# Research interests & activities

The webpage for the conference on Topology, Embeddings, and Attractors can be found here

The webpage for *The Navier-Stokes Equations in Venice* can be found here

You can find an English translation of the classical paper by Leray (1934) due to Robert Terrell here.

and a translation of the paper by Hopf due to Andreas Klockner here.

**Possible topics for essays, dissertations, or theses**

*Continuity of attractors under perturbation*. Global attractors of parametrised systems are continuous for a residual set of parameter values (Hoang et al., 2014). It would be interesting to have some "bad" examples where the set of discontinuities is large.

*Assouad dimension, equi-homogeneity, and the fine structure of sets. *The Assouad dimension in the largest of a sequence of measures of dimension; equi-homogeneity (Olson et al., 2014) is a related (but distinct) notion that encodes the property of a degree of uniformity at different scales. These ideas can be used to investigate in more detail the properties of sets arising in dynamical systems (e.g. self-similar sets) and in other contexts (e.g. the set of space-time singularities of a solution of the 3D Navier-Stokes equations).

*Lagrangian trajectories arising in 3D fluid flows.* For any suitable weak solution of the 3D Navier-Stokes equations, the solutions of the ODE $\dot X=u(X,t)$ are unique for almost every initial point $X(0)$ (Robinson & Sadowski, 2009). This gives an alternative way to view problems of uniqueness/regularity for the 3D Navier-Stokes equations. It would be instuctive to understand recent work by Jia & Sverak (2013) on possible non-uniqueness in this context.

*Magnetic relaxation and the equations of MHD*. A heuristic method proposed by Arnol'd and Moffatt for constructing stationary Euler flows involves studying the asymptotic behaviour of solutions of the equations of MHD in the case of zero magnetic diffusivity. There are partial results justifying this approach under the assumption of regularity of the magnetic field (Nunez, 2007) and local existence results for the MHD equations, but no satisfying general theory is currently available.

*A toy scalar model of the 3D Navier-Stokes equations.* The model of surface growth, $u_t-u_{xxxx}-\partial_x^2|u_x|^2$, shares many features in common with the 3D Navier-Stokes equations and provides an interesting testing ground for extending what we know for the NSE. Most NSE results have parallels for this equation, but so far not the $L^\infty(0,T;L^3)$ implies regularity result of Escauriaza et al. (2003).

*Semilinear parabolic equations and the heat equation. *The heat equation $u_t-\Delta u=0$ and its simplest nonlinear version $u_t-\Delta u=f(u)$ are classical problems but open questions still remain, e.g. given the distribution function of $u(0)$, what can be said about the solutions of these two problems? Another open question is whether one can characterise those $f$ for which the semilinear problem has a unique solution (those $f$ that yield local existence have been characterised only recently by Laister et al., 2014).

**Lecture notes**

- Some approaches to finite-dimensional behaviour in the Navier-Stokes Equations, lectures given at the University of Seville, Easter 1997.
- Topics in Mathematical Phsyics: a 4th year lecture course at Warwick based on my book, Infinite-Dimensional Dynamical Systems.
*Attractors and finite-dimensional behaviour in the Navier-Stokes equations*, notes for lectures given at the ICMS Instructional Conference on Mathematical Hydrodynamics, June 2003.-
*Finite-dimensional sets and finite-dimensional attractors**,*Lectures given in Palermo, November 2009 *The Navier-Stokes equations*, Lectures given at the AARMS Summer school at Memorial University, Newfoundland, July/August 2003*An introduction to the classical theory of the Navier-Stokes equations*, Lectures given at the IMECC-Campinas Fluid Summer School, January 2010