Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

Asymptotic stability of viscous contact wave to a radiation hydrodynamic limit model

This paper is concerned with the large time behavior of the solutions for 1D radiation hydrodynamic limit model without viscosity and its asymptotic stability of the viscous contact discontinuity wave under the smallness assumption of the strength of the contact wave and initial perturbations. The present pressure includes a fourth-order term about the absolute temperature from radiation effect which brings the main difficulty. Furthermore, the dissipative of the system is weaker for the lack of viscosity. All these make the problem more challenging. The prove is mainly based on the energy method, including normal and radial directions energy estimates.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity

We show that the solution of a semilinear transmission problem between an elastic and a thermoelastic material, decays exponentially to zero. That is, denoting by ?(t) the sum of the first, second and third order energy associated with the system, we show that there exist positive constants C and γsatisfying ?(t) ? C?(0)e^?γt Moreover, the existence of absorbing sets is achieved in the non‐homogeneous case. Copyright © 2002 John Wiley & Sons, Ltd.     

Asymptotic stability of solutions to the nonisentropic hydrodynamic model for semiconductors

In this paper, the asymptotic stability of smooth solutions to the multidimensional nonisentropic hydrodynamic model for semiconductors is established, under the assumption that the initial data are a small perturbation of the stationary solutions for the thermal equilibrium state, whose proofs mainly depend on the basic energy methods.     

Asymptotic analysis of a singular Sturm-Liouville boundary value problem

Asymptotic expansions are given for the eigenvalues λ_n and eigenfunctions u_n of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.     

Asymptotic analysis for a particle transport problem in a moving medium

In this paper we discuss the derivation of the diffusion theoryfor a linear particle transport in a moving gas. We performthe asymptotic analysis for a one-dimensional linear BGK modelproblem with a shifting Maxwellian which describes the timeevolution of the spatially dependent electron distribution functionin a moving weakly ionized host medium. The modified (compressed)Chapman-Enskog expansion procedure is applied to find the asymptoticsolution for small mean free path , showing that the differencebetween the exact and asymptotic solutions is of order ², uniformlyin time for arbitrary initial data.     

Towards the problem of hydrodynamic stability of plane-parallel dusty-gas flows

Several modifications of the classical Saffman formulation of the dusty-gas flow linear stability problem are considered. Dispersed flows are described by a two-fluid model. Linear stability problems are reduced to the solution of modified Orr-Sommerfeld equations which are solved by the orthogonalization method. It is shown, that the additional factors taken into account in the problem formulation affect significantly the flow stability limits. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > k _c (for some critical wave number k _c) implies the stability of the mode for a class of basic flows. Furthermore, if $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} , where U ₀ is the basic velocity, T ₀ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) , where the flow domain is 0 ≤ z ≤ D.     

Asymptotic analysis of a thin fluid layer-elastic plate interaction problem

We study the nonstationary flow of an incompressible fluid in a thin rectangle with an elastic plate as the upper part of the boundary. The flow is governed by a time-dependent pressure drop and an external force and it is modeled by Stokes equations. The dynamic of this fluid–structure interaction problem is studied in the limit when the thickness of the fluid domain tends to zero. Using the asymptotic techniques, we obtain for the effective plate displacement a sixth-order parabolic equation with a non standard boundary conditions. Results on existence, uniqueness and regularity of the solution are proved. The approximation is justified through a weak convergence theorem.     

Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k _s ?>?0; it is also shown that instability is possible only for 0?<?k?<?k _s and not for k?>?k _s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c _i, the imaginary part of the phase velocity as the wave number k →k _s ? to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.     

Periodic boundary-value problem for a fourth-order differential equation

In the present paper we obtain sufficient conditions for solvability of a periodic boundary-value problem for a fourth-order ordinary differential equation. The research technique is based on a solvability theorem for a quasi-linear operator equation in the resonance case. We formulate sufficient conditions for existence of periodic solutions in terms of the initial equation. The main result of the paper clarifies the existence theorem established by B. Mehry and D. Shadman in Sci. Iran. 15 (2), 182–185 (2008).     

Nodal solutions for a nonlinear fourth-order eigenvalue problem

We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa（x）f（y）,0〈x〈1,y（0）=y（1）=y″（0）=y″（1）=0where λ is a positive parameter, a ∈ C（[0, 1], （0, ∞）, f ∈C（R,R） satisfies f（u）u 〉 0 for all u ≠ 0. We give conditions on the ratio f（s）/s, at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.     

Eigenvalue problem of a class of fourth-order Hamiltonian operators

The eigenvalue problem of a class of fourth-order Hamiltonian operators is studied. We first obtain the geometric multiplicity, the algebraic index and the algebraic multiplicity of each eigenvalue of the Hamiltonian operators. Then, some necessary and sufficient conditions for the completeness of the eigen or root vector system of the Hamiltonian operators are given, which is characterized by that of the vector system consisting of the first components of all eigenvectors. Moreover, the results are applied to the plate bending problem.