Linearly strained flows with and without boundaries-the regularizing effect of the pressure term

Whether or not the Euler equations for incompressible flow admit solutions with finite-time singularities, it is clear that the nonlocal action of pressure (non-isotropic Hessian terms) plays a critical role. To address this question we contrast the boundary-free, linearly strained flow u=-(y+z,z+x,x+y) that has nonunique solutions including some which blow up in finite time, and some bounded flows with similar behavior near the origin, e.g., u=-(siny+sinz,sinz+sinx,sinx+siny). Using both pseudospectral and power series computed in time, it is found that there is no evidence for blowup of the latter sine flow. The nonuniqueness in the boundary-free flow is interpreted as an arbitrariness of the homogeneous solution of the pressure Poisson equation. The (1-t)-1 blowup follows from the inclusion of the particular solution only. In expanding about the origin, it is found that only the first spherical harmonic contributes to the non-isotropic Hessian. Strong growth in this mode, required for desingularization, is exhibited in the solution of the sine flow. This shows that the so-called Vieillefosse assumption fails quickly. Higher-order approximations are introduced in the initial conditions to mimic the linearly strained flow more faithfully. Their calculations behave similarly as the linearly strained flow at first, but in the end there is no evidence of singularity formation. It is also found from the corresponding Navier-Stokes problems that the total enstrophy is peaked at much later than t=1, consistent with absence or substantial delay of singularity formation in the sine flow.