×

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

Citations:

Zbl 1113.53002
PDF BibTeX XML Cite
Full Text: DOI arXiv

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.