Grain boundary interface evolution, from KG, Physica D, 2006.

Research

I am a blue-collar applied mathematician; a generalist in a world of specialists. I use a wide variety of tools to understand phenomenon related to nonlinear partial differential equations, patterns and interfaces, including asymptotic analysis and numerical simulation. Some of my interests include:

  • Pattern formation: dynamics, interfaces, computation
  • Materials science: block copolymers, grain boundaries, phase field modelling
  • Fluid Dynamics: thin films, contact line dynamics, microfluidics
  • Optics and nonlinear waves: Self-focusing, singular behavior and scattering

Publications

Patterns and Interfaces

  • The Stability and Evolution of Curved Domains Arising from One-Dimensional Localized Patterns , with Alan Lindsay, SIAM Journal on Applied Dynamical Systems 12.2 (2013): 650-673.
  • Regularized model of post-touchdown configurations in electrostatic MEMS: interface dynamics , with A. Lindsay, J. Lega, IMA J. Appl. Math (2015).
  • Regularized model of post-touchdown configurations in electrostatic MEMS: equilibrium configurations , with A. Lindsay, J. Lega, Physica D, Vol. 280 (2014).
  • Grain boundary motion arising from the gradient flow of the Aviles-Giga functional , Physica D,Volume 215, pp. 80-98 (2006).
  • A boundary integral formulation of quasi-steady fluid wetting, J. Comp. Phys., 207, pp. 529-541 (2005).

    • Materials Science

  • Theoretical prediction of morphological selection in amphiphilic systems, Phys. Rev. E (2019).
  • Evolution and Competition of Block Copolymer Nanoparticles, SIAM J. Appl. Math (2019).
  • Multilayered equilibria and their stability in copolymer-solvent mixtures, w/ S. Orizaga, Physica D (2018).
  • Multilayered equilibria in a density functional model of copolymer-solvent mixtures, SIAM J. Math. Analysis (2017).
  • Instability and reorientation of block copolymer microstructure by imposed electric fields , w/ S. Orizaga, Phys. Rev. E, vol. 93 (2016).
  • Hexagonal phase ordering in strongly segregated copolymer films , Phys. Rev. E, vol. 92 (2015).
  • Regularized model of post-touchdown configurations in electrostatic MEMS: interface dynamics , with A. Lindsay, J. Lega, IMA J. Appl. Math (2015).
  • Regularized model of post-touchdown configurations in electrostatic MEMS: equilibrium configurations , with A. Lindsay, J. Lega, Physica D, Vol. 280 (2014).
  • Characterising the disordered state of block copolymers: Bifurcations of localised states and self-replication dynamics , Euro. J. Appl. Math., Vol. 23 (2012).
  • Spatially locallized structures in diblock copolymer mixtures , SIAM J. Appl. Math, Vol. 70 (2010).
  • Coarsening and self-organization in dilute diblock copolymer melts and mixtures , with R. Choksi, Physica D, Vol. 238 (2009).
  • Solute Trapping and the Non-equilibrium Phase Diagram for Solidification of Binary Alloys , Physica D , 151 (1-4) (2001) pp. 253-270.
  • Rapid Growth and Critical Behavior in Phase Field Models of Solidification, European J. Appl. Math, Vol. 12, pp. 39-56, (2001).
  • Dual Fronts in a Phase Field Model, with R. Almgren, Physica D , 146, pp. 328-340 (2000).
  • Traveling Waves in Rapid Solidification, Elec. J. Diff. Eq. , 2000, #16, pp. 1-28 (2000).

    • Fluids

  • Viscosity solutions for a model of contact line motion , with I. Kim, Interfaces and Free boundaries (2009).
  • Ostwald Ripening in Thin Film Equations , SIAM J. Appl. Math., Vol. 69 (2008).
  • Ostwald Ripening of Droplets: The Role of Migration , w/ F. Otto, T. Rump. D. Slepcev, Euro. J. Appl. Math. (2008).
  • The dynamics of pendant droplets on a one dimensional surface, Phys. Fluids, Vol. 19, (2007).
  • A Diffuse Interface Model for Electrowetting Droplets in a Hele-Shaw Cell , w/ H.-W. Lu, A. Bertozzi, C.-J. Kim, JFM, Vol. 590, 411-435 (2007).
  • Homogenization of contact line dynamics, Int. Free Bound., Volume 8, (2006).
  • A boundary integral formulation of quasi-steady fluid wetting, J. Comp. Phys., 207, pp. 529-541 (2005).
  • Collision vs. collapse of droplets in coarsening of dewetting thin films, with T. P. Witeslski, Physica D, 209, 80-104 (2005).
  • Variational models for moving contact lines and the quasi-static approximation , Euro. J. Appl. Math, 16, pp. 1-28 (2005).
  • Spreading of droplets under the influence of intermolecular forces, Phys. of Fluids, Vol.15, pp. 1837-1842 (2003).
  • Coarsening Dynamics of Dewetting Films, with T. P. Witelski, Phys. Rev. E, Vol. 67, 016302 (2003).
  • A Diffuse Interface Approach to Hele-Shaw Flow , Nonlinearity , Vol. 16, 1-18 (2003).

    • Numerical Methods

  • Improving the accuracy of convexity splitting methods for gradient flow equations , w/ S. Orizaga, J. Comp. Physics (2016).
  • Nonlinear Preconditioning for Diffuse Interfaces , Journal of Computational Physics, Vol. 174, No. 2, Dec 2001, pp. 695-711.

    • Optical Science

  • Nonlinearity saturation as a singular perturbation of the nonlinear Schrodinger equation , w/ J. Allen-Flowers, SIAM J. Appl. Math., vol. 76 (2016).
  • Transient behavior of collapsing ring solutions in the critical nonlinear Schrödinger equation , with J. Allen-Flowers, Physica D, Vol. 284 (2014).
  • Canonical and singular propagation of ultrashort pulses in a nonlinear medium , w/ M. Kolesik, J. Moloney, A. Newell, International Journal of Optics (2011).

    • Etc...

  • Long time evolution of aeolian sand dune fields: Influence of dune formation and collision, with S. Diniega and S. Byrne, Geomorphology (2009).