Research

My research is centred on rigorous analysis of nonlinear partial differential equations and variational problems motivated by continuum mechanics, materials science and mathematical physics. A recurring aim is to understand how qualitative physical phenomena—defects, pattern formation, fluid–structure interactions and effective macroscopic behaviour—emerge from nonlinear PDE models.

Fluids and fluid–structure interaction

I study analytical questions for Newtonian and non-Newtonian fluid equations and systems in which fluids interact with rigid particles or structures. Recent work has addressed the asymptotic effect of small or numerous rigid bodies, compressible and incompressible flows, and non-Newtonian fluids.

These problems involve questions of existence, stability, asymptotic limits and the rigorous derivation of effective equations.

Representative papers:

• Dynamics of a Large System of Heavy Particles in a Newtonian Fluid — 2026

• On the effect of a large cloud of rigid particles on the motion of an incompressible non-Newtonian fluid — 2025

• On the motion of a large number of small rigid bodies in a viscous incompressible fluid — 2023

• Global existence of weak solutions for a model of nematic liquid crystal-colloidal interactions — 2024

Homogenisation and multiscale analysis

Another strand of my work concerns the emergence of effective macroscopic models from microscopic structure. Particular interests include colloidal suspensions, effective bulk and surface energies, homogenised descriptions of liquid-crystal materials, and singular limits in fluid systems.

An important motivation is mathematical material design: understanding how microscopic geometry and interactions may be used to generate prescribed effective behaviour.

Representative papers:

• Colloidal homogenization for the hydrodynamics of nematic liquid crystals

• Effective surface energies in nematic liquid crystals as homogenized rugosity effects

• Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation

Variational problems and nonlinear PDE

I am interested more broadly in nonlinear elliptic and evolutionary PDE, including phase-transition systems, harmonic-map-type problems, nonlinear fluid equations and variational structures associated with physical models.

Typical questions concern existence, regularity, uniqueness or multiplicity, symmetry, stability and singular or asymptotic limits.

Liquid crystals and complex materials

A major part of my research concerns mathematical theories of nematic liquid crystals, particularly Landau–de Gennes and Q-tensor models. Questions of interest include defects and their stability, symmetry and multiplicity of equilibria, singular limits, coupled liquid-crystal flows, and the mathematical design of effective material properties.

This work combines techniques from nonlinear PDE, calculus of variations, topology, asymptotic analysis and mathematical physics.

Representative papers:

• Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals

• Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals

• Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation

Newer directions

Recent work has also explored two complementary directions:

• Variational approaches to incompressible fluids, including dual variational formulations related to Euler and Navier–Stokes equations.

• Probability, numerical approximation and neural networks for PDEs, including constructive probabilistic and deep-neural-network approximations of boundary-value problems.

An experiment of Oleg D. Lavrentovich (Kent State University) with defect patterns in a nematic liquid crystal sample.