### A Glimm type functional for a special Jin–Xin relaxation model

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In [9], the author considers a sequence of invertible maps ${T}_{i}:S\xb9\to S\xb9$ 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) ${\sum}_{i=1}^{n}(Tot.Var.({T}_{i}-I)+Tot.Var.({T}_{i}^{-1}-I))\ge Clog1/h$. 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...

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial}_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with bounded divergence....

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