Abstract
We continue to consider a system of ordinary differential equations (1.1), but we are more interested in the map \(X \mapsto x(t,X)\), which is often called a flow map generated by a vector field b. If the initial value problem for (1.1) admits a unique local-in-time solution in a time interval \(I=(0,a)\) with some \(a>0\) independent of X, the flow map is well defined. In Sect. 1.1, we gave a few sufficient conditions so that the flow map is uniquely determined assuming the existence of solutions to (1.1) with a given initial datum.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is one of the fundamental properties of the Lebesque measure. It states that \(\lim _{|y|\to 0}\|\tau _y f-f\|{ }_{L^\alpha ({\mathbf {T}}^N)}=0\) for \(\alpha \in [1,\infty )\), where \((\tau _y f)(x)=f(x+y)\).
- 2.
If \(f_\varepsilon \to f\) in \(L^p({\mathbf {T}}^N)\), there is a subsequence \(f_{\varepsilon _k}\) that converges to f a.e.
References
L. Ambrosio, Transport equation and Cauchy problem for BV  vector fields. Invent. Math. 158, 227–260 (2004)
L. Ambrosio, M. Lecumberry, S. Maniglia, Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow. Rend. Sem. Mat. Univ. Padova 114, 29–50 (2005)
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext (Springer, New York, 2011)
T. Buckmaster, V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. Math. (2) 189, 101–144 (2019)
G. Crippa, C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)
R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
T.D. Drivas, T.M. Elgindi, G. Iyer, I.-J. Jeong, Anomalous dissipation in passive scalar transport. Arch. Ration. Mech. Anal. 243, 1151–1180 (2022)
L.C. Evans, Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. (American Mathematical Society, Providence, RI, 2010)
E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, 2nd edn. Advances in Mathematical Fluid Mechanics (Birkhäuser/Springer, Cham, 2017)
M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions. Progress in Nonlinear Differential Equations and Their Applications, vol. 79 (Birkhäuser Boston, Boston, MA, 2010)
P. Hartman, Ordinary Differential Equations. Reprint of the second edition. (Birkhäuser, Boston, MA, 1982)
L. Huysmans, E.S. Titi, Non-uniqueness and inadmissibility of the vanishing viscosity limit of the passive scalar transport equation (2023). arXiv: 2307.00809
S. Modena, L. Székelyhidi, Jr., Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4, Paper No. 18, 38 pp. (2018)
T. Tsuruhashi, T. Yoneda, Microscopic expression of anomalous dissipation in passive scalar transport (2022). arXiv: 2212.06395
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Giga, MH., Giga, Y. (2023). Ordinary Differential Equations and Transport Equations. In: A Basic Guide to Uniqueness Problems for Evolutionary Differential Equations. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-34796-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-34796-2_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-34795-5
Online ISBN: 978-3-031-34796-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)