Steady three-dimensional rotational flows: An approach via two stream functions and Nash-Moser iteration

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Steady three-dimensional rotational flows : An approach via two stream functions and Nash-Moser iteration. / Buffoni, Boris; Wahlén, Erik.

I: Analysis and PDE, Vol. 12, Nr. 5, 2019, s. 1225-1258.

Forskningsoutput: TidskriftsbidragArtikel i vetenskaplig tidskrift

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TY - JOUR

T1 - Steady three-dimensional rotational flows

T2 - An approach via two stream functions and Nash-Moser iteration

AU - Buffoni, Boris

AU - Wahlén, Erik

PY - 2019

Y1 - 2019

N2 - We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) × ℝ2. 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 ∂D. The Bernoulli equation states that the "Bernoulli function" H := 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 ∂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 = ∇ f × ∇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 (2006). Since we can allow H to be nonconstant on ∂D, our theory includes three-dimensional flows with nonvanishing vorticity.

AB - We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) × ℝ2. 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 ∂D. The Bernoulli equation states that the "Bernoulli function" H := 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 ∂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 = ∇ f × ∇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 (2006). Since we can allow H to be nonconstant on ∂D, our theory includes three-dimensional flows with nonvanishing vorticity.

KW - Boundary conditions

KW - Incompressible flows

KW - Nash-Moser iteration method

KW - Vorticity

U2 - 10.2140/apde.2019.12.1225

DO - 10.2140/apde.2019.12.1225

M3 - Article

VL - 12

SP - 1225

EP - 1258

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 5

ER -