Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

Large deviation principle of occupation measure for stochastic Burgers equation

In this paper we obtain a Large Deviation Principle for the occupation measure of the solution to a stochastic Burgers equation which describes the exact rate of exponential convergence. This Markov process is strongly Feller and has a unique invariant measure. Moreover, the rate function is explicit: it is the level-2 entropy of Donsker-Varadhan.     

Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier-Stokes-Poisson system converges strongly to the strong solution of the incompressible Navier-Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained.     

Global structure and asymptotic behavior of weak solutions to flood wave equations

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.     

Thermally coupled quasi-Newtonian flows: Analysis and computation

In this paper, we consider an incompressible quasi-Newtonian flow with a temperature dependent viscosity obeying a power law, and the thermal balance includes viscous heating. Some mathematical results such as the existence and uniqueness are established, finite element approximation based on an iterative solution scheme is proposed, and convergence analysis is presented.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Convergence analysis of trial free boundary methods for the two-dimensional filtration problem

Summary In this paper, we present a scheme of convergence analysis of trial free boundary methods for the two-dimensional filtration (or dam) problem. For the purpose we present a new variational principle of the filtration problem. This variational principle is defined on the set of admissible domains (candidates of the solution) in the dam. Under mild assumptions on the configuration of the dam, we may assume that all admissible domains are mapped from the unit disk by conformal mappings. Thus, proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk, and it is done using a method in the theory of minimal surfaces. Numerical examples are given.     

On the nonlinear Schrödinger limit of the magnetic Zakharov system

In this paper, we study the limit behavior of a smooth solution for the magnetic type Zakharov system. As the parameters tend to infinity, we prove that solutions of the magnetic Zakharov system converge to the solution of the nonlinear Schrödinger equation. Moreover, the detailed convergence rate is also investigated.     

The spatial behavior of a strongly coupled non-autonomous elliptic system

This article is concerned with the spatial behavior of a strongly coupled non-autonomous elliptic system modeling the steady state of populations that compete in some region. As the competition rate tends to infinity, we obtain the uniform convergence result and prove that non-negative solution of the system converges to the positive and negative parts of a solution of a scalar limit problem.     

The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions

In this paper, we study the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the dissipative wave equation in multi-dimensions. Using the fixed point theorem, we obtain the global existence of the solution. In addition, the pointwise estimates of the solution are obtained by the method of the Green function. Furthermore, we obtain the Lp, 1?p?∞, convergence rate of the solution.     

Differential quadrature method based on the highest derivative and its applications

The numerical differentiation is often used when dealing with the differential equations. Using the numerical differentiation, the differential equations can be transformed into algebraic equations. Then we can get the numerical solution from the algebraic equations. But the numerical differentiation process is very sensitive to even a small level of errors. In contrast, it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, we provide a new method using the DQ method based on the interpolation of the highest derivative (DQIHD) for the differential equations. The original function is then obtained by integration. In this paper, the DQIHD method was applied to the buckling analysis of thin isotropic plates and Winkler plates, the numerical results agree well with the analytic solutions, and the results show that our method is of high accuracy, of good convergence with little computational efforts. And it is easy to deal with the boundary conditions.     

Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces

In this paper, we first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

Weak and Strong Convergence Theorems for New Demimetric Mappings and the Split Common Fixed Point Problem in Banach Spaces

In this paper, we consider the split common fixed point problem for new demimetric mappings in Banach spaces. Using the idea of Mann’s iteration, we prove a weak convergence theorem for finding a solution of the split common fixed point problem in Banach spaces. Furthermore, using the idea of Halpern’s iteration, we obtain a strong convergence theorem for finding a solution of the problem in Banach spaces. Using these results, we obtain well-known and new weak and strong convergence theorems in Hilbert spaces and Banach spaces.     

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in RN

In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution u_b for the given problem. Moreover, we show that the energy of u_b is strictly larger than twice the ground state energy. We also give a convergence property of u_b as b ↘ 0, where b is regarded as a positive parameter.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.