## Steady three-dimensional rotational flows : an approach via two stream functions and Nash-Moser iteration.(English)Zbl 1406.35244

Summary: We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $$D=(0, L)\times \mathbb{R}^2$$. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary $$\partial D$$. The Bernoulli equation states that the “Bernoulli function” $$H:=\frac{1}{2} |v|^2+p$$ (where $$v$$ is the velocity field and $$p$$ the pressure) is constant along stream lines, that is, each particle is associated with a particular value of $$H$$. We also prescribe the value of $$H$$ on $$\partial D$$. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form $$v=\nabla f\times \nabla g$$ and deriving a degenerate nonlinear elliptic system for $$f$$ and $$g$$. This system is solved using the Nash-Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see, e.g., the book by Q. Han and J.-X. Hong [Isometric embedding of Riemannian manifolds in Euclidean spaces. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1113.53002)]. Since we can allow $$H$$ to be nonconstant on $$\partial D$$, our theory includes three-dimensional flows with nonvanishing vorticity.

### MSC:

 35Q31 Euler equations 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76B47 Vortex flows for incompressible inviscid fluids 35G60 Boundary value problems for systems of nonlinear higher-order PDEs 58C15 Implicit function theorems; global Newton methods on manifolds 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations

Zbl 1113.53002
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### References:

 [1] 10.1002/cpa.3160170104 · Zbl 0123.28706 [2] 10.1007/BF01444632 · Zbl 0772.35049 [3] 10.1090/S0033-569X-2011-01215-2 · Zbl 1220.26008 [4] 10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3 · Zbl 0910.35098 [5] 10.1007/s00021-011-0077-7 · Zbl 1255.35180 [6] 10.1007/s11511-015-0123-z · Zbl 1317.35184 [7] ; Grad, Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Vol. 31 : Theoretical and experimental aspects of controlled nuclear fusion, 190, (1958) [8] 10.1090/S0273-0979-1982-15004-2 · Zbl 0499.58003 [9] 10.1090/surv/130 [10] 10.1007/s002200050804 · Zbl 0964.76096 [11] 10.1007/BF00920036 · Zbl 0887.76013 [12] 10.2140/pjm.1978.74.105 · Zbl 0383.58003 [13] 10.1002/cpa.3160180305 · Zbl 0125.33302 [14] 10.1073/pnas.47.11.1824 · Zbl 0104.30503 [15] 10.2307/1994022 · Zbl 0141.19407 [16] ; Plotnikov, Dokl. Akad. Nauk SSSR, 251, 591, (1980) [17] ; Serrin, Handbuch der Physik, 8/1 : Strömungsmechanik I, 125, (1959) [18] 10.1063/1.4907922 · Zbl 1308.76342 [19] 10.1002/cpa.3160280104 · Zbl 0309.58006
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