How to compute the pressure for viscous incompressible flow with no-slip boundaryFor the Navier-Stokes equations for incompressible flow with no-slip boundary conditions, we describe a formula for the pressure that involves the commutator of the Laplacian and Leray-Helmholtz projection operators. An estimate for this commutator shows that it is strictly dominated by the viscous term at leading order. This leads to a number of developments, including a simple well-posedness theorem for an extended Navier-Stokes dynamics unconstrained by the divergence-free condition, and improvements of methods of numerical computation and numerical analysis for such flows.
Self-similarity and the scaling attractor for models of coagulation and clusteringWe study limiting behavior of rescaled size distributions in several models of clustering or coagulation dynamics, `solvable' in the sense that the Laplace transform converts them into nonlinear PDE. The scaling analysis that emerges has many connections with the classical limit theorems of probability theory, and a surprising application to the study of shock clustering in the inviscid Burgers equation with random-walk initial data. I'll focus on recent progress regarding a `min-driven' clustering model related to domain coarsening dynamics in the Allen-Cahn equation.
Asymptotic stability of solitary waves in FPU and Toda latticesIn infinite Hamiltonian particle chains, energy can be `lost' through propagation to infinity. We use this effect to establish asymptotic stability results for solitary waves by a robust method that uses integrability in a very limited way, to establish a linear stability property. This is done (a) in the small-amplitude KdV limit for a general class of pair potentials, and (b) for large waves of the Toda lattice, using a linearized Baeckland transformation defined on a codimension-one set.
Spectral stability of solitary water wavesThis is a preliminary account of a proof, joint with Shu-Ming Sun, of spectral stability for solitary waves of the 2D Euler equations for water waves with zero surface tension over a flat bottom, in the small-amplitude regime when the wave shape is well-approximated by the KdV equation.