arXiv subjects

Nicholas Sharples

Publications and source records attributed to Nicholas Sharples.

3 recordsLinked to original sources

arXiv2020

On solutions of the transport equation in the presence of singularities

We consider the transport equation on $[0,T]\times \mathbb{R}^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \mathbb{R}^n$. 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 $S$ 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 H\"older exponent of the curves.

math.AP↗

arXiv2014

Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at each fixed length-scale any two neighbourhoods of the set have covers of approximately equal cardinality. Self-similar sets are notable in that they are Ahlfors-David regular, which implies that their Assouad and box-counting dimensions coincide. More generally, attractors of non-autonomous iterated functions systems (where maps are allowed to vary between iterations) can have distinct Assouad and box-counting dimensions. Consequently the familiar notion of Ahlfors-David regularity is too strong to be useful in the analysis of this important class of sets, which include generalised Cantor sets and possess different dimensional behaviour at different length-scales. We further develop the theory of equi-homogeneity showing that it is a weaker property than Ahlfors-David regularity and distinct from any previously defined notion of dimensional equivalence. However, we show that if the upper and lower box-counting dimensions of an equi-homogeneous set are equal and `attained' in a sense we make precise then the lower Assouad, Hausdorff, packing, lower box-counting, upper box-counting and Assouad dimensions coincide. Our main results provide conditions under which the attractor of a non-autonomous iterated function system is equi-homogeneous and we use this to compute the Assouad dimension of a certain class of these highly non-trivial sets.

math.CA↗

arXiv2014

Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of `equi-homogeneity' of a set, which requires a uniformity in the size of local covers at all lengths and at all points. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set `attains' these dimensions (analogous to `s-sets' when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any $\alpha\in(0,1)$ and any $\beta,\gamma\in(0,1)$ such that $\beta + \gamma\geq 1$ we can construct two generalised Cantor sets $C$ and $D$ such that $\text{dim}_{B}C=\alpha\beta$, $\text{dim}_{B}D=\alpha\gamma$, and $\text{dim}_{A}C=\text{dim}_{A}D=\text{dim}_{A}(C\times D)=\text{dim}_{B}(C\times D)=\alpha$.

math.CA↗