Showing papers on "Free boundary problem published in 2010"
TL;DR: In this article, the authors discuss properties of a variational interface problem involving the fractional Laplacian, including optimal regularity, non-degeneracy, and smoothness of the free boundary.
Abstract: We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.
297 citations
Book•
01 Dec 2010
TL;DR: In this paper, two-point boundary value problems and nonlinear boundary value problems are discussed. And the Continuation Method and Advanced Averaging for Evolution Equations with Boundary Layers are compared.
Abstract: Basic Material.- Approximation of Integrals.- Boundary Layer Behaviour.- Two-Point Boundary Value Problems.- Nonlinear Boundary Value Problems.- Elliptic Boundary Value Problems.- Boundary Layers in Time.- Evolution Equations with Boundary Layers.- The Continuation Method.- Averaging and Timescales.- Advanced Averaging.- Averaging for Evolution Equations.- Wave Equations on Unbounded Domains.
224 citations
TL;DR: The existence of positive solutions to the singular boundary value problem for fractional differential equation is considered and a fixed point theorem for the mixed monotone operator is relied on.
Abstract: In this paper, we consider the existence of positive solutions to the singular boundary value problem for fractional differential equation. Our analysis relies on a fixed point theorem for the mixed monotone operator.
201 citations
TL;DR: A new computational tool for computing equilibria based on an L^2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional is proposed.
188 citations
TL;DR: In this article, a review of free and forced fluvial patterns is presented, where the authors focus on the interaction of natural fluids with the erodible surface of Earth, which is ultimately responsible for the variety of sedimentary patterns observed in rivers, estuaries, coasts, deserts and the deep submarine environment.
Abstract: Geomorphology is concerned with the shaping of Earth's surface. A major contributing mechanism is the interaction of natural fluids with the erodible surface of Earth, which is ultimately responsible for the variety of sedimentary patterns observed in rivers, estuaries, coasts, deserts, and the deep submarine environment. This review focuses on fluvial patterns, both free and forced. Free patterns arise spontaneously from instabilities of the liquid-solid interface in the form of interfacial waves affecting either bed elevation or channel alignment: Their peculiar feature is that they express instabilities of the boundary itself rather than flow instabilities capable of destabilizing the boundary. Forced patterns arise from external hydrologic forcing affecting the boundary conditions of the system. After reviewing the formulation of the problem of morphodynamics, which turns out to have the nature of a free boundary problem, I discuss systematically the hierarchy of patterns observed in river basins at different scales.
136 citations
TL;DR: In this paper, the existence results for a boundary value problem of nonlinear impulsive differential equations of fractional-order q(1,2] with integral boundary conditions were proved by applying the contraction mapping principle and Krasnoselskii's fixed point theorem.
131 citations
TL;DR: A two-point boundary value problem for a second order fuzzy differential equation is interpreted by using a generalized differentiability concept and the problem of finding new solutions is investigated.
Abstract: In this paper, we interpret a two-point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept. We present a new concept of solutions and, utilizing the generalized differentiability, we investigate the problem of finding new solutions. Some examples are provided for which the new solutions are found.
127 citations
TL;DR: A high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh that has good performance when applied to one and two-dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition.
126 citations
TL;DR: In this article, the evolutionary Navier-Stokes equations with a Navier slip-type boundary condition were considered and the convergence of the solutions to the solution of the Euler equations under the zero-flux boundary condition was studied.
Abstract: We consider the evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, and study the convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We obtain quite sharp results in the 2-D and 3-D cases. However, in the 3-D case, we need to assume that the boundary is flat.
123 citations
TL;DR: In this paper, the authors consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set.
Abstract: We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.
120 citations
Posted Content•
TL;DR: In this article, the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with physical vacuum was established by considering the problem as a free boundary problem.
Abstract: An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only few mathematical results available near vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with physical vacuum by considering the problem as a free boundary problem.
TL;DR: In this paper, the Navier-Stokes free boundary problem is considered in a situation where the initial interface is close to a halfplane and the fluids are separated by an interface that is unknown and has to be determined as part of the problem.
Abstract: The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic. In this paper we consider a free boundary problem that describes the motion of two viscous incompressible capillary Newtonian fluids. The fluids are separated by an interface that is unknown and has to be determined as part of the problem. Let 1(0) ⊂ R n+1 (n ≥ 1) be a region occupied by a viscous incompressible fluid, fluid1, and let 2(0) be the complement of the closure of 1(0) in R n+1 , corre- sponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let 0 be the hypersurface that bounds 1(0) (and hence also 2(0)) and let ( t) denote the position of 0 at time t. Thus, ( t) is a sharp interface which separates the fluids occupying the regions 1(t) and 2(t), respectively, where 2(t) := R n+1 \ 1(t). We denote the normal field on ( t), pointing from 1(t) into 2(t), by ν(t, � ). Moreover, we de- note by V (t, � ) and κ(t, � ) the normal velocity and the mean curvature of ( t) with respect to ν(t, � ), respectively. Here the curvature κ(x, t) is assumed to be negative when 1(t) is convex in a neighborhood of x ∈ ( t). The motion of the fluids is governed by the following system of equations for i = 1,2 :
TL;DR: Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative.
Abstract: We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. As an application, some examples to illustrate our results are given.
TL;DR: Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known energy distribution functions is, as expected, negligible, which are consistent with the results from the derived discrete macroscopic energy equation.
Abstract: Consistent 2D and 3D thermal boundary conditions for thermal lattice Boltzmann simulations are proposed. The boundary unknown energy distribution functions are made functions of known energy distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definition of internal energy density. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. The discrete macroscopic energy equation is also derived for a steady and fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous energy distribution across the boundary, provided the boundary unknown energy distribution functions satisfy the macroscopic energy level. Four different local known energy distribution functions are experimented with to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating the 2D thermal Poiseuille flow, thermal Couette flow, thermal Couette flow with wall injection, natural convection in a square cavity, and 3D thermal Poiseuille flow in a square duct. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known energy distribution functions is, as expected, negligible, which are consistent with the results from the derived discrete macroscopic energy equation.
TL;DR: In this article, the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations was studied. But the authors considered the problem in the standard Riemann-Liouville sense.
Abstract: We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.
TL;DR: In this paper, the authors developed a rigorous mathematical approach to overcome these difficulties and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow.
Abstract: When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.
TL;DR: In this article, the boundary heat control problems formulated in the book of Duvaut and Lions were studied and the authors proved continuity of the solution with the appropriate modulus, and extended the results to the fractional order case and to the anomalous diffusion problems.
01 Jan 2010
TL;DR: In this paper, a well-posed discrete right-focal fractional boundary value problem was introduced, where the order ν of the fractional difference satisfies 1 < ν ≤ 2.
Abstract: In this paper, we introduce a well-posed discrete right-focal fractional boundary value problem in the case where the order ν of the fractional difference satisfies 1 < ν ≤ 2. We deduce Green’s function for this problem and prove certain properties about Green’s function. We show in the case ν = 2 that our results agree with the previously known results for second-order discrete boundary value problems but that new results are obtained if 1 < ν < 2. In particular, we show that in great contrast to the case when ν = 2, Green’s function is not monotone in the case when 1 < ν < 2. Finally, we deduce some conditions under which positive solutions to the boundary value problem exist as well as some conditions under which the boundary value problem will have a unique solution. AMS Subject Classifications: Primary: 26A33, 39A05, 39A12; Secondary: 33B15, 47H10.
TL;DR: The simultaneous-approximation-term (SAT) approach to applying boundary conditions for the compressible Navier-Stokes equations is analyzed with respect to the errors associated with the formulation’s weak enforcement of the boundary data and it is found that for curvilinear boundaries the SAT approach is superior to the locally one-dimensional inviscid characteristic approach.
Abstract: The simultaneous-approximation-term (SAT) approach to applying boundary conditions for the compressible Navier-Stokes equations is analyzed with respect to the errors associated with the formulation’s weak enforcement of the boundary data. Three numerical examples are presented which illustrate the relationship between the penalty parameters and the accuracy; two examples are fundamentally acoustic and the third is viscous. The viscous problem is further analyzed by a continuous model whose solution is known analytically and which approximates the discrete problem. From the analysis it is found that at early times an overshoot in the boundary values relative to the boundary data can be expected for all values of the penalty parameters but whose amplitude reduces with the inverse of the parameter. Likewise, the long-time behavior exhibits a t
−1/2 relaxation towards the specified data, but with a very small amplitude. Based on these data it is evident that large values of the penalty parameters are not required for accuracies comparable to those obtained by a more traditional characteristics-based method. It is further found that for curvilinear boundaries the SAT approach is superior to the locally one-dimensional inviscid characteristic approach.
TL;DR: In this paper, the scaling limits of three different aggregation models on Ω d were studied: internal DLA, where particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, where each site distributes its excess mass equally among its neighbors.
Abstract: We study the scaling limits of three different aggregation models on ℤ d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝ d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.
TL;DR: In this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and corresponding formulae of the IBEFM for 2D elasticity problem are obtained.
Abstract: The paper begins by discussing the interpolating moving least-squares (IMLS) method. Then the formulae of the IMLS method obtained by Lancaster are revised. On the basis of the boundary element-free method (BEFM), combining the boundary integral equation method with the IMLS method improved in this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained. In the IMLS method in this paper, the shape function satisfies the property of Kronecker δ function, and then in the IBEFM the boundary conditions can be applied directly and easily. The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables. Thus it gives a greater computational precision. Numerical examples are presented to demonstrate the method.
TL;DR: In this paper, the upper and lower solutions method was used to study the p-Laplacian fractional boundary value problem with respect to the boundary value of the p Laplacians.
Abstract: The upper and lower solutions method is used to study the p-Laplacian fractional boundary value problem D0
TL;DR: This work analytically establishes the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equation.
Abstract: In the light of the fixed point theorems, we analytically establish the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equation. Furthermore, we provide a representative example to illustrate a possible application of the established analytical results.
TL;DR: Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains in this paper were studied in the context of boundary behavior and Martin boundary problems.
Abstract: Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains
TL;DR: In this paper, it was shown that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility, and that this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces.
Journal Article•
TL;DR: In this paper, a well-posedness result for mixed boundary value/interface problems of second-order, positive, strongly elliptic operators in weighted Sobolev spaces was given.
Abstract: Let � 2 Z+ be arbitrary. We prove a well-posedness result for mixed boundary value/interface problems of second-order, positive, strongly elliptic operators in weighted Sobolev spaces K �() on a bounded, curvilinear polyhedral domain in a manifold M of dimension n. The typical weightthat we consider is the (smoothed) distance to the set of singular boundary points of @. Our model problem is Pu := −div(Ar u) = f, in , u = 0 on @D, and D P � u = 0 on @�, where the function A � � >0 is piece-wise smooth on the polyhedral decomposition ¯ = ( jj, and @ = @D ( @N is a decomposition of the boundary into polyhedral subsets corre- sponding, respectively, to Dirichlet and Neumann boundary condi- tions. If there are no interfaces and no adjacent faces with Neu- mann boundary conditions, our main result gives an isomorphism P : K �+1
TL;DR: In this paper, a quantitative phase-field model for two-phase solidification processes is developed based on the anti-trapping current approach with the free energy functional formulated to suppress the formation of an extra phase at the interface.
01 Jan 2010
TL;DR: In this article, the authors studied nonhomogeneous initial-boundary value prob- lems for quasilinear one-dimensional odd-order equations posed on a bounded interval.
Abstract: This paper studies nonhomogeneous initial-boundary value prob- lems for quasilinear one-dimensional odd-order equations posed on a bounded interval. For reasonable initial and boundary conditions we prove existence and uniqueness of global weak and regular solutions. Also we show the ex- ponential decay of the obtained solution with zero boundary conditions and right-hand side, and small initial data.
TL;DR: The dual problem's formulation is used, showing that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.
Abstract: We discuss optimal trading strategies for general utility functions in portfolios of cash and stocks subject to small proportional transaction costs. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utility's value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problem's formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.