A Blowup Criterion for Ideal Viscoelastic Flow

We establish an analog of the Beale–Kato–Majda criterion for singularities of smooth solutions of the system of PDE arising in the Oldroyd model for ideal viscoelastic flow. It is well known that smooth solutions of the initial value problem associated with Euler’s equations { ∂tu+ (u · ∇)u = −∇p ∇ · u = 0 , (x, t) ∈ R 3 × (0, T ) (1.1) exist for some finite time T > 0. Here u = u(x, t) ∈ R and p = p(x, t) ∈ R represent the local velocity and pressure of a perfect fluid, respectively. One of the major challenges in PDE theory is to deduce whether or not finite time singularities do indeed occur. A celebrated result of Beale, Kato, and Majda asserts that if a solution u of Euler’s equations possesses a singularity at a finite time T , then necessarily