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

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## Abstract

Given a vector field \(\rho (1,\mathbf {b}) \in L^1_\mathrm{loc}(\mathbb {R}^+\times \mathbb {R}^{d},\mathbb {R}^{d+1})\) such that \({{\,\mathrm{div}\,}}_{t,x} (\rho (1,\mathbf {b}))\) is a measure, we consider the problem of uniqueness of the representation \(\eta \) of \(\rho (1,\mathbf {b}) {\mathcal {L}}^{d+1}\) as a superposition of characteristics \(\gamma : (t^-_\gamma ,t^+_\gamma ) \rightarrow \mathbb {R}^d\), \(\dot{\gamma } (t)= \mathbf {b}(t,\gamma (t))\). 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 \(\mathbb {R}^{d+1}\) into disjoint trajectories \(\wp _\mathfrak {a}\), \(\mathfrak {a}\in \mathfrak {A}\), such that the PDE can be disintegrated into a family of ODEs along \(\wp _\mathfrak {a}\) with measure r.h.s. The decomposition \(\wp _\mathfrak {a}\) is essentially unique. We finally show that \(\mathbf {b}\in L^1_t({{\,\mathrm{BV}\,}}_x)_\mathrm{loc}\) satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible \({{\,\mathrm{BV}\,}}\) vector fields.

$$\begin{aligned} {{\,\mathrm{div}\,}}_{t,x} \big ( u \rho (1,\mathbf {b}) \big ) \in {\mathcal {M}}(\mathbb {R}^{d+1}), \quad u \in L^\infty (\mathbb {R}^+\times \mathbb {R}^{d}), \end{aligned}$$

## Mathematics Subject Classification

35F10 35L03 28A50 35D30## Notes

### Acknowledgements

The authors would like to thank the Center of Mathematical Sciences and Applications (CMSA) of Harvard University and the Institut des Hautes Études Scientifiques (IHES) where part of this work has been done. They are also grateful to Guido de Philippis for useful discussions. During the revision of the paper, the second author was supported by ERC Starting Grant 676675 FLIRT.

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