Symmetry and nonexistence of positive solutions to an integral system with weighted functions

Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable conditions.It is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 <...     

Symmetry of solutions to some systems of integral equations

In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.

     

带权函数的积分方程组正解的对称性和单调性

主要研究全空间上一类带权函数的积分方程组正解的径向对称性和单调性问题．在合适条件下,主要利用积分形式的移动平面方法,Hardy—Littlewwood—Sobolev（HLS）和Holder不等式给出了积分方程组正解的径向对称性和单调性的结论．这一结论很好的推广了已有的结果．     

On the regularity of solutions to a parabolic system related to Maxwell's equations

The goal of this paper is to establish Hölder estimates for the solutions of a certain parabolic system related to Maxwell's equations arising in a quasi-stationary electromagnetic field.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

Remark on the regularity for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic (MHD) equations. Some regularity criteria, which are related only with u+B or u?B, are obtained for weak solutions to the MHD equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Liouville type theorems for a system of integral equations on upper half space

In this paper,we consider the following system of integral equations on upper half space {u(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α) λ1up1(y) + μ1vp2(y) + β1up3(y)vp4(y) dy;v(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α)(λ2uq1(y) + μ2vq2(y) + β2uq3(y)vq4(y) dy,where Rn + = {x =(x1,x2,...,xn) ∈ Rn|xn 0}, =(x1,x2,...,xn-1,-xn) is the reflection of the point x about the hyperplane xn= 0,0 α n,λi,μi,βi≥ 0(i = 1,2) are constants,pi≥ 0 and qi≥ 0(i = 1,2,3,4).We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.     

A system of integral equations on half space

Let be the n-dimensional upper half Euclidean space, and let α be any real number satisfying 0<α<n, we study positive solutions of the following system of integral equations in :

     

Liouville type theorems for the system of integral equations

Consider the system of integral equations in Rⁿ

     

Constant‐sign solutions for singular systems of Fredholm integral equations

We consider the system of Fredholm integral equations where T>0 is fixed and the nonlinearities H_i(t, u₁, u₂, …, u_n) can be singular at t=0 and u_j=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of constant‐sign solutions, i.e. θ_iu_i(t)≥0 for t∈[0, 1] and 1≤i≤n, where θ_i∈{1,?1} is fixed. We also include an example to illustrate the usefulness of the results obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal time and space regularity for solutions of degenerate differential equations

We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.      

Remarks on regularity criteria for axially symmetric weak solutions to the Navier–Stokes equations

We examine the conditional regularity of the solutions to the Navier–Stokes equations in the entire three‐dimensional space under the assumption that the data are axially symmetric. We show that if a radial or angular component of velocity satisfies a weighted Serrin condition, then the solution is regular. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence of solutions for dual singular integral equations with convolution kernels in case of non-normal type

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.     

伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.