2014/08/13 by Stefano Bianchini, Bianchini, Stefano, Nikolay A. Gusev +1
Mathematics · #28A50 #35F10 #35L03 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:28A50 #msc:35F10 #msc:35L03
paper · pdf · doi:10.48550/arxiv.1408.2932
50 pages, 8 figures
arxiv created 2014/08/13 · arxiv updated 2014/08/14
Given bounded vector field b : \mathbb Rd → \mathbb Rd, scalar field u : \mathbb Rd → \mathbb R and a smooth function β: \mathbb R → \mathbb R we study the characterization of the distribution div(β(u)b) in terms of div b and div(u b). In the case of BV vector fields b (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions) in case when d=2 we provide complete characterization of div(β(u) b) in terms of div b and div(u b). Our approach relies on the structure of level sets of Lipschitz functions on \mathrm R2 obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution u of ∂t u + b ⋅ ∇ u=0 is renormalized, i.e. also solves ∂t β(u) + b ⋅ ∇ β(u)=0 for any smooth function β: \mathbb R → \mathbb R. As a consequence we obtain new uniqueness result for this equation.