Solutions of the divergence and analysis of the stokes equations in planar Hölder-α domains

• Autores: Duran, Ricardo Guillermo; Lopez Garcia, Fernando Alfonso
• Año de publicación: 2010
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: DIVERGENCE OPERATOR; HÓLDER-α DOMAINS; STOKES EQUATIONS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

If Ω ⊂ n is a bounded domain, the existence of solutions u∈ H10(Ω)n of div u = f for f ∈ L 2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution (u,p)∈ H10(Ω)n× L2(Ω ), where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u,p)∈ H10(Ω) n× Lr(Ω) for some r < 2 depending on the power of the cusp. © 2010 World Scientific Publishing Company.
Fil: Duran, Ricardo Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Lopez Garcia, Fernando Alfonso. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina