A uniqueness result for the decomposition of vector fields in (\R^{\MakeLowercase{d}}
Speaker 
Prof. Stefano Bianchini
SISSA, Trieste, Italy


When 
Dec 11, 2017
from 11:30 AM to 12:30 PM 
Where  LH 006 
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Abstract: Given a vector field \(\rho (1,\b) \in L^1_\loc(\R^+\times
\R^{d},\R^{d+1})\) such that \(\dive_{t,x} (\rho (1,\b))\) is a measure, we
consider the problem of uniqueness of the representation \(\eta\) of \(\rho
(1,\b) \mathcal L^{d+1}\) as a superposition of characteristics \(\gamma :
(t^_\gamma,t^+_\gamma) \to \R^d\), \(\dot \gamma (t)= \b(t,\gamma(t))\).
We give conditions in terms of a local structure of the representation
\(\eta\) on suitable sets in order to prove that there is a partition of
\(\R^{d+1}\) into disjoint trajectories \(\wp_\a\), \(\a \in \A\), such that
the PDE
\(\dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u
\in L^\infty(\R^+\times \R^{d}),
can be disintegrated into a family of ODEs along\($\wp_\a\) with measure
r.h.s.. The decomposition \(\wp_\a\) is essentially unique. We finally
show that \(\b \in L^1_t(\BV_x)_\loc\) satisfies this local structural
assumption and this yields, in particular, the renormalization property
for nearly incompressible \(\BV\) vector fields.