An Introduction to Navier-Stokes Equation and Oceanography (Lecture Notes of the Unione Matematica Italiana Book 1)

The __Introduction to Navier-Stokes Equation and Oceanography__ corresponds to a graduate course in mathematics, taught at Carnegie Mellon University in the spring of 1999. Comments were added to the lecture notes distributed to the students, as well as short biographical information for all scientists mentioned in the text, the purpose being to show that the creation of scientific knowledge is an international enterprise, and who contributed to it, from where, and when. The goal of the course is to teach a critical point of view concerning the partial differential equations of continuum mechanics, and to show the need for developing new adapted mathematical tools. Basic physical laws and units....Pages 1-5 Radiation balance of atmosphere....Pages 7-10 Conservations in ocean and atmosphere....Pages 11-14 Sobolev spaces I....Pages 15-22 Particles and continuum mechanics....Pages 23-30 Conservation of mass and momentum....Pages 31-36 Conservation of energy....Pages 37-41 One-dimensional wave equation....Pages 43-48 Nonlinear effects, shocks....Pages 49-56 Sobolev spaces II....Pages 57-62 Linearized elasticity....Pages 63-67 Ellipticity conditions....Pages 69-72 Sobolev spaces III....Pages 73-76 Sobolev spaces IV....Pages 77-81 Sobolev spaces V....Pages 83-86 Sobolev embedding theorem....Pages 87-93 Fixed point theorems....Pages 95-100 Brouwer's topological degree....Pages 101-105 Time-dependent solutions I....Pages 107-112 Time-dependent solutions II....Pages 113-117 Time-dependent solutions III....Pages 119-124 Uniqueness in 2 dimensions....Pages 125-127 Traces....Pages 129-135 Using compactness....Pages 137-141 Existence of smooth solutions....Pages 143-146 Semilinear models....Pages 147-154 Size of singular sets....Pages 155-159 Local estimates, compensated integrability....Pages 161-165 Coriolis force....Pages 167-169 Equation for the vorticity....Pages 171-172 Boundary conditions in linearized elasticity....Pages 173-176 Turbulence, homogenization....Pages 177-180 G-convergence and H-convergence....Pages 181-185 One-dimensional homogenization, Young measures....Pages 187-190 Nonlocal effects I....Pages 191-195 Nonlocal effects II....Pages 197-200 A model problem....Pages 201-204 Compensated compactness I....Pages 205-208 Compensated compactness II....Pages 209-211 Differential forms....Pages 213-217 The compensated compactness method....Pages 219-224 H-measures and variants....Pages 225-232 Biographical Information....Pages 233-236 Abbreviations and Mathematical Notation....Pages 237-240 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.

the Introduction To Navier-stokes Equation And Oceanography Corresponds To A Graduate Course In Mathematics, Taught At Carnegie Mellon University In The Spring Of 1999. Comments Were Added To The Lecture Notes Distributed To The Students, As Well As Short Biographical Information For All Scientists Mentioned In The Text, The Purpose Being To Show That The Creation Of Scientific Knowledge Is An International Enterprise, And Who Contributed To It, From Where, And When. The Goal Of The Course Is To Teach A Critical Point Of View Concerning The Partial Differential Equations Of Continuum Mechanics, And To Show The Need For Developing New Adapted Mathematical Tools.

"This volume corresponds to a graduate course in mathematics, taught at Carnegie Mellon University in the spring of 1999. Comments were added to the lecture notes distributed to the students, as well as short biographical information for all scientists mentioned in the text, the purpose being to show that the creation of scientific knowledge is an international enterprise, and who contributed to it, from where, and when."--BOOK JACKET This text corresponds to a graduate mathematics course taught at Carnegie Mellon University in the spring of 1999. Included are comments added to the lecture notes, a bibliography containing 23 items, and brief biographical information for all scientists mentioned in the text, thus showing that the creation of scientific knowledge is an international enterprise.