Initial-boundary Problem Research Articles

Correctness of the initial-boundary problem of the compressible fluid filtration in a viscous porous medium

The local solvability of initial-boundary value problem for the system of the equations of non stationary fluid motion in a viscous deformable medium in the field of gravity is proved.

PROBLEM WITH DYNAMIC BOUNDARY CONDITIONS FOR A HYPERBOLIC EQUATION

We consider an initial-boundary problem with dynamic boundary condition for a hyperbolic equation in a rectangle. Dynamic boundary condition represents a relation between values of derivatives with respect of spacial variables of a required solution and first-order derivatives with respect to time variable. The main result lies in substantiation of solvability of this problem. We prove the existence and uniqueness of a generalized solution. The proof is based on the a priori estimates obtained in this paper, Galyorkins procedure and theproperties of Sobolev spaces.

Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation

A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.

Some properties of solutions for a sixth-order Cahn–Hilliard type equation with inertial term

ABSTRACTIn this paper, we study a Cahn–Hilliard type equation with inertial term, which arises in dynamics of phase transitions in ternary oil–water–surfactant systems. We use regularity estimates for the semigroups and a classical existence theorem of global attractor to derive that the sixth-order Cahn–Hilliard equation with possesses a global attractor. Using a lemma on the ordinary differential inequality of a second order, we prove the blow-up of the solution for the initial-boundary problem with .

Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids

We have investigated an initial-boundary problem for the perturbation equations of rotating, incompressible, and viscous magnetohydrodynamic (MHD) fluids with zero resistivity in a horizontally periodic domain. The velocity of the fluid in the domain is non-slip on both upper and lower flat boundaries. We switch the analysis of the initial-boundary problem from Euler coordinates to Lagrangian coordinates under proper initial data, and get a so-called transformed MHD problem. Then, we exploit the two-tiers energy method. We deduce the time-decay estimates for the transformed MHD problem which, together with a local well-posedness result, implies that there exists a unique time-decay solution to the transformed MHD problem. By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.

Significance of non-normality-induced patterns: Transient growth versus asymptotic stability.

Reaction-diffusion models following the original idea of Turing are widely applied to study the propensity of a system to develop a pattern. To this end, an asymptotic analysis is typically performed via the so-called dispersion relation that relates the spectral properties of a spatial operator (diffusion) to the temporal behaviour of the whole initial-boundary value reaction-diffusion problem. Here, we amend this approach by studying the transient growth due to non-normality that can also lead to a pattern development in non-linear systems. We conclude by identification of the significance of this transient growth and by assessing the plausibility of the standard spectral approach. Particularly, the non-normality-induced patterns are possible but require fine parameter tuning.

Unique solvability for the density-dependent incompressible Navier–Stokes–Korteweg system

This article concerns the strong solutions for the incompressible fluid models of Korteweg type with a density-dependent viscosity and capillary coefficient in a bounded domain Ω⊂R3. It is proved that the initial boundary problem of the density-dependent incompressible fluid of Korteweg type admits a unique local strong solution, which allows the existence of initial vacuum provided the initial data satisfies a compatibility condition.

Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials

Abstract Composites with microstructure display nonlocal effects and can be effectively modeled through dipolar elasticity. A mixed initial boundary problem is addressed for dipolar thermoelasticity. The basic equations and conditions of the problem are set-up and uniqueness results are proven, which are obtained without introducing definiteness restrictions on the elastic coefficients and, also, without the usual condition imposed to the heat conductivity tensor of being positive definite. An additional result is finally obtained, namely, a reciprocal identity of the Betti's type which underlies another uniqueness result obtained again under weak restrictions.

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less than or greater than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

Determining the characteristics of diffracted waves of small amplitude around a vessel in shallow water

The vessels at outer anchorage are exposed to the influence of a complex system of waves. This system is formed by the incident and diffracted waves. In order to examine characteristics of a wave field around the vessel, which floats idling at outer anchorage, we solved two problems. The first problem is determining the velocity potential of diffracted wave motion, caused by the incidence at arbitrary angle of regular waves with small amplitude to a stationary vessel under conditions of shallow water. The solution is executed by the matched asymptotic expansion method (MAEM). The region filled with fluid is conditionally divided into the internal and external zones. In both zones, separate boundary problems are stated, and they are considerably simpler than the initial boundary problem for the entire region. The solutions of the problems are asymptotically merged at the border of zones. The second problem is defining the characteristics of waves near a vessel. From the expression for a speed potential we derived the equation of perturbed free surface. The calculations are conducted of the waves’ amplitudes in the assigned points around the vessel. The variable parameters are the depth of water area, wavelength and a course angle of waves. The examples of wave fields are given. Determining the characteristics of a wave field transformed by the vessel is important for planning the operation of auxiliary ships (tugboats, bunkering vessels, pilot and port roads boats, as well as oil/garbage collector boats and boom crafts). This work is connected with the safety of traffic in the open sea (transfer of people and loads from one vessel to another) and with ecological safety (elimination of oil and petroleum spills). Wave height is the limiting factor for operating the auxiliary vessels, which is why determining it properly is a necessary task.

Hamilton’s principle for dynamical elasticity

There exists NO classical variational principle for continuum dynamics so far. The well-known Hamilton’s principle is valid only for the conditions prescribed at the beginning and at the end of the motion and is therefore useless to deal with the usual initial-boundary condition problems. In this paper an exact classical variational theory is established, all initial conditions are converted into natural ones, furthermore, all natural final conditions automatically meet the physical requirements, leading to a vital innovation of Hamilton’s principle.

Numerical Solution of Partial Integro-Differential Equations by using Projection Method

In this paper we study the numerical solution of initial-boundary problem for parabolic Volterra integro-differential equations in one dimensional. These equations include the partial differentiation of an unknown function and the integral term containing the unknown function which is the memory of problem. We have made an attempt to develop a method for Wavelet Galerkin which provides the approximate solution. Some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

Generalised Dirichlet to Neumann maps for linear dispersive equations on half-line

AbstractA large class of initial-boundary problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit representations are presented for the generalised Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values when an essential set of initial and boundary data is given.

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Gr\"{u}nwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.

О начально-граничной задаче применительно к вещественному уравнению теплопроводности

О начально-граничной задаче применительно к вещественному уравнению теплопроводности

Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system involving a tensor-valued sensitivity with saturation

This paper is concerned with the following Keller–Segel–Navier–Stokes system(⋆){nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)∇c),x∈Ω, t>0,ct+u⋅∇c=Δc−c+n,x∈Ω, t>0,ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ,x∈Ω, t>0,∇⋅u=0,x∈Ω, t>0, where Ω⊂R3 is a bounded domain with smooth boundary ∂Ω, κ∈R and S denotes a given tensor-valued function fulfilling|S(x,n,c)|≤CS(1+n)α with some CS>0 and α>0. As the case κ=0 has been considered in [25], it is shown in the present paper that the corresponding initial–boundary problem with κ≠0 admits at least one global weak solution if α≥37.

APPLICATION OF THE METHOD OF BOUNDARY INTEGRAL EQUATIONS FOR NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY IN AXIALLY SYMMETRIC DOMAIN

The article considers the non-stationary initial-boundary problem of thermal conductivity in axially symmetric domain in Minkowski space, formulated as equivalent boundary integral equation. Using the representation of the solution in the form of a Fourier series expansion, the problem is reformulated as an infinite system of two-dimensional singular integral equations regarding expansion coefficients. The paper presents and investigates the explicit form for fundamental solutions used in the integral representation of the solution in the domain and on the border. The obtained results can be used in the construction of efficient numerical boundary element method for estimation of structures behavior under the influence of intense thermal loads in real-time.

Asymptotic behavior for the filtration equation in domains with noncompact boundary

ABSTRACTWe consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.

Initial Boundary and Inverse Problems for the Inhomogeneous Equation of a Mixed Parabolic-Hyperbolic Equation

Для неоднородного уравнения смешанного параболо-гиперболического типа в прямоугольной области изучена задача с неоднородными граничными и начальным условиями. Решение построено в виде суммы ортогонального ряда. Установлен критерий единственности решения. Показано, что единственность решения и сходимость ряда зависят от отношения сторон прямоугольника из гиперболической части смешанной области. На основе этой задачи впервые поставлены и изучены обратные задачи по определению сомножителей правых частей исходного уравнения смешанного типа, зависящих от времени. На основании теории интегральных уравнений для обратных задач доказаны соответствующие теоремы единственности и существования решений. Библиография: 16 названий.

Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum

This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient $\kappa(\theta)$ satisfies \begin{equation*} C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q) \end{equation*} for some constants $q>0$, and $C_1,C_2>0$.