Weak solutions in nonlinear poroelasticity with incompressible constituents

We consider quasi-static nonlinear poroelastic systems with applications in biomechanics and, in particular, tissue perfusion. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. The system under consideration represents a nonlinear, implicit, degenerate evolution problem, which falls outside of the well-known implicit semigroup monotone theory. Previous literature related to proving existence of weak solutions for these systems is based on constructing solutions as limits of approximations, and energy estimates are obtained only for the constructed solutions. In comparison, in this treatment we provide for the first time a direct, fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solutions of the associated linear system (where the permeability is given as a function of space and time). The uniqueness proof for the associated linear problem is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions. The results of this work provide a foundation for addressing strong solutions, as well as uniqueness of weak solutions for nonlinear poroelastic systems.     

Discretization of time-dependent quantum systems: real-time propagation of the evolution operator

We discuss time-dependent quantum systems on bounded domains. Our work may be viewed as a framework for several models, including linear iterations involved in time-dependent density functional theory, the Hartree-Fock model, or other quantum models. A key aspect of the analysis of the algorithms is the use of time-ordered evolution operators, which allow for both a well-posed problem and its approximation. The approximation theorems obtained for the time-ordered evolution operators complement those in the current literature. We discuss the available theory at the outset, and proceed to apply the theory systematically in later sections via approximations and a global existence theorem for a nonlinear system, obtained via a fixed point theorem for the evolution operator. Our work is consistent with first-principle real-time propagation of electronic states, aimed at finding the electronic responses of quantum molecular systems and nanostructures. We present two full 3D quantum atomistic simulations using the finite element method for discretizing the real space, and the FEAST eigenvalue algorithm for solving the evolution operator at each time step. These numerical experiments are representative of the theoretical results.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Monte Carlo Euler approximations of HJM term structure financial models

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.     

Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a non-uniform time grid, which generalizes the well-known Störmer-Verlet scheme, with an internal approximation scheme.The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces.Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

Time‐dependent photon statistics in variable media

We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an abstract operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information , we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.     

Heat kernels and regularity for rough metrics on smooth manifolds

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.     

耗散的量子Zakharov方程解的渐进性行为

主要研究量子Zakharov方程.在先验估计的基础上通过标准的Galerkin逼近方法得到了量子Zakharov方程解的存在唯一性.同时,也讨论了相应的解的渐进性行为,并且构造出在不同的能量空间上弱拓扑意义下的全局吸引子.     

Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.     

Estimates for normal solutions of problems with zone controls in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> for the wave equation

For the wave equation with variable coefficients and homogeneous boundary conditions of the first kind, we consider problems with regular zone controls and dual zone observation problems. For weak generalized solutions of the observation problem on sufficiently large time intervals, we obtain constructive estimates that imply the well-posed solvability of the observation operator. These estimates contain information that permits one to construct stable approximate solutions of both problems with the use of a variational method suggested earlier by the author for linear equations with nonuniformly perturbed operators.     

伪自伴量子系统的酉演化与绝热定理

经典量子系统的哈密尔顿是自伴算子.哈密尔顿算符的自伴性不仅确保了系统遵循酉演化,而且也保证了它自身具有实的能量本征值.但是,确实有一些物理系统,其哈密尔顿是非自伴的,但也具有实的能量本征值,这种具有非自伴哈密尔顿的系统就是非自伴量子系统.具有伪自伴哈密尔顿的系统是一类特殊的非自伴量子系统,其哈密尔顿相似于一个自伴算子.本文研究伪自伴量子系统的酉演化与绝热定理.首先,给出了伪自伴算子定义及其等价刻画;其次,对于伪自伴哈密尔顿系统,通过构造新内积,证明了伪自伴哈密尔顿在新内积下是自伴的,并给出了系统在新内积下为酉演化的充分必要条件.最后,建立了伪自伴量子系统的绝热演化定理及与绝热逼近定理.     

Stochastic McKean-Vlasov equations

We prove the existence and uniqueness of solution to the nonlinear local martingale problems for a large class of infinite systems of interacting diffusions. These systems, which we call the stochastic McKean-Vlasov limits for the approximating finite systems, are described as stochastic evolutions in a space of probability measures onR ^d and are obtained as weak limits of the sequence of empirical measures for the finite systems, which are highly correlated and driven by dependent Brownian motions. Existence is shown to hold under a weak growth condition, while uniqueness is proved using only a weak monotonicity condition on the coefficients. The proof of the latter involves a coupling argument carried out in the context of associated stochastic evolution equations in Hilbert spaces. As a side result, these evolution equations are shown to be positivity preserving. In the case where a dual process exists, uniqueness is proved under continuity of the coefficients alone. Finally, we prove that strong continuity of paths holds with respect to various Sobolev norms, provided the appropriate stronger growth condition is verified. Strong solutions are obtained when a coercivity condition is added on to the growth condition guaranteeing existence.The research has been partially supported by NSERC, Canada.     

External Approximation of Nonlinear Operator Equations

Based on an external approximation scheme for the underlying Banach space, a nonlinear operator equation is approximated by a sequence of coercive problems. The equation is supposed to be governed by the sum of two nonlinear operators acting between a reflexive Banach space and its dual. Under suitable stability assumptions and if the underlying operators can be approximated consistently, weak convergence of a subsequence of approximate solutions is shown. This also proves existence of solutions to the original equation.     

First-order differential-operator equation with variable domains of piecewise smooth operators

We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients in the equations of finite and infinite order and in the boundary conditions.     

Best Approximation from the Kuhn-Tucker Set of Composite Monotone Inclusions

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.     

Global weak solutions for a modified two-component Camassa-Holm equation

We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa-Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space-time higher integrability estimates on the first-order derivatives of approximation solutions.