Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter $$\tau \rightarrow \infty $$ τ → ∞ . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }({\mathbb {R}}^n)$$ B ˙ q , σ s ( R n ) and $$\dot{F}^s_{q,\sigma }({\mathbb {R}}^n)$$ F ˙ q , σ s ( R n ) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

Numerical analysis of the stability of nonequilibrium 3D evolutionary transfer processes in the network

Analytical methods for solving various problems of an applied nature (for example, non-stationary transfer problems over network hydro, gas and heat carriers), whose mathematical models use the formalisms of evolutionary differential systems, are possible with rare exceptions. That is why the construction of numerical and simulation models for the use of quantitative analysis methods becomes a universal research tool, if at the same time the implementation of these models on a computer is carried out – in other words, a complex of software engineering of the process under study is formed. The study uses the method of semidiscretization by a time variable of the mathematical model of the evolutionary non-equilibrium process of continuous medium transfer, which remains one of the most effective methods for analyzing applied problems. In this case, the elliptic operator of the mathematical model has a special basis (a system of eigenfunctions), which is why the analysis is reduced to the study of a boundary value problem for elliptic-type equations with a spatial variable changing on a network-like domain. The paper presents the conditions for unambiguous weak solvability of a differential-difference system, which is a difference analogue in the time variable of the original system, and the way of constructing an algorithm for finding an approximate solution is indicated. The study contains an analysis of the stability and convergence of difference schemes of evolutionary network-like nonequilibrium processes of continuous media transfer over network carriers and includes an analysis of the correctness of the mathematical model of this process. The results of the work are applicable in the framework of oil and gas engineering to the study of issues of stabilization and parametric optimization of the processes of transportation of liquid media through spatial networks.

Numerical modeling of boundary value problems for differential equations with random coefficients

This paper deals with the numerical modeling of differential equations with coefficients in the form of random fields. Using the Karhunen-Lo´eve expansion, we approximate these coefficients as a sum of independent random variables and real functions. This allows us to use the computational probabilistic analysis. In particular, we apply the technique of probabilistic extensions to construct the probability density functions of the processes under study. As a result, we present a comparison of our approach with Monte Carlo method in terms of the number of operations and demonstrate the results of numerical experiments for boundary value problems for differential equations of the elliptic type.

The Continuous Classical Boundary Optimal Control of Triple Nonlinear Elliptic Partial Differential Equations with State Constraints

Our aim in this work is to study the classical continuous boundary control vector problem for triple nonlinear partial differential equations of elliptic type involving a Neumann boundary control. At first, we prove that the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector have a unique "state" solution vector, by using the Minty-Browder Theorem. In addition, we prove the existence of a classical continuous boundary optimal control vector ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the triple adjoint equations related with the triple state equations. The Fréchet derivative is obtained. Finally we prove the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type through the Kuhn-Tucker-Lagrange's Multipliers theorem with equality and inequality constraints.

OPTIMIZATION PROBLEMS FOR A THERMOELASTIC FRICTIONAL CONTACT PROBLEM

In the present paper, we analyze and study the control of a static thermoelastic contact problem. We consider a model which describes a frictional contact problem between a thermoelastic body and a deformable heat conductor obstacle. We derive a variational formulation of the model which is in the form of a coupled system of the quasi-variational inequality of elliptic type for the displacement and the nonlinear variational equation for the temperature. Then, under a smallness assumption, we prove the existence of a unique weak solution to the problem. Moreover, we establish the dependence of the solution with respect to the data and prove a convergence result. Finally, we introduce an optimization problem related to the contact model for which we prove the existence of a minimizer and provide a convergence result.

Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) {(k,m)} -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.

Extended elliptic-type integrals with associated properties and Turán-type inequalities

Our aim is to study and investigate the family of $(p, q)$ ( p , q ) -extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with $(p, q)$ ( p , q ) -extended Gauss’ hypergeometric function and $(p, q)$ ( p , q ) -extended Appell’s double hypergeometric function $F_{1}$ F 1 . Turán-type inequalities including log-convexity properties are proved for these $(p, q)$ ( p , q ) -extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these $(p, q)$ ( p , q ) -extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with $(p, q)$ ( p , q ) -extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce $(p, q)$ ( p , q ) -extension of the Epstein–Hubbell (E-H) elliptic-type integral.

Asymptotic derivation and simulations of a non-local Exner model in large viscosity regime

The present paper deals with the modeling and numerical approximation of bed load transport under the action of water. A new shallow water type model is derived from the stratified two-fluid Navier-Stokes equations. Its novelty lies in the magnitude of a viscosity term that leads to a momentum equation of elliptic type. The full model, sediment and water, verifies a dissipative energy balance for smooth solutions. The numerical resolution of the sediment layer is not trivial since the viscosity introduces a non-local term in the model. Adding a transport threshold makes the resolution even more challenging. A schema based on a staggered discretization is proposed for the full model, sediment and water.

ON APPROXIMATION OF FUNCTIONS FROM ZYGMUND CLASSES BY BIHARMONIC POISSON INTEGRALS

The paper is devoted to a behavior investigation of the upper bound of deviation of functions from Zygmund classes from their biharmonic Poisson integrals. Systematic research in this direction was conducted by a number of Ukrainian as well as foreign scientists. But most of the known results relate to an estimation of deviations of functions from different classes from operators that were constructed based on triangular l-methods of the Fourier series summation (Fejer, Valle Poussin, Riesz, Rogozinsky, Steklov, Favard, etc.). Concerning the results relating to linear methods of the Fourier series summation, given by a set of functions of natural argument (Abel-Poisson, Gauss-Weierstrass, biharmonic and threeharmonic Poisson integrals), in this direction the progress was less notable. This may be due to the fact that the above-mentioned linear methods the Fourier series summation are solutions of corresponding integral and differential equations of elliptic type. And, therefore, they require more time-consuming calculations in order to obtain some estimates, that are suitable for a direct use for applied purposes. At the same time, in the present paper we investigate approximative characteristics of linear positive Poisson-type operators on Zygmund classes of functions. According to the well-known results by P.P. Korovkin, these positive linear operators realize the best asymptotic approximation of functions from Zygmund classes. Thus, the estimate obtained in this paper for the deviation of functions from Zygmund classes from their biharmonic Poisson integrals (the least studied and most valuable among all linear positive operators) is relevant from the viewpoint of applied mathematics.