The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kähler manifolds.
Author(s): Michael E. Taylor
Series: Applied Mathematical Sciences 117
Edition: 3
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 755
City: Cham
Tags: Nonlinear Partial Differential Equations, Nonlinear Analysis, Euler Equations, Navier-Stokes Equations, Einstein's Equations
Contents of Volumes I and II
Preface
Acknowledgments
Introduction to the Second Edition
Introduction to the Third Edition
Contents
13 Function Space and Operator Theory for Nonlinear Analysis
1 Lp-Sobolev spaces
2 Sobolev imbedding theorems
3 Gagliardo–Nirenberg–Moser estimates
4 Trudinger's inequalities
5 Singular integral operators on Lp
6 The spaces Hs,p
7 Lp-spectral theory of the Laplace operator
8 Hölder spaces and Zygmund spaces
9 Pseudodifferential operators with nonregular symbols
10 Paradifferential operators
11 Young measures and fuzzy functions
12 Hardy spaces
A Variations on complex interpolation
References
14 Nonlinear Elliptic Equations
1 A class of semilinear equations
2 Surfaces with negative curvature
3 Local solvability of nonlinear elliptic equations
4 Elliptic regularity I (interior estimates)
5 Isometric imbedding of Riemannian manifolds
6 Minimal surfaces
6B Second variation of area
7 The minimal surface equation
8 Elliptic regularity II (boundary estimates)
9 Elliptic regularity III (DeGiorgi–Nash–Moser theory)
10 The Dirichlet problem for quasi-linear elliptic equations
11 Direct methods in the calculus of variations
12 Quasi-linear elliptic systems
12B Further results on quasi-linear systems
13 Elliptic regularity IV (Krylov–Safonov estimates)
14 Regularity for a class of completely nonlinear equations
15 Monge–Ampere equations
16 Elliptic equations in two variables
17 Overdetermined elliptic systems
A Morrey spaces
B Leray–Schauder fixed-point theorems
C The Weyl tensor
References
15 Nonlinear Parabolic Equations
1 Semilinear parabolic equations
2 Applications to harmonic maps
3 Semilinear equations on regions with boundary
4 Reaction-diffusion equations
5 A nonlinear Trotter product formula
6 The Stefan problem
7 Quasi-linear parabolic equations I
8 Quasi-linear parabolic equations II (sharper estimates)
9 Quasi-linear parabolic equations III (Nash–Moser estimates)
References
16 Nonlinear Hyperbolic Equations
1 Quasi-linear, symmetric hyperbolic systems
2 Symmetrizable hyperbolic systems
3 Second-order and higher-order hyperbolic systems
4 Equations in the complex domain and the Cauchy–Kowalewsky theorem
5 Compressible fluid motion
6 Weak solutions to scalar conservation laws; the viscosity method
7 Systems of conservation laws in one space variable; Riemann problems
8 Entropy-flux pairs and Riemann invariants
9 Global weak solutions of some 22 systems
10 Vibrating strings revisited
References
17 Euler and Navier–Stokes Equations for Incompressible Fluids
1 Euler's equations for ideal incompressible fluid flow
2 Existence of solutions to the Euler equations
3 Euler flows on bounded regions
4 Euler equations on a rotating surface
5 Navier–Stokes equations
6 Viscous flows on bounded regions
7 Vanishing viscosity limits
8 From velocity field convergence to flow convergence
A Regularity for the Stokes system on bounded domains
References
18 Einstein's Equations
1 The gravitational field equations
2 Spherically symmetric spacetimes and the Schwarzschild solution
3 Stationary and static spacetimes
4 Orbits in Schwarzschild spacetime
5 Coupled Maxwell–Einstein equations
6 Relativistic fluids
7 Gravitational collapse
8 The initial-value problem
9 Geometry of initial surfaces
10 Time slices and their evolution
References
Index
