2,328 research outputs found
Steady nearly incompressible vector fields in 2D: chain rule and renormalization
Given bounded vector field , scalar field and a smooth function we study the characterization of the distribution
in terms of and . In the case of vector fields (and under some further assumptions)
such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal\'y,
up to an error term which is a measure concentrated on so-called
\emph{tangential set} of . We answer some questions posed in their paper
concerning the properties of this term. In particular we construct a nearly
incompressible vector field and a bounded function for which this
term is nonzero.
For steady nearly incompressible vector fields (and under some further
assumptions) in case when we provide complete characterization of
in terms of and . Our approach relies on the structure of level sets of Lipschitz functions
on obtained by G. Alberti, S. Bianchini and G. Crippa.
Extending our technique we obtain new sufficient conditions when any bounded
weak solution of is
\emph{renormalized}, i.e. also solves for any smooth function . As a
consequence we obtain new uniqueness result for this equation.Comment: 50 pages, 8 figure
A connection between viscous profiles and singular ODEs
We deal with the viscous profiles for a class of mixed hyperbolic-parabolic
systems. We focus, in particular, on the case of the compressible Navier Stokes
equation in one space variable written in Eulerian coordinates. We describe the
link between these profiles and a singular ordinary differential equation in
the form Here and the function F
takes values into and is smooth. The real valued function z is as well
regular: the equation is singular in the sense that z (V) can attain the value
0.Comment: 6 pages, minor change
- …