On Bressan's conjecture on mixing properties of vector fields

• Volume: 74, Issue: 1, page 13-31
• ISSN: 0137-6934

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Abstract

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In [9], the author considers a sequence of invertible maps ${T}_{i}:S¹\to S¹$ which exchange the positions of adjacent intervals on the unit circle, and defines as Aₙ the image of the set 0 ≤ x ≤ 1/2 under the action of Tₙ ∘ ... ∘ T₁, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) ${\sum }_{i=1}^{n}\left(Tot.Var.\left({T}_{i}-I\right)+Tot.Var.\left({T}_{i}^{-1}-I\right)\right)\ge Clog1/h$. We prove that (1) holds for general quasi incompressible invertible BV maps on ℝ, and that this estimate implies that the map Tₙ ∘ ... ∘ T₁ belongs to the Besov space ${B}^{0,1,1}$, and its norm is bounded by the sum of the total variation of T - I and ${T}^{-1}-I$, as in (2).

How to cite

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Stefano Bianchini. "On Bressan's conjecture on mixing properties of vector fields." Banach Center Publications 74.1 (2006): 13-31. <http://eudml.org/doc/282120>.

@article{StefanoBianchini2006,
abstract = {In [9], the author considers a sequence of invertible maps $T_i : S¹ → S¹$ which exchange the positions of adjacent intervals on the unit circle, and defines as Aₙ the image of the set 0 ≤ x ≤ 1/2 under the action of Tₙ ∘ ... ∘ T₁, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) $∑_\{i=1\}^\{n\} (Tot.Var.(T_i - I) + Tot.Var.(T_i^\{-1\} - I)) ≥ Clog 1/h$. We prove that (1) holds for general quasi incompressible invertible BV maps on ℝ, and that this estimate implies that the map Tₙ ∘ ... ∘ T₁ belongs to the Besov space $B^\{0,1,1\}$, and its norm is bounded by the sum of the total variation of T - I and $T^\{-1\} - I$, as in (2).},
author = {Stefano Bianchini},
journal = {Banach Center Publications},
keywords = {transport equation; Besov spaces},
language = {eng},
number = {1},
pages = {13-31},
title = {On Bressan's conjecture on mixing properties of vector fields},
url = {http://eudml.org/doc/282120},
volume = {74},
year = {2006},
}

TY - JOUR
AU - Stefano Bianchini
TI - On Bressan's conjecture on mixing properties of vector fields
JO - Banach Center Publications
PY - 2006
VL - 74
IS - 1
SP - 13
EP - 31
AB - In [9], the author considers a sequence of invertible maps $T_i : S¹ → S¹$ which exchange the positions of adjacent intervals on the unit circle, and defines as Aₙ the image of the set 0 ≤ x ≤ 1/2 under the action of Tₙ ∘ ... ∘ T₁, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) $∑_{i=1}^{n} (Tot.Var.(T_i - I) + Tot.Var.(T_i^{-1} - I)) ≥ Clog 1/h$. We prove that (1) holds for general quasi incompressible invertible BV maps on ℝ, and that this estimate implies that the map Tₙ ∘ ... ∘ T₁ belongs to the Besov space $B^{0,1,1}$, and its norm is bounded by the sum of the total variation of T - I and $T^{-1} - I$, as in (2).
LA - eng
KW - transport equation; Besov spaces
UR - http://eudml.org/doc/282120
ER -

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