Singularity formation for geometric flows

Given a nonlinear evolution equation, the basic question is whether solutions starting from smooth initial data can become singular (blowup) in finite time and, if so, how does the blowup occur. We have been studying this problem for the semilinear geometric wave equations, in particular equivariant wave maps into spheres. These equations are scale invariant, that is if $u(t,r)$ is the solution, so is $u(t/L,r/L)$, where $L$ is a positive constant. Under this scaling the conserved energy transforms as a homogeneous function $E\rightarrow L^{d-2} E$, where $d$ is the number of spatial dimensions. The critical case $d=2$ has been intensively studied for the past two decades leading to a good understanding of the mechanism of blowup. The understanding of the supercritical case is much less satisfactory. All known examples of blowup involve self-similar solutions; however, it is not known if singularities must be self-similar. For $d=3$, the self-similar blowup was proved to be stable [1] confirming the numerical observations in [2]. In higher dimensions almost nothing was known until recently (except for some self-similar solutions in $d=5$ constructed by a variational method in [3] and by a shooting method in [4] ). In [5] we found new explicit self-similar equivariant wave maps dimensions in all dimensions $d\geq 4$, established their linear stability, and finally gave numerical evidence that they play the role of universal attractors in the generic blowup. In the same paper we obtained corresponding results for the equivariant Yang-Mills fields in supercritical dimensions $d\geq 5$. On the other hand, it is well known that solutions starting from small initial data remain globally regular in time. The dichotomy between global regularity and blowup raises a natural question about the nature of a borderline between these two generic asymptotic behaviors. This question was first studied for $d=3$ in [2] which gave evidence that the threshold for blowup is determined by the codimension-one stable manifold of a self-similar solution with one instability (whose existence was established in [4] ). In the recent paper [6] we extended this analysis to higher dimensions $4\leq d\leq 6$.

R. Donninger, On stable self-similar blowup for equivariant wave maps, Commun. Pure Appl. Math. 64, 1095 (2011) ↩
P. Bizoń, T. Chmaj, Z. Tabor, Dispersion and collapse of wave maps, Nonlinearity 13, 1411 (2000) ↩
T. Cazenave, J. Shatah, A.S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and the development of singularities in wave maps and YangMills fields, Ann. Inst. Henri Poincar 68, 315 (1998) ↩
P. Bizoń, Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere, Comm. Math. Phys. 215, 45 (2000) ↩
P. Bizoń, P. Biernat, Generic self-similar blowup for equivariant wave maps and Yang-Mills fields in higher dimensions, Comm. Math. Phys. 338, 1443 (2015) ↩
P. Biernat, P. Bizoń, M. Maliborski, Threshold for blowup for equivariant wave maps in higher dimensions, Nonlinearity 30, 4 (2017) ↩