Regularity criteria for 3D shear thinning fluids via two velocity components

Bae & Choe in a 1997 paper established a regularity criteria for the incompressible Navier–Stokes equations in the whole space ${\mathbb{R}}^3$ based on two velocity components. The goal of this note is to extend their result to the non-Newtonian fluids with shear thinning viscosity.     

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere ${\mathbb{S}}^2$. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.     

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.     

Unconditional stability and error estimates of the modified characteristics FEMs for the time-dependent incompressible MHD equations

This paper focuses on the unconditional stability and convergence of characteristics type methods for the time-dependent incompressible MHD equations. For this purpose, we introduce a new characteristics time-discrete system. The optimal error estimates in $L^2$ and $H^1$ norms for the typical modified characteristics finite element method unconditionally can be deduced, while the whole previous works require certain time-step restrictions. Some numerical experiments document performance of the characteristics type methods for the time-dependent incompressible MHD equations.     

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).     

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.     

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

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

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

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.     

Finite-time blow-up of solutions to a cancer invasion mathematical model with haptotaxis effects

This paper considers the nonlinear system of cancer invasion model with haptotaxis effects in a bounded domain $\Omega ?{\mathbb{R}}^n$ under the homogeneous Neumann and Robin type boundary conditions. This model also includes the growth and decay effects of cancer cells, normal cells and matrix degrading enzymes. This study devoted to estimate the lower bounds for the finite time blow up of non-negative solutions of the given cancer invasion system using first order differential inequality techniques and certain inequalities.     

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].     

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.