On Bressan's conjecture on mixing properties of vector fields

• Autorzy: Stefano Bianchini
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 35F25: Initial value problems for nonlinear first-order equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2006
• Tom: 74
• Numer: 1
• Strony: 13-31

Daty

wydano

2006

Twórcy

autor

Stefano Bianchini

• SISSA-ISAS, via Beirut 2-4, I-34014, Trieste, Italy

Identyfikatory

DOI

10.4064/bc74-0-1