Theory of nonstationary flows of Kelvin-Voigt fluids

One proves the global unique solvability in class \(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system $$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$      

TheH-function and temperature in a non-homogeneous bar

We employ theH-function in obtaining the formal solution of the partial differential equation $$\frac{{\partial v}}{{\partial t}} = \lambda \frac{\partial }{{\partial u}}\left[ {(1 - u^2 )\frac{{\partial v}}{{\partial t}}} \right]$$ related to a problem of heat conduction by making use of the integral and orthogonality property of the Jacobi polynomials. The result generalizes a number of known particular case on specialization of the parameters.     

Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations

The uniqueness of solutions for Cauchy problem of the form $$\frac{{\partial u}}{{\partial t}} = \Delta A(u) + \sum\limits_{i = 1}^N {\frac{{\partial b^i (u)}}{{\partial x_i }} + c(u)} $$ is studied. It is proved that ifu ∈BVx and A(u) is strictly increasing, the solution is unique.     

On convergence and stability of difference schemes for derivative nonlinear evolution equations

We investigate the initial-boundary value problem for the nonlinear equation system $$\frac{{\partial u}}{{\partial t}} = A\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u) + g(u)\frac{{\partial u}}{{\partial x}},$$ whereA is a complex diagonal matrix,f andg are complex vector-functions. The convergence and stability in theW ₂ ² norm of the proposed Crank-Nicolson type difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed.     

On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier-Stokes equations,equation of motion of the Oldroyd fluids,and equations of motion of the Kelvin-Voight fluids

Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem 9 $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ 10 $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ . We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.     

On the modulus of continuity in connection with a problem of J. Szabados concerning strong approximation

Пустьf(x) — интегрируемая 2π-периодическая функция, aω(f,δ) иs _n(x)=s_n(f, x). соответственно, модуль непрерывности иn-ая сумма Фурье этой функции. В настоящей работе, продолжающей исследования Г. Фрейда, Л. Лейндлера—E. M. Никищина, И. Сабадоша и К. И. Осколкова, доказывается следующая теорема.Если Ω(u) — выпуклая или вогнутая непрерывная функция и если (1) 1 $$\left\| {\left. {\sum\limits_{k = 1}^\infty \Omega (|S_k (x) - f(x)|)} \right\|_C } \right.$$ то 1 $$\omega (f;\delta ) = O\left( {\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} } \right),$$ где ¯Ω(v) —функция, обратная к Ω(и). При этом существует функция f₀(х), удовлетворяющая условию (1), для которой $$\omega (f;\delta ) = c\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} (c > 0).$$ ЕслиΩ(u)— вогнутая функция, то интеграл \(\int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} \) можно заменить на \(\int\limits_{\bar \Omega (\delta )}^1 {\frac{{du}}{{\Omega (u)}}.} \) . Отсюда вытекает, что еслиΩ(u) — функция типа модуля непрерывности, то для того, чтобы (1) всегда влекло принадлежность f(x) классу Lip 1, необходимо и достаточно условие \(\int\limits_0^1 {\frac{{du}}{{\Omega (u)}}}< \infty .\)      

Another method for computing the densities of integrals of motion for the Korteweg-de Vries equation

In the first section of this article a new method for computing the densities of integrals of motion for the KdV equation is given. In the second section the variation with respect to q of the functional ∫ ₀ ^π w (x,t,x,;q)dx (t is fixed) is computed, where W(x, t, s; q) is the Riemann function of the problem $$\begin{gathered} \frac{{\partial ^z u}}{{\partial x^2 }} - q(x)u = \frac{{\partial ^2 u}}{{\partial t^2 }} ( - \infty< x< \infty ), \hfill \\ u|_{t = 0} = f(x), \left. {\frac{{\partial u}}{{\partial t}}} \right|_{t = 0} = 0. \hfill \\ \end{gathered} $$      

Global solutions of nonstationary kinetic equations

For the nonstationary Boltzmann equation $$\frac{{\partial F}}{{\partial t}} + \xi _d \frac{{\partial F}}{{\partial x_d }} = Q(F,F),t > 0,\xi \in R^3 ,x \in \Omega \subset R^3 ,$$ one proves the unique global solvability of the Cauchy problem under nondifferentiable initial data and the unique global solvability of initial-boundary-value problems with homogeneous boundary conditions; it is shown that the solutions of the initial-boundary-value problems decay exponentially as t → ∞.     

Evaluation of the limits of maximal means

It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ I_f =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf.     

On convergence and stability of difference schemes for nonlinear schrödinger type equations

The first and the second boundary value problems for a system of nonlinear equations of Schrödinger type $$\frac{{\partial u}}{{\partial t}} = A\frac{{\partial u}}{{\partial x}} + iB\frac{{\partial ^2 u}}{{\partial x^2 }} + f\left( {u, u*} \right)$$ are investigated. HereA andB are real and real positive definite, respectively, constant diagonal matrices, f is a polynomial complex vector function. We do not try to get rid of the addend A?u/?x. Using a new type ofa priori estimates, convergence and stability of difference schemes of Crank-Nicolson type for these problems in W ₂ ¹ norm are proved. No restrictions on the ratio of time and space grid steps are assumed.     

Spectral method for solving nonlinear wave equations

In this paper,we consider the following nonlinear wave equations:(■~2φ)/(■t~2)-(■~2φ)/(■x~2)+μ~2φ+v~2x~2φ+f(|φ|~2)φ=0,(■~2x)/(■t~2-(■~2X)/(■X~2)+α~2x+α~2x+v~2x|φ|~2+g(X)=0with the periodic-initial conditions:φ(x-π,t)=φ(x+π,t),x(x-π,t)=x(x+v,t),φ(x,0)=■_0(x),φ_t(x,0)=■_1(x),X(x,0)=■_0(x),x_t(x,0)=■_1(x),-∞相似文献   

A differential equation without a solution

A direct construction is given of a functionf(x₁, x₂) ∈ C°, such that the equation $$\frac{{\partial u}}{{\partial x_1 }} + ix_1^{2k - 1} \frac{{\partial u}}{{\partial x_2 }} = f$$ has no solution in any neighborhood of the origin; the functionf and all its derivatives vanish for x₁=0.     

Construction of a self-adjoint dilatation for a problem with impedance boundary condition

We study the spectral problem $$\left. {\Delta \mathcal{U} + K^2 \mathcal{U} = 0, \frac{{\partial u}}{{\partial n}} - iK\mathcal{O}\mathcal{U}} \right|_{\partial \Omega } = 0, \mathfrak{S} \geqslant 0$$ We construct a self-adjoint dilatation and the problem is reduced to the investigation of a dissipative operator in a space with energy metric.     

On a problem with time-nonlocal conditions for a system of equations of hyperbolic type

In the domain Ω={(x,t):0<x<l, 0<t<T} we consider a mixed problem with time-nonlocal conditions for a system of equations of the form $$\frac{{\partial u_i }}{{\partial _t }} - \lambda _i (x,t)\frac{{\partial u_i }}{{\partial x}} = fi(x,t,u),{\text{ }}i = \overline {1,n.} $$ On the bases of the method of characteristics and the method of contraction mappings we prove an existence and uniqueness theorem for the solution of this problem.     

On the order of the error function of the square-full integers

LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.: (1) $$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$ (2) $$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$ for any integer ν≥1. In fact, we prove some general results and deduce the above from them. On the basis of (1) and (2) above, we conjecture that \(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}} + \varepsilon } )\) under the assumption of R.H.     

Mean value theorem and a maximum principle for Kolmogorov's equation

For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(α^ij) is a constant nonnegative matrix andΒ=(Β _i ⁱ ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.     

On the density of image of differential operators generated by polynomials

The main result of this work is the following theorem: LetE denote the ring of entire functions on C². LetP∈C[x, y]. Forf∈E, set 1 $$D_P (f) = \frac{{\partial P}}{{\partial x}}\frac{{\partial f}}{{\partial y}} - \frac{{\partial P}}{{\partial y}}\frac{{\partial f}}{{\partial x}}$$ .Theorem.The image of D _p is dense in E if and only ifP=σ(x), where σ is an automorphism of C[x, y].     

On a strengthened extremal property of the Carathéodory-Fejér functions

Для заданной на едини чной окружности огра ниченной функцииω(ξ) рассматр ивается усложненная задача а ппроксимации аналит ическими функциями: $$\mathop {\inf }\limits_{\varphi \in H^\infty } \left[ {\left\| {\omega - \varphi } \right\| + \mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k \left| {\lambda _k } \right|} \right],$$ где ∥·∥ понимается вL ^∞,ε _k ≧0 — заданные чис ла, $$\mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k< + \infty ,\varphi (z) = \mathop \Sigma \limits_{k = 0}^\infty \lambda _k z^k .$$ Доказывается, что при всех достаточно малы хε _k экстремальной в этой задаче будет функция обычного наилучшего приближения (та же, что и приε _k =0,k=0, 1, ...). В частности, при $$\omega (\zeta ) = \frac{{\gamma _0 }}{{\zeta ^n }} + \frac{{\gamma _1 }}{{\zeta ^{n - 1} }} + ... + \frac{{\gamma _{n - 1} }}{\zeta }$$ экстремальной оказы вается дробь Каратео дори—Фейера. Переход к двойственн ой задаче позволяет получить т очные оценки для клас са интегралов типа Коши, выделяемого огранич ениями, наложенными на велич ины коэффициентов ря да Тейлора.     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

It is proved that the operator $$P \equiv - \frac{{\partial ^2 }}{{\partial x_1^2 }} - \sum\nolimits_{k = 2}^n {\frac{\partial }{{\partial x_k }}\varphi ^2 (x)} \frac{\partial }{{\partial x_k }},$$ where ? ε C^∞(Ω) (Ω is a domain in Rⁿ), {x: ?(x) = 0} is a compactun in Ω which is the closure of its internal points, has the property of global hypoellipticity in Ω, i.e., $$\begin{array}{*{20}c} {v \in D'(\Omega ),} & {Pv \in C^\infty } & {(\Omega ) \Rightarrow \upsilon \in C^\infty (\Omega ).} \\ \end{array} $$ . This operator is not hypoelliptic.