On strongly anisotropic type II blow up

Charles Collot; Frank Merle; Pierre Raphael

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On strongly anisotropic type II blow up

Charles Collot, Frank Merle, Pierre Raphael

Analysis of PDEs · Submitted 14 Sep 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.85 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.29 · mentions code or data availability 0.01.

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We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A 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 {\it 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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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A fundamental open problem on this canonical nonlinear model is to understand the possible blow up profiles appearing after renormalization of a singularity.	background 0.97	0.14	0.04	asserted 0.89
s1	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.	result 0.52	0.96	0.82	asserted 1.00
s2	A fundamental step of the analysis is to solve the {\it reconnection problem} in order to produce finite energy solutions which is the heart of the matter.	method 0.86	0.59	0.07	asserted 1.00
s3	The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics.	interpretation 0.96	0.09	0.06	hedged possibility 0.83
s4	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.	method 0.89	0.94	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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Original abstract

We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A 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 {\it 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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9 arXiv papers cite this in 17 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 10 sentences without a confident label

Uses its method or tool 1 paper

The core of our approach, following [ref] , [ref] , [ref] and [ref] , is to obtain the local expansion of the perturbation in the parabolic neighbourhood of the singularity, relying on a precise spectral analysis.

Refined description and stability for singular solutions of the 2D Keller-Segel system · Statement of the result · 0.96

Compares with it 1 paper

The nonlinear analysis is more involved too, as we cannot rely on signed quantities or parameters restriction (respectively equation (1.8) and Lemma A.1 in [ref] ).

Refined description and stability for singular solutions of the 2D Keller-Segel system · Statement of the result · 0.98

Cites it as background 4 papers

Concrete examples of initial data leading to type-II blowup for [ref] with a general nonlinearity $|a|^p-1a$ were exhibited in several works [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] and references therein.

Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches · Introduction · 0.81
The existence of Type II blowup solutions to [ref] was established in some special range of $p = p(d)$, see for example [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] [ref] , [ref] , [ref] and references therein.

Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$ · What is known · 0.91
Let us also mention that in [ref] anistropic blow-ups were constructed for the energy supercritical semi-linear heat equation.

Singularity formation for Burgers equation with transverse viscosity · Comments on the result and open problems · 0.87

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