Boundary value problem with a displacement for a hyperbolic equation of third order with a derivative under boundary conditions

В работе исследована краевая задача со смещением для гиперболического уравнения третьего порядка, которая содержит производную в граничных условиях. Доказана теорема единственности и существования регулярного решения исследуемой задачи. The paper investigates a boundary value problem with a shift for a third-order hyperbolic equation, which contains a derivative in the boundary conditions. A uniqueness and existence theorem for a regular solution of the problem under study is proved.

Nonlocal problem with the integral condition for a loaded heate equation

В работе рассмотрена нелокальная задача с интегральным условием для нагруженного уравнения теплопроводности, где нагруженное слагаемое представляет собой производную второго порядка от неизвестной функции в начале координат. Доказано существование и единственность регулярного решения. С помощью функции Грина и тепловых потенциалов доказанао существование регулярного решения исследуемой задачи. Доказательство основано на редукции поставленной задачи к интегральному уравнению Вольтерра второго рода со слабой особенностью. Из разрешимости полученных интегральных уравнений Вольтерра следует существование единственного решения поставленной задачи. In this paper, we consider a non-local problem with the integral condition for the loaded heat equation, where the loaded term is a derivative of the second order from an unknown function at the origin. The existence and uniqueness of a regular solution is proven. Using the Green’s functions and thermal potentials, the existence of a regular solution to this problem is proved. The proof is based on the reduction of the formulated problem to the second kind Volterra integral equation with a weak singularity. The solvability of the obtained Volterra integral equations implies the existence of a unique solution to the problem.

Mixed problem for the inhomogeneous Hallaire equation

A mixed boundary value problem is investigated for the inhomogeneous Hallaire equation. An explicit representation of the regular solution is found using the Fourier method.

On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term

A boundary value problem for a stationary nonlinear dispersive equation of 2l+1 order with a convective term in the form u^ku_x, k\in N was considered on an interval (0,L). The existence, uniqueness and continuous dependence of a regular solution as well as a relation between the order l and critical values of k of the equation have been established.

Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

<p style='text-indent:20px;'>For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least <inline-formula><tex-math id="M1">\begin{document}$ C^{2,1} $\end{document}</tex-math></inline-formula>). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely <i>Lipschitz domains</i> of physical interest, that we call <i>sectors</i>.</p>

The Stabilization of Asphaltenes in Different Crude Fractions: A Molecular Approach

The conditions of petroleum extraction may allow asphaltenes to precipitate, causing deposition that clogs wells, pipes and equipment, consequently reducing productivity. In this work, the solubility parameters and precipitation onset of polar fractions of heavy crudes from Brazilian fields were estimated using a simplified system of n-heptane/toluene mixtures. Asphaltenes were extracted by two different methods with regard to pressure and temperature. The samples were physically and chemically characterized, and both density (1053-1159 kg m-3) and molecular weight (1176-5316 g mol-1) were estimated based on the density of diluted asphaltenes in toluene solutions. The solubility of those fractions was studied as well as their solubility parameter (ca. 19-23 MPa0.5) based on regular solution theory, Flory-Hüggins theory and empirical correlation. The influence of asphaltene concentration (between 0.5 and 5.0 g L-1) on the solubility parameter and precipitation onset was studied, and a strong linear correlation between them was not found.