### Abstract

Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self-similar blowup profile. More generally, we study the relation between • the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and • the existence of a smooth blowup profile for the (hyperbolic) wave map problem. This has several applications to questions of regularity and uniqueness for the wave map equation.

Original language | English (US) |
---|---|

Pages (from-to) | 706-728 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 62 |

Issue number | 5 |

DOIs | |

State | Published - May 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**On the existence of smooth self-similar blowup profiles for the wave map equation.** / Germain, Pierre.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the existence of smooth self-similar blowup profiles for the wave map equation

AU - Germain, Pierre

PY - 2009/5

Y1 - 2009/5

N2 - Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self-similar blowup profile. More generally, we study the relation between • the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and • the existence of a smooth blowup profile for the (hyperbolic) wave map problem. This has several applications to questions of regularity and uniqueness for the wave map equation.

AB - Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self-similar blowup profile. More generally, we study the relation between • the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and • the existence of a smooth blowup profile for the (hyperbolic) wave map problem. This has several applications to questions of regularity and uniqueness for the wave map equation.

UR - http://www.scopus.com/inward/record.url?scp=68049135833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68049135833&partnerID=8YFLogxK

U2 - 10.1002/cpa.20266

DO - 10.1002/cpa.20266

M3 - Article

VL - 62

SP - 706

EP - 728

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 5

ER -