Anisotropic Spaces Research Articles

Basic results for fractional anisotropic spaces and applications

Basic results for fractional anisotropic spaces and applications

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u0∈Ḃp,qα−2p(R+n)∩Ḃn+22,n+221−4n+2(R+n)∩Ḃp,11+np(R+n) is given, we will find the weak solutions for the Eq. (1.1) in the function space Cb[0,∞;Ḃp,qα−2p(R+n)∩Cb(0,∞;Ḃn+221−4n+2(R+n))∩L∞(0,∞;Ẇ∞1(R+n)), n+2<p<∞,1≤q≤∞,1+n+2p<α<2. We show the existence of weak solutions in the anisotropic Besov spaces Ḃp,qα,α2(R+n×(0,∞)) (see Theorem 1.2) and we show the embedding Ḃp,qα,α2R+n×(0,∞)⊂Cb[0,∞;Ḃp,qα−2p(R+n) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S(A)=F(A)A, where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions.

Nonlinear elliptic unilateral problems with measure data in the anisotropic Sobolev space

Abstract In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i ( x , u , ∇ u ) + L ( x , u , ∇ u ) + N ( x , u , ∇ u ) = μ − div ϕ ( u ) in Ω . -\mathop{\sum }\limits_{i=1}^{N}{\partial }^{i}{\sigma }_{i}\left(x,u,\nabla u)+L\left(x,u,\nabla u)+N\left(x,u,\nabla u)=\mu -{\rm{div}}\phi \left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega . We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F \mu =f-{\rm{div}}F belongs to L 1 ( Ω ) + W − 1 , p ′ ( Ω ) {L}^{1}\left(\Omega )+{W}^{-1,{p}^{^{\prime} }}\left(\Omega ) . The nonlinear terms L ( x , s , ∇ u ) L\left(x,s,\nabla u) satisfy the sign and growth conditions, and N ( x , s , ∇ u ) N\left(x,s,\nabla u) verifies only the growth conditions.

Approximation of a two‐dimensional Gross–Pitaevskii equation with a periodic potential in the tight‐binding limit

AbstractThe Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the ‐direction and a harmonic oscillator potential in the ‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.

Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications

Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications

Neural Gaussian Scale-Space Fields

Gaussian scale spaces are a cornerstone of signal representation and processing, with applications in filtering, multiscale analysis, anti-aliasing, and many more. However, obtaining such a scale space is costly and cumbersome, in particular for continuous representations such as neural fields. We present an efficient and lightweight method to learn the fully continuous, anisotropic Gaussian scale space of an arbitrary signal. Based on Fourier feature modulation and Lipschitz bounding, our approach is trained self-supervised, i.e., training does not require any manual filtering. Our neural Gaussian scale-space fields faithfully capture multiscale representations across a broad range of modalities, and support a diverse set of applications. These include images, geometry, light-stage data, texture anti-aliasing, and multiscale optimization.

Variable exponents anisotropic nonlinear elliptic systems with L p ′ → ( ⋅ ) -data

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary W ˚ 1 , p → ( ⋅ ) ( Ω , R m ) for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with L p ′ → ( ⋅ ) data for the datum f (i.e. ∈ L p ′ → ( ⋅ ) ( Ω , R m ) ), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives ∂ i u , i = 1 , … , N .

Breaking the limitation of terahertz resonances in ferrites through 4D printing of metamaterials

4D printing allows the creation of 3D-printed structures with a variety of new functionalities that respond to external stimuli, including electromagnetic properties, for which research is limited, such as terahertz resonances in ferrite materials; and therefore holding great promise for a wide range of applications. However, the limited physical properties of ferrite materials, including their anisotropic properties and limited material manipulation space, pose significant challenges to the development of ferrite materials designed for progressively increasing terahertz frequencies. Therefore, this paper proposes a platform for overcoming the limitations of terahertz resonance in ferrites through the 4D printing of metamaterials. Both the elastic and viscous moduli of the slurry are sensitive to the applied shear stress; as this stress increases, these moduli sharply decrease by several orders of magnitude, resulting in the transformation of the slurry into a fluid state similar to the liquefaction of ink. Electromagnetic coupling modes between electric dipoles and magnetic dipoles endow these ferrite metamaterials with induction and switching of terahertz resonances due to orientation stimulation. Our work sheds light on the creation of electromagnetic properties and their control by 4D printing.

About unimprovability the embedding theorems for anisotropic Nikol’skii-Besov spaces with dominated mixed derivates and mixed metric and anisotropic Lorentz spaces

About unimprovability the embedding theorems for anisotropic Nikol’skii-Besov spaces with dominated mixed derivates and mixed metric and anisotropic Lorentz spaces

Analytical analysis of the interaction between cavities and a crack in bonded isotropic and anisotropic half spaces under SH wave

Interaction between two cavities and a linear crack subjected to elastic shear horizontally (SH) wave is investigated analytically in this work. Two cavities are buried in two bounded half spaces respectively and the cross-section of each cavity has a different shape. The media of two half spaces are isotropic and anisotropic respectively, and the complex function method is applied to first solve the wave equation in each medium. Then, with the help of image principle, Green’s functions and wave field expressions in each separated half space are constructed. Subsequently, utilizing “conjunction” technique, a series of Fredholm integral equations containing undetermined forces at the interface are obtained and reduced in order to be solved. Dynamic stress distribution is mainly considered to discuss the interaction between the cavities and the crack. Finally, a parametric study on dynamic stress concentration factor (DSCF) and dynamic stress intensity factor (DSIF) presents the interaction is affected by several dimensionless factors simultaneously.

Kinetic Alfvén waves in the temperature anisotropic space plasma with a kappa-Maxwellian distribution

The dispersion and damping rate of kinetic Alfvén waves are studied in temperature anisotropic space plasma with kappa-Maxwellian distribution. Employing a kinetic approach, the wave frequency and damping rate of kinetic Alfvén waves and the modified ion acoustic waves are derived in a low β plasma, which both depend on the parameters κ and T⊥e/T∥e. The numerical analyses show that the wave frequency ωr/k∥VA of kinetic Alfvén waves is larger in kappa-Maxwellian plasma than that in Maxwellian case. The wave frequency of the modified ion acoustic waves in kappa-Maxwellian plasma is larger in the short-wave region but smaller in the long-wave region than that in Maxwellian case. Again, we found that the damping rate of kinetic Alfvén waves in kappa-Maxwellian plasma is stronger than that in Maxwellian case. The damping rate of modified ion acoustic waves in kappa-Maxwellian plasma is stronger in the short-wave region but weaker in the long-wave region than that in Maxwellian case. The impact of the parameter T⊥e/T∥eon the two modes is relatively small because we consider the low β case. These results are helpful for us to understand better the characteristics of kinetic Alfvén waves in space plasma.

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Horizontal p-Kirchhoff equation on the Heisenberg group

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, via Krasnoselskii's genus. Finally, an interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

A semilinear non-homogeneous problem related to Korteweg-de Vries Equation

In this paper, we consider a non-homogeneous generalized Korteweg-de Vries problem with some hypotheses on the right-hand side, and we give a new regularity result of the solution in an anisotropic Sobolev space. Then we apply the obtained result to a non-homogeneous KdV problem. This work is an extension of solvability results for a right-hand side f in Lebesgue space.

Embeddings of a Multi-Weighted Anisotropic Sobolev Type Space

Parameters such as various integral and differential characteristics of functions, smoothness properties of regions and their boundaries, as well as many classes of weight functions cause complex relationships and embedding conditions for multi-weighted anisotropic Sobolev type spaces. The desire not to restrict these parameters leads to the development of new approaches based on the introduction of alternative definitions of spaces and norms in them or on special localization methods. This article examines the embeddings of multi-weighted anisotropic Sobolev type spaces with anisotropy in all the defining characteristics of the norm of space, including differential indices, summability indices, as well as weight coefficients. The applied localization method made it possible to obtain an embedding for the case of an arbitrary domain and weights of a general type, which is important in applications in differential operators’ theory, numerical analysis.

Dual-View Whitening on Pre-trained Text Embeddings for Sequential Recommendation

Recent advances in sequential recommendation models have demonstrated the efficacy of integrating pre-trained text embeddings with item ID embeddings to achieve superior performance. However, our study takes a unique perspective by exclusively focusing on the untapped potential of text embeddings, obviating the need for ID embeddings. We begin by implementing a pre-processing strategy known as whitening, which effectively transforms the anisotropic semantic space of pre-trained text embeddings into an isotropic Gaussian distribution. Comprehensive experiments reveal that applying whitening to pre-trained text embeddings in sequential recommendation models significantly enhances performance. Yet, a full whitening operation might break the potential manifold of items with similar text semantics. To retain the original semantics while benefiting from the isotropy of the whitened text features, we propose a Dual-view Whitening method for Sequential Recommendation (DWSRec), which leverages both fully whitened and relaxed whitened item representations as dual views for effective recommendations. We further examine the advantages of our approach through both empirical and theoretical analyses. Experiments on three public benchmark datasets show that DWSRec outperforms state-of-the-art methods for sequential recommendation.

On subordination conditions for systems of minimal di erential operators

In this paper, we provide a review of results on apriori estimates for systems of minimal differential operators in the scale of spaces \(L^p(\Omega),\) where \(p\in[1,\infty].\) We present results on the characterization of elliptic and \(l\)-quasielliptic systems using apriori estimates in isotropic and anisotropic Sobolev spaces \(W_{p,0}^l(\mathbb
 R^n),\)\(p\in[1,\infty].\) For a given set \(l=(l_1,\dots,l_n)\in\mathbb
 N^n\) we prove criteria for the existence of \(l\)-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in \(W_{p,0}^l(\mathbb
 R^n),\) \(p\in[1,\infty],\) nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the \(L^\infty(\mathbb R^n)\)-norm to the tensor product of two elliptic differential polynomials.

On the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients

This work is devoted to the study of the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.