Semigroup Approach Research Articles

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

This paper concerns an optimal control problem (P) related to a generalized Fokker–Planck equation. Basic properties of the solutions to the generalized FP equation are derived via a semigroup approach in the space H−1(Rd). Problem (P) is proven to be deeply related to a stochastic optimal control problem (PS) for a McKean–Vlasov equation. The existence of an optimal control is obtained for the deterministic problem (P). The existence of an optimal control is established for an approximating optimal control problem (Ph) related to a backward Euler approximation of the generalized FP equation (with a constant discretization step h). One proves under additional hypotheses that “(Ph) converges to (P)” in a certain sense. First order necessary conditions for (Ph) are derived as well.

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

In this article, we consider an integral Stein equation for Laplace distribution and solve it using the semigroup approach. Next, we derive regularity estimates for the solution of the Laplace Stein equation. Finally, we apply these estimates to obtain a convergence rate for Laplace approximation of the random geometric sums. We also compare our rate with the existing literature.

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

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

Exponential stability and numerical analysis of Timoshenko system with dual‐phase‐lag thermoelasticity

AbstractThis study aims to investigate the well‐posedness and stability of a thermoelastic Timoshenko system with non‐Fourier heat conduction. Specifically, we analyze the system using the dual‐phase‐lag (DPL) model, which incorporates two thermal relaxation times, and , to model non‐instantaneous heat propagation. Applying the semigroup approach, we demonstrate the existence and uniqueness of the solutions. Subsequently, we introduce a novel stability parameter using the multiplier method. Exponential decay is proven for the case of with . Using Gearhart–Prüss theorem, we show the lack of exponential stability when and . Numerically, we present a fully discrete approximation using the finite element method and the backward Euler scheme, and we provide some numerical simulations to show the discrete energy decay and the behavior of solutions.

A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach

A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach

Stabilization for weakly coupled string–riser system with partial frictional damping

In this paper, we consider the stabilization of a 1‐D coupled string–riser system with partial frictional damping. The frictional damping is only distributed in the part of the string's or riser's domain, and its dissipative effect is transmitted to the other equation through the coupling. Firstly, we prove the existence and uniqueness result for solution of coupled system. Next, using a semigroup approach combined with the multipliers technique, we obtain that the two coupled systems are not exponentially stable, provide precise polynomial decay rates for their energy functionals, respectively, regardless of whether the friction damping only acts on the string equation or the riser equation, and give the optimality of polynomial decay rates. Finally, we compare the case of friction damping acting on string and riser equations simultaneously with the previous two cases. The results show that the case of friction damping acting on string and riser equations simultaneously is exponential stable, while the case of friction damping only acting on string or riser equations is polynomial stable and the decay result of friction damping only acts on the string equation is better than that only acts on the riser equation.

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

In this work, we obtain quantitative convergence of moderately interacting particle systems towards solutions of nonlinear Fokker-Planck equations with singular kernels. In addition, we prove the well-posedness for the McKean-Vlasov SDEs involving these singular kernels and the trajectorial propagation of chaos for the associated moderately interacting particle systems. Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. This seems to be the first time that such quantitative convergence results are obtained in Lebesgue and Sobolev norms for the aforementioned kernels. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the Sobolev regularity of infinite-dimensional stochastic convolution integrals, and we also exploit the regularity of the solutions of the limiting equation.