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# A uniqueness result for the decomposition of vector fields in (\R^{\MakeLowercase{d}}

Prof. Stefano Bianchini SISSA, Trieste, Italy
 Speaker Prof. Stefano Bianchini SISSA, Trieste, Italy Dec 11, 2017 from 11:30 AM to 12:30 PM LH 006 vCal iCal

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.

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