arXiv subjects

To Fu Ma

Publications and source records attributed to To Fu Ma.

7 recordsLinked to original sources

arXiv2023

Fundamentals of thermoelasticity for curved beams

The purpose of this paper is twofold. Firstly, we conduct an in-depth analysis of mathematical modeling concerning thermal-mechanical curved beams, by taking into consideration three primary forces widely accepted in the literature: axial load, shear force, and bending moment. Additionally, we examine their appropriate thermal couplings, shedding light on the intricate interplay between stress-strain relationships and temperature variations. This analysis is situated within the well-recognized context of the Bresse governing model for arched beams. Secondly, drawing upon distinguished constitutive laws for heat flux of conduction, we compile a comprehensive list of thermoelastic curved beam systems in various scenarios. We introduce new categories of problems that exhibit specific features from the thermal point of view.

math.AP↗

arXiv2021

Attractors for locally damped Bresse systems and a unique continuation property

This paper is devoted to Bresse systems, a robust model for circular beams, given by a set of three coupled wave equations. The main objective is to establish the existence of global attractors for dynamics of semilinear problems with localized damping. In order to deal with localized damping a unique continuation property (UCP) is needed. Therefore we also provide a suitable UCP for Bresse systems. Our strategy is to set the problem in a Riemannian geometry framework and see the system as a single equation with different Riemann metrics. Then we perform Carleman-type estimates to get our result.

math.AP↗

arXiv2020

Longtime dynamics of a semilinear Lam\'e system

This paper is concerned with longtime dynamics of semilinear Lam\'e systems $$ \partial^2_t u - \mu \Delta u - (\lambda + \mu) \nabla {\rm div} u + \alpha \partial_t u + f(u) = 0, $$ defined in bounded domains of $\mathbb{R}^3$ with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to critical forcings $f(u)$. Writing $\lambda + \mu$ as a positive parameter $\varepsilon$, we discuss some physical aspects of the limit case $\varepsilon \to 0$. Then, we show the upper-semicontinuity of attractors with respect to the parameter when $\varepsilon \to 0$. To our best knowledge, the analysis of attractors for dynamics of Lam\'e systems has not been studied before.

math.AP↗

arXiv2019

Attractors for semilinear wave equations with localized damping and external forces

This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $\mathbb{R}^3$ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the existence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping.

math.AP↗

arXiv2018

Pullback dynamics of a 3D Navier-Stokes equation with nonlinear viscosity

This paper is concerned with pullback dynamics of 3D Navier-Stokes equations with variable viscosity and subject to time-dependent external forces. Our main result establishes the existence of finite-dimensional pullback attractors in a general setting involving tempered universes. We also present a sufficient condition on the viscosity coefficients that guarantees the attractors are nontrivial. We end the paper by showing the upper semi-continuity of pullback attractors as the non-autonomous perturbation vanishes.

math.DS↗

arXiv2018

Pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation on Lipschitz-like domain

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation with non-homogeneous boundary condition on Lipschitz-like domain. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimate on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is the sum of a stationary force and a non-autonomous perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we also investigate the regularity of these attractors when smoother initial data is given. Our results are new even in the case of smooth domains.

math.AP↗

arXiv2015

Singular limit and long-time dynamics of Bresse systems

The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature $\ell=0$. Our first result shows the Timoshenko system as a singular limit of the Bresse system as $\ell \to 0$. The remaining results are concerned with the long-time dynamics of Bresse systems. In a general framework, allowing nonlinear damping and forcing terms, we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of upper semicontinuity of their attractors as $\ell \to 0$.

math.AP↗