Weighted Function Spaces Research Articles

Characterizing Nonuniform Hyperbolicity by Mather-Type Admissibility

AbstractWe consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.

Global stability of traveling waves in monostable stream-population model

<p>The stability of monotone traveling waves to a stream-population model is established in a particular weighted function space via the method of upper and lower solutions and a squeezing technique. By analyzing the behaviors of the traveling wave for a large time period under a small perturbation, we obtain the results of the local stability. The comparison principle and the squeeze theorem also allows us to prove the global stability of the positive steady-state solutions in the special weighted function space by constructing suitable upper and lower solutions.</p>

Approximation of solutions to integro-differential time fractional order parabolic equations in L^{p}-spaces

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

Gradient bounds for non-uniformly quasilinear elliptic two-sided obstacle problems with variable exponents

We are concerned with a class of quasilinear elliptic variational inequalities, the so-called two-obstacle problems, that are driven by the double-phase operators with variable exponents. In this paper, by using the terminology of fractional maximal distribution functions, we prove decay estimates for level sets of the gradient of weak solutions in turn implying regularity estimates in some generalized weighted function spaces.

Stability of self-similar solutions to geometric flows

We show that self-similar solutions for the mean curvature flow, surface diffusion, and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm in 2012, the compactness arguments adapted by Asai and Giga in 2014, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.

Approximation of solutions to integro-differential time fractional wave equations in $ L^{p}- $space

<abstract><p>In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter $ \varepsilon $. The $ L^{p}-L^{q} $ estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the $ C_{0}- $semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices $ p\in[1, +\infty) $ and $ s\in(1, +\infty) $, the mild solution of the approximation problem converges to the solution of the associated limit problem in $ L^{p}((0, T), L^{s}({\bf R}^{n})) $ as $ \varepsilon\rightarrow 0^{+} $. The resolvent operator family and a set of kernel $ k(t) $ assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as $ \alpha\rightarrow 2^{-} $.</p></abstract>

Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

<abstract><p>This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on $ \mathbb{T}^d (d \ge 2) $. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.</p></abstract>

Confluent Appell polynomials

In this paper, we introduce the confluent Appell polynomials and prove a Sheffer type characterization theorem for them by means of the Stieltjes integral of hypergeometric polynomials. We investigate their several properties such as explicit representation, integral representation and finite summation formulas. Moreover, by proving a pure recurrence relation and deriving the lowering and the raising operators, in terms of differential and shift operators, we obtain the equation satisfied the confluent Appell polynomials by using the factorization method. And then, we define the confluent Bernoulli and Hermite polynomials and exhibit their main properties such as explicit representations, recurrence formulas (involving the corresponding usual Bernoulli and Hermite polynomials), finite summation formulas and equations involving differential and shift operators. Finally, we construct approximation operators by using confluent Appell polynomials which helps to approximate to a function defined on the semi infinite interval in a weighted function space. We call these as the confluent Jakimovski–Leviatan operators which includes the confluent version of the well-known Szász–Mirakyan operators. Also, an illustrative example in order to show convergence efficiency of the confluent Szász–Mirakyan operators is given.

The Images of Integration Operators in Weighted Function Spaces

Images of integration operators of natural orders are considered as elements of Besov and Triebel--Lizorkin spaces with local Muckenhoupt weights on $\mathbb{R}^N$. The results connect entropy and approximation numbers of embedding operators with the same characteristics of the integration operators.

A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the s-dimensional unit cube by using rank-1 lattice point sets only for the first d(<s) coordinates and random points for the remaining s−d coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as long as d scales at most linearly with the number of points. This result might be useful for numerical integration in extremely high dimensions, such as partial differential equations with random coefficients for which even the standard fast component-by-component algorithm is considered computationally expensive.

L p Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A_{infty }-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Linear Differential Equations on Some Classes of Weighted Function Spaces

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation h(k)+Kk−1(υ)h(k−1)+…+K1(υ)h′+K0(υ)h=0, will be discussed in this current research, whereas the considered coefficients K0(υ),…,Kk−1(υ) are holomorphic in the disc ΓR={υ∈C:|υ|&lt;R},0&lt;R≤∞. In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.

On approximation properties of a new construction of Baskakov operators

The purpose of this research is to construct sequences of Baskakov operators such that their construction consists of a function σ by use of two function sequences, xi _{n} and eta _{n} . In these operators, σ not only features the sequences of operators but also features the Korovkin function set lbrace 1,sigma ,sigma ^{2} rbrace in a weighted function space such that the operators fix exactly two functions from the set. Thereafter, weighted uniform approximation on an unbounded interval, the degree of approximation with regards to a weighted modulus of continuity, and an asymptotic formula of the new operators are presented. Finally, some illustrative results are provided in order to observe the approximation properties of the newly defined Baskakov operators. The results demonstrate that the introduced operators provide better results in terms of the rate of convergence according to the selection of σ.

The asymptotic profile of solutions to damped Euler equations

AbstractThis paper is concerned with the asymptotic behavior of solution to the damped Euler equation far away from vacuum. First of all, the global existence together with the decay rate of the solution and its derivatives are derived by the standard energy method. The decay rates of the solution operator are further improved by Green function method. Finally, the solution of damped Euler equation is confirmed to converge its best asymptotic profile when the initial data is in a certain weighted function space, which establishes the upper and lower bounds of the decay rate to the solution of the damped Euler equation. Here, we provide a new way to check the optimal decay rate of the solutions to the damped Euler equation when the initial data is set in L1, which also could be applied to some other diffusive evolution system.

Approximation by Phillips operators via q-Dunkl generalization based on a new parameter

In the present article we study the approximation properties of Phillips operators by q-Dunkl generalization. We construct the operators in a new q-Dunkl form and obtain the approximation properties in weighted function space. We give the rate of convergence in terms of Lipschitz class by initiate the modulus of continuity and finally, we present some direct theorems in Peetre’s K -functional.

Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an s-dimensional integral is fully specified by its generating vectorz∈Zs and its number of points N. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions for good generating vectors for dimensions s≥3. This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the generating vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on s. The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings.

The Hodge Laplacian on axisymmetric domains and its discretization

Abstract We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourier finite element methods (Fourier-FEMs) in weighted function spaces. Closed Hilbert complexes and commuting projectors are used as in the work of Arnold, Falk &amp; Winther, (2010, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.), 47, 281–354), by using the new family of finite element spaces for general axisymmetric problems introduced in Oh, (2015, de Rham complexes arising from Fourier-FEMs in axisymmetric domains. Comput. Math. Appl., 70, 2063–2073). In order to get stability results and error estimates for the discrete mixed formulation, we construct commuting projectors that can be applied to functions with low regularity.

F-Frequently hypercyclic $$C_{0}$$-semigroups on complex sectors

We analyze f-frequently hypercyclic, q-frequently hypercyclic ( $$q> 1$$ ), and frequently hypercyclic $$C_{0}$$ -semigroups ( $$q=1$$ ) defined on complex sectors,with values in separable infinite-dimensional Fréchet spaces. Some structural results are given for a general class of $${\mathcal F}$$ -frequently hypercyclic $$C_{0}$$ -semigroups, as well. We investigate generalized frequently hypercyclic translation semigroups and generalized frequently hypercyclic semigroups induced by semiflows on weighted function spaces. Several illustrative examples are presented.

Degenerate equations in a diffusion–precipitation model for clogging porous media

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.