A variational approach of stationary Boltzmann equation under a condition of Poisson type

Un enfoque variacional de la ecuación de Boltzmann bajo una condición de Poisson

 

Mario Enrique Almanza Caro¹, Rafael Galeano Andrade²

¹ Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Postal: 130015, Fax: (095) 6753718, malmanzac@unicartagena.edu.co .

² Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Postal: 130015, Fax: (095) 6753718, rgaleanoa@unicartagena.edu.co

Abstract

A functional is defined on a subset of L2 ( ), with bounded and so that the divergence theorem is valid. It shows that this functional is differentiable, coercive and weakly lower semi-continuous bound, and therefore has critical points which coincide with the solutions of the stationary Boltmann equation solutions under a condition of Poisson.

Key words: Boltzmann equitation, Kinetic Theory, existence critical points.

Resumen

Se define un funcional sobre un subconjunto de L2 ( ), con acotado y tal que el Teorema de la Divergencia sea válido. Se prueba que este funcional es diferenciable, coercivo y débilmente semicontinuo inferiormente y por tanto tiene puntos críticos que coinciden con las soluciones de las soluciones de la ecuación estacionaria de Boltmann bajo una condición de Poisson.

Palabras clave: ecuación de Boltzmann, Teoría Cinética, existencia de puntos críticos.

Recibido el 9 de Mayo de 2012 En forma revisada el 10 de Junio de 2013

 

3. Result

Theorem. Let Rn with the same hypothesis of Lemma 1 and J(u) defined in (3), u, v, u', v' belonging to B1(0) (unit ball), then J has a critical point which is solution of (1). Proof. The previous lemmas and the Theorem 5.5 of [21] guarantee that there exist z H, such that z is a global minimum and J' (z ) 0, i.e., is solution of (1).

4. Conclusion

In this article has been shown that a solution stationary Boltzmann equation in a bounded domain in L2( ) exists. The existence has been proved through variational methods, defining a functional and showing that it is differentiable, coercive and weakly lower semi-continuous bound and that therefore has critical points which are solutions of the stationary Boltzmann equation.

Although, we’ve been achieved satisfactory results, we could try to seek solutions to the Boltzmann equation in general domains in L1( ), as well as to reflect on whether the variational methods throw some result in this way and at the same time find methods that prove uniqueness of the solution.

Bibliography  

1. Ukay, S.; Yang, T. and Zhao, H. “Stationary solutions to the exterior problems for the Boltzmann Equation Existence. Discrete and Continuous Dynamical Systems”, Vol. 23, No.152 (2009).  

2. Arkeryd, L. and Nouri, A. “On the Stationary Povzner equation in three Space Variables”. Journal of Mathematics Kyoto University, Vol. 39. (1999), 115-153.  

3. Panferov, V. “On the existence of Stationary solutions to the Povzner equation in a Bounded Domain”, Preprint, 2000.  

4. Arkeryd, L. and Nouri, A. “The Stationary Boltzmann equation in the Slab with given weighted mass for hard and Soft Forces”. Annals Scuola Normal Superior. PISA Cl. Sci., Vol. 27 (1998), 533-556.  

5. Arkeryd, L. and Nouri, A. “L1 Solutions to the Stationary Boltzmann equation in a Slab”. Annals Faculty Science of Toulose math. Vol. 9 (2000), 375-413.  

6. Arkeryd, L. and Nouri, A. “On the Milve problem and the hydrodyinamic limit for a steady Boltzmann equation Model”, Journal of Stat. Phys., Vol. 99 (2000) 993-1019.  

7. Arkeryd, L. and Nouri, A. “The Stationary Boltzmann equation in Rn with given in data”. Annals Scuola Normal Sup. Pisa, Vol. 31 (2002) 1-28.  

8. Arkeryd, L. and Nouri, A. “The Stationary nonlinear Boltzmann equation in a Coutle setting; isolated solutions and non-uniqueness”, Preprint, 2003.

9. Costa, G. “An Invitation to Variational Methods in Differential Equations”, Birkhauser, 2007.  

10. Grad, H. “High Frequency Sound Recording According to Boltzmann equation”. SIAM J. Appl. Math., Vol.14 (1966), 935-955.  

11. Guirand, J.P. “Probléme Aux Limites interieur pour L`equation de Boltzmann en régime Stationaire, faiblement non lineaire. J. Mécanique”, Vol. 11 (1972), 183-231.  

12. Heintz, A. “Solvability of a Boundary problem for the non linear Boltzmann equation in a Bounded Domain, in molecular gas dynamics (in Russian). Aerodynamics of vare fied gases”, Vol. 10 (1980), 16-24.  

13. Maslova, N. “Non linear evolution equations Kinetic Approach. Series on Advances in Mathematics for Applied Sciences”, Vol. 10, (1993).  

14. Soney, Y. “Kinetic Theory and Fluid Dynamics”, Birkhauser, Boston, 2002.  

15. Ukay, S. and Asano, K. “Steady solutions of the Boltzmann equation for a Gas Flow Past an Obstacle Existence”. Arch. Rat. Mech. Anal., Vol. 84 (1983), 249-91.  

16. Peral, A. “Ecuaciones en Derivadas Parciales”, Addison-Wesley/ Universidad autónoma de Madrid, 1995.  

17. Brezis, H. “Functional Analysis, Sobolev Spaces and Partial Differential Equations”, Springer, 2010.  

18. Rudin, W. “Functional Analysis”, McGraw- Hill, 1991.  

19. Caicedo, J. “Cálculo Avanzado”, Universidad Nacional de Colombia, 2005.  

20. Munkres, J. “Topology”, Prentice Hall, 2000.  

21. Ambrosetti A. Malchiodi A. “Nonlinear Analysis and Semilinear Elliptic Problems”. Cambridge Studies in Advanced Mathematics, 2007.