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BLOW-UP OF THE NONEQUIVARIANT ($2+1$)-DIMENSIONAL WAVE MAP

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical problems in dynamical systems

Published online by Cambridge University Press:  18 March 2014

JÖRG FRAUENDIENER  and
RALF PETER
JÖRG FRAUENDIENER*
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email [email protected]
RALF PETER
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email [email protected]
*
✉[email protected]
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Abstract

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It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.

Keywords

wave mapblow-upequivariancesymplectic integrator

MSC classification

Secondary: 35L53: Initial-boundary value problems for second-order hyperbolic systems 65M20: Method of lines 65P10: Hamiltonian systems including symplectic integrators
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 2 , October 2013 , pp. 151 - 161
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Andersen, H. C., “Rattle: a ‘velocity’ version of the SHAKE algorithm for molecular dynamics calculations”, J. Comput. Phys. 52 (1983) 24–34 doi:10.1016/0021-9991(83)90014-1.CrossRefGoogle Scholar
Bartels, S., Feng, X. and Prohl, A., “Finite element approximations of wave maps into spheres”, SIAM J. Numer. Anal. 46 (2007) 61–87 doi:10.1137/060659971.CrossRefGoogle Scholar
Bartels, S., “Semi-implicit approximation of wave maps into smooth or convex surfaces”, SIAM J. Numer. Anal. 47 (2009) 3486–3506 doi:10.1137/080731475.CrossRefGoogle Scholar
Bizoń, P., Chmaj, T. and Tabor, Z., “Formation of singularities for equivariant ($2+1$)-dimensional wave maps into the 2-sphere”, Nonlinearity 14 (2001) 1041–1053 doi:10.1088/0951-7715/14/5/308.CrossRefGoogle Scholar
Isenberg, J. and Liebling, S. L., “Singularity formation in $2+1$ wave maps”, J. Math. Phys. 43 (2002) 678–683 doi:10.1063/1.1418717.CrossRefGoogle Scholar
Ovchinnikov, Yu. N. and Sigal, I. M., “On collapse of wave maps”, Phys. D 240 (2011) 1311–1324 doi:10.1016/j.physd.2011.04.014.CrossRefGoogle Scholar
Peter, R. and Frauendiener, J., “Free versus constraint evolution of the $2+1$ equivariant wave map”, J. Phys. A 45 (2012) 055201; doi:10.1088/1751-8113/45/5/055201.CrossRefGoogle Scholar
Peter, R., “Numerical studies of geometric partial differential equations with symplectic methods”, Ph. D. Thesis, University of Otago, 2012.Google Scholar
Raphaël, P. and Rodnianski, I., “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 doi:10.1007/s10240-011-0037-z.CrossRefGoogle Scholar
Sterbenz, J. and Tataru, D., “Regularity of wave maps in dimension $2+1$”, Comm. Math. Phys. 298 (2010) 231–264 doi:10.1007/s00220-010-1062-3.CrossRefGoogle Scholar
Struwe, M., “Equivariant wave maps in two space dimensions”, Comm. Pure Appl. Math. 56 (2003) 815–823 doi:10.1002/cpa.10074.CrossRefGoogle Scholar