Abstract
Nonlinear operations such as multiplication of distributions are not allowed in the classical theory of distributions. As a result, some ambiguities arise when we want to solve nonlinear partial differential equations such as differential equations of elasticity and multifluid flows, or some new cosmological models such as signature changing space‐times. Colombeau′s new theory of generalized functions can be used to remove these ambiguities. In this paper, we first consider a simplified model of elasticity and multifluid flows in the framework of Colombeau′s theory and show how one can handle such problems, investigate their jump conditions, and resolve their ambiguities. Then we consider as a new proposal the case of cosmological models with signature change and use Colombeau′s theory to solve Einstein equation for the beginning of the Universe.
Highlights
Classical theory of distributions, based on Schwartz-Sobolev theory of distributions, does not allow nonlinear operations of distributions [4]
We find that the jump conditions of (3.1) depend on an arbitrary parameter, the real number A
Since jump conditions have most of the information on the boundary surface, infinite number of jump conditions physically is not acceptable, so one have to resolve this ambiguity
Summary
Classical theory of distributions, based on Schwartz-Sobolev theory of distributions, does not allow nonlinear operations of distributions [4]. In Colombeau’s theory, a mathematically consistent way of multiplying distributions is proposed. Colombeau’s motivation is the inconsistency in multiplication and differentiation of distributions. Take, as it is given in the classical theory of distributions, θn = θ ∀n = 2, 3, . . where θ is the Heaviside step function. Differentiation of (1.1) gives nθn−1θ = θ. Multiplication by θ gives 2θ2θ = θθ. The difference θn − θ, being infinitesimal, is the essence of Colombeau’s theory of generalized functions. Colombeau considers θ(t) as a function with “microscopic structure” at t = 0, making θ not to be a sharp step function, but having a width . In the following we give a short formulation of Colombeau’s theory
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