Showing papers on "Conservation law published in 1980"
Book•
01 Nov 1980
TL;DR: In this article, the authors present a model of relativistic kinetic theory with spin-1/2 particles, and derive the Transport Equation and the first Chapman-Enskog approximation.
Abstract: Preface. Historical background. Part A. Basic Equations. I. Elements of relativistic kinetic theory. II. Conservation laws and H-theorem. Part B. Derivation of the Transport Equation. III. Scalar particles. IV. Spin-1/2 particles. Part C. Linear Theory. V. First Chapman-Enskog approximation. VI. Transport coefficients. VII. Moment method. VIII. Propagation of sound waves. IX. Mathematical aspects of the linearized transport equation. Part D. Applications. X. Lepton systems. XI. Systems of hadrons. A Model. XII. Photon-Electron System. XIII. Reduction of the collision brackets. Bibliography. Author index. Subject index.
653 citations
TL;DR: In this article, a simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems is presented, which is formulated on the basis of conservation laws of elasticity and of fundamental relationships in fracture mechanics.
Abstract: A simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems is presented. The analysis is formulated on the basis of conservation laws of elasticity and of fundamental relationships in fracture mechanics. The problem is reduced to the determination of mixed-mode stress-intensity factor solutions in terms of conservation integrals involving known auxiliary solutions. One of the salient features of the present analysis is that the stress-intensity solutions can be determined directly by using information extracted in the far field. Several examples with solutions available in the literature are solved to examine the accuracy and other characteristics of the current approach. This method is demonstrated to be superior in its numerical simplicity and computational efficiency to other approaches. Solutions of more complicated and practical engineering fracture problems dealing with the crack emanating from a circular hole are presented also to illustrate the capacity of this method
555 citations
TL;DR: In this paper, a complete self-contained treatment of the stability and convergence properties of conservation form, monotone difference approximations to scalar conservation laws in several space variables is developed.
Abstract: : A complete self-contained treatment of the stability and convergence properties of conservation form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunov's scheme, the upwind scheme (Differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations. (Author)
362 citations
TL;DR: The wave propagation problem on an idealized nonlinear string, admitting both forward and backward waves, leads to a closely related system of conservation laws which are also solved in this article.
Abstract: where r = q~(u, v). This system models the propagation of forward longitudinal and transverse waves in a stretched elastic string which moves in a plane. The wave propagation problem on an idealized nonlinear string, admitting both forward and backward waves, leads to a closely related system of four conservation laws which we also solve. The feature of interest in system (1) is that the equations are non-strictly hyperbolic in the following sense. Introduce vector notation U = (u, v), F = (~b u, r v); then the system (1) can be differentiated to produce
280 citations
TL;DR: For the spherically symmetric system, this paper proved the existence of a new locally finite flux which can be interpreted to represent the total energy flux of matter and gravitational field, and studied the relation between the behavior of the event horizon and the energy flux across it.
Abstract: For the spherically symmetric system, we prove the existence o{ a new locally cotEen·ecl flux which can be interpreted to represent the total energy flux of matter and gravitational field. ·with the aid of this conservation law, we study the relation between the behavior of the event horizon and the energy flux across it and look for constraints imposed on the total energy radiated to infinity. Some implications of the results of this study to the backreaction problem in the black hole evaporation are discussed.
259 citations
TL;DR: In this paper, a local theory of weak solutions of first-order nonlinear systems of conservation laws is presented, where the transonic small disturbance equation is an example of this class of systems.
208 citations
01 May 1980
TL;DR: In this article, the authors prove related estimates on nonlinear evolution equations which are governed by homogeneous nonlinearities and apply to classes of nonlinear diffusion equations and to conservation laws.
Abstract: : It is well-known that solving the initial-value problem for the heat equation forward in time takes a 'rough' initial temperature into a temperature which is smooth at later times t > greater than 0. One aspect of this is the validity of certain estimates on tut when u is a solution of the heat equation. In this paper we prove related estimates on nonlinear evolution equations which are governed by homogeneous nonlinearities. The results apply to classes of nonlinear diffusion equations and to conservation laws. The results are interesting from the point of view of identifying a new 'regularization' mechanism and the estimates themselves cast new light on the nature of the solutions of some initial-value problems with rough initial data.
147 citations
TL;DR: In this paper, a very simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems in rectilinear anisotropic solids is presented, where the analysis is formulated on the basis of conservation laws of the elasticity and fundamental relationships in fracture mechanics.
Abstract: A very simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems in rectilinear anisotropic solids is presented. The analysis is formulated on the basis of conservation laws of anisotropic elasticity and of fundamental relationships in anisotropic fracture mechanics. The problem is reduced to a system of linear algebraic equations in mixed-mode stress intensity factors. One of the salient features of the present approach is that it can determine directly the mixed-mode stress intensity solutions from the conservation integrals evaluated along a path removed from the crack-tip region without the need of solving the corresponding complex near-field boundary value problem. Several examples with solutions available in the literature are solved to ensure the accuracy of the current analysis. This method is further demonstrated to be superior to other approaches in its numerical simplicity and computational efficiency. Solutions of more complicated and practical engineering problems dealing with the crack emanating from a circular hole in composites are presented also to illustrate the capacity of this method.
143 citations
TL;DR: It is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition.
Abstract: The stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme
131 citations
TL;DR: In this article, conservation of angular momentum and conservation of energy were used to place an upper bound on the fraction of electrons that can ever reach the wall of a perfectly conducting and perfectly absorbing cylindrical wall.
Abstract: A plasma consisting solely of particles of a single species is initially in the shape of a long column. It is confined by an axial magnetic field in a region of space bounded by a perfectly conducting and perfectly absorbing cylindrical wall. Conservation of angular momentum and conservation of energy are used to place an upper bound on the fraction of electrons that can ever reach the wall.
124 citations
TL;DR: It is shown that the S matrices of these models must factorize, and at least in one case that is checked explicitly, an analogous conservation law in the CPN model does not appear to survive quantization.
TL;DR: In this paper, the decay of solution to parabolic conservation laws is discussed. But the decay is not discussed in the context of partial differential equations (PDE), as in this paper.
Abstract: (1980). Decay of solution to parabolic conservation laws. Communications in Partial Differential Equations: Vol. 5, No. 4, pp. 449-473.
01 Jul 1980
TL;DR: The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law as discussed by the authors, and a hierarchy of conserved quantities for the Korteweg-de Vries equation.
Abstract: The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikii on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.
TL;DR: In this article, it was shown that in a symmetric background gauge field these conservation laws persist, but in modified form, and a further contribution to the conserved quantity occurs.
TL;DR: In this paper, the basic properties of propagation of electromagnetic waves in nonlinear media are a direct consequence of classical conservation laws for energy, and for components of momentum and angular momentum.
Abstract: Basic properties of propagation of electromagnetic waves in nonlinear media are a direct consequence of classical conservation laws for energy, and for components of momentum and angular momentum.
TL;DR: In this paper, a parametric Backlund transformation for principal chiral fields is constructed, which can then be used to derive local conservation laws, which are then used to obtain the conservation laws.
TL;DR: The direct and inverse scattering solution for the full three dimensional three-wave resonant interaction was solved in this paper, and an infinite set of conservation laws were found for the three-dimensional resonant interactions.
TL;DR: The nonlinear Schrodinger equation as mentioned in this paper provides a model for Langmuir evolution at low energy density and wavenumber, and the model is studied using virial theorem techniques.
Abstract: The nonlinear Schrodinger equation provides a model for Langmuir evolution at low energy density and wavenumber. This equation is studied using virial theorem techniques. Stationary solitons and pulsating solitons (related to ’’breathers’’) are found in one dimension, as well as collapsing packets in two or more dimensions. Initial wave‐packet collapse thresholds and times are found, with and without constant collisional damping. In three dimensions, a narrow collapsing core is observed to break away from an initial Gaussian packet and become asymptotically self‐similar with time.
TL;DR: In this paper, a nonlinear, time-dependent, hydromagnetic model is developed, based on the eight partial differential equations of resistive magnetohydrodynamics (MHD), which are expressed as a set of conservation laws in general, orthogonal, curvilinear coordinates in two space dimensions.
TL;DR: In this article, the effects of perturbations on the soliton solutions of the K-dV equation are investigated, and a system of equations which governs the slow modulation of the solution is derived systematically from the nonsecular conditions of the multiple time scale expansion.
Abstract: Effects of perturbations on the soliton solutions of the K-dV equation are investigated. A system of equations which governs the slow modulation of the soliton solution is derived systematically from the non-secular conditions of the multiple time scale expansion. An equation which governs the time evolution of the non-soliton part is also obtained from the first order of the expansion. The results are applied to one soliton and two soliton problems. In addition, contributions of the non-soliton part to the modified conservation laws are investigated briefly.
01 Mar 1980
TL;DR: In this paper, it was shown that the nonlinear functional used to make the Glimm Scheme converge, depends only on properties of the equations at E = 0, and that certain interaction differences are bounded by E as well as by the approaching waves.
Abstract: The constraints under which a gas at a certain state will evolve can be given by three partial differential equations which express the conservation of momentum, mass, and energy. In these equations, a particular gas is defined by specifying the constitutive relation e = e(u, S), where e = specific internal energy, v = specific volume, and S = specific entropy. The energy function e = -In u + (S/R) describes a polytropic gas for the exponent y = 1, and for this choice of e(V, S), global weak solutions for bounded measurable data having finite total variation were given by Nishida in [lo]. Here the following general existence theorem is obtained: let e,(v, S) be any smooth one parameter family of energy functions such that at E = 0 the energy is given by e&v, S) = -In v + (S/R). It is proven that there exists a constant C independent of E, such that, if E . (total variation of the initial data) < C, then there exists a global weak solution to the equations. Since any energy function can be connected to e&V, S) by a smooth parameterization, our results give an existence theorem for all the conservation laws of gas dynamics. As a corollary we obtain an existence theorem of Liu, Indiana Univ. Math. J. 26, No. 1 (1977) for polytropic gases. The main point in this argument is that the nonlinear functional used to make the Glimm Scheme converge, depends only on properties of the equations at E = 0. For general n x n systems of conservation laws, this technique provides an alternate proof for the interaction estimates in Glimm’s 1965 paper. The new result here is that certain interaction differences are bounded by E as well as by the approaching waves.
01 Oct 1980
TL;DR: In this paper, an implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form, which is maintained by use of approximate factorization techniques, and the numerical algorithm is first order in time and second order in space.
Abstract: An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form Computational efficiency is maintained by use of approximate factorization techniques The numerical algorithm is first order in time and second order in space A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm
01 Jan 1980
TL;DR: In this article, a stack of pivoted thin blades are attached to each blade for initiating selective independent movement within a common, externally-produced magnetic field to efficiently convert electrical print signals to kinetic energy in each printing tip of a vertical array thereof, thereby facilitating printing of symbols, characters and other indicia on underlying media with high resolution.
Abstract: A printer head for use in an impact printer of the dotmatrix type utilizes a stack of pivoted thin blades, each having a printing tip at one end thereof. A pancake coil is attached to each blade for initiating selective independent movement thereof within a common, externally-produced magnetic field to efficiently convert electrical print signals to kinetic energy in each printing tip of a vertical array thereof thereby facilitating printing of symbols, characters and other indicia on underlying media with high resolution. The single magnetic-field-producing means interacts with all pancake coils of the stack of printer blades to facilitate close spacing of the printing tips for superior character printing. Resilient members are integrally formed in each blade to support the moving structure with negligible loss, thereby increasing the printing speed of the stacked blade head.
TL;DR: In this article, the SO(n + 1)/SO(n ) and SU( n + 1/SO( n ) cases are treated, and a new class of field theoretical 2D models associated with symmetric spaces G/H is presented.
TL;DR: In this article, a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variation to stochastically processes and generalizations of the Euler equation and Noether's theorem are obtained.
Abstract: A theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes. Generalizations of the Euler equation and Noether's theorem are obtained and several conservation laws are discussed. An application to Nelson's probabilistic framework of quantum mechanics is also given.
TL;DR: In this article, an infinite sequence of conserved quantities follows from the Lax representation in both the Korteweg-de Vries and sine-Gordon systems, and these two sequences are related by a simple substitution.
Abstract: An infinite sequence of conserved quantities follows from the Lax representation in both the Korteweg-de Vries and sine-Gordon systems. We show that these two sequences are related by a simple substitution. In an appendix, two different methods of deriving conservation laws from the Lax representation are presented.
TL;DR: Two one-sided conservation form difference approximations to a scalar one-dimensional convex conservation law are introduced, respectively of first- and second-order accuracy and each has the minimum possible band-width.
Abstract: Two one-sided conservation form difference approximations to a scalar one-dimensional convex conservation law are introduced. These are respectively of first- and second-order accuracy and each has the minimum possible band-width. They are nonlinearly stable, they converge only to solutions satisfying the entropy condition, and they have sharp monotone profiles. No such stable approximation of order higher than two is possible. Dimensional splitting algorithms are constructed and used to approximate the small-disturbance equation of transonic flow. These approximations are also nonlinearly stable and without nonphysical limit solutions.
TL;DR: In this article, generalized conservation laws and gain-spread relations in the free-electron laser were derived for variable wiggler configurations, and it was suggested that bandwidth limitations on the optimization of the small-signal gain-to-spread ratio of a storage ring FEL might be overcome by transverse velocity filtering of the electrons.
Abstract: Generalized conservation laws and gain-spread relations in the free-electron laser are derived for variable wiggler configurations. The derivation follows from the general equations of paper I in this series. It is suggested that bandwidth limitations on the optimization of the small-signal gain-to-spread ratio of a storage ring FEL might be overcome by transverse velocity filtering of the electrons.
TL;DR: In this paper, a theory of fermions that is calculationally identical to the single-particle Dirac spinor theory is presented. But it does not have charge or magnetic moment.
Abstract: The Clifford algebra ${C}_{4}$ is used to construct a theory of fermions that is calculationally identical to the single-particle Dirac spinor theory. The anomalies in the conservation laws of the spinor theory are removed. A verifiable prediction is made that massless fermions cannot have charge or magnetic moment.
TL;DR: In this paper, the Landau equation is obtained for a system of particles of arbitrary statistics by expanding the semiclassical Boltzmann collision integral in the scattering angle, and the equation is next cast into the Fokker-Planck form.
Abstract: The Landau equation is obtained, for a system of particles of arbitrary statistics, by expanding the semiclassical Boltzmann collision integral in the scattering angle. The equation is next cast into the Fokker-Planck form. Both the nonrelativistic and the relativistic cases are considered. In each case, the conservation laws and the H theorem for the obtained equation are demonstrated. Generalization of the equation involving inelastic binary collisions is investigated.