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By Kenneth B Stolarsky

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The analysis of the stability could be also done for other types of minimizers. Consider a convex function w +-G ( w ) and the minimizer of the functional s, W > d X under the constraints of mass equal t o 1 and energy given. 1) which ought t o be studied in the same way as the standard mean field equation. In this direction some related stability results can be found in [19]. However it is the mean field equation itself or its generalization given in [16, 14, 121 which seems really pertinent for the description of coherent STABLE AND UNSTABLE IDEAL PLANE FLOWS 45 structures.

1 is used and a positive constant qo such that p ( z ) 2 2q0 in R is introduced. Let 0 5 q 5 min(q0,e-l) and observe the inclusion {x R\lw,(z,O)I < q } c C. BARDOS 32 L rl Y . GUO 1 W. STRAUSS Iwn(z, 0) - p(2)ldx + IIH(P)llL=(n) lw + rll l o g 4 dx . 14) z,O)-~(z>l_>~ To complete the proof of the claim, choose q t o make ql log71 less than c/2 and then choose n large enough t o ensure, with the strong L2 convergence of wn(0) t o p , that the sum of the two other terms is less than ~ / 2 . Then there exists a subsequence n j denoted below by n such that w n ( t n ) converges weakly in L2(R) t o a function v ( x ) .

1. 10) makes sense for solutions in H1(a)n L"(R). This is what we are looking for to begin with. 1) where H ( s , p ) stands for the vector H I , - - ,H N . 3) Morevover one has l ~ E ( x , P )Il 1 ,; VX,P. 2. 10), with a priori estimates, assuming A. BENSOUSSAN 50 a solution in J. FREHSE ( V V ~ ~ ~exists. 7) where y,L > 0 will be defined later. 6) with v = 1 - E , which is a negative function in H1 n Loo. 4), together with the fact that 1 - E yields 5L(-A < 0, - XoIDii(2)(1 - E)dx. 2) we get Choosing yields + l ( i iL)(1 - E ) d x 5 0.

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