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
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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.
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