Search

We construct a new initial data to prove the ill-posedness of both Navier-Stokes and Euler equations in weaker Besov spaces in the sense that the solution maps to these equations starting from u0 are discontinuous at t = 0. [ABSTRACT FROM AUTHOR]

In this paper, the estimation of the quantile density function based on i.i.d biased observations is investigated. The bias function is assumed to be positive and bounded. Of the various smoothing methods for selecting the model parameters, hard and block thresholding methods are proposed and two adaptive estimators based on them are constructed. We evaluate these theoretical performances via the minimax approach over Besov balls. We show that these estimators obtain near-optimal and optimal convergence rates under some mild assumptions. Finally, with a simulation study and application on a real set of data, the performance quality of these estimators will be compared to other wavelet methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible magnetohydrodynamic equations and establish several logarithmically improved regularity criteria in the homogeneous Besov space. These criteria are based on certain assumptions made in the velocity field and pressure, respectively. The first result solely based on the velocity field indicates that the velocity field plays a dominant role in determining the regularity of weak solutions, when compared with the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a class of Besov spaces via the heat semigroup on Carnot groups, and provide some basic properties about these spaces. Especially, we reveal the relationship between the Besov space and the known Sobolev space on Carnot groups. Then some Sobolev type inequalities for Besov spaces are established. Finally, we introduce the Besov capacity associated with Besov spaces and investigate some measure theories and geometric properties for this capacity. Furthermore, several applications of the Besov capacity are also obtained. [ABSTRACT FROM AUTHOR]

Abstract.Deconvolution model is investigated when the design points ti are random and follow a bounded density h. We propose a wavelet hard-thresholding estimator, and show that it achieves asymptotically near-optimal convergence rates for the L2-risk when the response function f belongs to Besov space. We consider the problem under two noise structure; (1) i.i.d. Gaussian errors and (2) long-memory Gaussian errors. At a first stage, we assume h is known. For the i.i.d. Gaussian errors case, our convergence rates are similar to those found in the literature under regular design. For the long memory case, our convergence rates are new. However, at each elbow, the convergence rates will depend on the long-memory parameter only if it is small enough, otherwise, the rates will be identical to those we obtained under i.i.d. errors. At a second stage, we assume h is unknown, construct the usual hard-thresholding estimator, and derive the upper bounds for its L2-risk under the new settings. The rates seem to depend on the smoothness parameter, γ, of h when γ is relatively small. [ABSTRACT FROM AUTHOR]

A new wavelet-based estimator of the conditional density is investigated. The estimator is constructed by combining a special ratio technique and applying a non negative estimator to the density function in the denominator. We used a wavelet shrinkage technique to find an adaptive estimator for this problem. In particular, a block thresholding estimator is proposed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. Moreover, it is shown that convergence rates for the mean integrated squared error (MISE) of the adaptive estimator are optimal under some mild assumptions. Finally, a numerical example has been considered to illustrate the performance of the estimator. [ABSTRACT FROM AUTHOR]

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov Ḃ ∞ , ∞ − 1 space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato Ṁ 2 , 3 r space. More precisely, we show that the smooth solution (u , d) can be extended beyond T , provided that ∫ 0 T (∥ ∇ h u ̃ (⋅ , t) ∥ B ̇ ∞ , ∞ − 1 2 + ∥ ∂ 3 d (⋅ , t) ∥ Ṁ 2 , 3 r 2) d t < ∞. [ABSTRACT FROM AUTHOR]

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces. [ABSTRACT FROM AUTHOR]

This article considers wavelet estimations for density functions in Besov spaces based on negatively dependent biased random samples. More precisely, wavelet estimators are defined and their L p (1 ≤ p < ∞) convergence rates are provided when the parameter μ is known and unknown, respectively. [ABSTRACT FROM AUTHOR]

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case for the two-component Fornberg-Whitham system in Besov spaces Bs p,r (T) x B s−1 p,r (T) for s > max{2 + 1 p, 5 2 }. In particular, when p = 2 and r = 2, this proves the non-uniform dependence on initial data for the system in Sobolev spaces Hs (T) × H s−1 (T) for s > 5/2 . [ABSTRACT FROM AUTHOR]

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces Mp,qs,α(ℝn), α ∈ [0, 1], introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm α-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm α-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm α-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm α-modulation spaces. [ABSTRACT FROM AUTHOR]

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established. [ABSTRACT FROM AUTHOR]

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander's class on matrix weighted Besov and Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of boundedness properties of integration and differentiation operators of Riemann–Liouville type on the real axis and semi–axis. The operators are acting in smoothness function spaces of Besov type with Muckenhoupt weight functions (weights) of standard and local type. The problem posed is solved by decomposing elements of the function spaces with respect to spline wavelet systems, which are the main solution tools. The article presents in detail a scheme for constructing such systems. In their terms, the corresponding decomposition theorems are established in the paper. The main results of the study are conditions on weights for the fulfilment of inequalities connecting the norms of images and pre–images of Riemann–Liouville integration operators. [ABSTRACT FROM AUTHOR]

Let p (·) , q (·) and α (·) be variable exponents satisfying some Hölder continuous conditions. In this paper, the authors characterize the variable inhomogeneous Besov space B p (·) , q (·) α (·) (R n) and the variable inhomogeneous Triebel-Lizorkin space F p (·) , q (·) α (·) (R n) in terms of Hajłasz gradient, via establishing the boundedness of a kind of almost diagonal operator A on the corresponding sequence spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Besov space and the Besov capacity and obtain several important capacitary inequalities in a strictly local Dirichlet space, which satisfies the doubling condition and the weak Bakry–Émery condition. It is worth noting that the capacitary inequalities in this paper are proved if the Dirichlet space supports the weak (1 , 2) -Poincaré inequality, which is weaker than the weak (1 , 1) -Poincaré inequality investigated in the previous references. Moreover, we first consider the strong subadditivity and its equality condition for the Besov capacity in metric space. [ABSTRACT FROM AUTHOR]

The Koopman operator is defined as the pullback of a dynamical system on a function space. Koopman operators have been investigated as one of the most promising approaches for analyzing time series data generated by a nonlinear dynamical system. It is important to find data-driven methods to estimate the mathematical invariants of Koopman operators. Consequently, in this study, we explain the motivation and idea behind applying the Koopman operator theory to data analysis and introduce three topics pertinent to our recent progress on the theoretical aspect of Koopman operators with function space theory. We consider several types of function spaces in which Koopman operators act, for example, reproducing kernel Hilbert spaces and Besov spaces, and reveal the relationship between the boundedness of a Koopman operator and the behavior of the dynamical system. In addition, we explicitly compute the generalized spectrum of the Koopman operator of the one-sided full 2-shift. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ (x) = ∫ R n e i ⁢ ϕ ⁢ (x , ξ) ⁢ a ⁢ (x , ξ) ⁢ f ^ ⁢ (ξ) ⁢ d ξ with 푎 in the forbidden Hörmander class S ρ , 1 m and ϕ ∈ Φ 2 satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 , set m ⁢ (n , ρ , p) = − (n − ρ) ⁢ (1 2 − 1 p) + n p ⁢ (ρ − 1) . When 2 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ and s > m − m ⁢ (n , ρ , p) , we show that T ϕ , a is bounded from the Besov space B p , q s to B p , q s − m + m ⁢ (n , ρ , p) . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a with a ∈ S 1 , 1 m , and indices 푠 and m ⁢ (n , ρ , p) are sharp in some cases. [ABSTRACT FROM AUTHOR]

We introduce Besov space $ \dot {B}_{p}^{\alpha,q}(\partial \Omega _k) $ B ˙ p α , q (∂ Ω k) and Triebel-Lizorkin space $ \dot {F}^{\alpha,q}_{p}(\partial \Omega _k) $ F ˙ p α , q (∂ Ω k) on a family of model domains $ \partial \Omega _k=\left \{(\mathbf{z},z_{n+1})=(z_1,z_2,\ldots, z_{n+1}):\right. \left. \mbox {Im} (z_{n+1})=\phi (|\mathbf{z}|^2)\right \} $ ∂ Ω k = { (z , z n + 1) = (z 1 , z 2 , ... , z n + 1) : Im (z n + 1) = ϕ (| z | 2) } with $ \phi (x)=x^{k} $ ϕ (x) = x k in $ \mathbf {C}^{n+1} $ C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $ \partial \Omega _k $ ∂ Ω k in these spaces. [ABSTRACT FROM AUTHOR]

The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained. [ABSTRACT FROM AUTHOR]

Let Δ d be the discrete Laplacian defined on Z d by setting Δ d f (n →) = ∑ j = 1 d − [ f (n → + e → j) + f (n → − e → j) − 2 f (n →) ] , n → ∈ Z d , where { e → j : j = 1 , ... , d } is the standard basis for R d. In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation { u t + Δ d u = f , t ∈ [ 0 , T) u (0 , ⋅) = 0 , where T ∈ (0 , ∞). [ABSTRACT FROM AUTHOR]

This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on $ (0, T] $ provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty , \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (e+\left\Vert u(\cdot , t)\right\Vert _{\dot{B}_{\infty , \infty }^{-\delta }})}dt<\infty. \end{align*} $for some $ 0<\delta <1 $. As a consequence, this result is some improvements of recent works [11,12] established by Neustupa-Penel and Miller. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation. [ABSTRACT FROM AUTHOR]

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential. [ABSTRACT FROM AUTHOR]

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω ⟨ A ⁢ (x , D ⁢ u) , D ⁢ (φ - u) ⟩ ⁢ d x ≥ 0 for all ⁢ φ ∈ K ψ ⁢ (Ω) , where Ω is a bounded open subset of R n , ψ ∈ W 1 , p ⁢ (Ω) is a fixed function called obstacle and K ψ ⁢ (Ω) = { w ∈ W 1 , p ⁢ (Ω) : w ≥ ψ ⁢ a.e. in ⁢ Ω } is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution. [ABSTRACT FROM AUTHOR]

Estimating functions with varying degrees of smoothness is a challenging problem in nonparametric function estimation. In this paper, we propose the Lévy adaptive B-Spline regression (LABS) model, an extension of the Lévy adaptive regression kernels (LARK) models, for estimating functions with varying degrees of smoothness. The LABS model is a LARK model with B-spline basis functions as generating kernels. The B-spline basis functions consist of piecewise k-degree polynomials with k -- 1 continuous derivatives, and can systematically express functions with varying degrees of smoothness. By changing the order of the B-spline basis, the LABS model systematically adapts to the smoothness of the functions, for example, jump discontinuities, sharp peaks, and so on. The results of simulation studies and real-data examples show that the proposed model captures smooth areas, jumps, and sharp peaks of functions. The proposed model also perform best in almost all examples. Finally, we provide theoretical results that the mean function for the LABS model belongs to certain Besov spaces, based on the order of the B-spline basis, and that the prior of the model has full support on the Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove Noetherian solvability and obtain the index formula for a singular integral equation with a Carleman shift in Besov spaces. We also obtain the existence of the solution of the singular integral equation as well as their explicit representations in Besov spaces. [ABSTRACT FROM AUTHOR]

Let H = −Δ + ∣x∣2 be a Hermite operator, where Δ is the Laplacian on ℝn. In this paper, we give a new characterization of the Besov space associated with Hermite operator H by the generalized derivatives. As an application, we get some regularities for the Hermite operator H. [ABSTRACT FROM AUTHOR]

We consider a Keller–Segel model coupled to the incompressible Navier–Stokes system in 3-dimensional case. We prove that the system has a unique local solution when (u0, n0, c0) ∈ Φ011 × Φ012 × Φ013, where Φ011 × Φ012 × Φ013 is a subspace of bmo − 1 (R 3) × B ˙ p , ∞ − 2 + 3 / p (R 3) × ( B ˙ q , ∞ 3 / q (R 3) ∩ L ∞ (R 3)) . Furthermore, we obtain that the system exists a unique global solution for any small initial data (u 0 , n 0 , c 0) ∈ BMO − 1 (R 3) × B ˙ p , ∞ − 2 + 3 / p (R 3) × ( B ˙ q , ∞ 3 / q (R 3) ∩ L ∞ (R 3)) . For the difference between these spaces and known ones, our results may be regarded as a new existence theorem on the system. [ABSTRACT FROM AUTHOR]

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [EMS Surv. Math. Sci. 7 (2020), 207–251] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations (d=2$$ d=2 $$ or d=3$$ d=3 $$) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case. [ABSTRACT FROM AUTHOR]

In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the φ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in L p → (R d) and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces L p → (R d) . In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established. [ABSTRACT FROM AUTHOR]

We study the relation between the composition operators Cϕ with closed range on the weighted Bloch spaces and Cϕ with closed range on the weighted Dirichlet spaces Dαp. In particular, we study the boundedness from below of composition operators between Lpa and Lqa, between Lpa and Hardy space, and between Lpa and Besov space. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three‐dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that wθeθ∈L1(0,T;Ḃ∞,∞0)$w_{\theta }e_{\theta }\in L^{1}\big (0,T;\dot{B}_{\infty ,\infty }^{0}\big)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations. [ABSTRACT FROM AUTHOR]

For any p ∈ (0 , 1 ] and s ∈ R , the authors prove two types of characterizations of the pointwise multiplier space M (B p , p s (R n)) of the Besov space B p , p s (R n). One type is based on wavelet analysis and is an extension of a well-known argument of Yves Meyer. The other type works with Fourier analytic terms. As an application of the above two types of characterizations, the authors further obtain a characterization of bounded functions in the uniform space B p , p , unif s , τ (R n) via Haar wavelets in the critical index τ = 1 p − s n. [ABSTRACT FROM AUTHOR]

The topic of modulus of smoothness still gets the interest of many researchers due to its applicable usage in different fields, especially for function approximation. In this paper, we define a new modulus of smoothness of weighted type. The properties of our modulus are studied. These properties can be easily used in different fields, in particular, the functions in the Besov spaces Bpr when 0 < p < 1. [ABSTRACT FROM AUTHOR]

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution (μ, ω) is regular if the velocity u satisfies. [ABSTRACT FROM AUTHOR]

Consider the non-stationary Navier-Stokes equation in the upper half space of R n. We prove the decay estimates of the strong solution and its derivatives in the setting of Besov and Triebel–Lizorkin spaces. This is the first time that the decay estimates of solutions to the non-stationary Navier-Stokes equations on Besov and Triebel-Lizorkin spaces are established. Our results not only extend the known results from Hardy spaces to Besov and Triebel-Lizorkin spaces but also imply new estimates on Sobolev spaces and give better decay estimates on Hardy spaces. [ABSTRACT FROM AUTHOR]

We introduce the generalized Besov-type space over the Haar basis. We give the two-sided estimate for the norm of functions of the space in terms of their Fourier–Haar coefficients. Also, we establish a criterion for the embedding and some two-sided estimate for the approximation of in the metric of , with and . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study multiplication as well as Nemytskij operators in anisotropic vector‐valued Besov spaces Bps,ω$B^{s, \omega }_p$, Bessel potential spaces Hps,ω$H^{s, \omega }_p$, and Sobolev–Slobodeckij spaces Wps,ω$W^{s, \omega }_p$. Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar‐valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector‐valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs. [ABSTRACT FROM AUTHOR]

Let φ be an analytic self-map of the open unit disk 픻 in the complex plane ℂ and u be an analytic function on 픻. The weighted composition operator is defined on the space H(픻) of analytic functions on 픻 by u C φ f = u ⋅ (f ∘ φ) , f ∈ H (D). The boundedness and the compactness of weighted composition operators from the Besov space into nth weighted type spaces are given in this work. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to B ̇ 2 , ∞ − σ 1 (R 3). [ABSTRACT FROM AUTHOR]

Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space V B p , q δ , w (R d) whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of V B p , q δ , w (R d). Next, approximation formulas such as sparse grids are derived using the determined formula. The results of this study are expected to provide considerable insight into the application of exponentially weighted Besov spaces with mixed smoothness. [ABSTRACT FROM AUTHOR]

Anisotropic functional deconvolution model is investigated in the bivariate case when the design points ti, i = 1 , 2 , ⋯ , N , and xl, l = 1 , 2 , ⋯ , M , are irregular and follow known densities h1, h2, respectively. In particular, we focus on the case when the densities h1 and h2 have singularities, but 1 / h 1 and 1 / h 2 are still integrable on [0, 1]. We construct an adaptive wavelet estimator that attains asymptotically near-optimal convergence rates in a wide range of Besov balls. The convergence rates are completely new and depend on a balance between the smoothness and the spatial homogeneity of the unknown function f, the degree of ill-posed-ness of the convolution operator and the degrees of spatial irregularity associated with h1 and h2. Nevertheless, the spatial irregularity affects convergence rates only when f is spatially inhomogeneous in either direction. [ABSTRACT FROM AUTHOR]

We prove global well-posedness for the 3D incompressible Hall-MHD equations under the assumption that (1.5) holds. Our work improves some previous consequences. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the principle of the energy conservation for the weak solutions of the incompressible inhomogeneous Euler–Korteweg equations. We provide two sufficient conditions on the regularity of the weak solutions in Besov space to ensure the energy conservation. [ABSTRACT FROM AUTHOR]