Periodic solutions of second order non-autonomous singular dynamical systems

In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray-Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder's fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. Recent results in the literature are generalized and significantly improved.     

On the singularities of solutions to 4-D semilinear dispersive wave equations

In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal     

Positive solutions for third-order variable-coefficient nonlinear equation with weak and strong singularities

By application of Green's function and a fixed-point theorem, i.e. Leray–Schauder alternative principle, we establish some new existence results of positive periodic solutions for nonlinear third-order singular equation with variable-coefficient, these results can be applied to study the case of a strong singularity as well as the case of a weak singularity.     

Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff

We present the first global well-posedness result for the Boltzmann equation without angular cutoff in the framework of weighted Sobolev spaces, in a close to equilibrium framework, and for Maxwellian molecules. These solutions become smooth for any positive time. An important ingredient of the proof rests on the introduction of a new norm, encoding both the singularity and the dissipation properties of the linearized collision operator.     

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: Global existence of weak solutions

For general initial data we prove the global existence and weak stability of weak solutions of the Boltzmann equation for Fermi-Dirac particles in a periodic box for very soft potentials (−5<γ?−3) with a weak angular cutoff. In particular the Coulomb interaction (γ=−3) with the weak angular cutoff is included. The conservation of energy and moment estimates are also proven under a further angular cutoff. The proof is based on the entropy inequality, velocity averaging compactness of weak solutions, and various continuity properties of general Boltzmann collision integral operators.     

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

Stress singularities in the neighborhood of corners of orthotropic plates

Generalized complex variables are used to study the character of the stresses in the neighborhood of corner points on the boundary of orthotropic plates in a two-dimensional stressed state. In the case of the first and second main problems, a singularity in the stresses is found to develop for angles with apertures exceeding π, and for the mixed problem, the appearance of a singularity depends on the plate material, as well as on the angular aperture. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 78–83, 1999.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.     

Singular and Rarefactive Solutions to a Nonlinear Variational Wave Equation

Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.     

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

Modular Invariants and Singularity Indices of Hyperelliptic Fibrations

The modular invariants of a family of curves are the degrees of the pullback of the corresponding divisors by the moduli map. The singularity indices were introduced by Xiao (1991) to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In semistable case, the author shows that the modular invariants corresponding to the boundary divisor classes are just the singularity indices. As an application, the author shows that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in semistable case.     

On initial value problems related to p-Laplacian and pseudo-Laplacian

Summary We establish some qualitative properties in the sense of weak singularity and super singularity for a certain system of two nonlinear differential equations related to the radially symmetric solution of p-Laplacian and pseudo-Laplacian problems. For the transformed system of differential equations we carry out the classification in the sense of weak singularity, singularity and super singularity. The choice of initial values at the point of singularity for correct settings of Cauchy problem is also considered.     

边界元近奇异积分计算的迭代Sinh-Sigmoidal组合式变换法

精确有效地消除积分的近奇异性是三维边界元法在工程应用中的首要问题.当源点与三角形积分单元间的距离无限趋近于零时,会出现近奇异积分问题,积分单元的形状和投影点的位置都是影响近奇异积分计算精度的重要因素.现有的非线性变换法大多只关注径向上积分的近奇异性,而忽略了角度方向和积分单元形状的影响,在投影点接近三角形积分单元边界的情况下,无法获得令人满意的计算精度,并且对子三角形积分单元的形状非常敏感.因此提出了一种改进的基于自适应分块技术和不同坐标变换的迭代sinh sigmoidal组合式变换法,分别消除径向和角度方向积分的近奇异性,在确保计算精度的同时,大大减小了计算规模.数值算例验证了该方法的有效性.     

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L¹L^p. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L¹ of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result. Mathematics Subject Classifications (2000) 60K35, 35S10.     

Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier–Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are axisymmetric, where γ is the specific heat ratio. Moreover, we obtain a new integrability estimate of the density in any neighborhood of the symmetric axis (the singularity axis).     

Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier-Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are helically symmetric, where the constant γ is the specific heat ratio. Moreover, a new integrability estimate of the density in any neighborhood of the symmetry axis (the singularity axis) is obtained.     

Well‐posedness for the FENE dumbbell model of polymeric flows

We prove local and global well‐posedness for the FENE dumbbell model for a very general class of potentials. Indeed, in prior local or global well‐posedness results, conditions on the strength of the singularity (or on the parameter b) were made. Here we give a proof in the general case. We also prove global existence results if the data is small or if we restrict to the co‐rotational model in dimension 2. © 2008 Wiley Periodicals, Inc.     

Global weak solutions to a ferrofluid flow model

We are concerned with the global solvability of the differential system introduced by Shliomis to describe the flow of a colloidal suspension of magnetized nanoparticles in a nonconducting liquid, under the action of an external magnetic field. The system is a combination of the Navier–Stokes equations, the magnetization equation, and the magnetostatic equations. We prove, by using a method of regularization, the existence of global‐in‐time weak solutions with finite energy to an initial boundary‐value problem and establish the long‐time behaviour of such solutions. The main difficulty is due to the singularity of the gradient magnetic force and the torque. Copyright © 2007 John Wiley & Sons, Ltd.