On Strongly Anisotropic Type I Blowup

Merle, Frank; Raphaël, Pierre; Szeftel, Jérémie

doi:10.1093/imrn/rny012

Cited by 20 publications

(16 citation statements)

References 38 publications

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“…In the supercritical and critical range in the Sobolev sense, type I blow-up is also known to occur for some solutions which are neither radial nor increasing in time: see [10] for N = 3, p > p S , [38] for N = 4, p > 5, [7] for N ≥ 7 and p = p S and [8] for p = p S . (ii) Type II blow-up.…”

Section: Resultsmentioning

confidence: 99%

Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Mizoguchi

Souplet

2019

Advances in Mathematics

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Section: Resultsmentioning

confidence: 99%

Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Mizoguchi

Souplet

2019

Advances in Mathematics

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“…Structure of the reconnection profile. In the companion paper [28], we address a similar result in the context of type I blow up with decreasing at infinity self similar profile. The analysis of type I blow up is simpler, and the reconnection profile is universal 14) which is reminiscent to the stability of the ODE type I blow up [2,29].…”

mentioning

confidence: 83%

Strongly anisotropic type II blow up at an isolated point

Collot

Merle²,

Raphaël

2020

J. Amer. Math. Soc.

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We consider the energy super critical d + 1 dimensional semilinear heat equationA fundamental open problem on this canonical nonlinear model is to understand the possible blow up profiles appearing after renormalization of a singularity. We exhibit in this paper a new scenario corresponding to the first example of strongly anisotropic blow up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the reconnection problem in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilization of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow up speeds.

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“…Local asymptotic stability in the interior of the backward light (acoustic) cone from the singularity relies on an abstract spectral argument for compact perturbations of maximal accretive operators. Related arguments have been used in the literature for the study of self-similar solutions both in focusing and defocusing regimes, for example [8,16,21,45,47] for parabolic and [19] for hyperbolic problems. The key to the control of the nonlinear flow in the exterior of the light cone is the propagation of certain weighted scale invariant norms.…”

Section: Restriction On the Parametersmentioning

confidence: 99%

On blow up for the energy super critical defocusing nonlinear Schrödinger equations

Merle

Raphaël²,

Rodnianski

et al. 2021

Invent. math.

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We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

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On Strongly Anisotropic Type I Blowup

Cited by 20 publications

References 38 publications

Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Strongly anisotropic type II blow up at an isolated point

On blow up for the energy super critical defocusing nonlinear Schrödinger equations

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