Mass and momentum conservation of the least-squares spectral collocation method for the Navier–Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier–Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.     

Differential analysis of polarity: Polar Hamilton-Jacobi,conservation laws,and Monge Ampère equations

We develop a differential theory for the polarity transform parallel to that of the Legendre transform, which is applicable when the functions studied are “geometric convex”, namely, convex, non-negative, and vanish at the origin. This analysis establishes basic tools for dealing with this duality transform, such as the polar subdifferential map, and variational formulas. Another crucial step is identifying a new, non-trivial, sub-class of C ² functions preserved under this transform. This analysis leads to a new method for solving many new first order equations reminiscent of Hamilton–Jacobi and conservation law equations, as well as some second order equations of Monge–Ampère type. This article develops the theory of strong solutions for these equations which, due to the nonlinear nature of the polarity transform, is considerably more delicate than its counterparts involving the Legendre transform. As one application, we introduce a polar form of the homogeneous Monge–Ampère equation that gives a dynamical meaning to a new method of interpolating between convex functions and bodies. A number of other applications, e.g., to optimal transport and affine differential geometry are considered in sequels.     

The forward dynamics in energy markets – infinite-dimensional modelling and simulation

In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.     

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P4

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P ⁴ such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronesé variety V ²⊂P ⁵ and the theory of associativity equations of two-dimensional topological field theory. Received: 15 August 2001     

Self-adjointness and conservation laws for Kadomtsev–Petviashvili–Burgers equation

In this work we study the Kadomtsev–Petviashvili–Burgers equation, which is a natural model for the propagation of the two-dimensional damped waves. We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent form. By using Ibragimov’s theorem on conservation laws we find some conservation laws for this equation.     

Lie symmetries and conservation laws of the Hirota–Ramani equation

In this paper, Lie symmetry method is performed for the Hirota–Ramani (H–R) equation. We will find the symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmetries are investigated. Finally conservation laws of the H–R equation are presented.     

How to find solutions,Lie symmetries,and conservation laws of forced Korteweg–de Vries equations in optimal way

It is shown that the forced Korteweg–de Vries (KdV) equation studied in the recent papers [A.H. Salas, Computing solutions to a forced KdV equation, Nonlinear Anal. RWA 12 (2011) 1314–1320] and [M.L. Gandarias, M.S. Bruzón, Some conservation laws for a forced KdV equation, Nonlinear Anal. RWA 13 (2012) 2692–2700] is reduced to the classical (constant-coefficient) KdV equation by point transformations for all values of variable coefficients. The equivalence-based approach proposed in [R.O. Popovych, O.O. Vaneeva, More common errors in finding exact solutions of nonlinear differential equations: part I, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3887–3899] allows one to obtain more results in a much simpler way.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Cycles,chaos, and noise in predator–prey dynamics

In contrast to the single species models that were extensively studied in the 1970s and 1980s, predator–prey models give rise to long-period oscillations, and even systems with stable equilibria can display oscillatory transients with a regular frequency. Many fluctuating populations appear to be governed by such interactions. However, predator–prey models have been poorly studied with respect to the interaction of nonlinear dynamics, noise, and system identification. I use simulated data from a simple host–parasitoid model to investigate these issues. The addition of even a modest amount of noise to a stable equilibrium produces enough structured variation to allow reasonably accurate parameter estimation. Despite the fact that more-or-less regular cycles are generated by adding noise to any of the classes of deterministic attractor (stable equilibrium, periodic and quasiperiodic orbits, and chaos), the underlying dynamics can usually be distinguished, especially with the aid of the mechanistic model. However, many of the time series can also be fit quite well by a wrong model, and the fitted wrong model usually misidentifies the underlying attractor. Only the chaotic time series convincingly rejected the wrong model in favor of the true one. Thus chaotic population dynamics offer the best chance for successfully identifying underlying regulatory mechanisms and attractors.     

Hamilton–Jacobi equations in space of measures associated with a system of conservation laws

We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects. In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock. The paper develops the method of estimating attraction domain for traveling waves we have studied before.     

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ?² as follows:

$\{\begin{array}{l}{?}_{tt}u-\Delta u=-u^3,\\ u\left(0,x\right)=u_0\left(x\right),{?}_{t}u\left(0,x\right)=u_1\left(x\right),\end{array}$

where the initial data (u₀, u₁) ? H^s(?²) × H^s?1(?²). It is shown that the IVP is global well-posedness in H^s(?²) × H^s?1(?²) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

Riemannian submersions, δ-invariants, and optimal inequality

In this study, we establish a sharp relation between δ-invariants and Riemannian submersions with totally geodesic fibers. By using this relationship, we establish an optimal inequality involving δ-invariants for submanifolds of the complex projective space CP ^m (4) via Hopf’s fibration ${\pi:S^{2m+1}\to CP^{m}(4)}$ . Moreover, we completely classify submanifolds of complex projective space which satisfy the equality case of the inequality.     

“Evolutionary” selection dynamics in games: Convergence and limit properties

This paper discusses convergence properties and limiting behavior in a class of dynamical systems of which the replicator dynamics of (biological) evolutionary game theory are a special case. It is known that such dynamics need not be well-behaved for arbitrary games. However, it is easy to show that dominance solvable games are convergent for any dynamics in the class and, what is somewhat more difficult to establish, weak dominance solvable games are as well, provided they are small in a sense to be made precise in the text. The paper goes on to compare dynamical solutions with standard solution concepts from noncooperative game theory.This paper is a revision of Chapter 1 of my Ph.D. thesis. It owes much to the guidance of Andreu Mas-Colell, Eric Maskin, Vijay Krishna, and Dilip Abreu. I wish also to express my thanks for the comments of an anonymous referee. Naturally, all remaining shortcomings are my responsibility.     

Copolymer–homopolymer blends: global energy minimisation and global energy bounds

We study a variational model for a diblock copolymer–homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta–Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.     

Bäcklund transformation,infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation

Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study the integrability of the equation, including its bilinear representation, soliton solutions, periodic wave solutions, Bäcklund transformation and Lax pairs, respectively. Furthermore, by virtue of its Lax equations, the infinite conservation laws of the equation are also derived with the recursion formulas. Finally, the asymptotic behavior of periodic wave solutions is shown with a limiting procedure.