Showing papers on "Free boundary problem published in 2016"
TL;DR: In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained based on some fixed point theorems.
Abstract: In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results.
109 citations
TL;DR: In this article, a diffusive logistic equation with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, in which the variable intrinsic growth rates may be "very negative" in a "suitable large region" was considered.
106 citations
TL;DR: In this paper, the authors formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers, using this foundation they then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary condition from nonlinear elasticity.
Abstract: Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.
88 citations
TL;DR: In this paper, a global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator is developed.
Abstract: In this paper, we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. There are no assumptions on the regularity of the boundary which is allowed to have arbitrary singularities. We give applications of the developed analysis to obtain a priori estimates for solutions of boundary value problems that are elliptic within the constructed calculus.
78 citations
TL;DR: In this paper, a boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.
Abstract: A boundary control problem for the viscous Cahn---Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.
78 citations
TL;DR: The nonlinear stability of Lane-Emden solutions is proved in this article for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, by establishing the global-in-time regularity uniformly up to the vacuum boundary.
Abstract: The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent $${\gamma}$$
lies in the stability regime $${(4/3, 2)}$$
, by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is $${C^{{1}/{2}}}$$
-Holder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent $${\gamma}$$
lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.
77 citations
TL;DR: In this article, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the selfsimilar solution, based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time.
Abstract: For the physical vacuum free boundary problem with the sound speed being C1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc.
75 citations
TL;DR: In this paper, a reaction diffusion model is investigated to understand infective environments in a man-environment-man epidemic model, and sufficient conditions for the bacteria to vanish or spread are given.
53 citations
TL;DR: In this article, it was shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C 1 / 2 -Holder continuous across the vacuum boundary provided that γ lies in ( 4 / 3, 2 ).
53 citations
TL;DR: This paper derives the efficient nonlinear absorbing boundary conditions (ABCs), which reformulates the problem on unbounded domain to an initial boundary value problem on bounded domain, and constructs a linearized finite difference scheme to solve the reduced nonlinear problem such that iterative methods become dispensable.
52 citations
TL;DR: A finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier–Stokes flow is developed, and it is observed that the conservation properties are fulfilled to a high precision.
Abstract: We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier---Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier---Stokes equation, taking surface viscosity effects of Boussinesq---Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (Numer Math 111:55-80, 2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision.
TL;DR: In this paper, the authors derived boundary conditions for phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation.
Abstract: We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean-free path to the characteristic system length scale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier description. We show that, in the bulk, the traditional heat conduction equation using Fourier's law as a constitutive relation is valid at least up to second order in the Knudsen number for steady problems and first order for time-dependent problems. However, this description does not hold within distances on the order of a few mean-free paths from the boundary; this breakdown is a result of kinetic effects that are always present in the boundary vicinity and require solution of a Boltzmann boundary layer problem to be determined. Matching the inner, boundary layer solution to the outer, bulk solution yields boundary conditions for the Fourier description as well as additive corrections in the form of universal kinetic boundary layers; both are found to be proportional to the bulk-solution gradients at the boundary and parametrized by the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Our derivation shows that the traditional no-jump boundary condition for prescribed temperature boundaries and the no-flux boundary condition for diffusely reflecting boundaries are appropriate only to zeroth order in the Knudsen number; at higher order, boundary conditions are of the jump type. We illustrate the utility of the asymptotic solution procedure by demonstrating that it can be used to predict the Kapitza resistance (and temperature jump) associated with an interface between two materials. All results are validated via comparisons with low-variance deviational Monte Carlo simulations.
TL;DR: In this article, a purely functional analytic framework for elliptic boundary value problems in a variational form is introduced, and a theory relating resolvents and spectra of these operators is developed.
Abstract: We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet-to-Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non-smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.
TL;DR: In this paper, an exact transparent boundary condition is introduced to reduce the scattering problem into a boundary value problem in a bounded domain, and the domain derivative is derived for the displacement with respect to the variation of the surface.
Abstract: Consider the scattering of a time-harmonic plane wave by a rigid obstacle which is embedded in an open space filled with a homogeneous and isotropic elastic medium. An exact transparent boundary condition is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is derived for the displacement with respect to the variation of the surface. A continuation method with respect to the frequency is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.
TL;DR: This paper designs a careful combination of the boundary treatments for the two regimes and obtains a stable and accurate boundary condition for general convection-diffusion equations.
TL;DR: In this article, the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another were studied and the authors established a sharp nonlinear global-in-time stability criterion and gave the explicit decay rates to the equilibrium.
Abstract: This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e., sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.
TL;DR: In this paper, a free discontinuity approach to a class of shape optimization problems involving Robin conditions on the free boundary is proposed. But this approach is restricted to a restricted class of domains on which the problem can be considered as a relaxed version of the corresponding one on regular domains.
TL;DR: In this article, the boundary layer problem for a hyperbolic system transformed via a Cole-Hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity was studied.
TL;DR: In this paper, a singular limit of the non-convex energy consisting of a bending energy, tension and an adhesion energy was derived for the one-dimensional case of a free boundary problem arising from materials science.
Abstract: A free boundary problem arising from materials science is studied in the one-dimensional case. The problem studied here is an obstacle problem for the non-convex energy consisting of a bending energy, tension and an adhesion energy. If the bending energy, which is a higher order term, is deleted then “edge” singularities of the solutions (surfaces) may occur at the free boundary as Alt–Caffarelli type variational problems. The main result of this paper is to give a singular limit of the energy utilizing the notion of $$\Gamma $$
-convergence, when the bending energy can be regarded as a perturbation. This singular limit energy only depends on the state of surfaces at the free boundary as seen in singular perturbations for phase transition models.
TL;DR: In this paper, a class of singular fractional differential equations with infinite-point boundary conditions is considered, and the nonlinearity is allowed to be singular with respect to not only the time variable but also to the space variable.
Abstract: In this paper, we consider a class of singular fractional differential equations with infinite-point boundary conditions. The fractional orders are involved in the nonlinearity of the boundary value problem, and the nonlinearity is allowed to be singular with respect to not only the time variable but also to the space variable. Firstly, we give Green’s function and establish its properties. Then, we utilize the sequential technique and regularization to investigate the existence of positive solutions. Finally, we give an example of application of our result.
TL;DR: In this article, the authors studied the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables.
Abstract: In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases.
TL;DR: In this article, the uniqueness of positive solution for a class of singular fractional differential systems with integral boundary conditions is considered and many types of equation system are contained in this equation system because there are many parameters which can be changeable in the equation system.
Abstract: The uniqueness of positive solution for a class of singular fractional differential system with integral boundary conditions is considered in this paper and many types of equation system are contained in this equation system because there are many parameters which can be changeable in this equation system. The fractional orders are involved in the nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable. The existence of uniqueness of positive solution is mainly obtained by fixed point theorem of mixed monotone operator and the positive solution of equation system is dependent on λ. An iterative sequence and convergence rate are given which are important for practical application and an example is given to demonstrate the validity of our main results.
TL;DR: In this paper, the Gibbs-Thomson relation is taken into account to model the free boundary problem of tumor growth, and it is shown that cell-to-cell adhesiveness is the key parameter which plays a crucial role on tumor invasion.
TL;DR: In this paper, the Morse index for a free boundary minimal submanifold is computed from data of two simpler problems: the first is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields.
Abstract: In this paper, we compute the Morse index for a free boundary minimal submanifold from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical catenoid.
TL;DR: In this article, the authors studied the free boundary problem arising from the theory of thermal insulation and showed that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have "locally flat" boundaries.
Abstract: We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have “locally flat” boundaries.
01 Jan 2016
TL;DR: In this paper, the following system of n first-order ODEs is considered, and the following ODE system is considered in terms of the first order ordinary differential equations (ODEs).
Abstract: Let us consider, the following system of n first-order ordinary differential equations (ODEs).
TL;DR: In this paper, a free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions, and a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad.
TL;DR: A new upper estimate is obtained for the Green’s function associated with a higher order fractional boundary value problem and criteria for the existence of positive solutions of the problem are established.
TL;DR: The results show the utility of the boundary treatments and very well support the theoretical predications on the second-order accuracy thereof, and can be directly generalized to 3-dimensional problems, multiple-relaxation-time models, and the Navier-Stokes equations.