Linear differential operators related to the Jacobian Conjecture have a closed image

The main result of this work is the following theorem: LetP,QɛC[x, y] satisfy the Jacobian identity

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 \(\int\limits_0^1 {\frac{{du}}{{\Omega (u)}}}< \infty .\)      

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 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.     

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 $$\begin{gathered} \frac{{\partial ^z u}}{{\partial x^2 }} - q(x)u = \frac{{\partial ^2 u}}{{\partial t^2 }} ( - \infty< x< \infty ), \hfill \\ u|_{t = 0} = f(x), \left. {\frac{{\partial u}}{{\partial t}}} \right|_{t = 0} = 0. \hfill \\ \end{gathered} $$      

An abstract Dirichlet problem in the Hilbert space

In the present paper, we consider an abstract partial differential equation of the form $\frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{{\partial ^2 u}}{{\partial x^2 }} + A\left( {x,t} \right)u = f\left( {x,t} \right)$ , where $\left\{ {A\left( {x,t} \right):\left( {x,t} \right) \in \bar G} \right\}$ is a family of linear closed operators and $\bar G = G \cup \partial G,G$ is a suitable bounded region in the (x, t)-plane with boundary?G. It is assumed thatu is given on the boundary?G. The objective of this paper is to study the considered Dirichlet problem for a wide class of operatorsA(x, t). A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considered.     

Construction of characteristic functions for the system of Navier-Stokes-Voigt equations and the BBM equation

For the system of Navier-Stokes-Voigt equations $$\frac{{\partial \vec v}}{{\partial t}} - v\Delta \vec v - \aleph \frac{{\partial \Delta \vec v}}{{\partial t}} + v_\kappa \frac{{\partial \Delta \vec v}}{{\partial x_\kappa }} + grad \rho = 0, div \vec v = 0$$ and the BBM equation $$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial \Delta v}}{{\partial x}} - \frac{{\partial ^3 v}}{{\partial t\partial x^2 }} = 0$$ characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) of the measure μ_t(ω)=μ(V _?1 ^t (ω)), describing the evolution in time of the probability measure μ(ω) defined on the set of initial conditions for the first initial boundary-value problem for system (1) or Eq. (2) are constructed and investigated. It is shown that the characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) constructed satisfy partial differential equations with an infinite number of independent variables (t; θ₁,θ₂,...) [the statistical equations of E. Hopf for the system (1) or Eq. (2)].     

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.     

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.     

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.     

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 optimization algorithm for the pile driver problem

In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL ²(0, 1) andJ(u;.) is a convex functional onH ¹(0, 1), for eachu, describing the soil-pile friction effect.     

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   

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 $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$      

Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation

We study the asymptotic behavior as t → ∞ of the solution of the initial-boundary value problem for the nonlinear integro-differential equation

$$\frac{{\partial U}}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {a\left( {\mathop \smallint \limits_0^t \left( {\frac{{\partial U}}{{\partial x}}} \right)^2 d\tau } \right)\frac{{\partial U}}{{\partial x}}} \right],$$

where a(S) = (1 + S) ^p , 0 < p ≤ 1. We consider problems with homogeneous boundary conditions as well as with a nonhomogeneous boundary condition on part of the boundary. The orders of convergence are established.

     

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 → ∞.     

Exact solutions for Belousov-Zhabotinskii reaction-diffusion system

Abstract. We consider the following simplified model for the Belousou-Zhabotinskii(B-Z)reaction:     

On the strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

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.     

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.