Book Review: Global classical solutions for nonlinear evolution equations

Walter A. Strauss
1993 Bulletin of the American Mathematical Society  
Most dynamical problems in physics are governed by evolution equations. Evolution equations are partial differential equations whose solutions one studies as functions of a distinguished independent variable t, called time. Examples are heat flow or diffusion governed by parabolic equations, vibration or electromagnetism governed by hyperbolic equations, and quantum mechanics governed by Schrödinger equations. These are the three types of equations studied in this book. Of course, there are
more » ... other famous evolution equations such as the Navier-Stokes equations of incompressible fluids, the Yang-Mills equations in Minkowski space, reaction-diffusion equations, hyperbolic conservation laws, the Korteweg-de Vries equation, and so on. The basic problem is to understand what happens when "arbitrary" initial data are specified, say at t = 0. The mathematician's job is to specify the word "arbitrary" and the properties of the solutions. Specifically, one asks for ( 1 ) local existence (to find classes of functions for which at least one solution exists for at least a nontrivial interval of time), (2) global existence (to find classes for which solutions exist for all time 0 < t < oo), (3) uniqueness (to find function classes in which there is at most one solution with given initial data), (4) regularity of the solutions (differentiability, etc.), (5) singularities (Which initial data lead
doi:10.1090/s0273-0979-1993-00415-4 fatcat:l6brapwlnzcbtbfhjjcoqfjjbq