In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2 + 1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from R 2 into S 2 .

In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity utt − ∆u = u p in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental selfsimilar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the exponent p.

We study numerically the Cauchy problem for equivariant wave maps from 3 + 1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.

We study the development of singularities for the spherically symmetric Yang-Mills equations in d + 1 dimensional Minkowski spacetime for d = 4 (the critical dimension) and d = 5 (the lowest supercritical dimension). Using combined numerical and analytical methods we show in both cases that generic solutions starting with sufficiently large initial data blow up in finite time. The mechanism of singularity formation depends on the dimension: in d = 5 the blowup is exactly self-similar while in d = 4 the blowup is only approximately self-similar and can be viewed as the adiabatic shrinking of the marginally stable static solution. The threshold for blowup and the connection with critical phenomena in gravitational collapse (which motivated this research) are also briefly discussed.Introduction: The Yang-Mills (YM) equations are the basic equations of gauge theories describing the fundamental forces of nature so understanding their solutions is an issue of great importance. This is not an easy task since, in contrast to Maxwell's equations or the Schrödinger equation, the YM equations are nonlinear which opens up the possibility that solutions which are initially smooth become singular in future. Actually such a spontaneous breakdown of solutions of YM equations cannot occur in the physical 3+1 dimensional Minkowski spacetime as was shown in a classic paper by Eardley and Moncrief [1] who proved that solutions starting from smooth initial data remain smooth for all future times. A natural question is: how does the property of global regularity depend on the dimension of the underlying spacetime, in particular can singularities develop in d + 1 dimensions for d > 3? We hope that our letter is a step towards answering this question. As we argue below, the problem of singularity formation for YM equations in higher dimensions is not only interesting in its own right but in addition it sheds some light on our understanding of Einstein's equations in the physical dimension.Despite intensive research the problem of global regularity for YM equations in 4 + 1 dimensions is entirely open [2]. A lot of progress has been made to prove local existence for "rough" initial data, yet the attempts of proving global regularity by establishing local wellposedness in the energy norm fail to achieve the goal by "epsilon" [3] (nota bene such a local proof of global existence has been obtained in 3 + 1 dimensions [4], thereby improving the theorem of Eardley and Moncrief). In this letter we report on numerical simulations which in combination with analytic results strongly suggest that generic solutions with sufficiently large energy do, in fact, blow up in finite time. Hence, we believe that the above mentioned epsilon in the optimal local well-posedness result is not a technical shortcoming but is indispensable. We

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