Global well-posedness of the d-D Oldroyd-B type models with fractional Laplacian dissipation

This paper focuses on the Cauchy problem of the d-dimensional incompressible Oldroyd-B type models for viscoelastic flow with fractional Laplacian dissipation, namely, with ${(?\Delta )}^{{\eta }_1}u$ and ${(?\Delta )}^{{\eta }_2}\tau$. For ${\eta }_1\ge \frac{1}{2}+\frac{d}{4}$, ${\eta }_2>0$ and ${\eta }_1+{\eta }_2\ge 1+\frac{d}{2}$, we obtain the global regularity of strong solutions when the initial data $(u_0,{\tau }_0)$ are sufficiently smooth.     

Note on nonparametric spectral analysis of wideband spectrum with missing data via sample-and-hold interpolation and deconvolution

In [1] a procedure for bias-free estimation of the autocorrelation function is introduced for equidistantly sampled data with randomly occurring samples being invalid. The method incorporates sample-and-hold interpolation of the missing data points. The occurring dynamic error of the primary estimate of the correlation function is treated by a deconvolution procedure with two parameters $c_0$ and $c_1$ with $c_0+2c_1=1$, which are the on-diagonal and the aside-diagonal parameters of a specific correction matrix (at all lag times except zero). The parameters $c_0$ and $c_1$ were obtained as a function of the probability α of a sample to be valid by numerical simulation. However, explicit expressions for the parameters $c_0(\alpha )=1-\frac{2}{\alpha }+\frac{2}{{\alpha }^2}$ and $c_1(\alpha )=\frac{1}{\alpha }-\frac{1}{{\alpha }^2}$ can be derived, which might improve the usability of the deconvolution procedure in [1].     

Dynamic behavior of droplet through a confining orifice:A lattice Boltzmann study

The motion of gravity-driven deformable droplets passing through a confining orifice in two-dimensional ($2D$) space is numerically studied by the phase-field-based multiple-relaxation-time (MRT) lattice Boltzmann (LB) model, and the ratio of orifice-to-droplet diameter is less than 1. Droplets are placed just above a sink with an orifice in the middle, accelerate under gravity and encounter the orifice plate. In this work, we mainly consider the effects of the Bond number ($Bo$), orifice-to-droplet diameter ratio ($r=d∕D$), plate thickness ($H_t$), wettability (or contact angle) and the diameter ratio of two droplets ($r_d=D_1∕D_2$) on the dynamic behavior of droplet through the orifice. The results show that these issues have great influences on the typical flow patterns (i.e., release and capture). With the decrease of contact angle, the droplet is more easily captured, and there exists a critical equilibrium contact angle ${\theta }^{eq}$ when the Bond number and the orifice-to-droplet diameter ratio as well as the thickness of the plate are specified. For the case with $\theta >{\theta }^{eq}$, the droplet can finally pass through the orifice, otherwise, the droplet cannot pass through the orifice. In addition, the droplet is more likely to pass through the orifice as the thickness of the obstacle increases. Actually, when the obstacle thickness is large enough, droplet breaks into three segments and a liquid slug is formed in a hydrophilic orifice. Finally, for the evolution of two droplets with a larger diameter ratio ($r_d=1.0$), the combined droplet finally passes through the orifice due to greater inertia than the cases with $r_d=0$ and $r_d=0.43$. Besides, we also establish the relation $r=0.57^{\frac{2}{3}}B{o}^{?\frac{1}{3}}$ which can be used to separate droplet release from capture at $H_t=1.2mm$.     

Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities

In this paper, we study the fractional Choquard equation

${(?\Delta )}^{s}u+u=\left(\right|x{|}^{?\mu }1F\left(u\right))f\left(u\right),\text{in}{\mathbb{R}}^N,$

where $N\ge 3$, $0<s<1$, $0<\mu <min\{N,4s\}$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz?aev type for the above equation. The result improves some ones in Shen et al. (2016).     

Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order $0<\epsilon <<1$ and we study the asymptotic behaviour of the flow as $\epsilon$ tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure $(v^0,p^0)$ on one hand and the limit micro-rotation $Z^0$ on the other hand. Moreover we prove that $v^0$, $p^0$ and $Z^0$ are uniquely determined via auxiliary well-posed problems.     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

A fast singular boundary method for 3D Helmholtz equation

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of $O\left(N^3\right)$ with direct solvers or $O\left(N^2\right)$ with iterative solvers, as well as the memory requirement of $O\left(N^2\right)$. To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from $O\left(N^2\right)$ to $O\left(NlogN\right)$ or $O\left(N\right)$. Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.     

Global well-posedness of the incompressible fractional Navier–Stokes equations in Fourier–Besov spaces with variable exponents

We study the Cauchy problem of the fractional Navier–Stokes equations in critical variable exponent Fourier–Besov spaces $F{\stackrel{?}{B}}_{p\left(?\right),q}^{4?2\alpha ?\frac{3}{p\left(?\right)}}$. We discuss some properties of variable exponent Fourier–Besov spaces and prove a general global well-posedness result which covers some recent works about classical Navier–Stokes equations.     

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system

$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x\right){(?△)}^{\alpha }u+\lambda u=f\left(u\right)+\gamma v,\text{in}{\mathbb{R}}^3,\hfill \\ \left(c+d{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}v{|}^2\mathrm{d}x\right){(?△)}^{\alpha }v+\mu v=g\left(v\right)+\gamma u,\text{in}{\mathbb{R}}^3,\hfill \\ u,v\in H^{\alpha }\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$

where $a,c,\lambda ,\mu >0$, $b,d\ge 0$ are constants, $\alpha \in [\frac{3}{4},1)$ and $\gamma >0$ is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms $f$ and $g$, we prove the existence of positive vector ground state solutions of Poho?aev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as $\gamma \to 0^{+}$ is explored as well. Recent results from the literature are generally improved and extended.     

An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems

The quadratic eigenvalue problem (QEP) $({\lambda }^{2}M+\lambda G+K)x=0$, with $M^T=M$ being positive definite, $K^T=K$ being negative definite and $G^T=?G$, is associated with gyroscopic systems. In Guo (2004), a cyclic-reduction-based solvent (CRS) method was proposed to compute all eigenvalues of the above mentioned QEP. Firstly, the problem is converted to find a suitable solvent of the quadratic matrix equation (QME) $M{X}^2+GX+K=0$. Then using a Cayley transformation and a proper substitution, the QME is transformed into the nonlinear matrix equation (NME) $Z+A^T{Z}^{?1}A=Q$ with $A=M+K+G$ and $Q=2(M?K)$. The problem finally can be solved by applying the CR method to obtain the maximal symmetric positive definite solution of the NME as long as the QEP has no eigenvalues on the imaginary axis or for some cases where the QEP has eigenvalues on the imaginary axis. However, when all eigenvalues of the QEP are far away from or near the origin, the Cayley transformation seems not to be the best one and the convergence rate of the CRS method proposed in Guo (2004) might be further improved. In this paper, inspired by using a doubling algorithm to solve the QME, we use a Möbius transformation instead of the Cayley transformation to present an accelerated CRS (ACRS) method for solving the QEP of gyroscopic systems. In addition, we discuss the selection strategies of optimal parameter for the ACRS method. Numerical results demonstrate the efficiency of our method.     

A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations

The construction of finite element approximations in $\mathbf{H}(\text{div},\Omega )$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region $\Omega$. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide $L^2$-errors with accuracy of order $k+1$ for sufficiently smooth flux functions, and of order $r+1$ for flux divergence. An example is $R{T}_k$ spaces on quadrilateral meshes, where $r=k$ or $k?1$ if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy $r+n+1$ as desired, for any $n\ge 1$. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level $k+n$, while keeping the original border fluxes at level $k$. The case $n=1$ has been discussed in the mentioned publication for two particular examples. General stronger enrichment $n>1$ shall be analyzed and applied to Darcy’s flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.     

GPU implementation of Borůvka’s algorithm to Euclidean minimum spanning tree based on Elias method

We present both sequential and data parallel approaches to build hierarchical minimum spanning forest (MSF) or tree (MST) in Euclidean space (EMSF/EMST) for applications whose input $N$ points are uniformly or boundedly distributed in Euclidean space. Each iteration of the sequential approach takes $O\left(N\right)$ time complexity through combining Borůvka’s algorithm with an improved component-based neighborhood search algorithm, namely sliced spiral search, which is a newly proposed improvement to Bentley’s spiral search for finding a component graph’s closest outgoing point on the plane. It works based on the uniqueness property in Euclidean space, and allows $O\left(1\right)$ time complexity for one search from a query point to find the component’s closest outgoing point at different iterations of Borůvka’s algorithm. The data parallel approach includes a newly proposed two-direction breadth-first search (BFS) implementation on graphics processing unit (GPU) platform, which is specialized for selecting a spanning tree’s shortest outgoing edge. This GPU two-direction parallel BFS enables a tree traversal operation to start from any one of its vertex acting as root. These GPU parallel implementations work by assigning $N$ threads with one thread associated to one input point, one thread occupies $O\left(1\right)$ local memory and the whole algorithm occupies $O\left(N\right)$ global memory. Experiments are conducted on both uniformly distributed data sets and TSPLIB database. We evaluate computation time of the proposed approaches on more than 80 benchmarks with size $N$ growing up to $10^6$ points on personal laptop.     

On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Global solution of a diffusive predator–prey model with prey-taxis

We consider the prey-taxis system:

$\left\{\begin{array}{cc}{u}_t=d_1\Delta u?\chi ??(u?v)+u(a?\mu u)+buf\left(v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_2\Delta v+v(c?\beta v)?uf\left(v\right),\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n$, with zero-flux boundary condition, where $a,d_1,d_2,\chi ,\mu ,b,c$ are positive constants and $\beta$ is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case $\beta =0$. Moreover, when $\beta >0$, we show that the solution exists globally and is uniformly bounded provided $\mu$ is large enough.     

Alzheimer’s disease multiclass diagnosis via multimodal neuroimaging embedding feature selection and fusion

Alzheimer’s disease (AD) will become a global burden in the coming decades according to the latest statistical survey. How to effectively detect AD or MCI (mild cognitive impairment) using reliable biomarkers and robust machine learning methods has become a challenging problem. In this study, we propose a novel AD multiclass classification framework with embedding feature selection and fusion based on multimodal neuroimaging. The framework has three novel aspects: (1) An $l_{2,1}$-norm regularization term combined with the multiclass hinge loss is used to naturally select features across all the classes in each modality. (2) To fuse the complementary information contained in each modality, an $l_p$-norm $(1<p<\infty )$ regularization term is introduced to combine different kernels to perform multiple kernel learning to avoid a sparse kernel coefficient distribution, thereby effectively exploiting complementary modalities. (3) A theorem that transforms the multiclass hinge loss minimization problem using the $l_{2,1}$-norm and $l_p$-norm regularizations to a previous solvable optimization problem and its proof are given. Additionally, it is theoretically proved that the optimization process converges to the global optimum. Extensive comparison experiments and analysis support the promising performance of the proposed method.     

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional $p$-Laplacian

$\begin{array}{c}\hfill \left\{\begin{array}{c}M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)?u\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p?2}u=F_u(x,u,v)+\lambda g\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)?v\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|v\right|}^{p?2}v=F_v(x,u,v)+\lambda h\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,v\left(x\right)\to 0,\text{as}\left|x\right|\to +\infty ,\hfill \end{array}\right.\end{array}$

where ${(?\Delta )}_p^s$ denotes the fractional $p$-Laplacian of order $s\in (0,1)$, $2\le p<\infty$, $ps<N$, $F_u=\frac{?F}{?u}$, $F_v=\frac{?F}{?v}$, $V\left(x\right)$ is allowed to be sign-changing, $\lambda >0$ and $g,h:{\mathbb{R}}^N\to \mathbb{R}$ is a perturbation. Under some certain assumptions on $f$, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.