An Introduction to Navier’Stokes Equation and Oceanography

Luc Tartar (auth.)3540357432, 9783540357438, 9783540365457

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.

Table of contents :
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