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基本説明
Corresponds to a graduate course in mathematics, taught at Carnegie Mellon University in the spring of 1999.
Full Description
In the spring of 1999, I taught (at CARNEGIEMELLON University) a graduate course entitled Partial Di?erential Equations Models in Oceanography, and I wrote lecture notes which I distributed to the students; these notes were then made available on the Internet, and they were distributed to the participants of a Summer School held in Lisbon, Portugal, in July 1999. After a few years, I feel it will be useful to make the text available to a larger audience by publishing a revised version. To an uninformed observer, it may seem that there is more interest in the Navier-Stokes equation nowadays, but many who claim to be interested show such a lack of knowledge about continuum mechanics that one may wonder about such a super?cial attraction. Could one of the Clay Millennium Prizes bethereasonbehindthisrenewedinterest?Readingthetextoftheconjectures to be solved for winning that particular prize leaves the impression that the subject was not chosen by people interested in continuum mechanics, as the selected questions have almost no physical content. Invariance by translation or scaling is mentioned, but why is invariance by rotations not pointed out 1 andwhyisGalileaninvariance omitted,asitistheessentialfactwhichmakes 1 Velocities involved for ordinary ?uids being much smaller than the velocity of light c, no relativistic corrections are necessary and Galilean invariance should then be used, but one should be aware that once the mathematical equation has been written it is not automatic that its solutions will only use velocities bounded by c.
Contents
Basic physical laws and units.- Radiation balance of atmosphere.- Conservations in ocean and atmosphere.- Sobolev spaces I.- Particles and continuum mechanics.- Conservation of mass and momentum.- Conservation of energy.- One-dimensional wave equation.- Nonlinear effects, shocks.- Sobolev spaces II.- Linearized elasticity.- Ellipticity conditions.- Sobolev spaces III.- Sobolev spaces IV.- Sobolev spaces V.- Sobolev embedding theorem.- Fixed point theorems.- Brouwer's topological degree.- Time-dependent solutions I.- Time-dependent solutions II.- Time-dependent solutions III.- Uniqueness in 2 dimensions.- Traces.- Using compactness.- Existence of smooth solutions.- Semilinear models.- Size of singular sets.- Local estimates, compensated integrability.- Coriolis force.- Equation for the vorticity.- Boundary conditions in linearized elasticity.- Turbulence, homogenization.- G-convergence and H-convergence.- One-dimensional homogenization, Young measures.- Nonlocal effects I.- Nonlocal effects II.- A model problem.- Compensated compactness I.- Compensated compactness II.- Differential forms.- The compensated compactness method.- H-measures and variants.- Biographical Information.- Abbreviations and Mathematical Notation.
