Uniform Estimates Research Articles

Approximation results of Phillips type operators including exponential function

<p>The current article deals with a study on some moderation of the Phillips operators, including constant and exponential functions. Here, we derive the moments applying the notion of moment-generating function for the well-known Phillips operators. The authors also establish uniform convergence estimates for the improved form of these operators. Additionally, some direct estimates involving the asymptotic-type results are discussed.</p>

The TRUNC element in any dimension and application to a modified Poisson equation

AbstractWe introduce a novel TRUNC finite element in dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we identify it as crucial for error estimate. This element is utilized to approximate a modified Poisson equation defined on a convex polytope, originating from the nonlocal electrostatics model. We have substantiated a uniform error estimate and conducted numerical tests on both the smooth solution and the solution with a sharp boundary layer, which align with the theoretical predictions.

The role of larval transport on recruitment dynamics of red mullet (Mullus barbatus) in the Central Mediterranean Sea

Recruitment success depends on external forcing mechanisms such as ocean currents that affect the transport of eggs and larvae to favorable habitats. In this study, we investigated the role of larval transport in the recruitment of Mullus barbatus in the Central Mediterranean Sea by modeling the recruits' abundance as a function of both spawning stock size and dispersal rates of the species’ early life stages. Our analysis involved twenty years of data on recruits and spawners abundance obtained from scientific trawl surveys, and data on larval dispersal rates derived from a combination of actualized published sources and original data. By calculating in the Sicilian nurseries, the estimates of retention, import and uniformity of the contribution of the spawning areas distributed among different Geographical Sub Areas (GSAs), we assessed their contribution to recruitment using modified Ricker stock size-recruits models. In particular, our results show that a higher uniform contribution from spawning areas within GSA16, mainly related to the oceanographic patterns promoting larval retention, together with spawners abundance, significantly reduced the variability of red mullet recruitment. We further highlighted that when switching from a higher to a lower level of evenness of contribution to the recruit population from different spawning areas in the GSA16, the expected spawning stock abundance per recruit for a given fishing pattern can suffer a rapid short-term decline, which is likely to have negative consequences for stock assessment and management decisions. Our results suggest that larval transport plays a crucial role in explaining the interannual variability of recruitment, thereby contributing to a better understanding of stock size variation. Additionally, our study enhances the understanding of the spatial dynamics involved in the recruitment of this species, which is of increasing interest within fisheries management frameworks.

Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

Numerical Ergodicity and Uniform Estimate of Monotone Spdes Driven by Multiplicative Noise

Numerical Ergodicity and Uniform Estimate of Monotone Spdes Driven by Multiplicative Noise

Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia

This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of Dk−absorbing sets in the stochastic dynamical system Φk, and the asymptotic compactness of Φk is proved via decomposition. Finally, the family of Dk−random attractors is acquired for the stochastic dynamical system Φk in Vm+k(Ω)×Vm+kb0(Ω). These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

On functions of finite analytical complexity

We construct examples of polynomials and analytic functions of any predetermined finite analytical complexity n n . We obtain an estimate of the order of derivative of the differential-algebraic criteria for membership in the class C l n Cl_{n} of functions of analytical complexity not higher than n n . We find uniform estimates for finite values d n d_{n} of the analytic spectrum { d n } \{d_{n}\} for systems of differential-algebraic equations of fixed order of derivative δ \delta .

Numerical Analysis for a Non-isothermal Incompressible Navier–Stokes–Allen–Cahn System

In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension N=2,3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2, 3,$$\\end{document} that describes the movement of a mixture of two incompressible viscous fluids. This model consists of modified Navier–Stokes equations coupled with a phase-field equation given by a convective Allen–Cahn equation, and energy transport equation for the temperature; which admits a dissipative energy inequality. We propose an energy stable numerical scheme based on the Finite Element Method, and we analyze optimal weak and strong error estimates, as well as convergence towards regular solutions. In order to construct the numerical scheme, we introduce two extra variables (given by the gradient of the temperature and the variation of the energy with respect to the phase-field function) which allows us to control the strong regularity required by the model, which is one of the main difficulties appearing from the numerical point of view. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed, energy stable and satisfies a set of uniform estimates which allow to analyze the convergence of the scheme. Finally, we present some numerical simulations to validate numerically our theoretical results.

Existence and uniqueness result for reaction-diffusion model of diffusive population dynamics

The present work investigates a convolution type nonlinear integrodifferential equation with diffusion. This type of equations represents not only pure mathematical interest, but also is widely used in various applications, especially in wide range of problems on population dynamics arising in biology. The existence of a parametric family of travelling wave solutions as well as the uniqueness of the solution in certain class of bounded continuous functions on R \mathbb {R} is proved. The study investigates also some important properties as well as asymptotic behaviour of constructed solutions. These results are then used to derive a new uniform estimate for the deviation between successive iterations, which provides us with a strong tool to control the number of iterations on our way of computing the desired numerical approximation of the exact solution. Finally, we apply our theoretical results to two well-known population problems modelled by delayed reaction-diffusion equation: Diffusion model for spatial-temporal spread of epidemics and stage structured population model.

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial Model

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} that are constrained to a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-sphere, equivalently a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.

Stationary solutions to stochastic 3D Euler equations in Hölder space

We establish the existence of infinitely many global and stationary solutions in C(R,Cϑ) space for some ϑ>0 to the three-dimensional Euler equations driven by an additive stochastic forcing. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in Hofmanová et al. (2022) and pathwise estimates to derive uniform moment estimates independent of time.

Schauder estimates for parabolic equations with degenerate or singular weights

We establish some C0,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{0,\\alpha }$$\\end{document} and C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} as a power a>-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a > -1$$\\end{document} of the distance to Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

Controllability of some semilinear shadow reaction-diffusion systems

The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove that the original model is null-controllable and that the shadow limit preserves this important feature.

Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion Equations on Unbounded Domains

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document}. Since the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not compact and the spectrum of the Laplace operator on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

Measure Attractors of Stochastic Fractional Lattice Systems

This paper seeks to establish the measure attractors in stochastic fractional lattice systems. First, the presence of these attractor measures is proven by the uniform estimates of the solution. Subsequently, the study also looks at the upper semicontinuous dependence of the measure attractors on the noise intensity as the latter goes to zero. The given asymptotic compactness for the family of probability measures occurring with the solution probability distributions is exhibited by a uniform prior estimation of the far-field solution values.

Compressible fluid limit for smooth solutions to the Landau equation

Although the compressible fluid limit of the Boltzmann equation with cutoff has been extensively investigated in Caflisch [Comm. Pure Appl. Math. 33 (1980), 651–666] and Guo, Jang, and Jiang [Comm. Pure Appl. Math. 63 (2010), 337–361], obtaining analogous results in the case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, still remains largely open, essentially due to the velocity diffusion effect of the collision operator such that L^{\infty} estimates are hard to obtain without using Sobolev embeddings. In this paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number \varepsilon>0 , yielding the O(\varepsilon) convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians, together with techniques for viscous compressible fluid and properties of Burnett functions, we design an \varepsilon -dependent energy functional to capture the dissipation in the compressible fluid limit with the feature that only the highest-order derivatives are most singular.