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June  2021, 41(6): 2739-2776. doi: 10.3934/dcds.2020384

## Forward untangling and applications to the uniqueness problem for the continuity equation

 1 S.I.S.S.A., via Bonomea 265, 34136 Trieste, Italy 2 Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051, Basel, Switzerland

* Corresponding author

Received  May 2020 Published  June 2021 Early access  November 2020

Fund Project: The work of the second author was supported by ERC Starting Grant 676675 (FLIRT)

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure $\rho(1, {\mathit{\boldsymbol{b}}})$, where $\rho \in \mathcal{M}^+( \mathbb{R}^{d+1})$ and ${\mathit{\boldsymbol{b}}} \colon \mathbb{R}^{d+1} \to \mathbb{R}^d$ is a $\rho$-integrable vector field with ${\rm{div}}_{t,x}(\rho(1, {\mathit{\boldsymbol{b}}})) = \mu \in \mathcal M( \mathbb{R} \times \mathbb{R}^d)$: forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE ${\rm{div}}_{t,x}(\rho(1, {\mathit{\boldsymbol{b}}})) = \mu$ on a partition of $\mathbb{R}^+ \times \mathbb{R}^d$ obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation.

Citation: Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384
##### References:
 [1] G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 863-902. [2] G. Alberti, S. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc. (JEMS), 16 (2014), 201-234.  doi: 10.4171/JEMS/431. [3] L. Ambrosio, Transport equation and Cauchy problem for ${\rm{BV}}$ vector fields, Inventiones Mathematicae, 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2. [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. [5] S. Bianchini and P. Bonicatto, Failure of the chain rule in the non steady two-dimensional setting, Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg, 33-60, Springer Optim. Appl., 135, Springer, Cham, 2018. [6] S. Bianchini and P. Bonicatto, A uniqueness result for the decomposition of vector fields in $\Bbb R^d$, Invent. Math., 220 (2020), 255-393.  doi: 10.1007/s00222-019-00928-8. [7] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Differential Equations, 36 (2011), 777-796.  doi: 10.1080/03605302.2010.534224. [8] S. Bianchini and A. Stavitskiy, Forward untangling in metric measure spaces and applications., [9] P. Bonicatto, Untangling of Trajectories for non-Smooth Vector Fields and Bressan's Compactness Conjecture, PhD thesis, SISSA, 2017. [10] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282.  doi: 10.1142/S0219891613500100. [11] G. Crippa, C. Nobili, C. Seis and S. Spirito, Eulerian and Lagrangian solutions to the continuity and Euler equations with $L^1$ vorticity, SIAM J. Math. Anal., 49 (2017), 3973-3998 doi: 10.1137/17M1130988. [12] H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete, 67 (1984), 399-432.  doi: 10.1007/BF00532047. [13] H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010, https://books.google.it/books?id=0Y5fAAAACAAJ. [14] S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents, St. Petersburg Math. J., 5 (1994), 841-867. [15] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, Springer, 1998, https://books.google.it/books?id=FhYGYJtMwcUC. doi: 10.1007/978-3-642-85473-6.

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##### References:
 [1] G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), 863-902. [2] G. Alberti, S. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc. (JEMS), 16 (2014), 201-234.  doi: 10.4171/JEMS/431. [3] L. Ambrosio, Transport equation and Cauchy problem for ${\rm{BV}}$ vector fields, Inventiones Mathematicae, 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2. [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. [5] S. Bianchini and P. Bonicatto, Failure of the chain rule in the non steady two-dimensional setting, Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg, 33-60, Springer Optim. Appl., 135, Springer, Cham, 2018. [6] S. Bianchini and P. Bonicatto, A uniqueness result for the decomposition of vector fields in $\Bbb R^d$, Invent. Math., 220 (2020), 255-393.  doi: 10.1007/s00222-019-00928-8. [7] S. Bianchini and M. Gloyer, An estimate on the flow generated by monotone operators, Comm. Partial Differential Equations, 36 (2011), 777-796.  doi: 10.1080/03605302.2010.534224. [8] S. Bianchini and A. Stavitskiy, Forward untangling in metric measure spaces and applications., [9] P. Bonicatto, Untangling of Trajectories for non-Smooth Vector Fields and Bressan's Compactness Conjecture, PhD thesis, SISSA, 2017. [10] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282.  doi: 10.1142/S0219891613500100. [11] G. Crippa, C. Nobili, C. Seis and S. Spirito, Eulerian and Lagrangian solutions to the continuity and Euler equations with $L^1$ vorticity, SIAM J. Math. Anal., 49 (2017), 3973-3998 doi: 10.1137/17M1130988. [12] H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete, 67 (1984), 399-432.  doi: 10.1007/BF00532047. [13] H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010, https://books.google.it/books?id=0Y5fAAAACAAJ. [14] S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents, St. Petersburg Math. J., 5 (1994), 841-867. [15] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, Springer, 1998, https://books.google.it/books?id=FhYGYJtMwcUC. doi: 10.1007/978-3-642-85473-6.
Two curves $\gamma,\gamma'$ with $(\gamma, \gamma') \in NF$ and visual depiction of the exchanging map $\tilde{\gamma}_{\gamma'}$
Concatenated families of trajectories and an example of set $F^t_x$
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