Customer Reviews

Partial Differential Equations III: Nonlinear Equations (Applied Mathematical Sciences)

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If you have a nonlinear problem and you are looking for results on local and global existence, uniqueness and regularity of solutions you should check this book as a primary reference, and let me give you two reasons for that.

1. The statements of the theorems are general enough to fit into a large number of nonlinear PDE problem such as elliptic, parabolic or hyperbolic equations. In fact, many of M. Taylor's theorems have two version, one in a euclidean space and another on a manifold.

2. The Fixed point methods in Banach spaces. This is my favorite part because Taylor shows in a very clear and emphatic manner why Fixed point theorems are important and how to use them properly.

Just a warning: the level of this book is for graduate students in mathematics.

For a more basic (or even first) approach to PDEs my call is always Evans' book Partial Differential Equations. For a second opinion on nonlinear equations I strongly recomend Smoller's book Shock Waves and Reaction-Diffusion Equations.

One of the most important and subtle features of mathematics is the great difference existing between linearity and non-linearity. One always tries to linearize whenever possible, because linear problems are easier to solve, but unfortunately the world is not linear, and we have to learn how to deal with non-linear problems.

This volume discusses thoroughly some very important cases of non-linear PDE's which are important in mathematical physics or that possess intrinsic theoretical interest.

The contents are: Function space and operator theory for nonlinear analysis; nonlinear elliptic equations; nonlinear parabolic equations; nonlinear hyperbolic equations; Euler and Navier-Stokes equations for incompressible fluids; Einstein's equations.

A little bit more advanced and speciallized than the other two volumes, but still useful for a broad community. A graduate student with a solid background will find it perfectly acquaintable. Includes lots of excercises and references.

Please read my other reviews (just click on my name above).