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

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$$\mathbb {R}^{{d}}$$ R d 中向量场分解的唯一性结果

给定一个向量场 $$\rho (1,\mathbf {b}) \in L^1_\mathrm{loc}(\mathbb {R}^+\times \mathbb {R}^{d},\mathbb { R}^{d+1})$$ 使得 $${{\,\mathrm{div}\,}}_{t,x} (\rho (1,\mathbf {b}))$$ 是一个度量，我们将 $$\rho (1,\mathbf {b}) {\mathcal {L}}^{d+1}$$ 的表示的唯一性问题考虑为 $$\eta $$ 作为叠加特征 $$\gamma : (t^-_\gamma ,t^+_\gamma ) \rightarrow \mathbb {R}^d$$, $$\dot{\gamma } (t)= \mathbf {b }(t,\gamma(t))$$。我们根据表示 $$\eta $$ 在合适集合上的局部结构给出条件，以证明 $$\mathbb {R}^{d+1}$$ 存在不相交轨迹 $ $\wp _\mathfrak {a}$$, $$\mathfrak {a}\in \mathfrak {A}$$，使得 PDE $$\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}^+\次 \mathbb {R}^{d})，\end{aligned}$$ 可以分解为沿 $$\wp _\mathfrak {a}$$ 的 ODE 族，测量 rhs 分解 $$\wp _ \mathfrak {a}$$ 本质上是独一无二的。我们最终证明 $$\mathbf {b}\in L^1_t({{\,\mathrm{BV}\,}}_x)_\mathrm{loc}$$ 满足这个局部结构假设，这产生，在特别是几乎不可压缩的 $${{\,\mathrm{BV}\,}}$$ 向量场的重整化属性。