Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.     

Projective modules and Hilbert spaces with a Nevanlinna-Pick kernel

In this paper we solve a mapping problem for a particular class of Hilbert modules over an algebra multipliers of a diagonal Nevanlinna-Pick (NP) kernel. In this case, the regular representation provides a multiplier norm which induces the topology on the algebra. In particular, we show that, in an appropriate category, a certain class of Hilbert modules are projective. In addition, we establish a commutant lifting theorem for diagonal NP kernels.

     

Profiles of bounded radial solutions of the focusing, energy-critical wave equation

In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence.     

Monotonicity properties of the Neumann heat kernel in the ball

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U⊂Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball.     

Higher-order quadratures for circulant preconditioned Wiener-Hopf equations

In this paper, we consider solving matrix systems arising from the discretization of Wiener-Hopf equations by preconditioned conjugate gradient (PCG) methods. Circulant integral operators as preconditioners have been proposed and studied. However, the discretization of these preconditioned equations by employing higher-order quadratures leads to matrix systems that cannot be solved efficiently by using fast Fourier transforms (FFTs). The aim of this paper is to propose new preconditioners for Wiener-Hopf equations. The discretization of these preconditioned operator equations by higher-order quadratures leads to matrix systems that involve only Toeplitz, circulant and diagonal matrix-vector multiplications and hence can be computed efficiently by FFTs in each iteration. We show that with the proper choice of kernel functions of Wiener-Hopf equations, the resulting preconditioned operators will have clustered spectra and therefore the PCG method converges very fast. Numerical examples are given to illustrate the fast convergence of the method and the improvement of the accuracy of the computed solutions with using higher-order quadratures.Research supported by the Cooperative Research Centre for Advanced Computational Systems.Research supported in part by Lee Ka Shing scholarship.     

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.     

弹性矩形薄板受迫振动的功的互等定理法(Ⅲ)—悬臂矩形板

本文应用功的互等定理法给出了在均布谐载和在任意点受集中谐载作用下悬臂矩形板受迫振动的稳态解,并给出了有关弯矩和挠度幅值的图表.     

Polynomial stability without polynomial decay of the relaxation function

We consider a linear viscoelastic problem and prove polynomial asymptotic stability of the steady state. This work improves previous works where it is proved that polynomial decay of solutions to the equilibrium state occurs provided that the relaxation function itself is polynomially decaying to zero. In this paper we will not assume any decay rate of the relaxation function. In case the kernel has some flat zones then we prove polynomial decay of solutions provided that these flat zones are not too big. If the kernel is strictly decreasing then there is no need for this assumption. Copyright © 2008 John Wiley & Sons, Ltd.     

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

In this paper, we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants χand κ defined on a 3D contact structure.     

High-frequency limit of the Helmholtz equation with variable refraction index

We study the high-frequency limit of the Helmholtz equation with variable refraction index and a source term concentrated near a p-dimensional affine subspace. Under some conditions, we first derive uniform estimates in Besov spaces for the solutions. Then, we prove that the semi-classical measure associated with these solutions satisfies the stationary Liouville equation with an explicit source term and has certain radiation property at infinity.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

Preconditined semilinear stochastic singular and hypersingular integral equations

In this paper, three classes of preconditioners are proposed for solving some stochastic integral equations with the weakly singular kernel and the hypersingular kernel. The first and the second class of preconditioners are based on circulant operators, but the third class of preconditioners is based on iterative substructuring. It is proved that substructuring preconditioners can be better than other preconditioners. Also, the spaces of solutions are discussed such that the solutions of these equations are smooth, therefore, we give special Banach spaces for these integral equations. Finally, numerical results which support our theories are presented     

Regularity of Stationary Weak Solutions in the Theory of Generalized Newtonian Fluids with Energy Transfer

In this paper, we prove regularity results for weak solutions to some stationary problems arising in the theory of generalized Newtonian fluids with energy transfer. Namely, we prove that these solutions are strong. In the two-dimensional case, we prove the Hölder continuity of the first gradient of a solution. Bibliography: 30 titles.     

Stability of stationary solutions in models of the Calvin cycle

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et al. (2009) for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of Grimbs et al. (2011), for which it was known that more than one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

Stationary perfect equilibria of an n-person noncooperative bargaining game and cooperative solution concepts

This paper characterizes the stationary (subgame) perfect equilibria of an n-person noncooperative bargaining model with characteristic functions, and provides strategic foundations of some cooperative solution concepts such as the core, the bargaining set and the kernel. The contribution of this paper is twofold. First, we show that a linear programming formulation successfully characterizes the stationary (subgame) perfect equilibria of our bargaining game. We suggest a linear programming formulation as an algorithm for the stationary (subgame) perfect equilibria of a class of n-person noncooperative games. Second, utilizing the linear programming formulation, we show that stationary (subgame) perfect equilibria of n-person noncooperative games provide strategic foundations for the bargaining set and the kernel.     

Stationary solutions for a multi-dimensional nonisentropic hydrodynamic model for semiconductors

In this paper, we discuss a multi-dimensional stationary flow model for semiconductor devices which is based on the nonisentropic hydrodynamic equations with non-constant lattice temperature. We establish the existence of the irrotational non-thermal equilibrium subsonic steady state solutions with positive particle density and positive carrier temperature. The proofs are based on the Schauder fixed-point principle, Stampacchia’s truncation methods and the careful energy estimates. Meanwhile, we also show that these strong stationary solutions are unique.     

半导体中扩散方程初始边界问题的解的渐近性

In this paper, we study the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.     

Optimal feedback control in the stationary mathematical model of low concentrated aqueous polymer solutions

In this article we will consider the feedback control problem in the stationary model of motion of low concentrated aqueous polymer solutions. We demonstrate the solvability of an approximating problem, using some a priori estimates and the topological degree theory. Then the convergence (in some generalized sense) of solutions of approximating problems to a solution of the given problem is proved. Moreover, we show the existence of a solution minimizing a given convex, lower semicontinuous functional.