Liouville Theorem for 2d Navier-stokes Equations

(One may modify the question by putting various other restrictions on (L); for example, one can consider only steady-state solutions, or solutions with finite rate of dissipation or belonging to various other function spaces, etc.) We have proved a positive result for dimension n = 2 which we will discuss below, but let us begin by mentioning why the basic problem is interesting. Generally speaking, it is known that Liouville properties are related to regularity (see, e.g., [6] and [9]), and we expect that progress in 3D Liouville problems for NSE, even in the steady-state case, will lead to progress in regularity theory. The difficulties which come up in the 3D regularity theory for NSE appear in the Liouville problem in a slightly different light, so hopefully studying the 3D Liouville problem will motivate some new ideas for the 3D regularity problem. We note that our 2D Liouville theorem is closely related to the fact that in 2D we have regularity for NSE (see e.g., [7]). We will highlight now some particular instances in the literature relating Liouville theorems to regularity; one of particular interest is in the 1987 paper of Giga and Kohn ([5]). They are interested in characterizing blow-up for non-negative solutions to the non-linear heat equation, ut −4u− |u|p−1u = 0 in Rn × (0, T ). (Note that for p = 3, the equation has the same scaling symmetries as does NSE.) Their result limits the rate of blow up at time