I am working on differential and integral equations modelling compressible and incompressible fluids. A particular theme is rigorous construction and numerical discovery of special exact solutions, which I use study transitions and pathological behaviour in physics and numerics.
I have worked on shock waves that interact with each other or with solid boundaries, in regular or Mach reflections. More recently I have begun to study vortex sheets -- curves of discontinuity in tangential velocity, with continuous pressure and normal velocity -- which have interesting and theoretically important spiral rollup and unstable behaviours.
Equations in this area are typically nonlinear and hyperbolic/parabolic, with steady or self-similar solutions leading to elliptic/mixed type problems. Free boundaries and corners with singularities are common. Since solutions of non-degenerate PDE tend to be smooth and "featureless", in absence of singularities induced by the data, many of the physically and theoretically interesting phenomena involve degeneracy, such as PDE that lose ellipticity as we approach a domain boundary, or oscillatory integral equations that do not have convergent series solutions at infinity.
