Abstract
We show that there are continuous, W^{1,p} (p<d-1), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.
Highlights
In this paper we consider the continuity equation∂t ρ + div = 0, (1)div u = 0, in a d-dimensional periodic domain, d ≥ 3, for a time-dependent incompressible vector field u : [0, 1] × Td → Rd and an unknown density ρ : [0, 1] × Td → R
We will always assume, without loss of generality, that the time interval is [0, 1]. We prove in these notes the following theorem
We proved in [16] the analog of Theorem 1.1, for fields and densities in the class ρ ∈ C [0, 1]; Lr (Td ), u ∈ C [0, 1]; Lr (Td ) ∩ C [0, 1]; W 1,p(Td ), with r ∈ (1, ∞), p ∈ [1, ∞), (8)
Summary
We would like to briefly comment about the continuity of the vector field u produced by Theorem 1.1 It was observed by Caravenna and Crippa [7] that the boundedness or the continuity of the vector field (in addition to some Sobolev regularity) could play a key role in the uniqueness problem in the class of integrable densities ρ ∈ L1((0, 1) × Td ). 1. The idea that the boundedness or the continuity of u can play a crucial role in the uniqueness problem is confirmed by the fact that the majority of the result concerning existence and uniqueness of the regular Lagrangian flow associated to a Sobolev or BV vector field u assume that u ∈ L∞ (see, for instance, the recent survey [2]).
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