New versions of the Colombeau algebras

Abstract

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half-space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions. The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one-dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]- [16], [49] and [50].

Сингулярные решения систем законов сохранения типа $\delta$- и $\delta'$-ударных волн и процессы переноса и концентрации

“…Тем не менее, в рамках подхода Коломбо можно решать задачи, которые допускают сингулярные решения [54]- [58]. В работах автора [60]- [62] были введены версии алгебр Коломбо, которые легко использовать для решения задач, возникающих в теории сингулярных решений нелинейных уравнений и систем законов сохранения.…”