Onsager relations and Eulerian hydrodynamics for systems with several conservation laws (original) (raw)

Mini-Workshop: Particle Systems with Several Conservation Laws: Fluctuations and Hydrodynamic Limit

Oberwolfach Reports, 2000

The Mini-Workshop is concerned with the large-scale description of microscopic many-particle systems with two or more conservation laws. This is topic of common interest for statistical mechanics, probability theory and PDE theory. The main difficulty lies in the proof of the hydrodynamic limit in terms of a system of (generically hyperbolic) PDE's which includes a proper treatment of shock and boundary discontinuities that result from the microscopic dynamics. Moreover, fundamental properties of current-carrying stationary states of such systems (which are not Gibbs states) are studied in terms of fluctuations of macroscopic quantities. Many powerful tools developed for particle systems (or PDE's respectively) with one conservation law have no obvious generalization to systems with two or more conservation laws and hence new mathematical ideas need to be developed. : 35L65, 60K35, 82C22.

Large-time behavior of entropy solutions of conservation laws

1999

We are concerned with the large-time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws. We present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions. These approaches allow the solutions of arbitrarily large oscillation without apriori assumption on the ways from which the solutions come. The relation between the large-time behavior of entropy solutions and the uniqueness of Riemann solutions leads to an extensive study of the uniqueness problem. We use a direct method to show the large-time behavior of large L ∞ solutions for a class of m × m systems including a model in multicomponent chromatography; we employ the uniqueness of Riemann solutions and the convergence of self-similar scaling sequence of solutions to show the asymptotic behavior of large BV solutions for the 3 × 3 system of Euler equations in thermoelasticity. These results indicate that the Riemann solution is the unique attractor of large discontinuous entropy solutions, whose initial data are L ∞ ∩ L 1 or BV ∩ L 1 perturbation of the Riemann data, for these systems. These approaches also work for proving the large-time behavior of approximate solutions to hyperbolic conservation laws.

On the structure of solutions of nonlinear hyperbolic systems of conservation laws

Communications on Pure and Applied Analysis, 2011

We are concerned with entropy solutions u in L ∞ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function η and any hyperplane t = const., if u satisfies a vanishing mean oscillation property on the half balls, then η(u) has a trace H d -almost everywhere on the hyperplane. For the general case, given any set E of finite perimeter and its inner unit normal ν : ∂ * E → S d and assuming the vanishing mean oscillation property of u on the half balls, we show that the weak trace of the vector field (η(u), q(u)), defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair (η, q). We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.

The entropy principle: from continuum mechanics to hyperbolic systems of balance laws

2005

We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. Examples are given in the case of extended thermodynamics for rarefied gases and in the case of a multi-temperature mixture of fluids.

Euler Equations and Related Hyperbolic Conservation Laws

Handbook of Differential Equations Evolutionary Equations, 2005

Some aspects of recent developments in the study of the Euler equations for compressible fluids and related hyperbolic conservation laws are analyzed and surveyed. Basic features and phenomena including convex entropy, symmetrization, hyperbolicity, genuine nonlinearity, singularities, BV bound, concentration and cavitation are exhibited. Global well-posedness for discontinuous solutions, including the BV theory and the L ~ theory, for the one-dimensional Euler equations and related hyperbolic systems of conservation laws is described. Some analytical approaches including techniques, methods and ideas, developed recently, for solving multidimensional steady problems are presented. Some multidimensional unsteady problems are analyzed. Connections between entropy solutions of hyperbolic conservation laws and divergence-measure fields, as well as the theory of divergence-measure fields, are discussed. Some further trends and open problems on the Euler equations and related multidimensional conservation laws are also addressed.