Abstract

This work is devoted to the stationary solutions of the 2D Euler equations describing the time-independent flows of an ideal incompressible fluid. There exists an infinite-dimensional set of such solutions; however, they do not form a smooth manifold in the space of all divergence-free vector fields tangent to the boundary of the flow domain. This circumstance hinders the efforts to understand the structure of the set of stationary flows, and to further study other classes of solutions such as the time-periodic or quasiperiodic flows. The previous authors considered the solutions in the Frechet space of smooth functions and used powerful methods such as the Nash-Moser-Hamilton implicit function theorem. However, in their approach they overlook a surprising feature of the stationary flows which makes the picture much more transparent, and opens the way to further progress. This is the observation that the particle trajectories in the flow described by arbitrary solutions of the Euler equations in domains with analytic boundary are analytic curves, even if the velocity field has a finite regularity (say, belongs to the Sobolev or Holder space). In particular, for any stationary solution, the flow lines are analytic curves, despite limited regularity of the velocity field.
To study the stationary flows we change the viewpoint and consider the flow field as a family of analytic flow lines non-analytically depending on parameter. We quantify the analyticity by introducing spaces of functions which have an analytic continuation to some strip containing the real axis such that on the boundary of the strip the function belongs to the Sobolev space. Further, we introduce the class of Sobolev functions of two variables which are analytic (in the above sense) with respect to one variable. Such functions describe the families of flow lines of stationary flows. These partially-analytic functions form a complex Banach space. The stationary solutions satisfy (in the new coordinates) a quasilinear elliptic equation whose local solvability is proved by using the Banach Analytic Implicit Function Theorem (BAIF Theorem). Thus we prove that the set of stationary flows is an analytic
manifold in the complex Banach space of the flows (i.e. families of flow lines).
In our previous work ([9]), we realized this idea in the case of stationary flows in a periodic channel with analytic boundaries. In the present work we study a more complicated case of flows in a domain close to the disc, having one stagnation point. We use polar coordinates centered at the (unknown) stagnation point. This results in an elliptic quasilinear equation in the annulus which is degenerate at one component of the boundary. This makes the analysis more difficult. We introduce function spaces which are adaptations of the Kondratev spaces to the partially-analytic setup, and prove that the problem is Fredholm in those spaces. Further we use the BAIF Theorem, and prove that in our spaces, the set of stationary flows
is locally a complex-analytic manifold.