Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

BV-Regularity of the Boltzmann Equation in Non-Convex Domains

We consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new W^1,1-trace estimate for the diffuse boundary condition and a delicate construction of an \({\varepsilon}\)-tubular neighborhood of the singular set.     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.     

On Nonlinear Instability and Stability for Stratified Shear Flow

An example of stratified shear flow is presented in which an explicit construction is given for unstable eigenvalues with smooth eigenfunctions for the Taylor--Goldstein equation. It is proved for any stratified, plane parallel shear flow that the unstable spectrum of the linear operator is purely discrete. A general theorem is then invoked to prove that the specific example is nonlinearly unstable. A sufficient condition for nonlinear stability for stratified shear flow is discussed.     

Stability and Instability of Fourth-Order Solitary Waves

We study ground-state traveling wave solutions of a fourth-order wave equation. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. The analysis depends on the variational characterization of the ground states rather than information about the linearized operator.     

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.     

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.     

Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, u _t  = J*u−u := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on . When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behaviors can be presented in a unified way through a suitable global approximation.     

Stability of Solitary Waves for the Generalized Higher-Order Boussinesq Equation

This work studies the stability of solitary waves of a class of sixth-order Boussinesq equations.     

Convective Instability in a Horizontal Porous Channel with Permeable and Conducting Side Boundaries

The stability analysis of the motionless state of a horizontal porous channel with rectangular cross-section and saturated by a fluid is developed. The heating from below is modelled by a uniform flux, while the top wall is assumed to be isothermal. The side boundaries are considered as permeable and perfectly conducting. The linear stability of the basic state is studied for the normal mode perturbations. The principle of exchange of stabilities is proved, so that only stationary normal modes need to be considered in the stability analysis. The eigenvalue problem for the neutral stability condition is solved analytically, and a closed-form dispersion relation is obtained for the neutral stability. The Darcy–Rayleigh number is expressed as an implicit function of the longitudinal wave number and of the aspect ratio. The critical wave number and the critical Darcy–Rayleigh number are evaluated for different aspect ratios. The preferred modes under critical conditions are detected. It is found that the selected patterns of instability at the critical Rayleigh number are two-dimensional, for slender or square cross-sections of the channel. On the other hand, instability is three dimensional when the critical width-to-height ratio, 1.350517, is exceeded. Eventually, the effects of a finite longitudinal length of the channel are discussed.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.     

Modified Ginzburg–Landau Equation and Benjamin–Feir Instability

In this paper, a modulated wave train in a nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in the general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg–Landau Equation (MGLE). Benjamin–Feir instability for the MGLE is analyzed.     

Divergent Instability Domains of a Fluid-Conveying Cantilevered Pipe with a Combined Support

The divergent instability of a fluid-conveying pipe is analyzed numerically. The pipe is modeled by a cantilevered beam restrained at the left end and supported by a special device (a rotational elastic restraint plus a Q-apparatus) at the right end. Divergent instability domains of the pipe are obtained by varying the rotational elastic constant of the right restraint     

Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.     

On Stability and Strong Convergence for the Spatially Homogeneous Boltzmann Equation for Fermi-Dirac Particles

 The paper considers the stability and strong convergence to equilibrium of solutions to the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Under a cutoff condition on the collision kernel, we prove a strong stability in L ¹ topology at any finite time interval, and, for hard and Maxwellian potentials, we prove that the solutions converge strongly in L ¹ to equilibrium under a high temperature condition. The basic tools used are moment-production estimates and the strong compactness of the collision gain term. (Accepted 25, October 2002) Published online March 14, 2003 Communicated by P.-L. Lions     

Stability Region Control for a Parametrically Forced Mathieu Equation

We exhibit common features of how the size of parametric regions of stability for the Mathieu equation can be enlarged. The paper shows that the mechanisms for these changes via parametric forcing follow the pattern established earlier for the Arnold circle map which provides a discrete model for external forcing. The various types of behaviour of the standard Mathieu equation for a given set of parameters can be classified as having either (i) all solutions bounded, (ii) at least one unbounded solution, or (iii) periodic solutions of period -/-2 or -/-4. The marginal case (iii) forms the boundary of the regions of stability and instability. We consider a parametric method for changing the shapes of the stability regions and show how maximally stable regions can be produced.     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

On the Global Stability of One Nonlinear Difference Equation

We give exact sufficient conditions for the global stability of the zero solution of the difference equation x _{n + 1} = qx _n + f _n (x _n , ..., x _{n – k} ), n , where the nonlinear functions f _n satisfy the conditions of negative feedback and sublinear growth.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 487–494, October–December, 2004.     

Analysis of Stability and Instability Regions in Parametrically Excited Hamiltonian Systems

Qualitative analysis of parametrically excited linear Hamiltonian systems is carried out. It is proved that the stability and instability regions are convex in the excitation frequency. Lower bounds for the boundaries of some instability regions are obtained expressed in the natural frequencies of the system in the absence of a parametric excitation. It is shown that a dominant high-frequency excitation affects the stability regions similarly to an increase of the natural frequencies. Some of these findings extend known results, obtained by asymptotic methods under the assumption that the parametric excitation is small or its frequency is large, to finite values of the excitation and frequency.