title: Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations.
abstract: The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.
authors.affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
authors.forename: Radu
authors.initials: R
authors.lastname: Dascaliuc
authors.affiliation: Department of Mathematics, New Mexico State University, Las Cruces, New Mexico 88003, USA.
authors.forename: Nicholas
authors.initials: N
authors.lastname: Michalowski
authors.affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
authors.forename: Enrique
authors.initials: E
authors.lastname: Thomann
authors.affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
authors.forename: Edward C
authors.initials: EC
authors.lastname: Waymire