Showing papers in "Methods and applications of analysis in 2007"
TL;DR: In this paper, the authors deal with the mathematical justification of such an asymptotic process assuming a non zero surface tension coefficient and some constraints on the data, and discuss relation between lubrication models and shallow water systems with no surface tension coefficients necessity.
Abstract: The shallow water equations are widely used to model the flow of a thin layer of fluid submitted to gravity forces. They are usually formally derived from the full incompressible Navier-Stokes equations with free surface under the modeling hypothesis that the pressure is hydrostatic, the flow is laminar, gradually varied and the characteristic fluid height is small relative to the characteristics flow length. This paper deals with the mathematical justification of such asymptotic process assuming a non zero surface tension coefficient and some constraints on the data. We also discuss relation between lubrication models and shallow water systems with no surface tension coefficient necessity.
56 citations
TL;DR: In this paper, the existence and stability of N−peaked steady-state can be reduced to computing two matrices in terms of the coefficients D,N, p, q, r, s.
Abstract: We consider the following Gierer-Meinhardt system in R: At = 2A ′′ −A + A p Hq x ∈ (−1, 1), t > 0, τHt = DH ′′ −H + A r Hs x ∈ (−1, 1), t > 0, A ′ (−1) = A(1) = H (−1) = H (1) = 0, where (p, q, r, s) satisfy 1 < qr (s + 1)(p− 1) < +∞, 1 < p < +∞, and where 2 << 1, 0 < D < ∞, τ ≥ 0, D and τ are constants which are independent of 2. We give a rigorous and unified approach to show that the existence and stability of N−peaked steady-states can be reduced to computing two matrices in terms of the coefficients D,N, p, q, r, s. Moreover, it is shown that N−peaked steady-states are generated by exactly two types of peaks, provided their mutual distance is bounded away from zero..
46 citations
TL;DR: In this article, a log improvement of Prodi-Serrin criterion for global regularity to solutions to Navier-Stokes equations in dimension 3 is presented. And the regularity holds under the condition that |u|/(log(1+|u|)) is integrable in space time variables.
Abstract: This article is devoted to a Log improvement of Prodi-Serrin criterion for global regularity to solutions to Navier-Stokes equations in dimension 3. It is shown that the global regularity holds under the condition that |u|/(log(1+|u|)) is integrable in space time variables. keywords: Navier-Stokes, regularity criterion, a priori estimates MSC: 35B65, 76D03, 76D05
44 citations
TL;DR: In this article, the authors studied the large-time behavior of the solution to an initial boundary value problem on the half line for scalar conservation law, where the data on the boundary and also at the far field are prescribed, and proved that even for a quite wide class of flux functions which are not necessarily convex, such the superposition of the stationary solution and the rarefaction wave is asymptotically stable.
Abstract: We study the large-time behavior of the solution to an initial boundary value problem on the half line for scalar conservation law, where the data on the boundary and also at the far field are prescribed. In the case where the flux is convex and the corresponding Riemann problem for the hyperbolic part admits the transonic rarefaction wave (which means its characteristic speed changes the sign), it is known by the work of Liu-Matsumura-Nishihara (’98) that the solution tends toward a linear superposition of the stationary solution and the rarefaction wave of the hyperbolic part. In this paper, it is proved that even for a quite wide class of flux functions which are not necessarily convex, such the superposition of the stationary solution and the rarefaction wave is asymptotically stable, provided the rarefaction wave is weak. The proof is given by a technical $L^2$-weighted energy method.
30 citations
TL;DR: In this article, the sharp interface limit for diffusive interface models with the generalized Navier boundary condition was derived for the moving contact line problem, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
Abstract: Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in (9, 11) for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
29 citations
TL;DR: In this article, the authors present an asymptotic completeness result for relativistic kinetic equations with short-range interaction forces, using dispersive estimates and explicit construction of a Lyapunov functional.
Abstract: We present an $L^1$-asymptotic completeness results for relativistic kinetic equations with short range interaction forces. We show that the uniform phase space-time bound for nonlinear terms to the relativistic nonlinear kinetic equations yields the asymptotic completeness of the relativistic kinetic equations. For this space-time bound, we employ dispersive estimates and explicit construction of a Lyapunov functional.
21 citations
TL;DR: In this paper, a tree-method approach was developed to obtain a wide class of intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials.
Abstract: We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinski\u{i} and Suslov. Such relation is also extended to the multidimensional case.
12 citations
TL;DR: In this paper, the Sturm-Liouville boundary value problem on a graph is considered and the spectral properties of the large eigenvalues up to the O(1/n)term are derived for a graph consisting of d, d ≥ 2, d ∈ N, joint inhomogeneous smooth strings.
Abstract: Abstract. In this paper, we consider the spectral problem of small vibrations of a graph consisting of d, d ≥ 2, d ∈ N, joint inhomogeneous smooth strings which can be reduced to the SturmLiouville boundary value problem on a graph. This problem occurs also in quantum mechanics. For the Sturm-Liouville problem on the compact metric graph consisting of d segments of equal length with the Dirichlet or Neumann boundary conditions at the pendant vertices and Kirchhoff condition at the central vertex, we first derive the asymptotic expressions of its large eigenvalues and obtain precise descriptions for the formulae of the square root of the large eigenvalues up to the O(1/n)term. In addition, regularized trace formulae of operators are established with residue techniques in complex analysis.
11 citations
TL;DR: In this paper, it was shown that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise. But the existence of such solutions is not guaranteed.
Abstract: We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.
8 citations
TL;DR: In this paper, an iterative scheme by viscosity approximation method was introduced for obtaining a common element of the set of solutions of an equilibrium problem and the fixed points of a nonexpansive mapping.
Abstract: In this paper, we introduce an iterative scheme by viscosity approximation method for obtaining a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. We obtain a strong convergence which improves and extends S. Takahashi and W. Takahashi’s result [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515].
8 citations
TL;DR: In this article, the authors consider a semilinear parabolic equation and a finite difference approximation for it and discuss the way how the asymptotic profile of the blow-up solution is reproduced by the numerical solution.
Abstract: We consider a semilinear parabolic equation ut = uxx + f(u) (0 < x < 1, 0 < t), and a finite difference approximation for it. We discuss the way how the asymptotic profile of the blow-up solution is reproduced by the numerical solution. We will also determine qualitatively the influence of the definition of time mesh on the blow-up set of the numerical solution. Moreover, we show that explicit and implicit schemes may claim different blow-up sets.
TL;DR: In this paper, the authors review some recent results on the Boltzmann equation near the equilibrium states in the whole space and emphasize on the well-posedness of the solution in some Sobolev space without time derivatives and its uniform stability and optimal decay rates.
Abstract: In this paper, we review some recent results on the Boltzmann equation near the equilibrium states in the whole space $\mathbb R^n$. The emphasize is put on the well-posedness of the solution in some Sobolev space without time derivatives and its uniform stability and optimal decay rates, and also on the existence and asymptotical stability of the time-periodic solution. Most of results obtained here are proved by combining the energy estimates and the spectral analysis.
TL;DR: In this article, the authors performed an asymptotic study on slightly viscous flows between two immiscible incompressible fluids and obtained a second-order expansion with respect to viscosity by using the method of multiple scales.
Abstract: In this paper, we perform an asymptotic study on slightly viscous flows between two immiscible incompressible fluids. The motion is governed by linearized Navier-Stokes equations together with interfacial conditions. A second-order asymptotic expansion with respect to viscosity is obtained by using the method of multiple scales. In particular, viscous decay rate for the interfacial wave amplitude and viscous correction for the phase speed are explicitly identified.
TL;DR: In this paper, the Navier-Stokes approximations of the two-dimensional Euler equations with vortex-sheets initial data in half plane and in the domain Ω = {(x1, x2) : x2 ≥ γ(x 1)}, γ = 0 for |x1| ≥ x0, x0 is a fixed constant, and γ is a sufficient smooth and simple curve.
Abstract: This paper concerns the two-dimensional Euler equations with vortex-sheets initial data in half plane and in the domain Ω = {(x1, x2) : x2 ≥ γ(x1)}, γ(x1) = 0 for |x1| ≥ x0, x0 is a fixed constant, and γ(x1) is a sufficient smooth and simple curve. The Navier-Stokes approximations are constructed in this paper and by means of vanishing the viscosity, the global existence of weak solutions is obtained under the assumption that the initial vorticity is of one-sign. Navier boundary conditions are applied when constructing the Navier-Stokes approximations.
TL;DR: In this paper, the relationship between the solutions, their 1st and 2nd derivatives of some second order linear differential equations and a meromorphic function of finite order was investigated and some precise estimates were obtained.
Abstract: In this paper, we investigate the relationship between the solutions, their 1st and 2nd derivatives of some second order linear differential equations and meromorphic function of finite order. We obtain some precise estimates.
TL;DR: In this paper, the authors studied deformation of surfaces induced by adding one and two extra space variables to the motions of space curves in higher-dimensional similarity geometries, and showed that the 2+1-and 3+1 -dimensional nonlinear evolution equations including the 2 + 1-dimensional mKdV equation and a generalization to the m KdV-Burgers system arise from such motions.
Abstract: In this paper, we study deformation of surfaces induced by adding one and two extra space variables to the motions of space curves in higher-dimensional similarity geometries. It is shown that the 2+1- and 3+1-dimensional nonlinear evolution equations including the 2+1-dimensional mKdV equation and a generalization to the mKdV-Burgers system arise from such motions.
TL;DR: In this paper, the stability of moving invariant manifolds of nonlinear impulsiveintegro-differential equations was studied based on the method of piecewise continuous Lyapunov's functions and the comparison principle.
Abstract: . This paper study the stability of moving invariant manifolds of nonlinear impulsiveintegro-differential equations. The obtain results are based on the method of piecewise continuousLyapunov’s functions and the comparison principle.Key words. Uncertain impulsive integro-differential system, moving invariant manifold, stabil-ity theoryAMS subject classifications. 34A37 1. Preliminary notes. Impulsive integro-differential equations arise naturallyfrom a wide variety of applications such as aircraft control, inspection process inoperations research, drug administration, and threshold theory in biology. There hasbeen a significant development in the theory of impulsive differential equations in thelast years [1–3].Now there also exist a well developed qualitative theory for impulsive integro-differential equations [7, 8].The efficient applications of impulsive integro-differential equations to mathemat-ical simulation request the finding of criteria for stability of their solutions.In this paper we use piecewise continuous Lyapunov’s functions to study thestability of moving invariant manifolds for general class of uncertain impulsive integro-differential equations. In the few publications dedicated to the subject of movinginvariant manifold for differential equations without impulses, earlier works were doneby [5, 6-7, 10].Our results are obtained by means of the comparison principle which permitsus to reduce the study of impulsive integro-differential equations to the study of ascalar differential equation.Let R
TL;DR: In this article, the existence of maximizers for a variational problem in R+Solutions was proved for steady geophysical flows over a surface of variable height which is bounded from below.
Abstract: We prove existence of maximizers for a variational problem in R+ Solutions represent steady geophysical flows over a surface of variable height which is bounded from below
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TL;DR: Refinable sets as mentioned in this paper are a generalization of self-affine tiles, but unlike the latter, the refinement equations defining refinable sets may have negative coefficients, and a refinable set may not tile.
Abstract: A refinable set is a compact set with positive Lebesgue measure whose characteristic function satisfies a refinement equation. Refinable sets are a generalization of self-affine tiles. But unlike the latter, the refinement equations defining refinable sets may have negative coefficients, and a refinable set may not tile. In this paper, we establish some fundamental properties of these sets.
TL;DR: In this article, the authors show that L 2 energy estimates combined with Cauchy integral formula for holomorphic functions can provide bounds for higher-order derivatives of smooth solutions of Navier-Stokes equations.
Abstract: We show that L 2 energy estimates combined with Cauchy integral formula for holomorphic functions can provide bounds for higher-order derivatives of smooth solutions of Navier- Stokes equations. We then extend this principle to weak solutions to improve regularization rates obtained by standard energy methods.
TL;DR: In this paper, the authors considered the nonlinear stability of a boundary layer of the Boltzmann equation with the cut-off hard potential when Mach number at far-field is greater than 1.
Abstract: In this paper, we consider the nonlinear stability of a boundary layer of the Boltzmann equation with the cuto hard potential when Mach number at far-field is greater than 1. Based on the Green's function for the Cauchy problem constructed in (10) and the weighted energy method, we obtain the estimates for the Green's function of the initial boundary problem and use it to obtain the nonlinear stability with an almost exponential convergent rate to the nonlinear Knudsen layer.
TL;DR: In this article, the asymptotic behavior of the ratio of contiguous polynomials is analyzed under the assumption that the coefficients of the recurrence formula are unbounded but vary regularly and have different behaviour for even and odd indices.
Abstract: Polynomials satisfying a certain twin asymptotic periodic recurrence relation are considered. It is assumed that the coefficients of the recurrence formula are unbounded but vary regularly and have different behaviour for even and odd indices. The asymptotic behaviour of the ratio of contiguous polynomials is analyzed.
TL;DR: In this paper, Kim et al. proved the central limit theorem for an m-dimensional linear process of the form Xt = P∞ j=0 AjZt−j, where {Zt} is a sequence of stationary mdimensional negatively associated random vectors with EZt = O and E||Zt|| < ∞.
Abstract: HYUN-CHULL KIM , MI-HWA KO , AND TAE-SUNG KIM Abstract. Let Aj be an m × m matrix such that P∞ j=0 ‖Aj‖ < ∞ and P∞ j=0 Aj 6= Om×m where for any m × m, m ≥ 1, matrix A = (aij), ‖A‖ = Pm i=1 Pm j=1 |aij | and Om×m denotes the m × m zero matrix. For an m-dimensional linear process of the form Xt = P∞ j=0 AjZt−j , where {Zt} is a sequence of stationary m-dimensional negatively associated random vectors with EZt = O and E||Zt|| < ∞, we prove the central limit theorems.