Uniqueness for the coagulation-fragmentation equation with strong fragmentation

The existence of at least one mass-conserving solution for continuous coagulation-fragmentation equation has been established by Escobedo et?al. (J Differ Equ 195:143?C174, 2003) for a large class of coagulation kernels under strong binary fragmentation. In this work, uniqueness of mass-conserving solutions is demonstrated with some additional restrictions on the fragmentation kernels.     

Mass-conserving self-similar solutions to coagulation–fragmentation equations

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.     

The steady spectra of particles in dispersible systems with coagulation and fragmentation

The formation of a steady dimensional distribution of particles (particle spectra) is dispersible systems with coagulation and fragmentation is considered. The relation between versions of the kinetic equation that defines these processes is traced. An analytical solution is obtained for the parametric set of coagulation coefficients and the velocities of paired fragmentation. The steady spectrum of particles is investigated in the case when the fragmentation is of the multiple type.     

Coagulation and fragmentation with discrete mass loss

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel K is bounded and the fragmentation rate function a satisfies a linear growth condition, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is governed by power-law kernels, an explicit formula is given for the substochastic semigroup associated with the resulting mass-loss fragmentation equation.     

Fast Reaction Limit of the Discrete Diffusive Coagulation–Fragmentation Equation

Abstract

The local mass of weak solutions to the discrete diffusive coagulation–fragmentation equation is proved to converge, in the fast reaction limit, to the solution of a nonlinear diffusion equation, the coagulation and fragmentation rates enjoying a detailed balance condition.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

Existence,Uniqueness and Mass Conservation for the Coagulation–Fragmentation Equation

We prove global existence and uniqueness to the initial value problem for the coagulation–fragmentation equation for an unbounded coagulation kernel with possible linear growth at infinity and a fragmentation kernel from a very large class of unbounded functions. We show that the solutions satisfy the mass conservation law.     

A short remark on a growth–fragmentation equation

An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in [5] about the existence of global solutions in the so-called self-similar case is not satisfied. The solution is local and blows up in finite time.     

Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with product kernel of homogeneity 2λ ∈ (0, 1). We establish rigorously that such solutions exhibit a singular behavior of the form x ^?(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now.     

Approach to Equilibrium for the Coagulation–Fragmentation Equation via a Lyapunov Functional

It is shown that the non-linear coagulation–fragmentation equation with constant kernels has a unique equilibrium solution. This equilibrium solution is given explicitly in terms of the initial data and the kernels. Weak L¹ convergence of time-dependent solutions to the unique equilibrium is demonstrated via an invariance principle employing a suitable lower semicontinuous Lyapunov functional.     

A finite element collocation method for variably saturated flows in porous media

One common formulation of Richard's equation for variably saturated flows in porous media treats pressure head as the principal unknown and moisture content as a constitutive variable. Numerical approximations to this “head-based” formulation often exhibit mass-balance errors arising from inaccuracies in the temporal discretization. This article presents a finite-element collocation scheme using a mass-conserving formulation. The article also proposes a computable index of global mass balance.     

Existence and uniqueness theorem for the Safronov–Dubovski coagulation equation

The solutions of the discrete Safronov–Dubovski coagulation equation are investigated. We prove global existence for a class of unbounded coagulation kernels. We also show that for sub-linear unbounded kernels, the mass conservation law holds. Finally, we show that for bounded kernels, this equation has a unique global solution that is continuously dependent on the initial data.     

Weak solutions to the continuous coagulation equation with multiple fragmentation

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.     

Convergence of a finite volume scheme for coagulation-fragmentation equations

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

     

On the Cauchy Problem of the Camassa-Holm Equation

The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L ^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.     

一类共形不变摄动积分方程正解的存在性

本文讨论了一类共形不变摄动积分方程正解的存在性. 我们证明了:当参数对(p, q) 属于集合(-n, 0) × (0,∞) 且pq + p + 2n = 0 时, 对应摄动积分方程存在正解; 而当参数对(p, q) 属于集合(0,∞)×(-∞, 0) 也满足pq +p+2n = 0 时, 摄动积分方程不存在非负解. 这与原共形不变积分方程有着本质的不同, 此结果隐含着这类积分方程正解的存在性取决于解在无穷远处的性态.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

Necessary conditions of nondominated solutions in multicriteria decision making

A partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay. By means of abstract regularity results, global existence and uniqueness of a strict solution are obtained; moreover a characterization of the infinitesimal generator of the solution operator is given.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

A Positive and Monotone Numerical Scheme for Volterra-Renewal Equations with Space Fluxes

We study a numerical method for solving a system of Volterra-renewal integral equations with space fluxes, that represents the Chapman-Kolmogorov equation for a class of piecewise deterministic stochastic processes. The solution of this equation is related to the time dependent distribution function of the stochastic process and it is a non-negative and non-decreasing function of the space. Based on the Bernstein polynomials, we build up and prove a non-negative and non-decreasing numerical method to solve that equation, with quadratic convergence order in space.