Partial Differential Equations III [electronic resource] : Nonlinear Equations /
Taylor, Michael E.author.
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. In this second edition, there are seven new sections including Sobolev spaces on rough domains, boundary layer phenomena for the heat equation, an extension of complex interpolation theory, and Navier-Stokes equations with small viscosity. In addition, several other sections have been substantially rewritten, and numerous others polished to reflect insights obtained through the use of these books over time. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC. Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(SIAM Review, June 1998).
Description based upon print version of record.
Partial Differential Equations III; Contents; Contents of Volumes I and II; Preface; 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 Equations1 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 systems13 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; A Morrey spaces; B Leray-Schauder fixed-point theorems; 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 systems10 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 Navier-Stokes equations; 5 Viscous flows on bounded regions; 6 Vanishing viscosity limits; 7 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
Includes bibliographical references and index.
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. In this second edition, there are seven new sections including Sobolev spaces on rough domains, boundary layer phenomena for the heat equation, an extension of complex interpolation theory, and Navier-Stokes equations with small viscosity. In addition, several other sections have been substantially rewritten, and numerous others polished to reflect insights obtained through the use of these books over time. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC. Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(SIAM Review, June 1998).
Partial differential equations.
Partial Differential Equations.
Applied mathematical sciences