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

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

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

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.

Glossary

${\mathrm {Cyl}}_{t,x}^{r,L}$ :

$\rho (1,\mathbf {b})$-proper cylinder

$\nu \ll \mu$ :

$\nu$ is absolute continuous w.r.t. $\mu$

$\fint _A f \, \mu$ :

Average integral on the sets A

$D^\mathrm {a.c.}\mathbf {b}$ :

Absolutely continuous part of Df

$\phi ^\ell _\gamma$ :

Approximate cylinder of flow

$J_{\mathbf {b}}$ :

Approximate jump set of $\mathbf {b}$

$\nu ^\mathrm {a.c.}$ :

Absolutely continuous part of $\nu$

$Q_{\ell ^-_{1,\gamma },\ell _{1,\gamma }^+,\ell }$ :

Approximate cylinder with shape determined by $\ell ^\pm _{1,\gamma },\ell$

$B^d_r(x)$ :

Balls of radius r centered at $x \in \mathbb {R}^d$

${{\,\mathrm{Fr}\,}}A$ :

Boundary of a set A

$\partial \Omega$ :

Boundary of a set in $\mathbb {R}^d$

${\bar{Q}}$ :

Base of the cylinder $Q_{\ell ^\pm _{1,\gamma },\ell }$

$A^c$ :

Complementary set of A

$C({\mathbf {e}}, a)$ :

Closed convex cone about $\varepsilon$ of vertex 0 ant openint a

$C({\mathbf {e}}, a; x)$ :

Closed convex cone about $\varepsilon$ of vertex x ant openint a

${\mathbb {1}}_A$ :

Characteristic function of the set A

${{\,\mathrm{clos}\,}}A$ :

Closure of the set A

$C_c^\infty (\Omega )$ :

Compactly supported smooth functions defined in the open set $\Omega \subset \mathbb {R}^d$

$*$ :

Convolution in $\mathbb {R}^d$

$D^{\mathrm {cantor}}\mathbf {b}$ :

Cantor part of $D^{\mathrm {sing}}f$

$\gamma$ :

Curve defined in an interval of time

$g \circ f$ :

Composition of two functions

$K_{{\bar{r}}}^{\varepsilon ,\varepsilon '} \subset K^{\varepsilon }$ :

Compact subset of $\partial \Omega$ defined in Lemma 4.16

$K^\tau _{\delta _c,{\bar{r}}}$ :

Compact set with suitable local covering

$K^{\tau ,\pm }$ :

Compact sets where the untangling functionals are controlled

${\mathcal {K}}^{n}$ :

Compact subset of $\varGamma$ of trajectories with existence interval $\ge 2^{1-n}$

$Q^\ell _\gamma$ :

Cylinders of approximate flow

$\varphi$ :

Convolution kernel

$\varpi$ :

Constant controlling the flux across the lateral boundary of approximate cylinders of flows

$x_{{\mathbf {n}}}$ :

Coordinate along ${\mathbf {n}}$

$x_{{\mathbf {n}}}^\perp$ :

Coordinates orthogonal to ${\mathbf {n}}$

$\mathrm{dist}(x, E)$ :

Distance of the point $x \in X$ from the set $E \subset X$ in a metric space X

$C_d$ :

Dimensional constant

$D_{\mathbf {e}} f$ :

Directional derivative of f along ${\mathbf {e}}$

$\delta _x$ :

Dirac mass at x

$D f$ :

Differential of the function f

$\mu = \int \mu _\alpha \, f_\sharp \mu (d\alpha )$ :

Disintegration of $\mu$ w.r.t. the partition $\{A_\alpha \}_\alpha$

${{\,\mathrm{div}\,}}{\mathbf {b}}$ :

Divergence of the vector field ${\mathbf {b}}$

${\mathcal {H}}^d$ :

d-Dimensional Hausdorff measure

$\langle f,\psi \rangle$ :

Distribution f evaluated on $\psi$

$\textsf {m}$ :

Direction of the variation in the rank-one property

$\overline{\textsf {m} }$ :

Direction of the variation in the rank-one property at the point $(\bar{t},{\bar{x}})$

$\texttt {M}$ :

Deformation factor

$\mu ^\beta _\mathfrak {a}$ :

Disintegration of the measure ${{\,\mathrm{div}\,}}(\beta (\rho ) (1,\mathbf {b}))$

$\mathbb {R}^d$ :

d-Dimensional real vector space

$\textsf {Tr} ^{{\mathrm {in}}}(B, \Omega ) \cdot \mathbf {n}$ :

Distributional inner normal trace

$\textsf {Tr} ^{{\mathrm {out}}}(B, \Omega ) \cdot \mathbf {n}$ :

Distributional outer normal trace

$w_\mathfrak {a}(t)$ :

Density of the disintegration of ${\mathcal {L}}^{d+1}$ w.r.t. $\{\wp _\mathfrak {a}\}_\mathfrak {a}$

$\mathbf {b}_t$ :

Equivalent to $\mathbf {b}(t)$ for time dependent vector fields

$E_\mathfrak {a}$ :

Equivalence classes of $\sim$

$\ell ^\pm _{1,\gamma }$ :

Evolution of the ${\mathbf {e}}_1$-boundary of the approximate cylinder

$\gamma \sim \gamma '$ :

Equivalent relation among untangled trajectories

$\rho ^i_\Omega$ :

Evaluation of the measure $\eta ^i_\Omega$

$\sigma (f(t))$ :

Evaluation of the function f w.r.t. the measure $\rho (t) {\mathcal {L}}^d$

$t^{i,-}_\gamma$ :

Entrance time of $\gamma$ in $\Omega$

$t^{i,+}_\gamma$ :

Exit time of $\gamma$ in $\Omega$

$\wp _\mathfrak {a}$ :

Evaluation of the equivalence class $E_\mathfrak {a}$

$\Phi ^{\ell }_{{\mathrm {enter}}}(\gamma )$ :

Functional computing intersecting curves across $\phi ^\ell _\gamma$

$\Phi ^{\ell }_{{\mathrm {exit}}}(\gamma )$ :

Functional computing the curves exiting the cylinder $\phi ^\ell _\gamma$

$I_{2,\gamma }$ :

Flow across $L_{2,\gamma }$

$I^-_{1,\gamma }$ :

Flow across $L^-_{1,\gamma }$

$I^+_{1,\gamma }$ :

Flow across $L^+_{1,\gamma }$

$t^+_\gamma$ :

Final time of the interval $I_\gamma$

${\mathcal {U}}$ :

Function locally approximating $\mathbf {b}$

$A$ :

Generic set

$B$ :

Generic vector field in $\mathbb {R}^{d+1}$

$C$ :

Generic constant

${{\,\mathrm{Graph}\,}}f$ :

Graph of the function f

${{\,\mathrm{Graph}\,}}\gamma$ :

Graph of the a.c. curve $\gamma$ in the closed interval of definition

$\mu$ :

Generic signed Radon measure

$\Omega$ :

Generic open set

$X$ :

Generic metric space

$\texttt {T}$ :

Hitting point map

$\phi ^{\delta ,\pm }$ :

Inner/outer distance functions from a set

${\mathrm {id}}$ :

Identity function

$I_\gamma = (t^-_\gamma ,t^+_\gamma )$ :

Interval of definition of the curve $\gamma$

${{\,\mathrm{int}\,}}A$ :

Interior of the set A

$\int f \, dx$ :

Integral of a Borel function f w.r.t. ${\mathcal {L}}^d$

$\int f \, \mu$ :

Integral of a Borel function f w.r.t. $\mu$

$\texttt {R} ^i_\Omega$ :

ith restriction operator

$t^-_\gamma$ :

Initial time of the interval $I_\gamma$

$\textsf {Tr} ^{\mathrm {in}}({\mathbf {b}},\Omega )$ :

Inner trace of the vector fields $\mathbf {b}$

$D^{\mathrm {jump}}\mathbf {b}$ :

Jump part of $D^{\mathrm {sing}}f$

$\eta$ :

Lagrangian representation

${\tilde{\eta }}$ :

Lagrangian representation of $\rho (1,\mathbf {b}) \llcorner _\Omega$, only used in Sect. 7

${\bar{L}}_{2}$ :

Lateral boundary of ${\bar{Q}}$ with normal ${\mathbf {e}}^\perp _1$

$L_{2,\gamma }$ :

Lateral boundary of $Q_{\ell ^\pm _{1,\gamma },\ell }$ with normal ${\mathbf {e}}^\perp _1$

${\mathcal {L}}^d$ :

Lebesgue measure in ${\mathbb {R}^d}$

$L_{1,\gamma }^\pm$ :

Lateral boundary of $Q_{\ell ^\pm _{1,\gamma },\ell }$ given by the graph of $\ell ^\pm _{1,\gamma }$

$\partial ^l Q$ :

Lateral boundary of the set Q

$u$ :

$L^\infty$-solution to ${{\,\mathrm{div}\,}}(u\rho (1,\mathbf {b})) \in {\mathcal {M}}$

$\varsigma _x$ :

Local representation of a Lipschitz boundary

$\delta _1$ :

Maximal shrinking coefficient of an approximate cylinder of flow

$\mu ^\beta$ :

Measure ${{\,\mathrm{div}\,}}(\beta (\rho ) (1,\mathbf {b}))$

$M(x)$ :

Matrix derivative of the absolutely continuous part of a ${{\,\mathrm{BV}\,}}$ vector field

$\texttt {T} ^{i,\pm }_\Omega$ :

Mapping of $\gamma$ to its $\Omega$ entering/exiting point

$\varpi ^\tau$ :

Measure controlling the untangling functional

$\zeta _C^\tau$ :

Measure locally controlling the untangling functionals

${\mathcal {O}}(f)$ :

Notation for constant of the order f

$N$ :

Negligible set w.r.t. some measure

$\textsf {n}$ :

Normal to the rank-one property

$\overline{\textsf {n} }$ :

Normal to the rank-one property at the point $(\bar{t},{\bar{x}})$

$\Vert \cdot \Vert$ :

Norm in a generic Banach space

$|\cdot |$ :

Norm in $\mathbb {R}^d$

$o(f)$ :

Notation for constant infinitesimal w.r.t. f

$U_\texttt {x}$ :

Neighborhood of $\texttt {x}$

$\mathbf {n}$ :

Outer unit normal to the $\rho (1,\mathbf {b})$-proper set $\Omega \subseteq \mathbb {R}^{d+1}$

$\nu ^\perp$ :

Orthogonal component of $\nu$ w.r.t. to another given measure

$\mu \perp \nu$ :

Orthogonal measures

$\eta ^i_\Omega$ :

Push forward of $\eta$ by $\texttt {R} ^i_\Omega$

$\phi$ :

Particular functions used in the paper, usually with an index/apex

$f_\sharp \mu$ :

Push-forward of the measure $\mu$ through f

$K^n$ :

Projection of ${\mathcal {K}}^n$

$\Omega ^\varepsilon$ :

Perturbation of a proper set constructed in Theorem 4.18

$\texttt {p} _X$ :

Projection on the space X

$\rho$ :

Positive solution to transport equation

$(\texttt {R} _\Omega )_\sharp \eta$ :

Push forward of $\eta$ by the multivalued map $R_\Omega$

$S_2$ :

Partition of the set $\partial (\Omega ^\varepsilon \setminus \Omega )$, Theorem 4.18

$S_3^-$ :

Partition of the set $\partial (\Omega ^\varepsilon \setminus \Omega )$, Theorem 4.18

$S_3^+$ :

Partition of the set $\partial (\Omega ^\varepsilon \setminus \Omega )$, Theorem 4.18

$S_4$ :

Partition of the set $\partial (\Omega ^\varepsilon \setminus \Omega )$, Theorem 4.18

$\varUpsilon$ :

Product space of intervals in $\mathbb {R}$ and curves in $\mathbb {R}^d$

$\hat{\texttt {f} }$ :

Quotient map for $\{E_\mathfrak {a}\}_\mathfrak {a}$

$\texttt {f}$ :

Quotient map for $\{\wp _\mathfrak {a}\}_\mathfrak {a}$

${{\,\mathrm{Fr}\,}}(A,B)$ :

Relative boundary of A in B

${{\,\mathrm{clos}\,}}(A,B)$ :

Relative closure of the set A in B

${\tilde{\eta }}^{\mathrm {cr}}$ :

Restriction of ${\tilde{\eta }}$ to $\varGamma ^{\mathrm {Cr}}$

$\eta ^\varXi$ :

Restriction of $\eta$ to $\varXi$

$f \llcorner _A$ :

Restriction of the function f to the set A

$f^r_x$ :

Rescaled f about $x \in \mathbb {R}^d$

$f({\bar{x}}\pm )$ :

Right left limit of a 1d function at ${\bar{x}}$

${{\,\mathrm{int}\,}}(A,B)$ :

Relative interior of the set A in B

$\mu ^r_x$ :

Rescaled $\mu$ about $x \in \mathbb {R}^d$

$\mu \llcorner _A$ :

Restriction of the measure $\mu$ to the set A

$\partial ^\star F$ :

Reduced boundary of the set of finite perimeter F

$\frac{d\nu }{d\mu }$ :

Radon–Nikodym derivative of $\nu$ w.r.t. $\mu \ge 0$

${\mathcal {R}}(f)$ :

Range of the function f

$\texttt {R} ^{0,\pm }_\Omega$ :

0th restriction operators

$\texttt {R} _\Omega$ :

Restriction operator

$(\texttt {R} _\Omega )_\sharp \eta ^{{\mathrm {out}}}$ :

Restriction of $(\texttt {R} _\Omega )_\sharp \eta$ to the exiting trajectories

$C(X,Y)$ :

Space of continuous functions over X

$A \Subset B$ :

Set A whose compact closure is contained in B

${\textsf {Adm}}(\{\mu _i\}_{i \in I})$ :

Sets of admissible transference plans between the measures $\mu _i$

$A^{\ell }_{\gamma }$ :

Set of intersecting curves

$A^{\pm }$ :

Subset of $\partial \Omega$ where trajectories are exiting or entering, respectively

$C^k(\mathbb {R}^d,\mathbb {R}^{d'})$ :

Space of functions on $\mathbb {R}^d$ with continuous derivatives up to order k

$\mathscr {D}^\prime (\Omega )$ :

Space of distributions over the open set $\Omega \subset \mathbb {R}^d$

$D^{\mathrm {sing}}\mathbf {b}$ :

Singular part of Df

$E\mathbf {b}$ :

Symmetric part of the derivative $D\mathbf {b}$

$E^{\ell }_\gamma$ :

Set of curves not contained in ${{\,\mathrm{supp}\,}}\phi ^\ell _\gamma$

${\bar{\ell }}_1$ :

Starting shape of the approximate cylinder of flow in the BV case

${\mathfrak {A}}$ :

Suitable set of indexes

$L$ :

Scale factor

$L^1(\mu ,Y)$ :

Space of functions whose modulus is $\mu$-integrable

$L^\infty (\mu ,Y)$ :

Space of functions with $\mu$-essentially bounded Y-norm

${\mathcal {M}}(X)$ :

Set of Radon measures

${\mathcal {S}}$ :

Sets of curves with the same initial point

${\mathcal {M}}_b(X)$ :

Set of bounded Radon measures

${\mathcal {M}}^+(X)$ :

Set of positive Radon measures

$\partial \Omega ^\varepsilon _1$ :

Subset of $\partial \Omega$ defined in Theorem 4.18

$\partial \Omega ^\varepsilon _2$ :

Subset of $\partial \Omega$ defined in Theorem 4.18

$\partial _{x_i} f$ :

Spatial partial derivative along the ith direction

$Q$ :

Sets of particular shape, with some index/apex

$S_1$ :

Subset of $\partial (\Omega ^\varepsilon \setminus \Omega )$ defined in Theorem 4.18

${{\,\mathrm{supp}\,}}f$ :

Support of a function f

$\varDelta$ :

Set of untangled trajectories

$\varGamma$ :

Space of characteristics

$\varGamma ^{\mathrm {cr}}$ :

Set of trajectories crossing a domain

$\varGamma ^{\mathrm {cr}}(\Omega )$ :

Set of $\Omega$-crossing trajectories

$\varGamma ^{\mathrm {in}}(\Omega )$ :

Set of $\Omega$-entering trajectories

$\varXi$ :

Set of uniqueness of $\eta$

$W$ :

Set of trajectories with good intersection properties

$W_1$ :

Set of disjoint trajectories

$W_2$ :

Set of trajectories whose intersection is still a trajectory

$x$ :

Space coordinate

$|\mu |$ :

Total variation measure of $\mu$

$\partial _t f_t$ :

Time partial derivative

$\pi$ :

Transference plan

$\rho ^{\mathrm {cr}}$ :

(t, x)-evaluation of $\eta ^{\mathrm {Cr}}$

$t$ :

Time coordinate

$\mathring{W}$ :

Trajectories with good intersection properties in the open graph

$\mathbf {e}$ :

Unit vector

$E^f_h$ :

Upper level set of the function f

$E_h$ :

Upper level set of a function

:

Untangling functional for $\eta ^{\mathrm {in}}$

:

Untangling functional for $\eta ^{\mathrm {out}}$

${\mathbb {S}}^d$ :

Unit sphere of dimension d

$\mathbf {b} = (b_i)_{i=1}^d$ :

Vector

$\omega _d$ :

Volume of the unit ball in $\mathbb {R}^d$

$A(x)$ :

x section of $A \subset X \times Y$