Expanding Domain Limit for Incompressible Fluids in the Plane

The general class of problems we consider is the following: Let Ω1 be a bounded domain in R d for d ≥ 2 and let u be a velocity field on all of R. Suppose that for all R ≥ 1 we have an operator TR that projects u restricted to RΩ1 (Ω1 scaled by R) into a function space on RΩ1 for which the solution to some initial value problem is well-posed with TRu 0 as the initial velocity. Can we show that as R → ∞ the solution to the initial value problem on RΩ1 converges to a solution in the whole space? We answer this question when d = 2 for weak solutions to the NavierStokes and Euler equations. For the Navier-Stokes equations we assume the lowest regularity of u for which one can obtain adequate control on the pressure. For the Euler equations we assume the lowest feasible regularity of u for which uniqueness of solutions to the Euler equations is known (thus, we allow “slightly unbounded” vorticity). In both cases, we obtain strong convergence of the velocity and the vorticity as R → ∞ and, for the Euler equations, the flow. Our approach yields, in principle, a bound on the rates of convergence. Recompiled on January 12, 2011 to add active links