We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under small perturbations of the initial data. The key ingredient is the introduction of a novel coordinate system that allows one to track the evolution past the blowup time and almost up to the Cauchy horizon of the singularity. As a consequence, we also obtain a result on continuation beyond blowup.
The problem of finite-time blowup for wave maps attracted a lot of interest in the recent past. The bulk of the literature focuses on the two-dimensional case which is energy-critical. The existence of finite-time blowup for energy-critical wave maps into the two-sphere has first been observed numerically in the work of Bizo 'n-Chmaj-Tabor @cite_46 . Rigorously, the existence of blowup solutions was proved by Krieger-Schlag-Tataru @cite_60 , Rodnianski-Sterbenz @cite_45 , and Rapha "el-Rodnianski @cite_58 , see also @cite_5 @cite_28 . We remark that the blowup in the energy-critical case is of type II and proceeds by dynamical rescaling of a soliton, cf. @cite_56 . In fact, there are by now powerful nonperturbative techniques for energy-critical equations which allow one to prove versions of the celebrated , see the work by C ^ote @cite_25 , C ^ote-Kenig-Lawrie-Schlag @cite_49 @cite_30 , and, very recently, Jia-Kenig @cite_40 , Duyckaerts-Jia-Kenig-Merle @cite_4 , see also @cite_52 @cite_21 . Large-data global well-posedness and scattering is addressed in the fundamental work by Tataru-Sterbenz @cite_19 @cite_61 and Krieger-Schlag @cite_47 .
@cite_30 @cite_61 @cite_47 @cite_4 @cite_60 @cite_28 @cite_21 @cite_52 @cite_56 @cite_19 @cite_40 @cite_45 @cite_49 @cite_5 @cite_46 @cite_58 @cite_25
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We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S 2 of the form (u(t,r)=Q( (t)r)+ R (t,r) ) where u is the polar angle on the sphere, (Q(r)=2 r ) is the ground state harmonic map, λ(t)=t -1-ν, and ( R (t,r) ) is a radiative error with local energy going to zero as t→0. The number ( > 1 2 ) can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis. In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere ( S ^ n-1 R ^ n ) with ( n ), and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Cote, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Cote, Commun. Pure Appl. Math. 69(4):609–612, 2016; Cote, Am. J. Math. 137(1):139–207, 2015; , Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Riviere (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times @math (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities. We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k ≥ 4. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.