Liapunov functions and monotonicity in the Navier-Stokes equation

Here u = u(t): R'" --+ R'" is the velocity field; v is the kinematic viscosity; B =grad; and P denotes the projection operator onto solenoidal (= divergence free) vectors along gradients (so that the pressure term has been eliminated). It is the purpose of this note to show that (1.1) has a large number of Liapunov functions, which decrease monotonically in time for every solution with small L'" (Rm)_ norm. It is known (see [3]) that if ¢> E L'"; then a unique strong solution u(t) with u(O) = ¢> exists for short time, and that it exists for all time if v-111¢>llm is small. (II lip denotes the LP-norm.) For this reason, we may call R[u] = v-1llullm a Reynolds number for the flow u, Of course other Reynolds numbers exist with a similar property, but it seems that R[u] is the simplest among them. For m = 3, it appears already in Leray's paper (see [5, p.231). Actually it turns out that if R[¢>] is small, then R[u(t)] decreases in t monotonically. Therefore it may also be called a Liapunov function for (1.1). It seems that such a monotone decay has so far been known only for Ilu112' In fact we shall show that there are many other Liapunov functions. They include the Ilull"p = 11(1-6)'/2ullp for certain sand p. All p E (1,00) are allowed if s = O. In particular, the Reynolds number R[u] = Ilulim is at the same time a Liapunov function. It is obvious that if L.:[u] is a Liapunov function, then so is (L.:[u]) for any monotone increasing function . To illustrate the situation in the simplest case, let us recall the L 2 -theory for (1.1) given some time ago [2]. If one works in the Hilbert space H = L2(Rm;Rm), -6 is a nonnegative selfadjoint operator, and P becomes an orthogonal projection of H onto the subspace H" consisting of solenoidal vectors. -6 is reduced by H", so that its part A in H" is nonnegative selfadjoint, and (1.1) can be regarded as an abstract equation