Pointwise Estimates Research Articles

Pointwise Estimate of Cahn-Hilliard Equation with Inertial Term in ℝ3

Cauchy problem of Cahn-Hilliard equation with inertial term in three-dimensional space is considered. Using delicate analysis of its Green function and its convolution with nonlinear term, pointwise decay rate is obtained.

Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div(A(x,∇u))=div(f)inΩ,andu=gon∂Ω,that related to Riesz potentials. Here, in our setting, Ω⊂Rn, n≥2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

Nonlinear Potential Estimates for Generalized Stokes System

In this paper, we consider the generalized stationary Stokes system with p -growth and Dini-\({\text {BMO}}\) regular coefficients. The main purpose is to establish pointwise estimates for the shear rate and the associated pressure to such Stokes system in terms of an unconventional nonlinear Havin–Maz’ya–Wolff type potential of the nonhomogeneous term in the plane. As a consequence, a symmetric gradient \(L^{\infty }\) estimate is obtained. Moreover, we derive potential estimates for the weak solution to the Stokes system without additional regularity assumptions on the coefficients in higher dimensional space.

Refined Pointwise Estimates for Solutions to the 1D Barotropic Compressible Navier–Stokes Equations: An Application to the Long-Time Behavior of a Point Mass

We study the long-time behavior of a point mass moving in a one-dimensional viscous compressible fluid. The author previously showed that the velocity of the point mass V(t) satisfies a decay estimate $$V(t)=O(t^{-3/2})$$ (Koike in J. Differ. Equ. 271:356–413, 2021). However, whether this decay estimate is optimal or not was not completely understood. In this paper, we answer this question by giving a simple necessary and sufficient condition for the validity of a lower bound of the form $$C^{-1}t^{-3/2}\le |V(t)|$$ for large t ( $$C>1$$ is a constant independent of t). This is achieved by refining the previously obtained pointwise estimates of solutions. We introduce inter-diffusion waves that, together with the classical diffusion waves, give an improved approximation of the fluid behavior around the point mass; this then leads to a sharper understanding of the long-time behavior of the point mass.

Almost Price's law in Schwarzschild and decay estimates in Kerr for Maxwell field

We consider the asymptotics of a Maxwell field in Schwarzschild and Kerr spacetimes. In a Kerr spacetime, we utilize the basic energy and Morawetz estimates proven in earlier work to derive pointwise estimates where the total power of decay rate for all components of the Maxwell field towards a stationary solution is −7/2. Moreover, on a Schwarzschild background, if the constructed Newman–Penrose constant vanishes (or does not vanish), we prove almost sharp Price's law decay v−2−sτ−3+s+ (or v−2−sτ−2+s+) for Maxwell field and v−2−sτ−2+s−ℓ0+ (or v−2−sτ−1+s−ℓ0+) for the ℓ≥ℓ0 modes of the field towards a static solution, with s being the spin weight of the Newman–Penrose components of Maxwell field. All estimates are uniform in the exterior of the black hole.

Constrained least-squares based adaptive-order finite-volume WENO scheme for the simulation of viscous compressible flows on unstructured grids

We develop a novel constrained least-squares technique for fourth-order weighted essentially non-oscillatory (WENO) polynomial reconstruction on general unstructured grids composed of curved-edged quadrilateral elements. Our technique relies on a segregation of the conditions that match the spatial averages of the candidate WENO polynomial and the reconstructed state variable over stencil-forming cells into two sets. The first set consists of exactly enforced linearly independent constraints that are applied only over the central cell and its edge-sharing neighbors. The second set consists of the remaining conditions and are enforced only approximately in a least-squares sense. We develop a specialized technique that reduces the computational burden associated with the numerical solution of the constrained least-squares system for the WENO interpolation coefficients. For a comparable computational expense, our constrained reconstruction approach yields pointwise estimates that are consistently more accurate than the conventional least-squares based estimates over a wide spectrum of spatial wavenumbers. This enhanced spectral resolution yields a significant, over two-fold increase in the accuracy with which unsteady smooth flow features, such as an advecting isentropic vortex, are captured over highly distorted curvilinear unstructured grids. Our constraint based reconstruction framework is generic, facilitates implementation of high-order boundary closures, and is amenable to large scale parallelization over several thousand CPU cores. Numerical tests over a range of inviscid and viscous flow configurations are presented to demonstrate the accuracy and robustness of our technique. Our constrained least-squares based adaptive WENO discretization faithfully captures intricate features that arise in strongly shocked, and highly unsteady and separated complex geometry flows. Our results suggest exact preservation of the match between the cell averages of the interpolant and the reconstructed state variable over the nearest neighbors to be quite an effective strategy that dramatically enhances the accuracy and spectral resolution of the high-order finite-volume reconstruction operation.

Approximation Properties of a New Type of Gamma Operator Defined with the Help of k -Gamma Function

With the help of the k -Gamma function, a new form of Gamma operator is given in this article. Voronovskaya type theorem, weighted approximation, rates of convergence, and pointwise estimates have been found for approximation features of the newly described operator. Finally, numerical examples have been provided to demonstrate that the operator is approaching the function.

Generalized Volterra‐type operators on generalized Fock spaces

AbstractLet φ and g be entire functions on the complex plane . The generalized Volterra‐type operators and induced by φ and g are defined by and where f is an entire function and .In this paper, we characterize the boundedness and compactness of the generalized Volterra‐type operators and acting between the generalized Fock spaces , induced by smooth radial weights ϕ that decay faster than the classical Gaussian ones. In addition, we obtain a upper pointwise estimate for the Bergman kernel for .

Using Simultaneous Confidence Bands to Calculate the Margin of Error in Estimating Typical Biomechanical Waveforms.

Studies of human movement usually collect data from multiple repetitions of a task and use the average of all movement trials to approximate the typical kinematics or kinetics pattern for each individual. Few studies report the expected accuracy of these estimated mean kinematics or kinetics waveforms for each individual. The purpose of this study is to demonstrate how simultaneous confidence bands, which is an approach to quantify uncertainty across an entire waveform based on limited data, can be used to calculate margin of error (MOE) for waveforms. Bilateral plantar pressure data were collected from 70 participants as they walked over 4 surfaces for an average of at least 300 steps per surface. The relationship between MOE and the number of steps included in the analysis was calculated using simultaneous confidence bands, and 3 methods commonly used for pointwise estimates: intraclass correlation, sequential averaging, and T-based MOE. The conventional pointwise approaches underestimated the number of trials required to estimate biomechanical waveforms within a desired MOE. Simultaneous confidence bands are an objective approach to more accurately estimate the relationship between the number of trials collected and the MOE in estimating typical biomechanical waveforms.

Estimates for matrix coefficients of representations

Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of ${\rm SL}(2,\Bbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schr\"odinger equation in $\Bbb{R}^n$.

Commutators of local bilinear maximal operator with Sobolev symbols

In the present paper, we introduce the commutators of local bilinear maximal operator, local bilinear maximal commutators and their fractional variants. Under the conditions that the symbol function belongs to the first-order Sobolev spaces, the bounds for the above commutators on the Sobolev spaces are established. These results are based on several new pointwise estimates for the weak gradients of the above commutators.

One step forward towards the full integration of BEM and CAD software: An effective adaptive approach

The advent of Isogeometric analysis enabled advances towards the straightforward connection between geometric design and mechanical modelling phases. 3D approaches of the Isogeometric Boundary Element Method (IGABEM) stand out in this context, because it requires information only from the boundary as well as the Computer-Aided Design (CAD) models. Thus, the 3D IGABEM best fulfils the isogeometric paradigm, since the geometric representation provided by CAD packages can be interpreted directly by BEM as mesh. However, there is a lack of information regarding convergence and accurate mesh refinement for the proper mechanical fields representation in the 3D IGABEM. Although the IGA models from CAD are geometrically exact in various problems, they usually are not refined enough for the accurate mechanical fields representation. This study proposes mesh adaptivity strategies for the 3D IGABEM formulation in elastostatics, which provide accurate geometric representation and mechanical fields description and contribute towards the full coupling of BEM and CAD schemes. The proposed strategy utilises the error based upon the hypersingular residuals, which provides point-wise error estimates at the boundary. Then, errors based on displacements/tractions or strains can be assessed. The adaptive scheme utilises mesh optimality criteria for both local and global conditions. The refinement strategy applies the knot insertion process, which makes the adaptivity procedure robust and accessible since it does not require iterative communication with the CAD system. The proposed adaptive strategy based on strains error provides good convergence rates in comparison to globally uniform refinement for homogeneous and nonhomogeneous bodies. Additionally, the mesh adaptivity strategy can be applied for fibre-reinforced IGABEM formulations, for which a different error estimator is proposed accounting for the coupling 1DBEM/BEM and FEM/BEM. The strain-based error estimator identifies the required mesh refinement for minimising the oscillating adherence forces surrounding the fibre discontinuity regions. Five applications demonstrate the accuracy of the proposed adaptive schemes, in which the globally homogeneous refinement is a reference.

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

Long-time behavior of several point particles in a 1D viscous compressible fluid

We study the long-time behavior of several point particles in a 1D viscous compressible fluid. It is shown that the velocities of the point particles all obey the power law $$t^{-3/2}$$ . This result extends author’s previous works on the long-time behavior of a single point particle. New difficulties arise in the derivation of pointwise estimates of Green’s functions due to infinite reflections of waves in-between the point particles. In particular, the differential equation technique used in previous works alone does not suffice. We overcome this by carefully analyzing the structure of Green’s functions in the Laplace variable, especially their asymptotic and analyticity properties.

On discrete q-derivatives of q-Bernstein operators

In the present paper, we shall investigate the pointwise approximation properties of the q analog of the Bernstein operators and estimate the rate of pointwise convergence of these operators to the functions f whose q-derivatives are bounded variations on the interval [0, 1]. We give an estimate for the rate of convergence of the operator (B n, q f) at those points x at which the one-sided q- derivatives Dq+ f(x), Dq− f(x) exists. We shall also prove that the operator's B n, q f converge to the limit f. As a continuation of the very recent study of the author on the q-Bernstein Durrmeyer operators [10], the present study will be the first study on the approximation of q analogous of the discrete type operators in the space of DqBV.

A pointwise estimate for pseudo-differential operators

Let [Formula: see text] be a pseudo-differential operator defined by the symbol [Formula: see text] with [Formula: see text] [Formula: see text]. It is shown that if [Formula: see text], then the operator satisfies the following pointwise estimate: [Formula: see text] for all [Formula: see text] and all Schwartz function [Formula: see text] Here, [Formula: see text] denotes the John-Nirenberg sharp function of [Formula: see text] and [Formula: see text] stands for the Hardy–Littlewood maximal operator.

Reliable Inlier Evaluation for Unsupervised Point Cloud Registration

Unsupervised point cloud registration algorithm usually suffers from the unsatisfied registration precision in the partially overlapping problem due to the lack of effective inlier evaluation. In this paper, we propose a neighborhood consensus based reliable inlier evaluation method for robust unsupervised point cloud registration. It is expected to capture the discriminative geometric difference between the source neighborhood and the corresponding pseudo target neighborhood for effective inlier distinction. Specifically, our model consists of a matching map refinement module and an inlier evaluation module. In our matching map refinement module, we improve the point-wise matching map estimation by integrating the matching scores of neighbors into it. The aggregated neighborhood information potentially facilitates the discriminative map construction so that high-quality correspondences can be provided for generating the pseudo target point cloud. Based on the observation that the outlier has the significant structure-wise difference between its source neighborhood and corresponding pseudo target neighborhood while this difference for inlier is small, the inlier evaluation module exploits this difference to score the inlier confidence for each estimated correspondence. In particular, we construct an effective graph representation for capturing this geometric difference between the neighborhoods. Finally, with the learned correspondences and the corresponding inlier confidence, we use the weighted SVD algorithm for transformation estimation.Under the unsupervised setting, we exploit the Huber function based global alignment loss, the local neighborhood consensus loss and spatial consistency loss for model optimization. The experimental results on extensive datasets demonstrate that our unsupervised point cloud registration method can yield comparable performance.

Riesz potential estimates for problems with Orlicz growth

In this paper, we consider the solutions of the non-homogeneous quasilinear elliptic equations with Dini-BMO coefficients involving measure data. At first, we prove pointwise gradient estimates for solutions by Riesz potentials; as a consequence, we obtain the borderline gradient regularity and extend a classical theorem of Stein for Poisson equations. Then we establish oscillation estimates of solutions via Riesz potentials, and these yield Hölder continuity of solutions.

Optimal rates of convergence for nonparametric regression estimation under anisotropic Hölder condition

In the multidimensional setting, we consider the nonparametric regression estimation with errors-in-variables. Both ordinary smooth noise and super smooth one are assumed for errors in the covariates. An anisotropic kernel estimator is provided based on a deconvolution technique. We study the pointwise estimation and obtain the optimal rates of convergence under the anisotropic Hölder condition.

Point-Wise Phase Estimation Method in Fringe Projection Profilometry under Non-Sinusoidal Distortion.

In fringe projection profilometry, high-order harmonics information of distorted fringe will lead to errors in the phase estimation. In order to solve this problem, a point-wise phase estimation method based on a neural network (PWPE-NN) is proposed in this paper. The complex nonlinear mapping relationship between the gray values and the phase under non-sinusoidal distortion is constructed by using the simple neural network model. It establishes a novel implicit expression for phase solution without complicated measurement operations. Compared with the previous method of combining local image information, it can accurately calculate each phase value by point. The comparison results show that the traditional method is with periodic phase errors, while the proposed method can effectively eliminate phase errors caused by non-sinusoidal phase shifting.