Anisotropic Spaces Research Articles

Dynamic analysis of anisotropic half space containing an elliptical inclusion under SH waves

Based on the methods of complex function, conformal mapping, and multipolar coordinate system, dynamic response of an elliptical inclusion embedded in an anisotropic half space is investigated. In order to find the solution of SH waves, the governing equation is transferred into its normalized form. Then, the scattering wave induced by the inclusion and the standing wave in the inclusion is deduced. Different incident wave angles and the corresponding anisotropy of the half space are considered to obtain the reflected waves. The elliptical inclusion is transferred into a unit circle by conformal mapping method, and then the undetermined coefficients in scattering wave and standing wave are solved by using the continuous condition at the boundary of the inclusion. Subsequently, the dynamic stress concentration factor (DSCF) around the inclusion is calculated and analyzed. Numerical results demonstrate that the distribution of the DSCF is mainly influenced by the incident wave angle and the incident wave number. It is affected by anisotropic parameters as well.

Blowup Criterion via Only the Middle Eigenvalue of the Strain Tensor in Anisotropic Lebesgue Spaces to the 3D Double-Diffusive Convection Equations

In this paper, we consider the three-dimensional (3D) incompressible double-diffusive convection equations. By making use of a anisotropic Sobolev’s inequality, we obtain a blowup criterion for the strong solution to the 3D double-diffusive convection equations involving only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces. This result improves a result established in a recent work by Miller (Arch Ration Mech Anal 235:99–139, 2019).

An eigenvalue problem for the anisotropic Φ-Laplacian

We study an eigenvalue problem involving a fully anisotropic elliptic differential operator in arbitrary Orlicz-Sobolev spaces. The relevant equations are associated with constrained minimization problems for integral functionals depending on the gradient of competing functions through general anisotropic N-functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called Δ2-condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.

Order of the trigonometric widths of the Nikol’skii-Besov classes with mixed metric in the metric of anisotropic Lorentz spaces

Order of the trigonometric widths of the Nikol’skii-Besov classes with mixed metric in the metric of anisotropic Lorentz spaces

THE CONVOLUTION IN ANISOTROPIC TRIEBEL–LIZORKIN SPACES

In this paper, we investigate the boundedness of the norm of the convolution operator in anisotropic Triebel-Lizorkin spaces. The Triebel-Lizorkin spaces are based on the Lorentz spaces pq L . In the anisotropic case, we take the anisotropic Lorentz space pq L as the base. The main goal of the paper is to solve the following problem: let f and g be functions from some classes of the Triebel-Lizorkin space scale. It is necessary to determine which conditions on the parameters of the spaces from f and g are taken and study which space belongs to their convolution gf  . An analogue of the O'Neil theorem was obtained for the Triebel-Lizorkin space scale αq pτF , where α , τ, p , q are vector parameters. Relations of the form γξ hν βη rμ F F  ↪ αq pτF are obtained, with the corresponding ratios of vector parameters γ βα  , hrp 11 1 1   , νμτ 111  , ηξq 111  . The research method is the functional spaces theory and inequalities of functional and harmonic analysis.

Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.

Three Dimensional Fibre-Reinforce Anisotropic Half Space with Lagging Behavior in the Presence of Heat Source and Gravity

A three dimensional fibre-reinforce thermoelastic model is proposed here for an anisotropic medium with lagging behaviour in the presence of heat source and gravity. In the context of memory dependent derivative, the basic governing equations and heat conduction equation are adopted here. The analytical solutions are obtained by forming the vector–matrix differential equation in the transform domain. The graphical representation and numerical analysis of diffrerent physical variables with respect to different space variables have been done to verify the initial and boundary conditions.

Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space

Given an Anosov vector field X-0, all sufficiently close vector fields are also of Anosov type. In this note, we check that the anisotropic spaces described by Faure and Sjostrand and by Dyatlov and Zworski can be chosen adapted to any smooth vector field sufficiently close to X-0 in C-1 norm

Study of 2m-th order parabolic equation in non-symmetric conical domains

This article is devoted to the study of a $N$-space dimensional linear high-order parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-symmetric conical domain. The analysis is performed in the framework of weighted anisotropic Sobolev spaces by using the domain decomposition method.

A nonlinear and anisotropic data-driven litho-petroelastic inversion for single-loop subsurface characterization: Theory, uncertainties, and a case study

A new data assimilation algorithm for joint litho-petroelastic (LPE) inversion using nonlinear anisotropic Zoeppritz reflectivity operators, together with iterative sequential filtering, is used to provide joint elastic, lithology, and petrophysical estimates. The sequential filtering approach is an attractive choice because integration of petroelastic rock models with seismic data for multiparameter estimation can become quickly ill posed. Data assimilation techniques integrate the data and model in a manner that tracks uncertainties and manages the flow of data, which helps control this difficult, ill-posed estimation problem. For anisotropic formations, joint LPE inversion makes it possible to use anisotropy information associated with the lithology units or petrophysical parameters, either as fixed lithology-dependent trends by means of prior averages or as a direct function of the petrophysical parameters. This proves beneficial for amplitude variation with angle/amplitude variation with azimuth inversion where the azimuthal invariant anisotropic parameter space, containing the vertical transverse isotropy (VTI) components of the reflectivity operators, is not resolvable independently from shear and density information. In many cases, neglecting to account for anisotropy will lead to inferior characterization of anisotropic plays. Anisotropic LPE inversion enables anisotropic characterization by identifying the anisotropic formation as part of the inversion process. LPE inversion also provides estimates of the petrophysical parameter space and, therefore, honors the relation between the anisotropic elasticity, lithology, and petrophysical parameters. This study reviews the LPE inversion workflow and shows how anisotropic LPE inversion in VTI media is used to derive jointly elastic properties, lithology, and petrophysical parameters of the Wolfcamp Permian Basin shale play in Texas. We also demonstrate the importance of including anisotropy in the inversion and show that isotropic inversion poorly delineated the lithology of the play.

Approximation by sampling‐type operators in Lp‐spaces

Approximation properties of the sampling‐type quasi‐projection operators for functions from anisotropic Besov spaces are studied. Error estimates in ‐norm are obtained for a large class of tempered distributions and a large class of functions under the assumptions that has enough decay, satisfies the Strang‐Fix conditions, and has a compatibility condition with . The estimates are given in terms of moduli of smoothness and best approximations.

Physics-Based Simulation of Ocean Scenes in Marine Simulator Visual System

The realistic simulation of ocean scenes is of great significance in many scientific fields. We propose an improved Smoothed Particle Hydrodynamics (SPH) framework to simulate the ocean scenes. The improved SPH combines nonlinear constant density constraints and divergence-free velocity field constraint. Density constraints adjust the particle distribution on position layer, so that the density is constrained to a constant state. The addition of the divergence-free velocity field constraint significantly accelerates the convergence of constant density constraint and further reduces the density change. The simulation results show that the improved SPH has high solution efficiency, large time steps, and strong stability. Then, we introduce a unified boundary handling model to simulate coupling scenes. The model samples the boundary geometry as particles by means of single layer nonuniform sampling. The contribution of the boundary particles is taken into account when the physical quantities of fluid particles are computed. The unified model can handle various types of complex geometry adaptively. When rendering the ocean, we propose an improved anisotropic screen space fluid method, which alleviates the discontinuity problem near the boundary and maintains the anisotropy of particles. The research provides a theoretical reference for the highly believable maritime scene simulation in marine simulators.

Local Limit Theorem for Randomly Deforming Billiards

We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a Central Limit Theorem for the cell index of planar motion, as well as a mixing Local Limit Theorem for the cell index with piecewise H\"older continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.

Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator

Using the Mountain Pass Theorem we show that the problem{ddtLv(t,u(t),u˙(t))=Lx(t,u(t),u˙(t)) for a.e. t∈[a,b]u(a)=u(b)=0 has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian L=F(t,x,v)+V(t,x)+〈f(t),x〉 with growth conditions determined by anisotropic G-function and some geometric conditions of Ambrosetti-Rabinowitz type.

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_{\lambda}^\ast$-functions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$. In addition, the authors also obtain the duality between $H_A^{\vec{p}}(\mathbb{R}^n)$ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $\delta$-type and non-convolutional $\beta$-order Calder\'{o}n-Zygmund operators from $H_A^{\vec{p}}(\mathbb{R}^n)$ to itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. As a corollary, the boundedness of anisotropic convolutional $\delta$-type Calder\'on-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is also presented.

Analytical techniques for linear response formula of equilibrium states

In this paper, we expose some of the analytical framework used to prove the differentiability of equilibrium states probabilities with respect to the dynamics. We explore the relationship between Birkhoff metrics in cones and their corresponding anisotropic spaces and prove some useful folklore theorems. We apply that framework to revisit several examples recently studied in the research works about equilibrium states.

Эквивалентность энтропийных и ренормализованных решений анизотропной эллиптической задачи в неограниченных областях с данными в виде меры

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

AbstractWe are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic-Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.

On the Well-Posedness of the Dissipative Kadomtsev–Petviashvili Equation

The well-posedness of the initial-value problem associated with the dissipative Kadomtsev–Petviashvili equation in the case of two-dimensional space is studied. It is proved by using a dyadic partition of unity in Fourier variables that the Cauchy problem associated with this equation is globally well posed in the anisotropic Sobolev space Hs0(ℝ2) for all s>−1/2. It is also shown that this result is sharp in a certain sense.

Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space

The global well-posedness for the KP-Ⅱ equation is established in the anisotropic Sobolev space \begin{document}$ H^{s, 0} $\end{document} for \begin{document}$ s>-\frac{3}{8} $\end{document} . Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymptotic behavior of the solution by the aid of the \begin{document}$ I $\end{document} -method in which Colliander, Keel, Staffilani, Takaoka, and Tao introduced a series of modified energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-Ⅱ equation in the weak topology of the Sobolev space when \begin{document}$ s>-\frac{1}{8} $\end{document} .