Uniform Stability Estimates Research Articles

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

Stability estimate for a nonlinear coupled heat transfer model

This paper focuses on a coupled model of heat conduction and heat radiation within bilayer textiles, which is decoupled to a parabolic model with nonlinear source term of functional type. Uniform boundedness and stability estimate of the solution to the nonlinear bilayer model are derived by energy estimates and Young’s inequality skills.

Uniform stability and uniform-in-time mean-field limit of the thermodynamic Kuramoto model

We consider the thermodynamic Kuramoto model proposed in \cite{H-P-R-S}. For each oscillator in thermodynamic Kuramoto model, there is a coupling effect between the phase and the temperature field. For such a model, we study a uniform stability and uniform-in-time mean-field limit to the corresponding kinetic equation. For this, we first derive a uniform ℓ p \ell ^p -stability of the thermodynamic Kuramoto model with respect to initial data by directly estimating the temporal evolution of ℓ p \ell ^p -distance between two admissible solutions to the particle thermodynamic Kuramoto model. In a large-oscillator limit, the Vlasov type mean-field equation can be rigorously derived using the BBGKY hierarchy, uniform stability estimate, and particle-in-cell method. We construct unique global-in-time measure-valued solutions to the derived kinetic equation and also derive a uniform-in-time stability estimate and emergent estimates.

Uniform stability estimate for the Vlasov-Poisson-Boltzmann system

<p style='text-indent:20px;'>This paper is concerned with the uniform stability estimate to the Cauchy problem of the Vlasov-Poisson-Boltzmann system. Our analysis is based on compensating function introduced by Kawashima and the standard energy method.

Convergence analysis of a coefficient inverse problem for the semi-discrete damped wave equation

ABSTRACT In this paper, we study the convergence of a coefficient inverse problem for the two dimensional semi-discrete damped wave equation. Based on the new discrete Carleman estimates for the semi-discrete wave operator, the uniform stability estimate is proved. Then we prove the convergence of the semi-discrete scheme with respect to the discretization grid step h>0.

Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term

The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore, we construct some convex function properties, and we prove the uniform stability estimate.

Global Existence and Decay of Solutions for Coupled Nondegenerate Kirchhoff System with a Time Varying Delay Term

This paper deals with the global existence of solutions in a bounded domain for nonlinear viscoelastic Kirchhoff system with a time varying delay by using the energy and Faedo–Galerkin method with respect to the delay term weight condition in the feedback and the delay speed. Furthermore, by using some convex functions properties, we prove a uniform stability estimate.

Two-Level Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics

In this paper, Newton iteration and two-level finite element algorithm are combined for solving numerically the stationary incompressible magnetohydrodynamics (MHD) under a strong uniqueness condition. The method consists of solving the nonlinear MHD system by $$m$$m Newton iterations on a coarse mesh with size $$H$$H and then computing the Stokes and Maxwell problems on a fine mesh with size $$h\ll H$$h?H. The uniform stability and optimal error estimates of both Newton iterative method and two-level Newton iterative method are given. The error analysis shows that the two-level Newton iterative solution is of the same convergence order as the Newton iterative solution on a fine grid with $$h=O(H^2)$$h=O(H2). However, the two-level Newton iterative method for solving the stationary incompressible MHD equations is simpler and more efficient than Newton iterative one. Finally, the effectiveness of the two-level Newton iterative method is illustrated by several numerical investigations.

Uniform Ergodicity and Strong Stability Estimates of Homogeneous Markov Chains*

The aim of this paper is to investigate some uniform ergodicity and strong stability estimates for homogeneous markov chains, which may be considered as a refinement of those established by the authors with respect to a given weight norm. As a general rule, the initial parameter values of the most complex systems are approximately known (they are defined on the basis of statistical methods), which results in errors for the calculus of research characteristics for each studied system. For this, the uniform ergodicity and stability inequalities obtained in this paper make it possible to use them in order to estimate numerically the error of definition for the considered characteristics for small perturbations of the system’s parameters. As an example of application, we study the well-known Lindley process, and a comparison with some results obtained by Kartashov is established.

Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum

We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.

Uniform stability estimates for the discrete Calderón problems

In this paper, we focus on the analysis of discrete versions of the Calderón problem in dimension d ⩾ 3. In particular, our goal is to obtain stability estimates for the discrete Calderón problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that in our case, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical complex geometric optics (CGO) approach, we can thus derive discrete CGO solutions, but with a limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderón problems.

Global existence and stability of mild solutions to the inelastic Boltzmann system for gas mixtures

We study the global existence and uniform stability estimate of mild solutions to the inelastic Boltzmann system for gas mixtures, when initial data are small and decay exponentially in phase space, and we also provide a general multi-dimensional Bony-type potential which yields a priori weighted two-point correlation estimates in phase space to the mild solutions with finite mass and energy without any smallness restriction.

Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach

As to the Cauchy problem for the spatially inhomogeneous Boltzmann equation with cut-off, we prove uniform stability estimates for solutions and their gradients in a unified and elementary way.

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the time-dependent Stokes equations with a source term in L p (0, T; L q (Ω)) and prove uniform estimates on the time derivative and discrete Laplacian of the discrete velocity that are similar to those in Sohr and von Wahl [20].

Uniform [formula omitted]-stability estimate of the Boltzmann equation near a local Maxwellian

We consider the Boltzmann equation describing the spacial–temporal evolution of the velocity distribution function in a near exponential local Maxwellian regime. We present two robust nonlinear functionals measuring possible future collisions and L 1 -distance between two mild solutions. Through the explicit local cancellation of the collision mechanism due to the special structure of an exponential local Maxwellian solution, we show that the nonlinear functionals in [S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005) 178–205] satisfy uniform stability estimates. These functionals can be employed to the study of L 1 -scattering and uniform L 1 -stability estimate of mild solutions for the Boltzmann equation with inverse power intermolecular potentials.

Stability estimates of the Boltzmann equation with quantum effects

We study uniform stability estimates to the Boltzmann equation for quantum particles such as Bose-Einstein particles and Fermi-Dirac particles. When the small amount of particles expands toward the vacuum, we show that continuous mild solutions are L1-stable and also satisfy BV-type estimates using a nonlinear functional approach.

Formula omitted] and BV-type stability of the Boltzmann equation with external forces

Based on the existence theory on the Boltzmann equation with external forces in infinite vacuum, in this paper, we will study the L 1 and BV-type stability of the classical solutions for small initial data. The stability results generalize those for the Boltzmann equation without force to the case with external force. In particular, we show that the stability can be established for the soft potentials directly, while the stability for the hard potentials and hard sphere model is obtained through the construction of some nonlinear functionals. The functionals thus constructed generalize those constructed in [S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005) 178–205] for the case without force to capture the effect of the force term on the time evolution of the solutions.

Nonlinear functionals of the Boltzmann equation and uniform stability estimates

We present nonlinear functionals measuring physical space variation and L 1 -distance between two classical solutions for the Boltzmann equation with a cut-off inverse power potential. In the case that initial datum is a small, smooth perturbation of vacuum and decays fast enough in the phase space, we show that these functionals satisfy stability estimates which lead to BV-type estimates and a uniform L 1 -stability.

Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and f(t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.

Boundary conditions for hyperbolic systems with stiff source terms

This work is concerned with boundary conditions for multi-dimensional first-order hyperbolic systems with stiff source terms (also called relaxation). It is observed that usual relaxation stability conditions and the uniform Kreiss condition are not enough for the existence of the zero relaxation limit. To remedy this, we propose a so-called generalized Kreiss condition for initial-boundary value problems (henceforth, IBVPs) of the relaxation systems. By assuming that the relaxation system admits the quasi-stability condition and the prescribed boundary condition satisfies the generalized Kreiss condition, we derive a reduced boundary condition, for the corresponding equilibrium system, satisfying the uniform Kreiss condition and show the existence of boundary-layers. Moreover, if the relaxation system admits a more restrictive relaxation stability condition, then Friedrichs’ strictly dissipative boundary conditions, which induce certain uniform stability estimates, are shown to satisfy the generalized Kreiss condition. The present results are expected to be used as theoretical criteria to construct relaxation approximations for IBVPs of conservation laws, which are of practical interest.