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Meccanica

, Volume 12, Issue 1, pp 15–18 | Cite as

Absence of turbulence in a unidimensional model of fluid motion (Burgers model)

  • Carlo Boldrighini
  • Livio Triolo
Article

Summary

We study a quasilinear partial differential equation which is a classical unidimensional model of fluid motion (Burgers equation). We consider the problem in a finite interval with stationary boundary conditions. The aim is to see whether the model shows transitions in the asymptotic behaviour as viscosity varies. We show that there is always a unique stationary solution, which is explicitely exhibited, and, by using the Hopf-Cole transformation, that it is globally attractive for any value of the viscosity, both in the homogeneous and inhomogeneous case.

Keywords

Viscosity Boundary Condition Differential Equation Mechanical Engineer Civil Engineer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sommario

In questo lavoro si considera un classico modello unidimensionale della fluidodinamica, consistente in un'equazione alle derivate parziali parabolica quasilineare (equazione di Burgers). Si considera il problema in un intervallo con condizioni al contorno stazionarie. L'intento è quello di indagare se il modello presenta transizioni nel comportamento asintotico al variare della viscosità. Facendo uso della trasformazione di Hopf-Cole si trova che la soluzione stazionaria (che viene calcolata esplicitamente) è globalmente attrattiva per ogni valore della viscosità, sia nel caso omogeneo che in presenza di una forza.

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References

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    HOPF E., Comm. Pure and Applied Mathematics3, 201, 1950.Google Scholar
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    BARDOS C., PENEL P., FRISCH U., SULEM P. L., in the Proceedings of “Journées Mathématiques sur la Turbulence”, Orsay, June 1975.Google Scholar
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    LANFORD O., III in the same Proceedings as ref. [2].Google Scholar
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    SOBOLEV S. L.,Partial Differential Equations in Mathematical Physics, Pergamon Press, Oxford, 1964, pp. 304.Google Scholar
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    COLE J. D., Quart. of Applied Mathematics9, 225, 1951.Google Scholar
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    JORGENS K.,Spectral Theory of Second Order Ordinary Differential Equations, Lecture Notes Series n. 2, Aarhus Universitet, 1964.Google Scholar

Copyright information

© Tamburini Editore s.p.a. Milano 1977

Authors and Affiliations

  • Carlo Boldrighini
    • 1
  • Livio Triolo
    • 1
  1. 1.Istituto di Matematica dell'UniversitàCamerino

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