Generalized solutions for equations of physics and engineering

• Grant number: 13/04381-9
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: June 26, 2013
• End date: August 23, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Jean-François Claude Colombeau
• Grantee: Jean-François Claude Colombeau
• Visiting researcher: Vladimir G. Danilov
• Visiting researcher institution: National Research University Higher School of Economics, Russia
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The Brazilian group of IME-USP and DM-UNESP campus de Ilha Solteira (J. Aragona, JF. Colombeau, M. C-Fonteyne, SO. Juriaans, F. Villareal] and the Russian group of Prof. Danilov work with identical tools: "solutions in the association sense of generalized functions" and "weak asymptotic methods" respectively. With this tool we intend to study two basic equations of mathematical physics and engineering. The first one is a version of the Navier Stokes equation as a continuation of an article [J. Aragona, F. Villareal, "Colombeau s theory and shock waves in a problem of hydrodynamics J. Anal. Math. Jerusalem, 61,1993, p.113-144]. The second equation is an equation of engineering which is the standard model for a liquid-gas mixture used in off-shore petroleum exploitation whose study was started in the IME-USP last year at the theoretical level in [J. Aragona, JF. Colombeau, SO. Juriaans,"Nonlinear generalized functions applied to a nonconservative system: jump conditions for the one-pressure model for a mixture of a liquid and a gas", preprint, annexed] and from the numerical side as an adaptation of the scheme of M. C-Fonteyne, 3 recent articles quoted in the detailed project [SIAM J. Numer. Anal. 2011, Numer. Meth. for PDEs 2013, J. Comput. Appl. Math. 2013]. (AU)