Multiplications of distributions in elasticity and hydrodynamics.

*(English)*Zbl 0646.76007Summary: It is shown how a mathematical theory of generalized functions, in which the multiplications of distributions appearing in nonlinear equations of physics make sense, gives new formulas and new numerical results. The new methods shown here are quite general but since each particular problem requires its own study, this paper is limited to elasticity and hydrodynamics. In elasticity Hooke’s law gives systems in a nonconservative form; the study of shock waves for these systems gives nonclassical multiplications of distributions of the form \(Y\cdot \delta\) (Y \(=\) Heaviside function, \(\delta\) \(=\) Dirac mass at the origin). Using this new mathematical tool new formulas are obtained (more generally new numerical schemes): in a first step “ambiguous” results are obtained; then the ambiguity if removed. In hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes. Strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics, since the basic equations are recalled. All computations done in this paper are rigorous from the mathematical viewpoint.

##### MSC:

76A02 | Foundations of fluid mechanics |

74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |

76L05 | Shock waves and blast waves in fluid mechanics |

74B99 | Elastic materials |

##### Keywords:

mathematical theory of generalized functions; Hooke’s law; nonclassical multiplications of distributions
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\textit{J. F. Colombeau} and \textit{A. Y. Le Roux}, J. Math. Phys. 29, No. 2, 315--319 (1988; Zbl 0646.76007)

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##### References:

[1] | Cauret J. J., C. R. Acad. Sci. (Paris) 302 pp 435– (1986) |

[2] | Colombeau J. F., C. R. Acad. Sci. (Paris) 305 pp 453– (1987) |

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