# On Bressan's conjecture on mixing properties of vector fields

Banach Center Publications (2006)

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

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topStefano 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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