An Introduction to Navier-Stokes Equation and Oceanography (eBook)

Luc Tartar studied at Ecole Polytechnique in Paris, France, 1965-1967, where he was taught by Laurent Schwartz and Jacques-Louis Lions in mathematics, and by Jean Mandel in continuum mechanics.He did research at Centre National de la Recherche Scientifique, Paris, France, 1968-1971, working under the direction of Jacques-Louis Lions for his thèse d'état, 1971.He taught at Université Paris IX-Dauphine, Paris, France, 1971-1974, at University of Wisconsin, Madison, WI, 1974-1975, at Université de Paris-Sud, Orsay, France, 1975-1982.He did research at Commissariat à l'Energie Atomique, Limeil, France, 1982-1987.In 1987, he was elected Correspondant de l'Académie des Sciences, Paris, in the section Mécanique.Since 1987 he has been teaching at Carnegie Mellon University, Pittsburgh, PA, where he has been University Professor of Mathematics since 1994.Partly in collaboration with François Murat, he has specialized in the development of new mathematical tools for solving the partial differential equations of continuum mechanics (homogenization, compensated compactness, H-measures), pioneering the study of microstructures compatible with the partial differential equations describing the physical balance laws, and the constitutive relations.He likes to point out the defects of many of the models which are used, as a natural way to achieve the goal of improving our understanding of mathematics and of continuum mechanics.

Preface 7
Introduction 15
Detailed Description of Lectures 18
Contents 25
Basic physical laws and units 28
Radiation balance of atmosphere 33
Conservations in ocean and atmosphere 37
Sobolev spaces I 41
Particles and continuum mechanics 49
Conservation of mass and momentum 57
Conservation of energy 63
One-dimensional wave equation 68
Nonlinear e.ects, shocks 74
Sobolev spaces II 82
Linearized elasticity 88
Ellipticity conditions 93
Sobolev spaces III 97
Sobolev spaces IV 101
Sobolev spaces V 106
Sobolev embedding theorem 110
Fixed point theorems 117
Brouwer’s topological degree 123
Time-dependent solutions I 128
Time-dependent solutions II 134
Time-dependent solutions III 139
Uniqueness in 2 dimensions 145
Traces 148
Using compactness 155
Existence of smooth solutions 160
Semilinear models 164
Size of singular sets 172
Local estimates, compensated integrability 177
Coriolis force 182
Equation for the vorticity 185
Boundary conditions in linearized elasticity 187
Turbulence, homogenization 191
G-convergence and H-convergence 195
One-dimensional homogenization, Young 200
measures 200
Nonlocal e.ects I 204
Nonlocal e.ects II 209
A model problem 213
Compensated compactness I 217
Compensated compactness II 221
Di.erential forms 224
The compensated compactness method 229
H-measures and variants 235
Biographical Information 243
Abbreviations and Mathematical Notation 247
References 251
Index 253