Nonstationary plane flow of viscous and ideal fluids

Vortex methods. II. Higher order accuracy in two and three dimensions

“…It is an underlying assumption of the analysis that the flow being approximated is sufficiently smooth. It has long been known that, with mild regularity assumptions on the initial vorticity, the Euler equations of ideal flow in two dimensions have a classical solution for all time; for a recent treatment see McGrath [ 13]. Moreover, if the initial vorticity is smooth, the solution is also, and bounds for derivatives of the velocity on a time interval 0 < t *£ T, T arbitrary, can be obtained from bounds on the initial data (e.g., see Lemmas 3.1 and 3.2 of [3]).…”

A relaxation approximation of the incompressible Navier-Stokes system

“…Recalling the classical result 11,12 on the existence of sufficiently regular solutions of the incompressible Navier-Stokes (3), we have the following regularity result about (u , V , φ ). …”

Viscous splitting for the unbounded problem of the Navier-Stokes equations

“…Under that restriction we can prove by induction global existence for problem (1.6)-(1.9), for any i. In fact, if tok(ik -0) e Ll , then cok(t) eL[(Q) for te [ik, (i+l)k) (see [11]). For the operator 8 given in §2, (I-Q)uk((i+l)k-0) has compact support, so cok(ik) = -VA@ük{(i+ l)k-0) e L (Q), and cok satisfieŝ = vAtok -VA(7 -e)ük((i +l)k-0), a)\t=,k = 0)k('k)-Using the fundamental solution of the heat equation, it is easy to prove that cok(t)eL[(Q) for te [ik, (i + l)k).…”

“…A sufficient condition for global existence was given in [8,11], namely the initial value u0 and body force / should satisfy, in addition, VAw0 6 Ll(Q), VA/ e L (Í2x (0, T)). Under that restriction we can prove by induction global existence for problem (1.6)-(1.9), for any i.…”