Browsing by Author "Gusev, N.A."

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    Renormalization for autonomous nearly incompressible BV vector fields in 2D
    (SISSA, 2014-12) Bianchini, Stefano; Bonicatto, Paolo; Gusev, N.A.
    Given a bounded autonomous vector field $b \colon \R^d \to \R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Assuming that the ambient space has dimension $d=2$, we prove uniqueness of weak solutions to the transport equation. The starting point of the present work is the result which has been obtained in [7] (where the steady case is treated). Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [3], using the results on the structure of level sets of Lipschitz maps obtained in [1]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [4].
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    Steady nearly incompressible vector elds in 2D: chain rule and renormalization
    (SISSA, 2014-08) Bianchini, Stefano; Gusev, N.A.; Mathematics