Journal ArticleDOI
Equivariant wave maps in two space dimensions
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Singularities of wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into ℕ.Abstract:
Singularities of corotational wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into ℕ. In consequence, for noncompact target surfaces of revolution, the Cauchy problem for wave maps is globally well-posed. © 2003 Wiley Periodicals, Inc.read more
Citations
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MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Journal ArticleDOI
Liouville theorems for the Navier–Stokes equations and applications
TL;DR: In this article, Chen, Strain, Tsai and Yau showed that for axi-symmetric Navier-Stokes equations with bounded velocity, the solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions.
Journal ArticleDOI
Renormalization and blow up for charge one equivariant critical wave maps
TL;DR: In this paper, the authors prove the existence of equivariant finite-time blow-up solutions for the wave map problem from ℝ2+1→S petertodd 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}( t,r)$cffff where u is the polar angle on the sphere, $Q(r)=2\arctan r$cffff is the ground state harmonic map, λ(t)=t -1-ν, and $\mathcal {R} (t
Journal ArticleDOI
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
Pierre Raphaël,Igor Rodnianski +1 more
TL;DR: In this paper, stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem were derived.
Journal ArticleDOI
Global regularity of wave maps II. Small energy in two dimensions
TL;DR: In this article, it was shown that wave maps from Minkowski space to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space.
References
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Journal ArticleDOI
The existence of minimal immersions of $2$-spheres
Jonathan Sacks,Karen Uhlenbeck +1 more
Book
Geometric wave equations
Jalal Shatah,Michael Struwe +1 more
TL;DR: The wave equation conservation laws Function spaces The linear wave equation wellposedness Semilinear wave equations Wave maps Wave maps with symmetry as discussed by the authors The wave equation Conservation laws and function spaces.
Journal ArticleDOI
Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds
J. Shath,A. Tahvildar-Zadeh +1 more
Journal ArticleDOI
Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere
TL;DR: In this article, numerical studies of the Cauchy problem for equivariant wave maps from (2+1)-dimensional Minkowski spacetime into the 2-sphere were conducted.
Journal ArticleDOI
Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
TL;DR: In this article, numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere were conducted.
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