On stable self-similar blowup for equivariant wave maps

Abstract

Abstract. We consider co-rotational wave maps from (3 + 1) Minkowski space into the threesphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blow up to a linear ODE spectral problem. Although we are unable, at the moment, to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0.

Stable self-similar blowup in the supercritical heat flow of harmonic maps

“…• From now on we follow the argument introduced in our earlier works [11][12][13][14][15][16] on selfsimilar blowup for wave-type equations. We first show that the nonlinearity is locally Lipschitz on X .…”

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“…Outline of the proof. We use the method developed in the series of papers [14,17,18,15,19,20,9,16]. First, we introduce the rescaled variables…”

On Type I Blow-Up Formation for the Critical NLW

“…To get this, we revisit the bad cases (2), (4) in the proof of Proposition 3.1, which are responsible for the linear growth ofǫ. Considering for example case (4), and dividing into the casesξ…”

“…It is in regard to the second step, the control of the remaining error, where the present work is striving to achieve a different approach, based on a constructive parametrix approach to the linear operator arising upon linearisation around the bulk part of the blow up solution. Our approach may be seen as somewhat in the spirit of the recent remarkable work by Donninger and Donninger and Schorkhuber on the stability of self-similar blow up solutions, for example in [4], [6], which also completely avoids Lyapounov/Morawetz type estimates. It may be hoped that constructive methods like those employed in this paper may shed further light on the precise features of the solutions obtained.…”

“…uniformly in τ ě τ 0 . Moreover, there is a splitting 4) and such that △x p1q 0,1 pξq in turn satisfy the vanishing conditions (3.1). Here we shall let κ " 2p1`ν´1qδ 0 throughout, with δ 0 ą 0 as in the definition ofS , S , see…”