Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Fefferman, Charles; Pooley, Benjamin C.; Rodrigo, José Luis

doi:10.1007/s00205-021-01708-6

Cited by 3 publications

(2 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer to the articles by Caprino et al [6] for the two-dimensional case and by Crippa, Ligabue, Saffirio [12] for the threedimensional case. Finally, in connection with the study of singular velocities generated by singular measure-valued vorticities in fluid dynamics, we mention the recent work by Fefferman, Pooley and Rodrigo [20] on the construction of velocity fields for active scalar systems with measure-valued solutions whose support does not satisfy conservation of the Hausdorff dimension.…”

Section: The Resulting Velocity Fieldmentioning

confidence: 99%

On solutions of the transport equation in the presence of singularities

Miot

Sharples

2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We consider the transport equation on [ 0 , T ] × R n [0,T]\times \mathbb {R}^n in the situation where the vector field is B V BV off a set S ⊂ [ 0 , T ] × R n S\subset [0,T]\times \mathbb {R}^n . We demonstrate that solutions exist and are unique provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result improves upon recent results of Ambrosio who requires the vector field to be of bounded variation everywhere. In addition, we demonstrate that under these conditions almost every trajectory of the associated regular Lagrangian flow does not intersect the set S S of singularities. Finally, we consider the particular case of an initial set of singularities that evolve in time so the singularities consists of curves in the phase space, which is typical in applications such as vortex dynamics. We demonstrate that solutions of the transport equation exist and are unique provided that the box-counting dimension of the singularities is bounded in terms of the Hölder exponent of the curves.

show abstract

Section: The Resulting Velocity Fieldmentioning

confidence: 99%

On solutions of the transport equation in the presence of singularities

Miot

Sharples

2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…They constructed a divergence-free vector field u ∈ W 1,p (T d , R d ) with p < d − 1 and showed that the nonuniqueness of trajectories could occur on a set of positive measure. We also note that explicit examples of vector fields for which nonuniqueness happens on a set of full Hausdorff dimension but of measure zero are known [FPR21].…”

Section: Definition 13 (Regular Lagrangian Flow) a Map X Umentioning

confidence: 96%

Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields

Kumar¹

2023

Preprint

View full text Add to dashboard Cite

In the theory of DiPerna-Lions for Sobolev vector fields W 1,p , an important question was whether the uniqueness of regular Lagrangian flow could be implied by proving almost everywhere uniqueness of trajectories. In this work, we construct an explicit example of divergence-free vector fields in W 1,p with p < d such that the set of initial conditions for which trajectories are not unique is a set of full measure. To prove this, we build a vector field u and a corresponding flow map X u such that after finite time T > 0, the flow map takes the whole domain T d to a Cantor set CΦ, i.e., X u (T, T d ) = CΦ and the Hausdorff dimension of this Cantor set is strictly less than d. The flow map X u constructed as such is not a regular Lagrangian flow. The nonuniqueness of trajectories on a full measure set is then deduced from the existence of the regular Lagrangian flow in the DiPerna-Lions theory.

show abstract

Nonuniqueness of Trajectories on a Set of Full Measure for Sobolev Vector Fields

Kumar

2024

Arch Rational Mech Anal

View full text Add to dashboard Cite

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

Abstract: How to cite:Please refer to published version for the most recent bibliographic citation information.

Cited by 3 publications

References 16 publications

On solutions of the transport equation in the presence of singularities

On solutions of the transport equation in the presence of singularities

Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields

Nonuniqueness of Trajectories on a Set of Full Measure for Sobolev Vector Fields

Contact Info

Product

Resources

About