On an estimate,uniform on the semiaxis t ≥ 0, for the rate of convergence of galerkin approximations for the equations of motion of Kelvin-Voight fluids

An estimate, uniform on the semiaxis t 0, is obtained for the rate of convergence of spectral Galerkin approximations for the solutions of an initial-boundary value problem for the Eqs. (1), (2) of motion of Kelvin-Voight fluids. Namely, under the condition that the solution v is asymptotically stable for t in the norm of the Dirichlet integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 123–130, 1990.     

Asymptotic behavior as t→∞ of the solutions of initial-boundary value problems for the equations of motions of linear viscoelastic fluids

It is shown that for the nonstationary equations of motion of the linear viscoelastic fluids, whose defining equation has the form the stationary system is the Navier-Stokes stationary system with viscosity coefficient v: It is proved that for small Reynolds numbers the solutions of the initial-boundary value problems for the equations of motion of the Oldroyd fluids (M=L=1, 2, ...) and Kelvin-Voight fluids (M=L + 1, L=0, 1, 2, ...) converge for t to the solution of the first boundary value problem for the stationary Navier-Stokes system (*).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 171, pp. 174–181, 1989.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.Received: October 31, 2002; revised: September 17, 2003     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.     

Solvability of the basic initial-boundary problem for the equations of motion of an Oldroyd fluid on (0,∞) and the behavior of its solutions as t→+∞

The solvability in the large on (0,) of the first initial-boundary problem for the equations of motion of an Oldroyd fluid with two spatial variables is proved and the connection as t of the solutions of this problem with the solution of the analogous problem for the Navier-Stokes equation is investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 48–52, 1986.     

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.     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ε-approximations

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (?⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω??³. Bibliography: 25 titles.     

Interior Hölder and gradient estimates for the homogenization of the linear elliptic equations

Hölder and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius’s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O(?) in the Hölder norm; it is O(?) in W ^1,∞ based on the Avellaneda-Lin’s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.     

A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.     

Solvability in the large on (0, ∞) of the fundamental initial-boundary value problem for the two-dimensional equations of the motion of Oldroyd fluids

The unqiue global solvability for t (0, ) is proved for the system describing the two-dimensional motion of an Oldroyd fluid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 69–72, 1986.     

A comment on “Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations”

In a recent paper Alikhani and Bahrami (in press) [1], we found some defects about the exact solutions of given examples. Also, the main result (Theorem 4.5) is not fulfilled. For this purpose, some examples are given.     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.     

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.     

A direct integral equation method for the solution of dual-or triple-series equations with applications to heat conduction and diffusion–reaction systems

A direct integral equation method is presented for the solution of dual- or triple-series equations obtained from separation-of-variables solutions to mixed boundary- value problems. The approach is based upon transformation of the dual- or triple- series to a single or set of Fredholm integral equations of the first kind whose kernel and forcing function aren an infinite series that can be systematically obtained from generalized formulas. Solution values for the integral equation are ontained by application of an appropriate quadrature method that accounts for the presense of logarithmic singularities in the kernel.The integral equation method is applied to several application-type problems such as heat conduction and simultaneous diffusion with chemical reaction. Comparisons are made to exact where available and also to other approximate solutions based upon the method of wieghted residuals. The results of various numerical experiments suggest that the integral equation method can yield results of the same or superior accuracy with less computational effort than those based upon MWR.     

Towards a theory of global solvability on [0, ∞) of initial-boundary value problems for the equations of motion of oldroyd and Kelvin—Voight fluids

Classical global solvability on [0, ) is proved for initial-boundary value problems (30), (32), (33), and (31), (32), (33) which describe two-dimensional motion of Oldroyd fluids and three-dimensional motion of Kelvin—Voight fluids of orders L = 2, 3, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 121–141, 1990.     

Regularity results for the gradient of solutions linear elliptic equations with L 1, λ data

We prove that if f belongs to the Morrey space , with λ ∊ [0, n−2], and u is the solution of the problem

Global existence of weak solutions for the Navier�CStokes equations with capillarity on the half-line

In this paper, we construct the global weak solutions to the initial-boundary problem for the Navier–Stokes system with capillarity in the half space \mathbbR₊¹{\mathbb{R}_+^1}. The result extends Eugene Tsyganov’s existence theorem which considered the problem in the finite region published in J. Differential Equaions 245:3936–3955, 2008.     

A priori estimates and blow-up behavior for solutions of −QNu = Veu in bounded domain in ℝN

Let Q_N be an N-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, N-Laplacian operator and the anisotropic Laplacian operator. We firstly obtain the properties of Q_N, which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of Q_Nu in bounded domain in ?^N, N ≥ 2 are established. Finally, the blow-up behavior of the only singular point is also considered.