Authors:J. A. Carrillo, M. G. Delgadino, F. S. Patacchini Pages: 1 - 31 Abstract: Analysis and Applications, Ahead of Print. We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not. Citation: Analysis and Applications PubDate: 2018-09-19T02:42:13Z DOI: 10.1142/S0219530518500276

Authors:Lili Fan, Guiqiong Gong, Shaojun Tang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system has the solution consisting of a contact discontinuity and rarefaction waves, we show that if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends to the corresponding composition of a viscous contact wave with rarefaction waves, which extended the results by Huang–Li–Matsumura [Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system, Arch. Ration. Mech. Anal. 197 (2010) 89–116.], where they treated the viscous and heat-conductive ideal gas. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:50Z DOI: 10.1142/S0219530518500239

Authors:Claudianor O. Alves, Vincenzo Ambrosio, César E. Torres Ledesma Pages: 1 - 27 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type (−Δ)αu + a(ðœ–x)V′(u) = 0,x ∈ ℝ,limx→−∞u(x) = −1andlimx→+∞u(x) = 1, where [math], [math] are continuous functions verifying some technical conditions. For example [math] can be asymptotically periodic and potential [math] can be the Ginzburg–Landau potential, that is, [math]. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:50Z DOI: 10.1142/S0219530518500252

Authors:Gerlind Plonka, Kilian Stampfer, Ingeborg Keller Pages: 1 - 32 Abstract: Analysis and Applications, Ahead of Print. We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:49Z DOI: 10.1142/S0219530518500240

Authors:Christoph Schwab, Jakob Zech Pages: 1 - 37 Abstract: Analysis and Applications, Ahead of Print. We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps [math] on the parameter domain [math]. Dimension-independent rates of best [math]-term truncations of generalized polynomial chaos (gpc for short) approximations depend only on the summability exponent of the sequence of their gpc expansion coefficients. So-called [math]-holomorphic maps [math], with [math] for some [math], are known to allow gpc expansions with coefficient sequences in [math]. Such maps arise for example as response surfaces of parametric PDEs, with applications in PDE uncertainty quantification (UQ) for many mathematical models in engineering and the sciences. Up to logarithmic terms, we establish the dimension independent approximation rate [math] for these functions in terms of the total number [math] of units and weights in the DNN. It follows that certain DNN architectures can overcome the curse of dimensionality when expressing possibly countably-parametric, real-valued maps with a certain degree of sparsity in the sequences of their gpc expansion coefficients. We also obtain rates of expressive power of DNNs for countably-parametric maps [math], where [math] is the Hilbert space [math]. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:59Z DOI: 10.1142/S0219530518500203

Authors:Marc Briant, Sara Merino-Aceituno, Clément Mouhot Pages: 1 - 32 Abstract: Analysis and Applications, Ahead of Print. We study the Boltzmann equation on the [math]-dimensional torus in a perturbative setting around a global equilibrium under the Navier–Stokes linearization. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a [math]-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently of the Knudsen number. Finally, we prove well-posedness of the Cauchy problem for the nonlinear Boltzmann equation in perturbative setting and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal. Furthermore, this result only requires derivatives in the space variable and allows to connect solutions to the incompressible Navier–Stokes equations in these spaces. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:58Z DOI: 10.1142/S021953051850015X

Authors:Robert J. Martin, Ionel-Dumitrel Ghiba, Patrizio Neff Pages: 1 - 13 Abstract: Analysis and Applications, Ahead of Print. Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [math] which is equal to the classical Hencky strain energy WH(F) = μ∥devnlogU∥2 + κ 2 [tr(logU)]2 = μ∥logU∥2 + Λ 2 [tr(logU)]2 in a neighborhood of the identity matrix [math]; here, [math] denotes the set of [math]-matrices with positive determinant, [math] denotes the deformation gradient, [math] is the corresponding stretch tensor, [math] is the principal matrix logarithm of [math], tr is the trace operator, [math] is the Frobenius matrix norm and [math] is the deviatoric part of [math]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [math] in the so-called Valanis–Landel form WVL(F) =∑i=1nw(λ i) with [math], where [math] denote the singular values of [math]. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:57Z DOI: 10.1142/S0219530518500173

Authors:Suzhen Mao, Huoxiong Wu, Dongyong Yang Pages: 1 - 34 Abstract: Analysis and Applications, Ahead of Print. Let [math] and [math] be the Bessel operator on [math]. In this paper, the authors show that [math] (or [math], respectively) if and only if the Riesz transform commutator [math] is bounded (or compact, respectively) on Morrey spaces [math], where [math], [math] and [math]. A weak factorization theorem for functions belonging to the Hardy space [math] in the sense of Coifman–Rochberg–Weiss in Bessel setting, via [math] and its adjoint, is also obtained. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:57Z DOI: 10.1142/S0219530518500227

Authors:Nian Yao, Mingqing Xiao Pages: 1 - 27 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider a generalized stochastic model associated with affine point processes based on several classical models. In particular, we study the asymptotic behavior of the process when the initial intensity is large, i.e. the intensity of arriving events observed initially is considerably larger, which appears in many real applications. For our generalized model, we establish (i) the large deviation principle; (ii) the corresponding functional law of large numbers; (iii) the corresponding central limit theorem, that reflect the fundamentals of the process asymptotic behavior. Our obtained results include existing results as special cases with a more general structure. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:55Z DOI: 10.1142/S0219530518500197

Authors:Mircea Sofonea Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interface law which involves unilateral constraints and subdifferential conditions. We describe the model and derive its weak formulation, which is in the form of an elliptic variational–hemivariational inequality for the displacement field. Then, we establish the existence of a unique weak solution to the problem. Next, we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a particular version of the model for which we prove the continuous dependence of the solution on the bounds which govern the locking and the normal displacement constraints, respectively. We apply this convergence result in the study of an optimization problem associated to the contact model. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:55Z DOI: 10.1142/S0219530518500215

Authors:Fangchao He, Qiang Wu Pages: 1 - 17 Abstract: Analysis and Applications, Ahead of Print. We propose a bias corrected regularization kernel ranking (BCRKR) method and characterize the asymptotic bias and variance of the estimated ranking score function. The results show that BCRKR has smaller asymptotic bias than the traditional regularization kernel ranking (RKR) method. The variance of BCRKR has the same order of decay as that of RKR when the sample size goes to infinity. Therefore, BCRKR is expected to be as effective as RKR and its smaller bias favors its use in block wise data analysis such as distributed learning for big data. The proofs make use of a concentration inequality of integral operator U-statistic. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:54Z DOI: 10.1142/S0219530518500161

Authors:Jianbin Yang Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces [math]. Under some decay conditions of the generator [math], we approximate [math] with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of [math] when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by [math]. In the end, we give a reconstruction algorithm for the random sampling of functions in [math]. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:53Z DOI: 10.1142/S0219530518500185

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: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: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