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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000300002 

CUBO A Mathematical Journal Vol.12, N° 03, (13–32). October 2010

 

A Family of Stationary Solutions to the Euler Equations and Generalized Solutions

 

JULIANA CONCEIÇÃO PRECIOSO

Departamento de Matemática, Universidade Estadual Paulista, 15054-000, S.J.R. Preto, SP, Brazil email: precioso@ibilce.unesp.br


ABSTRACT

In this work, we present a interesting family of stationary solutions for the Euler equations, which behaves in the same way that the approximated solutions presented in [6].

Key words and phrases: Euler equations, incompressible flows, generalized solutions.


RESUMEN

En este trabajo, presentamos una familia interesante de soluciones estacionarias para las ecuaciones de Euler, que se comportan de la misma manera que las soluciones aproximadas presentadas en [6].

Math. Subj. Class.: 35D99.


Acknowledgement.

The author would like to thank Lucas C. F. Ferreira, Helena J. Nussenzveig Lopes and Milton C. Lopes Filho for their useful comments and Weber F. Pereira for many fruitful discussions. The author gratefully acknowledge FAPESP Thematic Project #2007/51490-7.

References

[1] ARNOLD, V.I., Sur la Géométrie Différentielle des Groupes de Lie de Dimension Infine et ses Applications à L’Hidrodynamique, Ann. Inst. Fourier, 16, 319–361, 1966.

[2] ARNOLD, V.I. AND KHESIN, B., Topological Methods in Hidrodynamics, Annu. Rev. Fluid Mech., 24, 145–166, 1992.

[3] BENSOW, R.E., LARSON, M.G. AND VESTERLUND, P., Vorticity-strain residual-based turbulence modelling of the Taylor-Green vortex, Int. J. Numer. Meth. Fluids, 54, 745– 756, 2007.

[4] BRENIER, Y., The Least Action Principle and the Related Concept of Generalized Flows for Incompressible Perfect Fluids, Journal of the American Mathematical Society, Vol. 2, Number 2, 225–255, 1989.

[5] BRENIER, Y., The Dual Least Action Problem for an Ideal, Incompressible Fluid, Arch. Rational Mech. Anal., 122, Number 4, 323–351, 1993.

[6] BRENIER, Y., Minimal Geodesics on Groups of Volume-Preserving Maps and Generalized Solutions of the Euler Equations, CPAM 52, 411–452, 1999.

[7] DIPERNA, R. AND MAJDA A., Oscillations and Concentrations in Weak Solutions of the Incompressible Fluid Equations, Comm. Math. Phys., 108, 667–689, 1987.

[8] DON, W.-S, GOTTLIEB, D., JAMESON, L., SCHILLING, O. AND SHU, C.-W., Numerical Convergence Study of Nearly-Incompressible, Inviscid Taylor-Green Vortex Flow, Journal of Scientific Computing, 24, 569–595, 2005.

[9] EBIN, D.G. AND MARSDEN J., Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, Ann. of Math., 92, 102–163, 1970.

[10] LEE,H. AND SOHRAB S.H., Hydrodynamic Aspects of Premixed Flame Stripes in Two- Dimensional Stagnation-Point Flows, Combustionand Flame, 101, 441–451, 1995.

[11] PRECIOSO, J.C., Equações relaxadas para hidrodinâmica ideal, não homogênea, Tese de doutorado, IMECC-UNICAMP, 2005.        [ Links ]

[12] SHNIRELMAN, A.I., The Geometry of the Groups of Diffeomorphisms and the Dynamics of an Ideal Incompressible Fluid, Mat. Sb (N.S.) 128(170), 82–109, 144, 1985.

[13] SHNIRELMAN, A.I., Generalized Fluid Flows, their Aproximation and Applications, Geom. Funct. Anal., 4, 586–620, 1994.

[14] SHU, C.-W. AND WEINAN, E., A Numerical Resolution Study of High Order Essentially Non-Oscillatory Schemes Applied to Incompressible flow, Journal of Computational Physics, 110, 39–46, 1994.

[15] TARTAR, L., The Compensated Compactness Method Applied to Systems of Conservation Laws. Systems of Nonlinear Partial Differential Equations, (Oxford, 1982), 263–285. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 111. Reidel, Dordrecht-Boston, 1983.

[16] WAN, D.C., WEI, G.W. AND ZHOU, Y.C., Numerical solution of incompressible flows by discrete singular convolution, Int. J. Numer. Meth. Fluids, 38, 789–810, 2002.

[17] WEI, D.C., WEI, A new algorithm for solving some mechanical problems, Comput. Methods Appl. Mech. Engrg., 190, 2017–2030, 2001.

[18] YOUNG, L.C., Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, New York, 1980        [ Links ]