Showing papers on "Free boundary problem published in 2012"
TL;DR: In this article, the existence of positive solutions for a class of nonlinear boundary value problems with integral boundary conditions was studied, based on the known Guo-Krasnoselskii fixed point theorem.
259 citations
TL;DR: In this article, the authors prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγργ for γ > 1.
Abstract: We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγργ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.
187 citations
TL;DR: By the critical point theory, the boundary value problem is discussed for a fractional differential equation containing the left and right fractional derivative operators, and various criteria on the existence of solutions are obtained.
Abstract: In this paper, by the critical point theory, the boundary value problem is discussed for a fractional differential equation containing the left and right fractional derivative operators, and various criteria on the existence of solutions are obtained. To the authors' knowledge, this is the first time, the existence of solutions to the fractional boundary value problem is dealt with by using critical point theory.
177 citations
TL;DR: In this paper, a weak competition model with a free boundary in a one-dimensional habitat was studied and the authors provided sufficient conditions for spreading success and failure, respectively, and provided an estimate to show that the spreading speed cannot be faster than the minimal speed of traveling wavefront solutions.
Abstract: We study a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary.
174 citations
TL;DR: In this paper, the authors considered the initial-boundary value problem for under homogeneous Neumann boundary conditions in a bounded domain, n ≥ 1, with smooth boundary, and showed that given any such a1 and a2 and any positive diffusivities d1 and d2 and cross-diffusivities χ1 and χ2, this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients μ1 and μ2.
Abstract: We study a system of three partial differential equations modelling the spatio-temporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. More precisely, we consider the initial-boundary value problem for under homogeneous Neumann boundary conditions in a bounded domain , n ≥ 1, with smooth boundary.When 0 ≤ a1 < 1 and 0 ≤ a2 < 1, this system possesses a uniquely determined spatially homogeneous positive equilibrium (u, v). We show that given any such a1 and a2 and any positive diffusivities d1 and d2 and cross-diffusivities χ1 and χ2, this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients μ1 and μ2.
152 citations
TL;DR: In this paper, the authors considered the Fisher-KPP problem with a free boundary governed by a one-phase Stefan condition, and established the existence and uniqueness of the weak solution, and through suitable comparison arguments, extended some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) to this general case.
128 citations
TL;DR: In this article, the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure was proved.
Abstract: For every $${k \in \mathbb{N}}$$
, we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.
123 citations
Journal Article•
TL;DR: In this paper, the authors present a set of partial differential equations with Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods for science, engineering, and applied mathematics.
Abstract: This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods. This text is ideal for readers interested in science, engineering, and applied mathematics.
107 citations
TL;DR: In this paper, the authors considered the Stokes equations under some nonlinear boundary conditions, which are described in terms of subdifferentials of maximal monotone graphs and are called leak and slip boundary conditions of friction type.
Abstract: We consider the Stokes equations under some nonlinear boundary conditions, which are described in terms of subdifferentials of maximal monotone graphs and are called leak and slip boundary conditions of friction type. The main objective is to show the existence of strong solutions, say u ∈ H 2 and p ∈ H 1 , to these problems. We start with weak solutions to variational inequalities, and then study the regularity of weak solutions. Our main theorems imply the maximality of Stokes operators with such nonlinear boundary conditions in a suitable Hilbert space and they are of use in analysis of time-dependent problems. Linear boundary conditions of Neumann type, such as slip and penetration conditions, are also discussed.
TL;DR: In this paper, a discrete fractional three-point boundary value problem (BVP) was studied, and it was shown that the range of admissible boundary conditions depends upon the order of the difference equation.
Abstract: In this paper, we analyse a ν-th order, , discrete fractional three-point boundary value problem (BVP). We show that Green's function associated to this problem satisfies certain conditions. We demonstrate that the range of admissible boundary conditions depends upon the order ν of the difference equation, and we give explicit formulae for this dependence. By using both the Brouwer fixed point theorem and the Krasnosel'skiĭ fixed point theorem, we then show that a solution to this problem exists. Our results extend recent results on discrete fractional BVPs (FBVPs), and they also provide an initial set of results on the theory of multipoint FBVPs on the time scale of integers.
TL;DR: Under certain nonlinear growth conditions of the nonlinearity, a new existence result is obtained by using Schaefer’s fixed point theorem for the fractional p -Laplacian equation.
TL;DR: This boundary method could be considered as a competitive alternative for boundary treatment in LBM simulations, especially for particulate and porous flows with large fluid-solid interfacial areas and exhibits a comparable or even better accuracy in describing flow field and flow-structure interaction.
TL;DR: The expression for Green's function of the associated problem involving the classical gamma function and the generalized incomplete gamma function is obtained and some existence results are obtained by means of Banach's contraction mapping principle and Krasnoselskii's fixed point theorem.
Abstract: This paper studies a nonlinear three-point boundary value problem of sequential fractional differential equations of order @a+1 with 1<@a@?2. The expression for Green's function of the associated problem involving the classical gamma function and the generalized incomplete gamma function is obtained. Some existence results are obtained by means of Banach's contraction mapping principle and Krasnoselskii's fixed point theorem. An illustrative example is also presented. Existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results.
TL;DR: The attainable order of convergence of proposed algorithms is studied and it is shown how the convergence rate depends on the choice of the grid and collocation points.
TL;DR: It is shown that the method of the energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case.
TL;DR: In this paper, the boundary value problem is considered in the setting where ϕ may be strictly non-positive for some y > 0 and the existence of at least one positive solution is assumed.
Abstract: In this paper, we consider the boundary value problem y Δ Δ ( t ) = − λ f ( t , y σ ( t ) ) subject to the boundary conditions y ( a ) = ϕ ( y ) and y ( σ 2 ( b ) ) = 0 . In this setting, ϕ : C rd ( [ a , σ 2 ( b ) ] T , R ) → R is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on ϕ and the nonlinearity f , we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that ϕ may be strictly nonpositive for some y > 0 . Our results are achieved by appealing to the Krasnosel’skiĭ fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford.
TL;DR: This work presents a model for elastohydrodynamic (EHD) lubrication problems that is extended to account for non-Newtonian lubricant behaviour and thermal effects and artificial diffusion procedures are introduced to stabilize the solution at high loads.
TL;DR: In this article, a type of nonlinear fractional boundary value problem with non-homogeneous integral boundary conditions is studied, and the existence and uniqueness of positive solutions are discussed.
Abstract: The authors study a type of nonlinear fractional boundary value problem with non-homogeneous integral boundary conditions. The existence and uniqueness of positive solutions are discussed. An example is given as the application of the results.
TL;DR: In this article, a free boundary problem for a system of partial difierential equations, which arises in a model of tumor growth with a necrotic core, is considered.
Abstract: We consider a free boundary problem for a system of partial difierential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ‰ < R, there exists a radially symmetric stationary solution with tumor boundary r = R and necrotic core boundary r = ‰. The system depends on a positive parameter „, which describes the tumor aggressiveness. There also exists a sequence of values „2 < „3 < ¢¢¢ for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.
TL;DR: In this article, the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u⋅n = 0, [S(u)n]tan+Au=0, in a bounded domain in R3 when the viscosity, e>0, is studied.
TL;DR: It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust.
TL;DR: It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map.
Abstract: We study an inverse problem for nonlinear elliptic equations modeled after the $p$-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics-type solutions based on $p$-harmonic exponentials, while for the real case we use $p$-harmonic functions first introduced by Wolff.
TL;DR: In this article, critical point theory and variational methods were used to investigate the multiple solutions of boundary value problems for second order impulsive differential equations, and conditions for the existence of multiple solutions were established.
TL;DR: In this paper, a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions was studied and the existence and uniqueness of solutions of the problem was proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems.
Abstract: This paper studies a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the q-difference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of q-difference equations. The results corresponding to a second-order q-difference equation with anti-periodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical second-order boundary value problems with non-separated boundary conditions in the limit q → 1, which provides a useful check.
TL;DR: It is shown that the regularity condition on potentials of the Schrodinger equation in uniqueness results on the inverse boundary value problem is relaxed.
Abstract: We relax the regularity condition on potentials of the Schrodinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [O. Imanuvilov, G. Uhlmann, and M. Yamamoto, J. Amer. Math. Soc., 23 (2010), pp. 655–691] and [A. Bukhgeim, J. Inverse Ill-Posed Probl., 16 (2008), pp. 19–34].
TL;DR: In this article, an analytical approach for free vibration analysis of moderately thick functionally graded rectangular plates coupled with piezoelectric layers is presented, based on the first order shear deformation plate theory and using both the Maxwell equation and Hamilton principle, the governing equations are obtained.
Abstract: In this paper, an analytical approach for free vibration analysis of moderately thick functionally graded rectangular plates coupled with piezoelectric layers is presented. The transverse distribution of electric potential satisfies the Maxwell equation as well as the electrical boundary conditions for both closed and open circuit piezoelectric layers. Based on the first order shear deformation plate theory and using both the Maxwell equation and Hamilton principle, the governing equations are obtained. These equations, which are six coupled partial differential equations, are decoupled through introducing four auxiliary functions. The decoupled equations are solved analytically for the Levy type of mechanical boundary condition, two opposite edges simply supported and arbitrary boundary conditions at the other edges. The numerical results for the plate natural frequency are established for various plate dimensions, power law indices and electrical and mechanical boundary conditions. Finally, the effect of piezoelectric layer thickness on the natural frequency is discussed for various plate parameters. It is found that the effect of the piezoelectric layer on the plate natural frequencies strongly depends on the mechanical and electrical boundary conditions.
TL;DR: In this article, the symmetry property of the scattering data emerging from the presence of the boundary has been investigated in the Manakov model and two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions.
Abstract: We investigate the Manakov model or, more generally, the vector nonlinear Schrodinger equation on the half-line. Using a Backlund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by constructing a generating function for the conserved quantities. We apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions. The important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary. Particular attention is paid to the discrete spectrum. An interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated. This is specific to the vector nature of the model and is absent in the scalar case. For one-soliton solutions, we show that the boundary can be used to make certain components of the incoming soliton vanishingly small. This is reminiscent of the phenomenon of light polarization by reflection.
TL;DR: In this article, the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics was considered and an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives was obtained.
Abstract: We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.