By Luc Tartar

The* creation to Navier-Stokes Equation and Oceanography* corresponds to a graduate path in arithmetic, taught at Carnegie Mellon collage within the spring of 1999. reviews have been extra to the lecture notes allotted to the scholars, in addition to brief biographical details for all scientists pointed out within the textual content, the aim being to teach that the production of clinical wisdom is a global company, and who contributed to it, from the place, and while. The objective of the path is to coach a severe perspective about the partial differential equations of continuum mechanics, and to teach the necessity for constructing new tailored mathematical tools.

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**Example text**

The ideas of KOLMOGOROV have the same defect as the naive idea that the effective conductivity of a mixture of materials only depends upon the proportions used, which is false in more than one dimension. In the case of small amplitude variations of the conductivities of the materials used, I have introduced H-measures for computing the quadratic correction for the eﬀective conductivity, and in some exceptional cases the correction only depends upon proportions; one could hope that something similar would occur in the case of a weak turbulence created by small oscillations of the velocity, but there are some other diﬃculties which appear, and the tools for carrying this kind of analysis will be described at the end of this course.

He worked in Manchester and Cambridge, England, UK. Andrei Nikolaevich KOLMOGOROV, Russian mathematician, 1903–1987. He received the Wolf Prize in 1980. He worked in Moscow, Russia. The ideas of KOLMOGOROV have the same defect as the naive idea that the effective conductivity of a mixture of materials only depends upon the proportions used, which is false in more than one dimension. In the case of small amplitude variations of the conductivities of the materials used, I have introduced H-measures for computing the quadratic correction for the eﬀective conductivity, and in some exceptional cases the correction only depends upon proportions; one could hope that something similar would occur in the case of a weak turbulence created by small oscillations of the velocity, but there are some other diﬃculties which appear, and the tools for carrying this kind of analysis will be described at the end of this course.

He works at CNRS (Centre National de la Recherche Scientiﬁque) at Universit´e Paris VI (Pierre et Marie CURIE), Paris, France. ˇ Ivo M. BABUSKA , Czech-born mathematician, born in 1926. He works at University of Texas, Austin, TX. 5 Particles and continuum mechanics 25 solutions of some partial diﬀerential equations may behave like particles, it still faces a few theoretical obstacles and cannot explain yet how to interpret what physicists say when they use a discrete description, starting from atoms arranged in a crystalline way (or a polycrystalline way with important eﬀects at the grain boundaries), describing defects in the crystalline arrangement and how these defects move around in order to explain the eﬀects of plasticity, which limit the applicability of theories like nonlinear elasticity (Owen RICHMOND9 described a few years ago his program, very similar to mine, but which dealt precisely with the scales that I cannot explain at the moment, where there are dislocations and other defects).