Noncylindrical Domains Research Articles

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

Lamé's system in large size domains: Existence, asymptotic behaviour and explicit solution

This work is devoted to study the parabolic Lamé system defined in noncylindrical domains. We focus on the asymptotic behaviour of the solution as t and the state variable domain become very large. Different rates of convergence are established according to the growth of the domain. The “parabolic-elliptic regularization” method is used to treat the existence of the solution of such problems in time growing domains with multi-initial conditions. The steady state problem is a Lamé system defined on unbounded domains. In the cases of polynomial data and cylindrical symmetries, the limit solution will be given explicitly and some examples are provided.

Exact Null Controllability of String Equations with Neumann Boundaries

This article focuses on the exact null controllability of a one-dimensional wave equation in noncylindrical domains. Both the fixed endpoint and the moving endpoint are Neumann-type boundary conditions. The control is put on the moving endpoint. When the speed of the moving endpoint is less than the characteristic speed, we can obtain the exact null controllability of this equation by using the Hilbert uniqueness method. In addition, we get a sharper estimate on controllability time that depends on the speed of the moving endpoint.

New energy effect in non-cylindrical domains with a thermally insulated moving boundary

Objectives. To develop mathematical model representations of the energy effect in non-cylindrical domains having a thermally insulated moving boundary; to introduce a new boundary condition for thermal insulation of a moving boundary both for locally equilibrium heat transfer processes in the framework of classical Fourier phenomenology, as well as for more complex locally non-equilibrium processes in the framework of Maxwell–Cattaneo–Lykov–Vernott phenomenology, taking into account the finite rate of heat propagation into analytical thermophysics and applied thermomechanics; to consider an applied problem of analytical thermophysics according to the theory of thermal shock for a domain with a moving thermally insulated boundary free from external and internal influences; to obtain an exact analytical solution of the formulated mathematical models for hyperbolic type equations; to investigate the solutions obtained using a computational experiment at various values of the parameters included in it; to describe the wave nature of the kinetics of the processes under consideration.Methods. Methods and theorems of operational calculus, Riemann–Mellin contour integrals are used in calculating the originals of complex images with two branch points. A new mathematical apparatus for the equivalence of functional constructions for the originals of the obtained operational solutions, which considers the computational difficulties in finding analytical solutions to boundary value problems for equations of hyperbolic type in the domain with a moving boundary, is developed.Results. Developed mathematical models of locally nonequilibrium heat transfer and the theory of thermal shock for equations of hyperbolic type in a domain with a moving thermally insulated boundary are presented. It is shown that, despite the absence of external and internal sources of heat, the presence of a thermally insulated moving boundary leads to the appearance of a temperature gradient in the domain and, consequently, to the appearance of a temperature field and corresponding thermoelastic stresses in the domain, which have a wave character. A stochastic analysis of this energy effect forms the basis for a proposed transition of the kinetic energy of a moving thermally insulated boundary into the thermal energy of the domain. The presented model representations of the indicated effect confirmed the stated assumption.Conclusions. Mathematical models for locally nonequilibrium heat transfer processes and the theory of thermal stresses are developed and investigated on the basis of constitutive relations of the theory of thermal shock for equations of hyperbolic type in a domain with a thermally isolated moving boundary. A numerical experiment is presented to demonstrate the possibility of transiting from one form of analytical solution of a thermophysical problem to another equivalent form of a new type. The described energy effect manifests itself both for parabolic type equations based on the classical Fourier phenomenology, as well as for hyperbolic type equations based on the generalized Maxwell–Cattaneo–Lykov–Vernott phenomenology.

Exact Null Controllability of a One-Dimensional Wave Equation with a Mixed Boundary

In this paper, exact null controllability of one-dimensional wave equations in non-cylindrical domains was discussed. It is different from past papers, as we consider boundary conditions for more complex cases. The wave equations have a mixed Dirichlet–Neumann boundary condition. The control is put on the fixed endpoint with a Neumann boundary condition. By using the Hilbert Uniqueness Method, exact null controllability can be obtained.

Exact Null Controllability of a Wave Equation with Dirichlet–Neumann Boundary in a Non-Cylindrical Domain

In this paper, by applying the Hilbert Uniqueness Method in a non-cylindrical domain, we prove the exact null controllability of one wave equation with a moving boundary. The moving endpoint of this wave equation has a Neumann-type boundary condition, while the fixed endpoint has a Dirichlet boundary condition. We derived the exact null controllability and obtained an exact controllability time of the wave equation.

Construction of a Right Inverse for the Divergence in Non-cylindrical Time Dependent Domains

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. The most precise estimates are given in weighted spaces, where the weight depends on the distance to the boundary. This allows for the deficit to be captured precisely in the vicinity of irregularities of the boundary. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier–Stokes equations in time dependent domains.

Optimal tubes for non-cylindrical Navier-Stokes flows with Navier boundary condition

We consider a shape optimization problem where state equations are the time-dependent Navier-Stokes equations with Navier boundary condition occupying a non-cylindrical domain. In the first part of our work, we prove existence of weak solutions for the state equations using the elliptic regularization method. In the second part, we show stability of the state problem solutions with respect to admissible shape variations and derive an existence result for the considered shape minimization problem.

On the wave equation on moving domains: regularity, energy balance and application to dynamic debonding

We revisit some issues about existence and regularity for the wave equation in noncylindrical domains. Using a method of diffeomorphisms we show, through increasing regularity assumptions, how the existence of weak solutions, their improved regularity and an energy balance can be derived. As an application, we give a rigorous definition of dynamic energy release rate density for some problems of debonding, and we formulate a proper notion of solutions for such problems. We compare the consistence of such a formulation with that of previous ones, given in the literature for particular cases.

Boundary Value Problems for Parabolic Equations in Noncylindrical Domains

Boundary Value Problems for Parabolic Equations in Noncylindrical Domains

Finite element method for nonlocal problems of Kirchhoff-type in domains with moving boundary

This paper is devoted to the analysis of the finite element method for the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equations in a bounded noncylindrical domain with moving boundaries. With the use of the coordinate transformation which fixes the boundaries, the semidiscrete formulation is presented and the convergence and error bounds in the energy norm and for the first order derivative with respect to time in the L2-norm are established. In particular, the error in the energy norm and for the first order derivative with respect to time in the L2-norm is shown to converge with the optimal order O(hr) with respect to the mesh size h and the polynomial degree r≥1. To obtain the fully discrete solution, the generalized-α method is adapted to the semidiscrete formulation. The test problems used are designed to illustrate the behavior of the algorithms. Some numerical tests using Matlab are performed to confirm our theoretical findings.

Boundary Integral Operators for the Heat Equation in Time-Dependent Domains

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the C^2-diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

In this article, we study the homogenization of heat equations in a domain with randomly oscillating boundary parts. The random oscillating boundary is time‐dependent and confined by a stationary random field. Here, we follow a new homogenization technique that deals with the evolving domains, which covers many applications. We obtain the asymptotic limit as ε → 0 in the reference configuration, in which the heat equation becomes a parabolic equation with random oscillating coefficients in the reference domain. To the best of our knowledge, this is the first result of the homogenization of problems on the random evolving boundary domain. One of the major contributions is the corrector result which we establish in this article.

Carleson estimates for the singular parabolic $p$-Laplacian in time-dependent domains

We deal with the parabolic p -Laplacian in the so-called singular super-critical range \frac{2N}{N+1}<p<2 , and we prove Carleson estimates for non-negative solutions in suitable non-cylindrical domains \Omega \subset\mathbb{R}^{N+1} . The sets \Omega satisfy a proper NTA condition, tailored on the parabolic p -Laplacian. As an intermediate step, we show that in these domains non-negative solutions which vanish at the boundary, are Hölder continuous up to the same boundary.

Existence of global weak solutions for a p-Laplacian inequality with strong issipation in noncylindrical domains

Existence of global weak solutions for a p-Laplacian inequality with strong issipation in noncylindrical domains

Solution of the boundary value problem of heat conduction in a cone

In the paper we consider the boundary value problem of heat conduction in a non-cylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We use the Carleman-Vekua method of equivalent regularization to solve the obtained singular Volterra integral equation.

Support of solutions of the linear Klein-Gordon equation and exact boundary controllability in non-cylindrical domains

This paper address to show that it is possible to extend each pair of functions of H1(Ω)×L2(Ω) to H1(R2)×L2(R2), with Ω⊂R2, with Ω⊂R2, such that the solution of the Cauchy problem for linear Klein-Gordon equation, with such extended initial data, vanishes in certain conical region of R2×[0,∞). After, we show how use it for obtain exact boundary control on certain non-cylindrical domains for above considered equation. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data is taken. The control is square integrable and acts on whole boundary of the non-cylindrical domain and it is given by conormal derivative associated to linear Klein-Gordon operator.

A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams: Proof of the strong solvability

In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid’s absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy’s law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation). Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.

Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

<p style='text-indent:20px;'>In this paper, we deal with boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domains. By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of the solution are given explicitly.</p>

Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castroet al.[SIAM J. Control Optim.52(2014)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical supportqof the control of minimalL2(q)-norm. In this respect, we prove a uniform observability inequality for a class of domainsqsatisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.