Mathematical theory of incompressible nonviscous fluids
type: Item
instance of: scholarly article
title: Mathematical theory of incompressible nonviscous fluids
zbMATH Open document ID: 0789.76002
published in: Applied Mathematical Sciences
publication date: 1993-12-15T00:00:00Z
review text: Although being more than two centuries old, the theory of incompressible nonviscous flows (so-called perfect flows) is far from being complete and generates a permanent interest within mathematical community. Many basic questions in this theory remain still open; e.g., one may recall such famous challenging problems as development of singularities in a finite time or the long-time behaviour of smooth solutions. Among many books devoted to the subject, no recent one, in authors' opinion, is oriented toward mathematical physics. Thus, the purpose of the authors was to fill this gap and write a mathematically rigorous book presenting, on the other hand, the underlying physical ideas in a clear and natural manner. The text consists of seven chapters. The introductory chapter 1 starts out with the derivation of the equations of motion of incompressible nonviscous fluids (the Euler equations) and discusses such basic notions as potential and vorticity flows, stream function, conservation laws. Chapter 2 deals with existence, uniqueness, and regularity of the solution to the initial value problem for the Euler equation. In chapter 3, the concept of stability is introduced and the results obtained are applied to stationary solutions of the Euler equation. The vortex model is explained in chapter 4 both in two and three dimensions. In chapter 5, the spectral and vortex methods are considered as finite-dimensional approximation schemes for the Euler equation, and their convergence is investigated. Chapter 6 is devoted to the study of discontinuities arising in the perfect fluids. As examples, the evolution of vortex sheets and the evolution of density discontinuities (water waves) are presented. The remainder of the book, chapter 7, is essentially different from that of all the others. Namely, the Euler equations are replaced here by the Navier-Stokes equations in order to take into account dissipative effects omitted up to now. On this basis, the authors provide the descript…
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 76Bxx
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76F20
zbMATH DE Number: 482828
zbMATH Keywords: Euler equations
zbMATH Keywords: potential
zbMATH Keywords: vorticity
zbMATH Keywords: stream function
zbMATH Keywords: conservation laws
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: initial value problem
zbMATH Keywords: stability
zbMATH Keywords: vortex model
zbMATH Keywords: spectral and vortex methods
zbMATH Keywords: convergence
zbMATH Keywords: discontinuities
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: dissipative effects
zbMATH Keywords: turbulent flows
zbMATH Keywords: statistical theories
zbMATH Keywords: invariant measures
author: Carlo Marchioro
author: Mario Pulvirenti
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 482828