https://doi.org/10.1002/cpa.3160420102
  • [14] Atsushi Tachikawa, Rotationally symmetric harmonic maps from a ball into a warped product manifold, Manuscripta Math. 53 (1985), no. 3, 235-254. MR 807098 (87a:58051), https://doi.org/10.1007/BF01626399
  • [15] Neil Turok and David Spergel, Global texture and the microwave background, Phys. Rev. Lett. 64 (1990), no. 23, 2736-2739.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35L05

Retrieve articles in all journals with MSC (2010): 35L05


Additional Information

Elisabetta Chiodaroli
Affiliation: EPFL Lausanne, Station 8, CH-1015 Lausanne, Switzerland

Joachim Krieger
Affiliation: EPFL Lausanne, Station 8, CH-1015 Lausanne, Switzerland

DOI: https://doi.org/10.1090/tran/6805
Received by editor(s): April 22, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society