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基本説明
Will be of interest to researchers in partial differential equations, graduate level students, and engineers involved in the field of partial differential equations.
Full Description
This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider Partial Differential Equations, such as some nonlinear conservation laws, as linear kinetic (or semi-kinetic) equations acting on a nonlinear quantity. It also aids the reader with using Fourier transform, regularisation, and moments methods to provide new approaches for proving uniqueness, regularizing effects, and a priori bounds.
Special care has been given to introduce basic tools, including the classical Boltzmann formalism to derive compressible fluid dynamics, the study of oscillatons through the kinetic defect measure, and an elementary construction of solutions to scalar conservation laws. More advanced material contains regularizing effects through averaging lemmas, existence of global large solutions to isentropic gas dynamics, and a new uniqueness proof for scalar conservation laws. Sections are also devoted to the derivation of numerical approaches, the 'kinetic schemes', and the analysis of their theoretical properties.
Contents
Foreword ; 1. A brief overview of the kinetic approach ; 2. The function chi, entropies and representation of nonlinear functions ; 3. Kinetic formulation of multidimensional scalar conservation laws ; 4. Uniqueness of solutions to scalar conservation laws and consequences ; 5. Compactness, cancellation of oscillations and averaging lemmas ; 6. Kinetic schemes for SCL ; 7. Isentropic gas dynamics ; 8. Kinetic schemes for gas dynamics ; Appendices
