Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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Applied Mathematical Sciences
Volume 117
Editors
S.S. Antman
Department of Mathematics
and
Institute for Physical
Science and Technology
University of Maryland
College Park, MD 20742-4015
USA
ssa@math.umd.edu
L. Sirovich
Laboratory of Applied Mathematics
Department of Biomathematical
Sciences
Mount Sinai School of Medicine
New York, NY 10029-6574
lsirovich@rockefeller.edu
J.E. Marsden
Control and Dynamical Systems,
107-81
California Institute of Technology
Pasadena, CA 91125
USA
marsden@cds.caltech.edu
Advisors
L. Greengard P. Holmes J. Keener
J. Keller R. Laubenbacher B.J. Matkowsky
A. Mielke C.S. Peskin K.R. Sreenivasan A. Stevens A. Stuart
For further volumes:
http://www.springer.com/series/34
Michael E. Taylor
Partial Differential
Equations III
Nonlinear Equations
Second Edition
123
Michael E. Taylor
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599
USA
met@math.unc.edu
ISSN 0066-5452
ISBN 978-1-4419-7048-0 e-ISBN 978-1-4419-7049-7
DOI 10.1007/978-1-4419-7049-7
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010937758
Mathematics Subject Classification (2010): 35A01, 35A02, 35J05, 35J25, 35K05, 35L05, 35Q30,
35Q35, 35S05
c
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Springer is part of Springer Science+Business Media (www.springer.com)
To my wife and daughter, Jane Hawkins
and Diane Taylor
Contents
Contents of Volumes I and II .................................................. xi
Preface............................................................................ xiii
13 Function Space and Operator Theory for Nonlinear Analysis ....... 1
1 L
p
-Sobolev spaces .................................................. 2
2 Sobolev imbedding theorems ....................................... 4
3 Gagliardo–Nirenberg–Moser estimates ............................ 8
4 Trudinger’s inequalities ............................................. 14
5 Singular integral operators on L
p
.................................. 17
6 The spaces H
s;p
..................................................... 24
7 L
p
-spectral theory of the Laplace operator ........................ 31
8H¨older spaces and Zygmund spaces ................................ 40
9 Pseudodifferential operators with nonregular symbols ............ 50
10 Paradifferential operators ........................................... 60
11 Young measures and fuzzy functions ............................... 74
12 Hardy spaces ......................................................... 86
A Variations on complex interpolation ................................ 96
References ........................................................... 102
14 Nonlinear Elliptic Equations ............................................. 105
1 A class of semilinear equations ..................................... 107
2 Surfaces with negative curvature ................................... 119
3 Local solvability of nonlinear elliptic equations ................... 127
4 Elliptic regularity I (interior estimates) ............................. 135
5 Isometric imbedding of Riemannian manifolds.................... 147
6 Minimal surfaces .................................................... 152
6B Second variation of area............................................. 168
7 The minimal surface equation ...................................... 176
8 Elliptic regularity II (boundary estimates) ......................... 185
9 Elliptic regularity III (DeGiorgi–Nash–Moser theory) ............ 196
10 The Dirichlet problem for quasi-linear elliptic equations ......... 208
11 Direct methods in the calculus of variations ....................... 222
12 Quasi-linear elliptic systems ........................................ 229
12B Further results on quasi-linear systems ............................. 244
13 Elliptic regularity IV (Krylov–Safonov estimates) ................ 258
viii Contents
14 Regularity for a class of completely nonlinear equations.......... 273
15 Monge–Ampere equations .......................................... 282
16 Elliptic equations in two variables .................................. 294
A Morrey spaces ....................................................... 299
B Leray–Schauder fixed-point theorems .............................. 302
References ........................................................... 304
15 Nonlinear Parabolic Equations .......................................... 313
1 Semilinear parabolic equations ..................................... 314
2 Applications to harmonic maps ..................................... 325
3 Semilinear equations on regions with boundary ................... 332
4 Reaction-diffusion equations........................................ 335
5 A nonlinear Trotter product formula................................ 353
6 The Stefan problem.................................................. 362
7 Quasi-linear parabolic equations I .................................. 376
8 Quasi-linear parabolic equations II (sharper estimates) ........... 387
9 Quasi-linear parabolic equations III (Nash–Moser estimates) .... 396
References ........................................................... 407
16 Nonlinear Hyperbolic Equations ........................................ 413
1 Quasi-linear, symmetric hyperbolic systems ....................... 414
2 Symmetrizable hyperbolic systems ................................. 425
3 Second-order and higher-order hyperbolic systems................ 432
4 Equations in the complex domain and the Cauchy–
Kowalewsky theorem ................................................ 445
5 Compressible fluid motion .......................................... 448
6 Weak solutions to scalar conservation laws; the viscosity method 457
7 Systems of conservation laws in one space variable;
Riemann problems................................................... 472
8 Entropy-flux pairs and Riemann invariants......................... 498
9 Global weak solutions of some 2 2 systems ..................... 509
10 Vibrating strings revisited ........................................... 517
References ........................................................... 524
17 Euler and Navier–Stokes Equations for Incompressible Fluids ...... 531
1 Euler’s equations for ideal incompressible fluid flow.............. 532
2 Existence of solutions to the Euler equations ...................... 542
3 Euler flows on bounded regions ......................
.............. 553
4 Navier–Stokes equations ............................................ 561
5 Viscous flows on bounded regions.................................. 575
6 Vanishing viscosity limits ........................................... 586
7 From velocity field convergence to flow convergence ............. 599
A Regularity for the Stokes system on bounded domains............ 605
References ........................................................... 610
Contents ix
18 Einstein’s Equations....................................................... 615
1 The gravitational field equations.................................... 616
2 Spherically symmetric spacetimes and the
Schwarzschild solution .............................................. 626
3 Stationary and static spacetimes .................................... 639
4 Orbits in Schwarzschild spacetime ................................. 649
5 Coupled Maxwell–Einstein equations .............................. 656
6 Relativistic fluids .................................................... 659
7 Gravitational collapse ............................................... 670
8 The initial-value problem ........................................... 677
9 Geometry of initial surfaces......................................... 687
10 Time slices and their evolution ..................................... 699
References ........................................................... 705
Index .............................................................................. 711