With this book, the author gives an overview of the mathematical connections between kinetic theory and conservation laws. The main advantage of such a connection is that tools from linear theory can be applied to nonlinear problems because the kinetic approach allows to rewrite the original nonlinear equation as a linear equation acting on a nonlinear quantity. Tools like Fourier transform, moment methods, and regularization by convolution can then be applied and either yield new results (like regularity or a priori bounds for the solution of the nonlinear equation), or known results are recovered with a different proof (like existence and uniqueness for scalar conservation laws). Apart from giving a new perspective on the theory of conservation laws, the author also shows how the kinetic approach can be used to construct numerical methods with interesting features.
A nice feature of the book is the rapid introduction into kinetic formulations presented in the first chapter. By skipping many technical details, the reader quickly learns about typical tools and applications. This motivates to read further and to collect the missing details in the subsequent chapters. The different kinetic techniques are explained using the following examples: scalar conservation laws, isentropic gas dynamics and \(2\times 2\) systems, multidimensional Saint-Venant system, multidimensional gas dynamics, Ginzburg-Landau line energies.
The second chapter is devoted to the study of a particular indicator function which appears as equilibrium distribution in the kinetic approach to scalar conservation laws. It is shown how the indicator function can be used to characterize weak limits of nonlinear functions which links the approach to the concept of Young measures.
In chapters three to five, the paradigmatic example of multidimensional scalar conservation laws is carefully presented and examined. The central tool in chapters three and four is the hydrodynamical limit which connects the conservation law with a particular semilinear relaxation-type kinetic equation. This equation plays a similar role as the viscous extension of the conservation law but has the advantage that the type of the equation is not changed. It allows to obtain extensions of the classical Kruzkov existence and uniqueness results. In chapter five, the kinetic formulation is used to prove various regularity effects for conservation laws and chapter six concentrates on the extension of the kinetic formulation to the finite volume method.
In chapter seven, the approach is extended to the study of the \(2\times 2\) system of isentropic gas dynamics in a single space dimension. An important difference to the case of scalar conservation laws is the loss of a purely kinetic transport operator in the kinetic formulation which now depends also on the macroscopic velocity. Even though fewer tools are available for such semi-kinetic equations, it is still possible to use the approach to prove a priori bounds, regularizing effects and time decay. The presentation closes with a short chapter on kinetic schemes for equations of gas dynamics.
Altogether, the book is a good introduction into the theory of kinetic formulations of conservation laws written by the leading expert in the field. A list of open problems and the extensive bibliography also makes it a starting point for further research.