About a family of distributional products important in the applications. (English) Zbl 0664.46042

Sarrico, C. O. R.

Port. Math. 45, No. 3, 295-316 (1988).

The author defines a family of products of a distribution from \({\mathcal D}'\) by a distribution from \(C^{\infty}\oplus {\mathcal D}_ n'\) where \({\mathcal D}_ n'\) is the space of distributions with nowhere dense support. These products (which are dependent of the choice of a group G of unimodular transformations of \({\mathbb{R}}^ n\) and a function \(\alpha\in {\mathcal D}\) with \(\int \alpha =1\) which is G-invariant) are consistent with the usual product of a distribution by a \(C^{\infty}\) function, are distributive and verify the normal law of derivative of products. Products like \(\delta\) \(\cdot \delta\), \(H\cdot \delta\), (pf(1/t)\(\cdot \delta\), \(\delta\) \(\cdot \delta '\) \((H=the\) Heaviside function) are considered with some simple physical applications. Certain shock wave solutions of the differential equation \[ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0 \] are discussed.

Reviewer: L.Goras

MSC:

46F10	Operations with distributions and generalized functions
35D05	Existence of generalized solutions of PDE (MSC2000)
76L05	Shock waves and blast waves in fluid mechanics

Keywords:

distributional products; space of distributions with nowhere dense support; unimodular transformations; derivative of products; shock wave solutions