Showing papers on "Conservation law published in 2015"
TL;DR: In this paper, a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law is developed, where local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state.
Abstract: We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.
322 citations
TL;DR: In this paper, the authors consider the non-equilibrium dynamics in isolated systems, described by quantum field theories (QFTs), and demonstrate that in order to obtain a correct description of the stationary state, it is necessary to take into account conservation laws that are not (ultra-)local in the usual sense of QFT, but fulfil a significantly weaker form of locality.
Abstract: We consider the non-equilibrium dynamics in isolated systems, described by quantum field theories (QFTs) After being prepared in a density matrix that is not an eigenstate of the Hamiltonian, such systems are expected to relax locally to a stationary state In a presence of local conservation laws, these stationary states are believed to be described by appropriate generalized Gibbs ensembles Here we demonstrate that in order to obtain a correct description of the stationary state, it is necessary to take into account conservation laws that are not (ultra-)local in the usual sense of QFT, but fulfil a significantly weaker form of locality We discuss implications of our results for integrable QFTs in one spatial dimension
172 citations
TL;DR: In this article, a new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations.
Abstract: A new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations. The proposed approach is demonstrated on subdiffusion and diffusion-wave equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the equations with the Riemann–Liouville and Caputo time-fractional derivatives of order $$\alpha \in (0,2)$$
. Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-wave equations by their Lie point symmetries.
147 citations
TL;DR: Four time-fractional generalizations of the Kompaneets equation are considered and it is shown that all approximations have nontrivial symmetries and conservation laws.
116 citations
TL;DR: In this paper, a complex short pulse equation and a coupled complex short equation are proposed to describe ultra-short pulse propagation in optical fibers, which are integrable due to the existence of Lax pairs and infinite number of conservation laws.
107 citations
TL;DR: In this paper, the existence of a point transformation under which the WdW equation is invariant is shown to be equivalent to conservation laws for the field equations, which indicates the presence of analytical solutions.
Abstract: We give a general method to find exact cosmological solutions for scalar-field dark energy in the presence of perfect fluids. We use the existence of invariant transformations for the Wheeler De Witt (WdW) equation. We show that the existence of a point transformation under which the WdW equation is invariant is equivalent to the existence of conservation laws for the field equations, which indicates the existence of analytical solutions. We extend previous work by providing exact solutions for the Hubble parameter and the effective dark-energy equation of state parameter for cosmologies containing a combination of perfect fluid and a scalar field whose self-interaction potential is a power of hyperbolic functions. We find solutions explicitly when the perfect fluid is radiation or cold dark matter and determine the effects of nonzero spatial curvature. Using the Planck 2015 data, we determine the evolution of the effective equation of state of the dark energy. Finally, we study the global dynamics using dimensionless variables. We find that if the current cosmological model is Liouville integrable (admits conservation laws) then there is a unique stable point which describes the de-Sitter phase of the universe.
100 citations
TL;DR: This paper focuses on unstructured triangular meshes and proposes the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables in the Riemann problem.
100 citations
TL;DR: In this article, it was shown that the Kržkov entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law.
Abstract: We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1–Wasserstein topology (respectively in \({\mathbf{L^{1}_{loc}}}\)) to the unique Kružkov entropy solution of the conservation law. The initial data are taken in \({\mathbf{L}^\infty}\), nonnegative, and with compact support, hence we are able to handle densities with a vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, for example in the Lighthill-Whitham-Richards model for traffic flow) with a possible degenerate slope near the vacuum state. The proof of the result is based on discrete \({\mathbf{BV}}\) estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect \({\mathbf{L}^\infty \mapsto \mathbf{BV}}\) for nonlinear scalar conservation laws is intrinsic to the discrete model.
96 citations
TL;DR: In this article, conservation laws restricting the slow drift of energy between the different normal modes due to non-linearities were discovered in gravity-scalar field systems, and the nature of these conservation laws was best understood through an appeal to averaging methods which allow one to derive an effective Lagrangian or Hamiltonian description of the slow energy transfer between the normal modes.
Abstract: We continue our analytic investigations of non-linear spherically symmetric perturbations around the anti-de Sitter background in gravity-scalar field systems, and focus on conservation laws restricting the (perturbatively) slow drift of energy between the different normal modes due to non-linearities. We discover two conservation laws in addition to the energy conservation previously discussed in relation to AdS instability. A similar set of three conservation laws was previously noted for a self-interacting scalar field in a non-dynamical AdS background, and we highlight the similarities of this system to the fully dynamical case of gravitational instability. The nature of these conservation laws is best understood through an appeal to averaging methods which allow one to derive an effective Lagrangian or Hamiltonian description of the slow energy transfer between the normal modes. The conservation laws in question then follow from explicit symmetries of this averaged effective theory.
95 citations
87 citations
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TL;DR: In this article, a two-stage Lax-Wendroff (L-W) discretization scheme for hyperbolic conservation laws is proposed, which is based on the generalized Riemann problem (GRP) solver.
Abstract: In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge-Kutta (R-K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax-Wendroff (L-W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using well-developed four stages for R-K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L-W type schemes for the construction of a two-stage fourth order scheme.
TL;DR: In this article, a new computational framework for the analysis of large strain fast solid dynamics is introduced, where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables.
TL;DR: In this article, the number of independent tensor structures appearing in any three-point function is obtained by a simple counting, which is then determined using the Operator Product Expansion (OPE), which is consistent with crossing symmetry.
Abstract: We classify and compute, by means of the six-dimensional embedding formalism in twistor space, all possible three-point functions in four dimensional conformal field theories involving bosonic or fermionic operators in irreducible representations of the Lorentz group. We show how to impose in this formalism constraints due to conservation of bosonic or fermionic currents. The number of independent tensor structures appearing in any three-point function is obtained by a simple counting. Using the Operator Product Expansion (OPE), we can then determine the number of structures appearing in 4-point functions with arbitrary operators. This procedure is independent of the way we take the OPE between pairs of operators, namely it is consistent with crossing symmetry, as it should be. An analytic formula for the number of tensor structures for three-point correlators with two symmetric and an arbitrary bosonic (non-conserved) operators is found, which in turn allows to analytically determine the number of structures in 4-point functions of symmetric traceless tensors.
TL;DR: In this article, a connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian is made, which is a remarkable distinction with respect to the conventional quantum inverse scattering method, which requires non-unitary irreducible representations of Yang?Baxter algebra which are typically of infinite dimensionality.
Abstract: We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain?s ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin- chain, the Fermi?Hubbard chain, and the Lai?Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang?Baxter equation. However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang?Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of the anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
TL;DR: A new reduced basis technique for parametrized nonlinear scalar conservation laws in presence of shocks is presented, based on some theoretical properties of the solution to the problem.
Abstract: In this paper we present a new reduced basis technique for parametrized nonlinear scalar conservation laws in presence of shocks. The essential ingredients are an efficient algorithm to approximate the shock curve, a procedure to detect the smooth components of the solution at the two sides of the shock, and a suitable interpolation strategy to reconstruct such smooth components during the online stage. The approach we propose is based on some theoretical properties of the solution to the problem. Some numerical examples prove the effectiveness of the proposed strategy.
TL;DR: In this paper, a general class of 1D nonlocal conservation laws from a numerical point of view are studied and an algorithm to numerically integrate them and prove their convergence is presented.
Abstract: We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Burger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.
TL;DR: In this paper, the conservation laws of the Rosenau-KdV-RLW equation with power law nonlinearity were computed using Lie symmetry analysis, and the corresponding conserved quantities were computed from their respective densities.
Abstract: This paper computes the conservation laws of the Rosenau–KdV–RLW equation with power law nonlinearity by the aid of multiplier approach in Lie symmetry analysis. This equation models the dynamics of dispersive shallow water waves along lake shores and beaches. The usual conservation laws are reported earlier that are computed from basic mathematical principles. The conservation laws in this paper are extracted using Lie symmetry analysis. The corresponding conserved quantities are computed from their respective densities.
TL;DR: In this article, the authors discuss the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang-Baxter equation.
Abstract: We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain's ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin-1/2 chain, the Fermi-Hubbard chain, and the Lai-Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang-Baxter equation.
However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang-Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
TL;DR: A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced in this article, where a classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family.
Abstract: A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved H1 norm and it possesses N-peakon solutions when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case N = 2.
TL;DR: The convergence of the approximate solutions is proved, providing the existence of a solution in a slightly more general setting than in other results in the current literature.
Abstract: We present a Lax--Friedrichs-type algorithm to numerically integrate a class of nonlocal and nonlinear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved, also providing the existence of a solution in a slightly more general setting than in other results in the current literature. An application to a crowd dynamics model is considered.
TL;DR: In this paper, the authors studied the long-time behavior of periodic scalar first-order conservation laws with stochastic forcing in any space dimension and showed the existence of an invariant measure for sub-cubic fluxes.
Abstract: Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure. Also, since this invariant measure is supported by $$L^p$$
for some $$p$$
small, we are led to generalize to the stochastic case the theory of $$L^1$$
solutions developed by Chen and Perthame (Ann Inst H Poincare Anal Non Lineaire 20(4):645–668, 2003).
TL;DR: This work considers a class of density-flow systems, described by linear hyperbolic conservation laws, which can be monitored and controlled at the boundaries, which are open-loop unstable and subject to unmeasured flow disturbances.
TL;DR: It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form and can be drastically reduced and a significant gain in performance can be obtained.
TL;DR: In this article, the complete algebra of Lie point symmetries for the class of time-fractional nonlinear dispersive equation is derived by means of the classical Lie symmetry method, the associated vector fields are obtained which in turn are utilized for the reduction of the equation.
Abstract: In this paper, we derive the complete algebra of Lie point symmetries for the class of time-fractional nonlinear dispersive equation. By means of the classical Lie symmetry method, the associated vector fields are obtained which in turn are utilized for the reduction of the equation. In particular, the conservation laws of the equation are obtained.
TL;DR: This review compares the structure of solutions of Riemann problems for a conservation law with nonconvex, cubic flux regularized by two different mechanisms: (1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and (2) a combination of diffusion and disp immersion in the mKDV--Burgers equation.
Abstract: We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.
TL;DR: The evolving surface finite element method is used to solve a Cahn–Hilliard equation on an evolving surface with prescribed velocity and shows a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme.
Abstract: We use the evolving surface finite element method to solve a Cahn---Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples.
TL;DR: In this paper, the authors considered the dynamics of a spherically symmetric massless scalar field coupled to general relativity in anti-de Sitter spacetime in the small-amplitude limit.
Abstract: We consider the dynamics of a spherically symmetric massless scalar field coupled to general relativity in anti--de Sitter spacetime in the small-amplitude limit. Within the context of our previously developed two time framework (TTF) to study the leading self-gravitating effects, we demonstrate the existence of two new conserved quantities in addition to the known total energy $E$ of the modes: The particle number $N$ and Hamiltonian $H$ of our TTF system. Simultaneous conservation of $E$ and $N$ implies that weakly turbulent processes undergo dual cascades (direct cascade of $E$ and inverse cascade of $N$ or vice versa). This partially explains the observed dynamics of 2-mode initial data. In addition, conservation of $E$ and $N$ limits the region of phase space that can be explored within the TTF approximation and, in particular, rules out equipartition of energy among the modes for general initial data. Finally, we discuss the possible effects of conservation of $N$ and $E$ on late time dynamics.
TL;DR: A network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function is considered.
Abstract: The adjoint method provides a computationally efficient means of calculating the gradient for applications in constrained optimization In this article, we consider a network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function After discretizing the corresponding partial differential equation models via the Godunov scheme, we detail the computation of the gradient of the discretized system with respect to the control parameters and show that the complexity of its computation scales linearly with the number of discrete state variables for networks of small vertex degree The method is applied to the problem of coordinated ramp metering on freeway networks Numerical simulations on the I15 freeway in California demonstrate an improvement in performance and running time compared with existing methods In the context of model predictive control, the algorithm is shown to be robust to noise in the initial data and boundary conditions
TL;DR: In this paper, the authors presented a theory of high-order harmonic generation by a bichromatic elliptically polarized laser field which consists of two coplanar components having the frequencies and (r and s are integers) and is defined in the xy plane.
Abstract: We present a theory of high-order harmonic generation by a bichromatic elliptically polarized laser field which consists of two coplanar components having the frequencies and (r and s are integers) and is defined in the xy plane. Laser and harmonic fields are decomposed in the components having opposite helicities. Using the conservation laws for energy and projection of the total angular momentum of atom and laser and harmonic photons on the z axis we have derived a general selection rule. This rule reduces to the known result in the case of bicircular field which consists of two counter-rotating circularly polarized fields. We apply our results to explain recent experiment by Fleischer et al 2014 (Nature Photonics 8 543).