Anisotropic Spaces Research Articles

Rip behaviour in Brans–Dicke theory

In this work, we present little rip and pseudo rip scenarios in the framework of the generalised Brans–Dicke (GBD) theory where an evolving BD parameter along with a self-interacting potential is considered. The GBD field equations are derived for an anisotropic space time to provide a more general approach to the cosmic expansion. The evolutionary behaviour of the Brans–Dicke (BD) scalar field, dynamical Brans–Dicke parameter and the self interacting potential are studied. It is not possible to separate out the dynamical Brans–Dicke parameter into distinct evolving and non-evolving part for the rip scenarios as is obtained for constant deceleration parameter. While the little rip behaviour requires a positive value for the GBD parameter, a small negative value around −1.33 favours a pseudo rip behaviour.

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.

Anisotropic ball Campanato-type function spaces and their applications

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on $${\mathbb {R}}^n$$ , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces $$H_X^A({\mathbb {R}}^n)$$ associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of $$H_X^A({\mathbb {R}}^n)$$ . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm $$\Vert \cdot \Vert _X$$ under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

Anisotropic Prodi–Serrin regularity criteria for the 3D Navier–Stokes equations involving the gradient of one velocity component

In this paper, we prove some optimal regularity criteria for the 3D incompressible Navier–Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces satisfying Prodi–Serrin condition. These results cover the previous ones established by Wolf (2015) and O (2023).

Existence of a capacity solution to a nonlinear parabolic–elliptic coupled system in anisotropic Orlicz-Sobolev spaces

In this work, we show the existence of a capacity solution for a nonlinear coupled parabolic–elliptic system in inhomogeneous anisotropic Orlicz-Sobolev spaces without assuming the Δ2-condition on the N-functions. This system describes the heat surge generated within a semiconductor device by an electrical direct current. This may be seen as a generalization of the so-called thermistor problem.

Dual Spaces of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let A be a general expansive matrix on $$\mathbb {R}^n$$ and $$H_A^{p(\cdot ),q}(\mathbb R^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A, where $$p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty ]$$ denotes a variable exponent function satisfying the globally log-Hölder continuous condition and $$q\in (0,\infty )$$ . In this article, the authors give the appropriate dual space of $$H_A^{p(\cdot ),q}(\mathbb {R}^n)$$ with full range $$p(\cdot )$$ by introducing some anisotropic variable $$\eta $$ -type Campanato spaces. As an application, the Carleson measure characterizations of these $$\eta $$ -type Campanato spaces are established. To achieve these, the authors first deduce several equivalent characterizations of the anisotropic variable $$\eta $$ -type Campanato space, and then introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. All these results are new even for the isotropic Hardy–Lorentz space $$H^{p,q}(\mathbb {R}^n)$$ and the isotropic variable Hardy–Lorentz space $$H^{p(\cdot ),q}(\mathbb {R}^n)$$ .

Higher-order NLS with anisotropic dispersion and modulation spaces: a global existence and scattering result

Abstract In this paper we transfer a small data global existence and scattering result by Wang and Hudzik to the more general case of modulation spaces $M_{p, q}^s(\mathbb{R}^d)$ where q = 1 and $s \geq 0$ or $q \in (1, \infty]$ and $s \gt \frac{d}{q^{\prime}}$ and to the non-linear Schrödinger equation with higher-order anisotropic dispersion.

A partial differential equation characterization of anisotropic Hardy spaces

AbstractWe obtain a differential characterization for the anisotropic Hardy space by identifying it with a parabolic Hardy space associated with a general continuous group. This allows to be defined using a parabolic differential equation of Calderón and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.

Linear Orthosets and Orthogeometries

Anisotropic Hermitian spaces can be characterised as anisotropic orthogeometries, that is, as projective spaces that are additionally endowed with a suitable orthogonality relation. But linear dependence is uniquely determined by the orthogonality relation and hence it makes sense to investigate solely the latter. It turns out that by means of orthosets, which are structures based on a symmetric, irreflexive binary relation, we can achieve a quite compact description of the inner-product spaces under consideration. In particular, Pasch’s axiom, or any of its variants, is no longer needed. Having established the correspondence between anisotropic Hermitian spaces on the one hand and so-called linear orthosets on the other hand, we moreover consider the respective symmetries. We present a version of Wigner’s Theorem adapted to the present context.

Boundedness of the solutions of a kind of nonlinear parabolic systems

We deal with nonlinear systems of parabolic type satisfying componentwise structural conditions. The nonlinear terms are Carathéodory maps having controlled growth with respect to the solution and the gradient and the data are in anisotropic Lebesgue spaces. Under these assumptions we obtain essential boundedness of the weak solutions.

Efficient simulation of cables with anisotropic high‐order solid finite elements

AbstractCables are slender structures with a complex geometry, that can undergo large deformations during their installation or in dynamic environments. In combination with their complex inner structure they pose a challenging problem for finite element simulations. In this article a simplified approach is presented to model cables by using an effective material and a homogeneous cross‐section, which is anisotropic in the elastic and inelastic domain. For the high‐order hierarchic shape functions with quasi‐regional mapping, anisotropic ansatz spaces are applied in tension, torsion and bending simulations. The identification of material parameters is conducted based on experimental results for a tension and a torsion test using the particle swarm optimization algorithm. Finally, the identified parameters are validated for a combined load case with torsion and free bending, showing a promising approach regarding the simulation of cable structures.

On the algebraic approach to GUP in anisotropic space

Motivated by current searches for signals of Lorentz symmetry violation in nature and recent investigations on generalized uncertainty principle (GUP) models in anisotropic space, in this paper we identify GUP models satisfying two criteria: (i) invariance of commutators under canonical transformations, and (ii) physical independence of position and momentum on the ordering of auxiliary operators in their definitions. Compliance of these criteria is fundamental if one wishes to unambiguously describe GUP using an algebraic approach and, surprisingly, neither is trivially satisfied when GUP is assumed within anisotropic space. As a consequence, we use these criteria to place important restrictions on what or how GUP models may be approached algebraically.

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces dot{textbf{F}}^{alpha }_{p,q} for the endpoint case of p = infty and the full scale of parameters alpha in mathbb {R} and q in (0,infty ]. In particular, a Peetre-type characterization of the anisotropic Besov space dot{textbf{B}}^{alpha }_{infty ,infty } = dot{textbf{F}}^{alpha }_{infty ,infty } is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in dot{textbf{F}}^{alpha }_{infty ,q}.

Embeddings of anisotropic Besov spaces into Sobolev spaces

AbstractWe study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.

Magnetohydrodynamic Kelvin–Helmholtz instabilities of supersonic shear layers with finite interface thickness and heat flux in anisotropic space plasmas

ABSTRACT The linear magnetohydrodynamic Kelvin–Helmholtz instability (KHI) in an anisotropic plasma is studied. The governing equations obtained as the 16 moments of Boltzmann–Vlasov kinetic equations, including the heat flow, are applied. In the case of tangential discontinuity between the supersonic flows along the magnetic field, the calculated growth rates as functions of the anisotropic plasma properties allow us to conclude that quasi-transverse modes grow faster. Then, dispersion equations for the KHI of quasi-transverse modes are derived, considering the finite width of the transition zone with different velocity profiles. For these modes, when the role of heat flow is not important, the plasma parameters are controlled so that the fundamental plasma instabilities (firehose and mirror) do not affect the KHI. The problem is solved analytically, which will be helpful in verifying numerical simulations. In contrast to the tangential discontinuity, the finite width of the transition layer confines KHI excitation as the wavenumber increases. In the general case of oblique propagation (when heat flux complicates the problem), the boundary value problem is solved to determine the spectral eigenvalues. In particular, it is observed that the fundamental plasma instabilities that arise in the transition zone between flows with a finite width can modify and considerably enhance the KHI.

Cosmic evolution in the anisotropic space–time from modified f(R, T) gravity

Cosmic evolution in the anisotropic space–time from modified f(R, T) gravity

Globally Coupled Anosov Diffeomorphisms: Statistical Properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, h_varepsilon . Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map varepsilon mapsto h_varepsilon is Lipschitz continuous.

On regularity criteria for MHD system in anisotropic Lebesgue spaces

<abstract><p>This paper concerns the regularity criteria of the three-dimensional magnetohydrodynamic (MHD) system in anisotropic Lebesgue spaces. Two regularity results were proved under additional assumptions on the horizontal components of the velocity field $ {{\bf{u}}} $ and the magnetic field $ {{\bf{B}}} $, or directions of Elsässer's variables $ {{\bf{u}}}\pm{{\bf{B}}} $.</p></abstract>

Blow-up criteria for different fluid models in anisotropic Lorentz spaces

<abstract><p>This paper establishes new blow-up criteria, in anisotropic Lorentz spaces, via one-directional derivatives of the velocity and magnetic fields for the Cauchy problem to the 3D magneto-micropolar model and via one-directional derivative of velocity for the Cauchy problem to the 3D nonlinear dissipative system.</p></abstract>

Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.