Authors:Hai-Yang Jin, Zhi-An Wang Pages: 307 - 338 Abstract: Analysis and Applications, Volume 16, Issue 03, Page 307-338, May 2018. In this paper, we consider the following dual-gradient chemotaxis model ut = Δu −∇⋅ (χu∇v) + ∇⋅ (ξf(u)∇w),x ∈ Ω,t> 0,τ1vt = Δv + αu − βv, x ∈ Ω,t> 0,τ2wt = Δw + γu − δw, x ∈ Ω,t> 0, with [math] for [math] and [math] for [math], where [math] is a bounded domain in [math] with smooth boundary, [math] and [math]. The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer’s disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when [math], the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of [math], which means there is no pattern formation. In this paper, we shall show as [math], the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case [math]. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented. Citation: Analysis and Applications PubDate: 2018-05-11T02:41:10Z DOI: 10.1142/S0219530517500087

Authors:Sibei Yang, Der-Chen Chang, Dachun Yang, Zunwei Fu Pages: 339 - 361 Abstract: Analysis and Applications, Volume 16, Issue 03, Page 339-361, May 2018. In this paper, by applying the well-known method for dealing with [math]-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained. Citation: Analysis and Applications PubDate: 2018-05-11T02:41:08Z DOI: 10.1142/S0219530517500142

Authors:Xiaoping Zhai, Yongsheng Li, Wei Yan Pages: 363 - 405 Abstract: Analysis and Applications, Volume 16, Issue 03, Page 363-405, May 2018. In this paper, we investigate the Cauchy problem for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations in critical Besov spaces. We aim at proving the local and global well-posedness for respectively large and small initial data having critical Besov regularity, without assumptions of small density variation. Citation: Analysis and Applications PubDate: 2018-05-11T02:41:06Z DOI: 10.1142/S0219530517500014

Authors:Peter Greiner, Yutian Li Pages: 407 - 433 Abstract: Analysis and Applications, Volume 16, Issue 03, Page 407-433, May 2018. Let [math] denote the holomorphic tangential vector field to the generalized upper-half plane [math]. In our terminology, [math]. Consider the [math] operator on the boundary of [math], [math]; note that [math] is nowhere elliptic, but it is subelliptic with step three. The principal result of this paper is the derivation of an explicit fundamental solution [math] to [math]. Our approach is based on special functions and their properties. Citation: Analysis and Applications PubDate: 2018-05-11T02:41:12Z DOI: 10.1142/S0219530516500196

Authors:Blanca Bujanda, José L. López, Pedro J. Pagola Pages: 435 - 448 Abstract: Analysis and Applications, Volume 16, Issue 03, Page 435-448, May 2018. We consider the incomplete gamma function [math] for [math] and [math]. We derive several convergent expansions of [math] in terms of exponentials and rational functions of [math] that hold uniformly in [math] with [math] bounded from below. These expansions, multiplied by [math], are expansions of [math] uniformly convergent in [math] with [math] bounded from above. The expansions are accompanied by realistic error bounds. Citation: Analysis and Applications PubDate: 2018-05-11T02:41:03Z DOI: 10.1142/S0219530517500099

Authors:Philippe G. Ciarlet, Maria Malin, Cristinel Mardare Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. A nonlinear Korn inequality on a surface is any estimate of the distance, up to a proper isometry of [math], between two surfaces measured by some appropriate norms (the “left-hand side” of the inequality) in terms of the distances between their three fundamental forms measured by some appropriate norms (the “right-hand side” of the inequality). The first objective of this paper is to provide several extensions of a nonlinear Korn inequality on a surface obtained in 2006 by the first and third authors and Gratie, then measured by means of [math]-norms on the left-hand side and [math]-norms on the right-hand side. First, we extend this inequality to [math]-norms on the left-hand side and [math]-norms on the right-hand side for any [math] and [math] that satisfy [math]; second, we show how the third fundamental forms can be disposed in the right-hand side; and third, we show that there is no need to introduce proper isometries of [math] in the left-hand side if the surfaces satisfy appropriate boundary conditions. The second objective is to provide nonlinear Korn inequalities on a surface where the left-hand sides are now measured by means of [math]-norms while the right-hand sides are measured by means of [math]-norms, for any [math]. These nonlinear Korn inequalities on a surface themselves rely on various nonlinear Korn inequalities in a domain in [math], recently obtained by the first and third authors in 2015 and by the first author and Sorin Mardare in 2016. Citation: Analysis and Applications PubDate: 2018-03-29T09:55:40Z DOI: 10.1142/S0219530518500136

Authors:Elena Cordero, Fabio Nicola, Eva Primo Pages: 1 - 19 Abstract: Analysis and Applications, Ahead of Print. We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in [math] with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling–Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in [math] is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from time-frequency analysis. Citation: Analysis and Applications PubDate: 2018-03-14T07:21:30Z DOI: 10.1142/S0219530518500112

Authors:Ding-Xuan Zhou Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. Deep learning based on structured deep neural networks has provided powerful applications in various fields. The structures imposed on the deep neural networks are crucial, which makes deep learning essentially different from classical schemes based on fully connected neural networks. One of the commonly used deep neural network structures is generated by convolutions. The produced deep learning algorithms form the family of deep convolutional neural networks. Despite of their power in some practical domains, little is known about the mathematical foundation of deep convolutional neural networks such as universality of approximation. In this paper, we propose a family of new structured deep neural networks: deep distributed convolutional neural networks. We show that these deep neural networks have the same order of computational complexity as the deep convolutional neural networks, and we prove their universality of approximation. Some ideas of our analysis are from ridge approximation, wavelets, and learning theory. Citation: Analysis and Applications PubDate: 2018-03-14T07:21:30Z DOI: 10.1142/S0219530518500124

Authors:Xingwei Zhang, Guojing Zhang, Hai-Liang Li Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the stability of three-dimensional compressible viscous fluid around the plane Couette flow in the presence of a uniform transverse magnetic field and show that the uniform transverse magnetic field has a stabilizing effect on the plane Couette flow. Namely, for a sufficiently large Hartmann number, the compressible viscous plane Couette flow is nonlinear stable for small Mach number and arbitrary Reynolds number so long as the initial perturbation is small enough. Citation: Analysis and Applications PubDate: 2018-02-27T03:35:03Z DOI: 10.1142/S0219530518500100

Authors:Patrick Ciarlet, Charles F. Dunkl, Stefan A. Sauter Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. In this paper, we will develop a family of non-conforming “Crouzeix–Raviart” type finite elements in three dimensions. They consist of local polynomials of maximal degree [math] on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally, we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space. Citation: Analysis and Applications PubDate: 2018-02-27T03:35:02Z DOI: 10.1142/S0219530518500070

Authors:Erich Novak, Mario Ullrich, Henryk Woźniakowski, Shun Zhang Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. The standard Sobolev space [math], with arbitrary positive integers [math] and [math] for which [math], has the reproducing kernel Kd,s(x,t) =∫ℝd ∏j=1dcos(2π(x j − tj)uj) 1 +∑0< α 1≤s∏j=1d(2πuj)2αjdu for all [math], where [math] are components of [math]-variate [math], and [math] with non-negative integers [math]. We obtain a more explicit form for the reproducing kernel [math] and find a closed form for the kernel [math]. Knowing the form of [math], we present applications on the best embedding constants between the Sobolev space [math] and [math], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [math], whereas worst case integration errors of algorithms using [math] function values are also exponentially small in [math] and decay at least like [math]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion. Citation: Analysis and Applications PubDate: 2018-02-27T03:35:02Z DOI: 10.1142/S0219530518500094

Authors:Leevan Ling, Qi Ye Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimators from normal random variables, which are defined by kernel-based probability measures induced from symmetric positive definite kernels, to reconstruct the unknown partial derivatives from scattered noisy data. Our developed theories give rise to Tikhonov regularization methods with a priori parameter, but the shape parameters of the kernels remain tunable. For that, we propose an error measure that is computable without the exact values of the derivative. This allows users to obtain a quasi-optimal kernel-based estimator by comparing the approximation quality of kernel-based estimators. Numerical examples in two dimensions and three dimensions are included to demonstrate the convergence behavior and effectiveness of the proposed numerical differentiation scheme. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:33Z DOI: 10.1142/S021953051850001X

Authors:Bin Han, Michelle Michelle Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. Many problems in applications are defined on a bounded interval. Therefore, wavelets and framelets on a bounded interval are of importance in both theory and application. There is a great deal of effort in the literature on constructing various wavelets on a bounded interval and exploring their applications in areas such as numerical mathematics and signal processing. However, many papers on this topic mainly deal with individual examples which often have many boundary wavelets with complicated structures. In this paper, we shall propose a method for constructing wavelets and framelets in [math] from symmetric wavelets and framelets on the real line. The constructed wavelets and framelets in [math] often have a few simple boundary wavelets/framelets with the additional flexibility to satisfy various desired boundary conditions. To illustrate our construction method, from several spline refinable vector functions, we present several examples of (bi)orthogonal wavelets and spline tight framelets in [math] with very simple boundary wavelets/framelets. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:31Z DOI: 10.1142/S0219530518500045

Authors:Xin Zhong Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We study an initial boundary value problem for the nonhomogeneous heat conducting fluids with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the gradient of viscosity satisfies [math]. Then, under certain smallness condition, we prove that there exists a unique global strong solution to the 2D viscous nonhomogeneous heat conducting Navier–Stokes flows with variable viscosity. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:28Z DOI: 10.1142/S0219530518500069

Authors:Seung-Yeal Ha, Jeongho Kim, Chanho Min, Tommaso Ruggeri, Xiongtao Zhang Pages: 1 - 49 Abstract: Analysis and Applications, Ahead of Print. We present a hydrodynamic model for the ensemble of thermodynamic Cucker–Smale (TCS) particles in presence of a temperature field, and study its global-in-time well-posedness in Sobolev space. Our hydrodynamic model can be formally derived from the kinetic TCS model under the mono-kinetic ansatz, and can be viewed as a pressureless gas dynamics with non-local flocking forces. For the global-in-time well-posedness, we assume that communication weight functions are non-negative and non-increasing in their arguments and initial data satisfy non-vacuum conditions and suitable regularity in Sobolev space. In this setting, we use the method of energy estimates and obtain the global existence of classical solutions in any finite time interval. We also present an asymptotic flocking estimate using the Lyapunov functional approach. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:25Z DOI: 10.1142/S0219530518500033

Authors:Alexei Iantchenko Pages: 1 - 76 Abstract: Analysis and Applications, Ahead of Print. We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr–Newman–de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov in [Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole, Commun. Math. Phys. 306(1) (2011) 119–163; Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes, Ann. Henri Poincaré 13(5) (2012) 1101–1166] to the (uncharged) Kerr–de Sitter black holes. We show that the mass of the Dirac field does not have an effect on the two leading terms in the expansions of resonances. We give an expansion of the solution of the evolution equation for the Dirac fields in the outer region of the slowly rotating Kerr–Newman–de Sitter black hole which implies the exponential decay of the local energy. Moreover, using the [math]-normal hyperbolicity of the trapped set and applying the techniques from [Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys. 335 (2015) 1445–1485; Resonance projectors and asymptotics for [math]-normally hyperbolic trapped sets, J. Amer. Math. Soc. 28 (2015) 311–381], we give location of the resonance free band and the Weyl-type formula for the resonances in the band near the real axis. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:22Z DOI: 10.1142/S0219530518500057

Authors:Radu Ioan Boţ, Ernö Robert Csetnek, Szilárd Csaba László Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. In this paper, we investigate in a Hilbert space setting a second-order dynamical system of the form ẍ(t) + γ(t)ẋ(t) + x(t) − Jλ(t)A(x(t) − λ(t)D(x(t)) − λ(t)β(t)B(x(t))) = 0, where [math][math] is a maximal monotone operator, [math] is the resolvent operator of [math] and [math] are cocoercive operators, and [math], and [math] are step size, penalization and, respectively, damping functions, all depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy–Lipschitz–Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator [math] where [math] and [math] denotes the normal cone operator of [math]. To this end, we use Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show strong convergence for trajectories to the unique zero of [math], provided that [math] is a strongly monotone operator. Citation: Analysis and Applications PubDate: 2018-02-05T03:47:19Z DOI: 10.1142/S0219530518500021

Authors:Yi Gu, Yongping Liu Pages: 1 - 15 Abstract: Analysis and Applications, Ahead of Print. We consider the class of functions on a convex body in [math], which are uniformly bounded by some second-order differential operator in any direction, and the class of periodic functions with respect to a full-rank lattice [math], which have the same restriction in any direction. We study the optimal recovery problems of these classes and obtain the estimate of the optimal algorithm that recovers functions with their values and gradient values at [math] nodes, respectively. Citation: Analysis and Applications PubDate: 2017-12-27T07:55:31Z DOI: 10.1142/S0219530517500166

Authors:Seung-Yeal Ha, Hwa Kil Kim, Jinyeong Park Pages: 1 - 39 Abstract: Analysis and Applications, Ahead of Print. The synchronous dynamics of many limit-cycle oscillators can be described by phase models. The Kuramoto model serves as a prototype model for phase synchronization and has been extensively studied in the last 40 years. In this paper, we deal with the complete synchronization problem of the Kuramoto model with frustrations on a complete graph. We study the robustness of complete synchronization with respect to the network structure and the interaction frustrations, and provide sufficient frameworks leading to the complete synchronization, in which all frequency differences of oscillators tend to zero asymptotically. For a uniform frustration and unit capacity, we extend the applicable range of initial configurations for the complete synchronization to be distributed on larger arcs than a half circle by analyzing the detailed dynamics of the order parameters. This improves the earlier results [S.-Y. Ha, H. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015) 1441–1462; Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci. 26 (2016) 357–382.] which can be applicable only for initial configurations confined in a half circle. Citation: Analysis and Applications PubDate: 2017-07-05T07:46:04Z DOI: 10.1142/S0219530517500130

Authors:Philippe G. Ciarlet, Cristinel Mardare Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. We recast the displacement-traction problem of the Kirchhoff–Love theory of linearly elastic plates as a boundary value problem with the bending moments and stress resultants inside the middle section of the plate as the sole unknowns, instead of the displacement field in the classical formulation. To this end, we show in particular how to recast the Dirichlet boundary conditions satisfied by the displacement field of the middle surface of a plate as boundary conditions satisfied by the bending moments and stress resultants. Citation: Analysis and Applications PubDate: 2017-04-11T11:48:27Z DOI: 10.1142/S0219530517500105