Équation aux dérivées partielles
Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d'un hyperplan
Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 249-252.
Nous présentons ici un exemple de champ de vecteurs plan dépendant du temps, borné à divergence nulle, et une solution non nulle bornée du problème de Cauchy homogène pour l'équation de transport associée.
We present here an example of a plane time-dependent bounded divergence-free vector field and a bounded non-zero solution of the homogeneous Cauchy problem for the associated transport equation.
Reçu le :
Accepté le :
Publié le :
DOI :
10.1016/S1631-073X(03)00330-3
Accepté le :
Publié le :
Affiliations des auteurs :
Nicolas Depauw 1
@article{CRMATH_2003__337_4_249_0,
author = {Nicolas Depauw},
title = {Non unicit\'e des solutions born\'ees pour un champ de vecteurs {\protect\emph{BV}} en dehors d'un hyperplan},
journal = {Comptes Rendus. Math\'ematique},
pages = {249--252},
publisher = {Elsevier},
volume = {337},
number = {4},
year = {2003},
doi = {10.1016/S1631-073X(03)00330-3},
language = {fr},
}
Nicolas Depauw. Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d'un hyperplan. Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 249-252. doi : 10.1016/S1631-073X(03)00330-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00330-3/
[1] On vector fields as generators of flows: a counterexample to Nelson's conjecture, Ann. of Math., Volume 107 (1978) no. 2, pp. 287-296
[2] Uniqueness of continuous solutions for BV vector fields, Duke Math. J., Volume 111 (2002) no. 2, pp. 357-384
[3] F. Colombini, N. Lerner, Uniqueness of L∞ solutions for a class of conormal BV vector fields, Preprint de Rennes, 2003
[4] F. Colombini, J. Rauch, Unicity and nonunicity for nonsmooth divergence free transport, Séminaire X-EDP 2002-2003, École Polytechnique, in preparation
[5] Ordinary differential equations transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511-547
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- Non-uniqueness inadmissibility of the vanishing viscosity limit of the passive scalar transport equation, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 198 (2025), p. 51 (Id/No 103685) | DOI:10.1016/j.matpur.2025.103685 | Zbl:8014006
- On the lack of selection for the transport equation over a dense set of vector fields, Journal of Functional Analysis, Volume 289 (2025) no. 6, p. 110942 | DOI:10.1016/j.jfa.2025.110942
- On measure-preserving selection of solutions of ODEs, Proceedings of the American Mathematical Society, Volume 153 (2025) no. 5, pp. 2037-2051 | DOI:10.1090/proc/17097 | Zbl:8015185
- Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions, Analysis PDE, Volume 17 (2024) no. 6, pp. 2161-2177 | DOI:10.2140/apde.2024.17.2161 | Zbl:1547.35592
- Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, Archive for Rational Mechanics and Analysis, Volume 248 (2024) no. 6, p. 26 (Id/No 114) | DOI:10.1007/s00205-024-02063-y | Zbl:7949017
- A remark on selection of solutions for the transport equation, Journal of Evolution Equations, Volume 24 (2024) no. 3, p. 26 (Id/No 69) | DOI:10.1007/s00028-024-00996-1 | Zbl:1548.35167
- Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence, Annals of PDE, Volume 9 (2023) no. 2, p. 48 (Id/No 21) | DOI:10.1007/s40818-023-00162-9 | Zbl:1536.35261
- Bounds on the rate of enhanced dissipation, Communications in Mathematical Physics, Volume 399 (2023) no. 3, pp. 2071-2081 | DOI:10.1007/s00220-022-04588-3 | Zbl:1514.35324
- Properties of mixing BV vector fields, Communications in Mathematical Physics, Volume 402 (2023) no. 2, pp. 1953-2009 | DOI:10.1007/s00220-023-04780-z | Zbl:1532.35365
- A note on the Lagrangian flow associated to a partially regular vector field, Differential Equations and Dynamical Systems, Volume 31 (2023) no. 4, pp. 767-786 | DOI:10.1007/s12591-020-00530-y | Zbl:1527.35335
- Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift, Inventiones Mathematicae, Volume 234 (2023) no. 2, pp. 573-636 | DOI:10.1007/s00222-023-01207-3 | Zbl:1527.35425
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- Stochastic transport equation with bounded and Dini continuous drift, Journal of Differential Equations, Volume 323 (2022), pp. 359-403 | DOI:10.1016/j.jde.2022.03.038 | Zbl:1486.60085
- Mathematical advances in geophysical fluid dynamics. Abstracts from the workshop held November 13–19, 2022, Oberwolfach Rep. 19, No. 4, 2961-3003, 2022 | DOI:10.4171/owr/2022/51 | Zbl:1521.00016
- Nonuniqueness of weak solutions for the transport equation at critical space regularity, Annals of PDE, Volume 7 (2021) no. 1, p. 45 (Id/No 2) | DOI:10.1007/s40818-020-00091-x | Zbl:1469.35001
- Retracted: A uniqueness lemma with applications to regularization and incompressible fluid mechanics, Science China. Mathematics, Volume 64 (2021) no. 4, pp. 711-724 | DOI:10.1007/s11425-018-9477-3 | Zbl:1464.35185
- Stochastic regularization for transport equations, Stochastic and Partial Differential Equations. Analysis and Computations, Volume 9 (2021) no. 1, pp. 105-141 | DOI:10.1007/s40072-020-00171-w | Zbl:1477.60103
- Convex integration solutions to the transport equation with full dimensional concentration, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 37 (2020) no. 5, pp. 1075-1108 | DOI:10.1016/j.anihpc.2020.03.002 | Zbl:1458.35363
- Smooth approximation is not a selection principle for the transport equation with rough vector field, Calculus of Variations and Partial Differential Equations, Volume 59 (2020) no. 1, p. 21 (Id/No 13) | DOI:10.1007/s00526-019-1659-0 | Zbl:1428.35082
- Onsager-type conjecture and renormalized solutions for the relativistic Vlasov-Maxwell system, Quarterly of Applied Mathematics, Volume 78 (2020) no. 2, pp. 193-217 | DOI:10.1090/qam/1549 | Zbl:1431.35189
- Convergence of numerical approximations to non-linear continuity equations with rough force fields, Archive for Rational Mechanics and Analysis, Volume 234 (2019) no. 2, pp. 509-547 | DOI:10.1007/s00205-019-01396-3 | Zbl:1437.35475
- Non-renormalized solutions to the continuity equation, Calculus of Variations and Partial Differential Equations, Volume 58 (2019) no. 6, p. 30 (Id/No 208) | DOI:10.1007/s00526-019-1651-8 | Zbl:1428.35005
- Exponential self-similar mixing by incompressible flows, Journal of the American Mathematical Society, Volume 32 (2019) no. 2, pp. 445-490 | DOI:10.1090/jams/913 | Zbl:1456.35158
- Renormalization of active scalar equations, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 179 (2019), pp. 254-269 | DOI:10.1016/j.na.2018.08.018 | Zbl:1419.35135
- Polynomial mixing under a certain stationary Euler flow, Physica D, Volume 394 (2019), pp. 44-55 | DOI:10.1016/j.physd.2019.01.009 | Zbl:1451.76035
- Very weak solutions to the two-dimensional Monge-Ampére equation, Science China. Mathematics, Volume 62 (2019) no. 6, pp. 1041-1056 | DOI:10.1007/s11425-018-9516-7 | Zbl:1421.35185
- Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Annals of Mathematics. Second Series, Volume 188 (2018) no. 2, pp. 577-684 | DOI:10.4007/annals.2018.188.2.4 | Zbl:1405.35133
- Non-uniqueness for the transport equation with Sobolev vector fields, Annals of PDE, Volume 4 (2018) no. 2, p. 38 (Id/No 18) | DOI:10.1007/s40818-018-0056-x | Zbl:1411.35066
- A uniqueness lemma with applications to regularization and incompressible fluid mechanics, Communications in Contemporary Mathematics, Volume 20 (2018) no. 3, p. 18 (Id/No 1750048) | DOI:10.1142/s0219199717500481 | Zbl:1516.35007
- On Lipschitz vector fields and the Cauchy problem in homogeneous groups, Communications in Contemporary Mathematics, Volume 20 (2018) no. 5, p. 21 (Id/No 1750057) | DOI:10.1142/s0219199717500572 | Zbl:1394.53038
- On an anisotropic Serrin criterion for weak solutions of the Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 117 (2018), pp. 123-145 | DOI:10.1016/j.matpur.2018.06.004 | Zbl:1406.35230
- Lecture notes on the DiPerna-Lions theory in abstract measure spaces, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI, Volume 26 (2017) no. 4, pp. 729-766 | DOI:10.5802/afst.1551 | Zbl:1402.35003
- Mixing and un-mixing by incompressible flows, Journal of the European Mathematical Society (JEMS), Volume 19 (2017) no. 7, pp. 1911-1948 | DOI:10.4171/jems/709 | Zbl:1369.35071
- Cellular mixing with bounded palenstrophy, M
- Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications, Revista Matemática Complutense, Volume 30 (2017) no. 3, pp. 427-450 | DOI:10.1007/s13163-017-0244-3 | Zbl:1380.37039
- Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization, Archive for Rational Mechanics and Analysis, Volume 222 (2016) no. 2, pp. 451-505 | DOI:10.1007/s00205-016-1006-y | Zbl:1359.35145
- Critical non-Sobolev regularity for continuity equations with rough velocity fields, Journal of Differential Equations, Volume 260 (2016) no. 5, pp. 4739-4757 | DOI:10.1016/j.jde.2015.11.028 | Zbl:1347.35075
- Diperna-Lions flow for relativistic particles in an electromagnetic field, Archive for Rational Mechanics and Analysis, Volume 217 (2015) no. 3, pp. 1029-1067 | DOI:10.1007/s00205-015-0850-5 | Zbl:1317.35249
- Existence and uniqueness of maximal regular flows for non-smooth vector fields, Archive for Rational Mechanics and Analysis, Volume 218 (2015) no. 2, pp. 1043-1081 | DOI:10.1007/s00205-015-0875-9 | Zbl:1348.34039
- Exponential self-similar mixing and loss of regularity for continuity equations, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 352 (2014) no. 11, pp. 901-906 | DOI:10.1016/j.crma.2014.08.021 | Zbl:1302.35241
- Initial-boundary value problems for continuity equations with BV coefficients, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 102 (2014) no. 1, pp. 79-98 | DOI:10.1016/j.matpur.2013.11.002 | Zbl:1292.35092
- Lagrangian flows for vector fields with gradient given by a singular integral, Journal of Hyperbolic Differential Equations, Volume 10 (2013) no. 2, pp. 235-282 | DOI:10.1142/s0219891613500100 | Zbl:1275.35076
- Wiener chaos and uniqueness for stochastic transport equation, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 349 (2011) no. 11-12, pp. 669-672 | DOI:10.1016/j.crma.2011.05.006 | Zbl:1220.60037
- The flow associated to weakly differentiable vector fields: recent results and open problems, Nonlinear conservation laws and applications. Proceedings of the summer program, IMA, Minneapolis, MN, USA, July 13–31, 2009, New York, NY: Springer, 2011, pp. 181-193 | DOI:10.1007/978-1-4419-9554-4_7 | Zbl:1248.35048
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- Well-posedness of the transport equation by stochastic perturbation, Inventiones Mathematicae, Volume 180 (2010) no. 1, pp. 1-53 | DOI:10.1007/s00222-009-0224-4 | Zbl:1200.35226
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- Transport Equation and Cauchy Problem for Non-Smooth Vector Fields, Calculus of Variations and Nonlinear Partial Differential Equations, Volume 1927 (2008), p. 1 | DOI:10.1007/978-3-540-75914-0_1
- Transport equation and Cauchy problem forBVvector fields and applications, Journées équations aux dérivées partielles (2008), p. 1 | DOI:10.5802/jedp.1
- Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields, Transport Equations and Multi-D Hyperbolic Conservation Laws, Volume 5 (2008), p. 3 | DOI:10.1007/978-3-540-76781-7_1
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