Semigroup Approach Research Articles

A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions

A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate for singular modelled distributions

Exponential stability for a swelling porous-heat system with thermodiffusion effects and distributed delay term

This work is concerned with the study of the well-posedness and asymptotic behaviour of solutions to a swelling porous system with thermodiffusion effects and distributed delay term. We demonstrate the well-posedness of the solution of the system by using the semigroup approach under suitable assumption on the weight of the distributed delay. Then, we obtain the exponential decay result by using the energy method which is based on the multiplier technique in which we construct an appropriate Lyapunov functional.

An inverse semigroup approach to self-similar k-graph $$C^*$$-algebras and simplicity

An inverse semigroup approach to self-similar k-graph $$C^*$$-algebras and simplicity

Dynamical systems analysis of a reaction-diffusion SIRS model with optimal control for the COVID-19 spread

We examine an SIRS reaction-diffusion model with local dispersal and spatial heterogeneity to study COVID-19 dynamics. Using the operator semigroup approach, we establish the existence of disease-free equilibrium (DFE) and endemic equilibrium (EE), and derive the basic reproduction number, R 0 . Simulations show that without dispersal, reinfection and limited medical resources problems can cause a plateau in cases. Dispersal and spatial heterogeneity intensify localised outbreaks, while integrated control strategies (vaccination and treatment) effectively reduce infection numbers and epidemic duration. The possibility of reinfection demonstrates the need for adaptable health measures. These insights can guide optimised control strategies for enhanced public health preparedness.

Committor Guided Estimates of Molecular Transition Rates.

The probability that a configuration of a physical system reacts, or transitions from one metastable state to another, is quantified by the committor function. This function contains richly detailed mechanistic information about transition pathways, but a full parametrization of the committor requires the construction of a high-dimensional function, a generically challenging task. Recent efforts to leverage neural networks as a means to solve high-dimensional partial differential equations, often called "physics-informed" machine learning, have brought the committor into computational reach. Here, we build on the semigroup approach to learning the committor and assess its utility for predicting dynamical quantities such as transition rates. We show that a careful reframing of the objective function and improved adaptive sampling strategies provide highly accurate representations of the committor. Furthermore, by directly applying the Hill relation, we show that these committors provide accurate transition rates for molecular systems.

Weak and strong solutions for a fluid‐poroelastic‐structure interaction via a semigroup approach

A filtration system comprising a Biot poroelastic solid coupled to an incompressible Stokes free‐flow is considered in 3D. Across the flat 2D interface, the Beavers‐Joseph‐Saffman coupling conditions are taken. In the inertial, linear, and non‐degenerate case, the hyperbolic‐parabolic coupled problem is posed through a dynamics operator on a chosen energy space, adapted from Stokes‐Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. The generation of a strongly continuous semigroup for the dynamics operator is obtained via a non‐standard maximality argument. The latter employs a mixed‐variational formulation in order to invoke the Babuška‐Brezzi theorem. The Lumer‐Philips theorem then yields semigroup generation, and thereby, strong and generalized solutions are obtained. For the linear dynamics, density obtains the existence of weak solutions; we extend to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial linear Biot‐Stokes filtration system, we provide a clear elucidation of strong solutions and a construction of weak solutions, as well as their regularity through associated estimates.

Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems

We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic initial datum, then there exist exactly two families of periodic one-forms satisfying the structural equations for a surface. Each pair then defines a metric of constant Gaussian curvature and a corresponding Levi-Civita connection form. We prove the existence of universal connection forms giving rise to second fundamental forms compatible with the metric. The main tool to prove our geometrical results is the Kato’s semi-group approach, which is used to establish well-posedness of solutions of the Cauchy problem involved and ensure C 1 regularity for the first fundamental form and the Levi-Civita connection form.

A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay

Abstract In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

Semigroup approach to the sign problem in quantum Monte Carlo simulations

Semigroup approach to the sign problem in quantum Monte Carlo simulations

Asymptotic smoothness effects and global attractor for a peridynamic model with energy damping

AbstractIn this paper, we consider a peridynamic model with energy damping inspired by the works of Balakrishnan and Taylor on “damping models” based on the instantaneous total energy of the system. We study the asymptotic behavior of solutions, in the sense of attractors, of these peridynamic models in suitable phase space; more precisely, we prove a result of existence and characterization of compact global attractors with a nonlinear strongly continuous semigroup approach based in the asymptotic smoothness property thanks to Chueshov and Lasiecka and Nakao's lemma.

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto.

A new proof of the Gagliardo–Nirenberg and Sobolev inequalities: Heat semigroup approach

A new proof of the Gagliardo–Nirenberg and Sobolev inequalities: Heat semigroup approach

Existence and ergodicity for the two-dimensional stochastic Allen–Cahn–Navier–Stokes equations

We study in this article a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter, both endowed with suitable boundary conditions. We prove the existence of solutions via a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods.

Reaction-diffusion equations on metric graphs with edge noise

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.

Decay Properties for Transmission System with Infinite Memory and Distributed Delay

We consider a damped transmission problem in a bounded domain where the damping is effective in a neighborhood of a suitable subset of the boundary. Using the semigroup approach together with Hille–Yosida theorem, we prove the existence and uniqueness of global solution. Under suitable assumption on the geometrical conditions on the damping, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.

Controlling a generalized Fokker–Planck equation via inputs with nonlocal action

Controlling a generalized Fokker–Planck equation via inputs with nonlocal action

Equations Related to Stochastic Processes: Semigroup Approach and Fourier Transform

Equations Related to Stochastic Processes: Semigroup Approach and Fourier Transform

The influence of damping on the asymptotic behavior of solution for laminated beam

<p>This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.</p>

On Stein factors for Laplace approximation and their application to random sums

On Stein factors for Laplace approximation and their application to random sums

Spectral analysis for some third-order differential equations: a semigroup approach

Spectral analysis for some third-order differential equations: a semigroup approach