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On Bressan's conjecture on mixing properties of vector fields

Stefano Bianchini (2006)

Banach Center Publications

In [9], the author considers a sequence of invertible maps T i : S ¹ S ¹ 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 ( T o t . V a r . ( T i - I ) + T o t . V a r . ( T i - 1 - I ) ) C l o g 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...

On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We study the asymptotic behavior of λ v λ as λ 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) λ v λ + H ( x , D v λ ) = 0 , with H ( x , p ) : = min b B max a A { - f ( x , a , b ) · p - l ( x , a , b ) } . We discuss the cases in which the state of the system is required to stay in an n -dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary)...

On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the asymptotic behavior of λ v λ as λ 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) λ v λ + H ( x , D v λ ) = 0 , with H ( x , p ) : = min b B max a A { - f ( x , a , b ) · p - l ( x , a , b ) } . We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the...

On microlocal analyticity of solutions of first-order nonlinear PDE

Shif Berhanu (2009)

Annales de l’institut Fourier

We study the microlocal analyticity of solutions u of the nonlinear equation u t = f ( x , t , u , u x ) where f ( x , t , ζ 0 , ζ ) is complex-valued, real analytic in all its arguments and holomorphic in ( ζ 0 , ζ ) . We show that if the function u is a C 2 solution, σ Char L u and 1 i σ ( [ L u , L u ¯ ] ) < 0 or if u is a C 3 solution, σ Char L u , σ ( [ L u , L u ¯ ] ) = 0 , and σ ( [ L u , [ L u , L u ¯ ] ] ) 0 , then σ W F a u . Here W F a u denotes the analytic wave-front set of u and Char L u is the characteristic set of the linearized operator. When m = 1 , we prove a more general result involving the repeated brackets of L u and L u ¯ of any order.

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