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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  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 & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384
References:
[1]

G. AlbertiS. 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.   Google Scholar

[2]

G. AlbertiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Bianchini and A. Stavitskiy, Forward untangling in metric measure spaces and applications., Google Scholar

[9]

P. Bonicatto, Untangling of Trajectories for non-Smooth Vector Fields and Bressan's Compactness Conjecture, PhD thesis, SISSA, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete, 67 (1984), 399-432.  doi: 10.1007/BF00532047.  Google Scholar

[13]

H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010, https://books.google.it/books?id=0Y5fAAAACAAJ. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. AlbertiS. 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.   Google Scholar

[2]

G. AlbertiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Bianchini and A. Stavitskiy, Forward untangling in metric measure spaces and applications., Google Scholar

[9]

P. Bonicatto, Untangling of Trajectories for non-Smooth Vector Fields and Bressan's Compactness Conjecture, PhD thesis, SISSA, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete, 67 (1984), 399-432.  doi: 10.1007/BF00532047.  Google Scholar

[13]

H. Royden and P. Fitzpatrick, Real Analysis, Prentice Hall, 2010, https://books.google.it/books?id=0Y5fAAAACAAJ. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

Figure 1.  Two curves $ \gamma,\gamma' $ with $ (\gamma, \gamma') \in NF $ and visual depiction of the exchanging map $ \tilde{\gamma}_{\gamma'} $
Figure 2.  Concatenated families of trajectories and an example of set $ F^t_x $
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