In honor of Pi Day (3.14), the AMS Bookstore is offering a 25% discount on select AMS titles. Go to www.ams.org/bookstore/pi to view this list of Pi-related books. Use market code PI314 when ordering. Offer expires 4/14/14.
Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations
The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of a slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman-Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad's moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert's 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton's iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.
In the American Mathematical Society’s first-ever book for kids (and kids at heart!), mathematician and author Richard Evan Schwartz leads math lovers of all ages on an innovative and strikingly illustrated journey through the infinite number system. This book is enthusiastically recommended to every teacher, parent and grandparent, student, child, or other individual interested in exploring the vast universe of numbers.
Student Mathematical Library Series
The books presented in this series serve as pivotal stepping stones in a student’s journey from mathematical coursework to research. Beneficial to beginning and advanced mathematicians alike.
Graduate Studies in Mathematics Series
Volumes in this series are specifically designed as graduate studies texts, but are also suitable for supplemental course reading and make ideal independent study resources.