A new regularity criterion for weak solutions to the Navier–Stokes equations

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier–Stokes equations. We show that if any one component of the velocity field belongs to $L^{\alpha }([0,T);L^{\gamma }({\mathbb{R}}^3))$ with $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.⩽\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $6<\gamma ⩽\infty$, then the weak solution actually is regular and unique.     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {Omega_0} subset bigcuplimits_{k = 1}^N {{Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {left| {{v_r}} right|_{{L_infty }left( {0,T;{L_3}left( {{Omega_k}} right)} right)}} leq {alpha_1},or;{left| {frac{{{v_r}}}{r}} right|_{{L_infty }left( {0,T;{L_{3/2}}left( {{Omega_k}} right)} right)}} leq {alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) left( {v in W_2^{2,1}left( {Omega times left( {0,T} right)} right),nabla p in {L_2}left( {Omega times left( {0,T} right)} right)} right) . Bibliography: 28 titles.     

A unified approach to the integrals of Mellin–Barnes–Hecke type

In this paper, we provide a unified approach to a family of integrals of Mellin–Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by Hörmander. We derive by this method the integrals of Hecke and Sonine related to various types of Bessel functions which have found applications in analytic and algebraic number theory.     

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.     

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

We prove that any subcritical solution to the Becker–Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin and Niethammer (2003) [17]. Our approach is based on a careful spectral analysis of the linearized Becker–Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted ?¹${?}^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.     

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Study on existence of solutions for some nonlinear functional–integral equations

Using the technique of measures of noncompactness in Banach algebra, we employ abstract fixed point theorems such as Darbo’s theorem to study the existence solution in Banach algebra C[0,a]$C[0,a]$ for some functional–integral equations which include many key integral and functional equations that arise in nonlinear analysis.     

A logarithmically improved regularity criterion for the Navier–Stokes equations

In this note we prove a logarithmically improved regularity criterion in terms of the Besov space norm for the Navier–Stokes equations. The result shows that if a mild solution u satisfies ${\int_{0}^{T}\frac{\|u (t,\cdot)\|_{{\dot{B}}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}}{1+\ln(e+\| u(t,\cdot)\|_{H^{s}})}\text{d}t < \infty}$ for some 0?≤ r?<?1 and ${s\geq\frac{n}{2}-1}$ , then u is regular at t?=?T.     

Approximations of solutions to infinite–dimensional algebraic riccati equations with unbounded input operators

In this paper an approximation theory is provided for the solutions of infinite dimensional Algebraic Riccati Equations, which in particular includes convergence of the approximating Riccati operators as well as convergence of the approximating gain operators. The main features which distinguish this paper from other work existing in the literature of Riccati approximation theory are: (i) the original free system only generates a strongly continuous semigroup; (ii) the input (control) operator is generally unbounded; and (iii) no “smoothing” hypothesis on the observation operator is assumed. The abstract theory is illustrated by several examples arising in boundary control problems for wave and plate equations.     

A regularity criterion for the Navier–Stokes equations with mass diffusion

In this work, a regularity criterion is proved for local strong solutions of the Navier–Stokes equations in the presence of mass diffusion.     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.     

A note on the regularity criterion of the two-dimensional Newton–Boussinesq equations

In this paper, we consider the two-dimensional Newton–Boussinesq equations with the incompressibility condition. We obtain a regularity criterion for the Newton–Boussinesq equations by virtue of the commutator estimate.