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Publication# Steady Three-Dimensional Rotational Flows: An Approach Via Two Stream Functions And Nash-Moser Iteration

Abstract

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) x 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 aD. The Bernoulli equation states that the "Bernoulli function" H :=-1/2 vertical bar v vertical bar(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 derivative 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 = del f x del 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 partial derivative D, our theory includes three-dimensional flows with nonvanishing vorticity.

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System

A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and is expressed in its functioning. Systems are the subjects of study of systems theory and other systems sciences. Systems have several common properties and characteristics, including structure, function(s), behavior and interconnectivity.

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

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