Hyperbolic conservation laws with relaxation

The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman-Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.The paper was written at Mittag-Leffler Institute; the author wants to thank the Institute for the visiting position in 1986. This work was supported in part by an NSF grant     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.     

Torsion and conservation laws

The interaction of spinor and electromagnetic fields with the torsion of space-time is studied within the framework of the Einstein-Cartan theory. The equivalent nonlinear theory in Riemann space is obtained. The conservation laws for the vector and pseudovector currents are investigated in the nonlinear theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–52, February, 1986.     

Syzygies and conservation laws

We show how the notion of syzygies of a system of partial differential equations allows to derive some conservation laws, for the case of Maxwell and Proca systems. More in general, we apply some classical tools in algebraic analysis to derive properties of the solutions of the previous systems like their integral representations.     

Energy conservation laws and antimatter

In field theory, an energy-momentum tensor fails to be conserved if internal symmetries are broken. This revised version was published online in August 2006 with corrections to the Cover Date.     

Noether equations and conservation laws

The purpose of the paper is to present a rigorous derivation of the relation between conservation laws and transformations leaving invariant the action integral. The (space-)time development of a physical system is represented by a cross section of a product bundleM. A Lagrange function is defined as a mapping where is the bundle space of the first jet extension ofM. A symmetry transformation is defined as a bundle automorphism ofM, carrying solutions of the Euler-Lagrange equation into solutions of the same equation. An important class of symmetry transformations is that of generalized invariant transformations: they are defined by specifying their action on the Euler-Lagrange equation. The generators of generalized invariant transformations are solutions of a system of linear, homogeneous partial differential equation (Noether equations). The set of all solutions of these equations has a natural structure of Lie algebra. In a simple manner, the Noether equations give rise to differential conservation laws.Supported by Air Force Office of Scientifie Research and Aeronautical Research Laboratories.On leave of absence from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Duality for systems of conservation laws

Letters in Mathematical Physics - For one-dimensional systems of conservation laws admitting two additional conservation laws, we assign a ruled hypersurface of codimension two in projective space....     

Nonlocal integral conservation laws and tetradic currents in the Einstein-Cartan theory I. Nonlocal conservation laws

Nonlocal integral conservation laws in the Einstein-Cartan theory for the energy-momentum tensor of general-form sources, in essence being an integral equivalent of convoluted Bianchi identities, are obtained.     

Singular Lyapunov spectra and conservation laws

We give analytic arguments and numerical evidence to show that the presence of conservation laws can produce a singularity in the spectrum of Lyapunov exponents for extended dynamical systems of low spatial dimensionality. This phenomenon can be used, e.g., for finding hidden conservation laws. (c) 1995 American Institute of Physics.     

Toda lattice hierarchy and conservation laws

Time evolutions of the Toda lattice hierarchies of Ueno and Takasaki are induced by Hamiltonians which are conservation laws for the original (one and two dimensional) Toda lattice obtained by Olive and Turok. Moreover these Hamiltonians for two dimensional Toda lattice hierarchy are also conserved quantities of the two component KP hierarchy in which that system is embedded. The one dimensional Toda lattice hierarchy is characterized by the bilinear relations, and a new version of the one dimensional Toda lattice hierarchy is constructed. Generalized Toda lattice hierarchies associated to all affine Lie algebras are presented.     

Uncertainty quantification for systems of conservation laws

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers’ equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.     

Incompressible quantum liquids and new conservation laws

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this type of symmetry encapsulates two prominent examples of novel states of matter, namely, the fractional quantum Hall liquid and the quantum dimer liquid. We discuss the relevance of this class of Hamiltonian to the search for featureless Mott insulators.     

Covariant conservation laws from the palatini formalism

Covariant conservation laws in the Palatini formalism are derived. The result indicates that the gravitational part of conserved charges in general relativity should be calculated from a combination of Komar's strongly conserved current and the Einstein tensor. This implies that the set of complete diffeomorphism charges of a gravitating system consisting of scalar matter is described by Komar's vector density, and that the identification of gravitational energy and momentum reduces to two choices: a choice of relative weights of the contributions resulting from Komar's current and from the Einstein tensor, and a choice of preferred vector fields in space-time. A proposal is made which yields energy and momentum as scalars under diffeomorphisms and as a Lorentz vector in tangent space. Furthermore, the result can be used to identify covariant conservation laws holding separately for the matter contributions to diffeomorphism charges.     

Canonical connectivity and conservation laws in gravitation

The Riemann-canonical tetrad which defines, in the general case, six 2-directions of extreme values of the sectional Riemann curvature is introduced. It localizes the gravitational energy, defines the canonical 1-form of connectivity and canonical tetrad currents introduced by the author, which provide the conservation laws in the Einstein-Cartan theory.     

Symmetries and conservation laws in gauge theories

The relationship between conservation laws and symmetries of space-time is familiar. Here it is shown that in a symmetric background gauge field these conservation laws persist, but in modified form. A further contribution to the conserved quantity occurs. It is determined by the gauge transformation which, when acting together with some coordinate transformation, leaves the symmetric background gauge-potential invariant. The addition to the constant of motion can also be interpreted as arising from the dynamical interaction of the gauge field with the system. A calssical example is the angular momentum conservation law for a charged particle moving in the field of a magnetic monopole. Generalizations of this are here derived.