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