Modelling with PDEs

Abstract

Taking in consideration the audience, we explore a prospective point of view. We intend to comment on some qualitative aspects and issues concerning the mathematical work of modelization with Partial Differential Equations (PDEs). A class of evolution PDEs: \pa_t u+div f(u)=\eps div b(u,\grad u)+\del div \pa_(\xi) c(u,\grad u), which include generalized Korteweg-de Vries-Burgers equation (when \xi is a space variable) and Benjamin-Bona-Mahony-Burgers equation (when \xi is the time variable), or that of \pa_t u+div f(u)=\del div c(u,\grad u), which can present unexpected dissipative properties. As "\eps,\del-parameters tend to zero we can have failure (i.e., no limit solution at all for \del > O(\eps^\gamma)) or reliability (with different limits: classical-entropy weak-solutions inside an integrity region \del < O(\eps^\gamma) and nonclassicalentropy weak-solutions along \del = O(\eps^\gamma)), according to the \gamma-balance of \eps,\del-strengths and to the ratio between the growths of the b-dissipation and c-dispersion.