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By Javier Jiménez

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Some other examples using Theorem 2 will be considered in Chapter 3. CHAPTER 2 Topology of Two-Dimensional Flows §3. 1. Admissible stream functions. In this section we analyze the topological structure of the family of streamlines of a two-dimensional stationary flow of an incompressible fluid in the part V of a closed surface (two-dimensional compact connected smooth closed oriented manifold) of genus g = 0, 1, ... (g = 0 corresponds to the plane, g = 1 to the torus, g = 2 to the sphere with two handles, as is shown in Figure 4, etc.

Ofpairwise nonintersecting simple arcs with one endpoint on ¢c (arc endpoints may coincide on ¢c) separating the signs of the difference I/I - C and dividing Q into n + l Jordan domains (sectors) F if ¢c does not coincide with J, or into n sectors F if ¢c coincides with J, n being even in the latter case (Figure 6a); §3. ff 27 a J :-l (mtn) FIGURE 6. Boundary continua with relative neighborhoods. (ii) moreover, for any 0 < c < co, the simple arc dividing F into two Jordan domains and separating the signs of the difference If/ - ( C =F c) is a nonempty set ofpoints of the level If/= C =fc; (iii) if ¢'e is an isolated continuum where a relative minimum (maximum) of the function If/ is attained, then for any C < C' < C + co (for any C < C' < C - co, respectively) the set of points of the level If/ = C' in Q is an arc if ¢'e does not coincide with J, or a cycle if ¢'e coincides with J, that separate the signs of the difference If/ - C' and divide Q into two domains, one of which is again a relative neighborhood of the continuum ¢'e (Figure 6b).

Since for ¢e -:j:. J the endpoints of the arc A. ) and for ¢e = J the set S is closed, the set of points of the level C lying on S can always be covered by a finite number of widths of this level. By Proposition 5, any arc Ye ending on ¢e ends on rand, thus, intersects S. Hence, the total number of these arcs is finite (we additionally take into account the fact that at each saddle point of the level Con Sonly a finite number of maximal arcs can intersect according to statement (i) of Theorem 1).

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An introduction to turbulence by Javier Jiménez

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