J.F Colombeau

A multiplication of distributions

If Ω denotes an open subset of Rn (n = 1, 2,…), we define an algebra g (Ω) which contains the space D′(Ω) of all distributions on Ω and such that C∞(Ω) is a subalgebra of G (Ω). The elements of G (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in G(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and ƒ∈OM(R2q)—a classical Schwartz notation—for any G1,…,Gq∈G(σ), we define naturally an element ƒG1,…,Gq∈G(σ). These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.

Microscopic profiles of shock waves and ambiguities in multiplications of distributions

A (rigorous) mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form. The evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics.

Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations

Abstract Nonlinear nonstationary problems, arising in elastodynamics, have naturally a nonconservative form. For these systems it is not possible to define a weak solution according to distribution theory: in order to define the concept of discontinuous solutions for these systems, one is confronted with multiplications of distributions. This is unavoidable from a physical viewpoint since discontinuous solutions are needed to represent shock waves. The theory of generalized functions of the second author, in which one may multiply arbitrary distributions, gives a concept of generalized solutions that may be discontinuous functions. Existence of such global solutions of the Cauchy problem for a system of two equations is proved by a compactness argument from a convergent numerical scheme. In the case under consideration, the generalized solutions are bounded variation functions. The results thus obtained agree with the results expected by the engineers and lead to the development of new numerical schemes.

Holomorphic generalized functions

Abstract In the new theory of generalized functions introduced by one author we study the generalized functions G on open sets of C n solutions of the equation ∂ G = 0 . These generalized functions—which cannot be distributions except if they are usual holomorphic functions—have many properties of the usual holomorphic functions but they present also serious differences in relation with the analytic continuation.

Generalized solutions of nonlinear parabolic equations with distributions as initial conditions

Abstract We consider the nonlinear parabolic equation u t −Δu + u 3 = 0 in Q = Ω×]0, T[ ( T > 0, Ω open set in R d, d = 1, 2, …) with the boundary condition u(x, t) = 0 on ∂Ω × ]0, T[ and the initial condition u(x, 0) = δ(x) in Ω , where δ is the Dirac mass at the origin of R . It is known that this problem has no weak solution in any known classical sense (within the distribution theory). Using a theory of generalized functions we obtain existence, uniqueness, and consistence results, which describe mathematically the behaviour of the solutions ue, obtained with smooth initial conditions u e (x, 0) = δ e (x), δ e ∈D(Ω) , and δe → δ when e → 0.

Whitney's extension theorem for generalized functions

Abstract If X is a subset of R n we define generalized functions on X as a direct generalization of C ∞ functions on X in Whitneys sense and of generalized functions on X when X is open. Then we prove that, if X is closed, any generalized function on X may be extended as a generalized function on R n , which is a Whitneys extension theorem for generalized functions. This result generalizes Borels theorem for generalized functions already proved by the same authors. The proof is an adaptation of a classical proof.

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Frechet space (a DFN space for short) and let F be a closed C∞ form of type (0, 1) on E. Then there exists a C∞ function f on E as the solution of ∂f=F.” Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ∂ equation in Hilbert or DFN spaces.

Convergence of numerical schemes involving powers of the Dirac delta function

Abstract Developments of the Hull elastoplastic numerical method lead to nonconservative versions, which in the case of shock waves involve multiplications of distributions of the type of powers of the Dirac delta function. In the one dimensional case of the shock wave equation u t + uu x = 0, the numerical solutions will converge to the solution of a different equation, if the convergence and the latter equation are considered within the nonlinear theory of generalized functions introduced recently by the second author. The study of this phenomenon, presented here in one of its relevant particular cases, offers for the first time a rigorous understanding of important similar situations encountered in industrial applications, when numerical solutions may show either agreement with, or deviations from the expected solutions.

Holomorphic maps with a given asymptotic expansion at a boundary point

Abstract The following result has been known for a long time: let 0 α π and let S be the sector { z ≠ 0 and arg z ≠ α (+ 2 kπ )} of the complex plane; let ( u n ) be a given infinite sequence of complex numbers; then there exists a holomorphic function on S which admits the formal power series ∑ +∞ n = 0 u n z n as asymptotic expansion at the origin. A first generalization of this result to the infinite dimensional case is given by the author (A result of existence of holomorphic maps which admit a given asymptotic expansion, in “Advances in Holomorphy” (J. A. Barroso, Ed.), in press). We give here an improvement of this last result, based upon a different proof. Then we give two counterexamples showing that our assumptions on the spaces are essential.