Global Nonlinear Stability Research Articles

Magnetic effect on instability and nonlinear stability in a reacting fluid

We study the problem of convective movement of a reacting solute in a viscous incompressible fluid occupying a plane layer and subjected to a vertical magnetic field. The thresholds for linear instability are found and compared to those derived by a global nonlinear energy stability analysis.

On the Asymptotic Stability of an Hassell Predator-Prey Model with Mutual Interference

The longtime behaviour of a nonautonomous, bidimensional, Lotka-Volterra-Hassell model with mutual interference is investigated. The existence of an absorbing set is shown together with global nonlinear asymptotic stability of the positive critical points.

Shrinking Dimer Dynamics and Its Applications to Saddle Point Search

Saddle point search on an energy surface has broad applications in fields like materials science, physics, chemistry, and biology. In this paper, we present the shrinking dimer dynamics (SDD), a dynamic system which can be applied to locate a transition state on an energy surface corresponding to an index-1 saddle point where the Hessian has a negative eigenvalue. By searching for the saddle point and the associated unstable direction simultaneously in a single dynamic system defined in an extended space, we show that unstable index-1 saddle points of the energy become linearly stable steady equilibria of the SDD which makes the SDD a robust approach for the computation of saddle points. The time discretization of the SDD is connected to various iterative algorithms, including the popular dimer method used in many practical applications. Our study of these discretization schemes lays a rigorous mathematical foundation for the corresponding iterative saddle point search algorithms. Both linear local asymptotic stability analysis and optimal error reduction (convergence) rates are presented and further confirmed by numerical experiments. Global convergence and nonlinear asymptotic stability are also illustrated for some simpler systems. Applications of the SDD in both finite- and infinite-dimensional energy spaces are discussed.

Triple -Diffusive Convection in a Magnetized Ferrofluid With MFD Viscosity: A Nonlinear Stability Analysis

A nonlinear stability analysis is performed for a triple- diffusive convection in a magnetized ferrofluid with magnetic field -dependent viscosity (MFD) for stress- free boundaries. The major mathematical emphasis is on how to control the non-linear terms caused by magnetic body force and inertia forces. A suitable generalized energy functional is introduced to perform the nonlinear energy stability analysis. It is found that nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability, and thus indicate that the subcritical instabilities are possible. However, it is noted that in case of non-ferrofluid global nonlinear stability Rayleigh number is exactly same as that of linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effects of magnetic parameter �� 3 , solute gradients �� 1 & �� 2and MFD viscosity parameter �� , on the subcritical instability region have also been analyzed.

Global nonlinear stability for a triply diffusive convection in a porous layer

A triply convective-diffusive fluid mixture saturating a porous horizontal layer in the Darcy–Oberbeck–Boussinesq scheme is studied. The nonlinear stability analysis of the conduction solution is performed when the layer is heated from below and salted from above by one salt and below by another salt. Denoting by Pi, (i = 1, 2), the salts Prandtl numbers, it is shown that in the cases {P1 = 1; P2 = 1; P1 = P2} do not exist subcritical instabilities and the thermal Rayleigh critical number of global stability in a simple closed form is given. The methodology used and the results obtained appear to be new in the existing literature and useful for the applications.

Global non-linear stability in double diffusive convection via hidden symmetries

Via a new strategy, the global non-linear stability in double diffusive convection in porous layers (heated from below and salted from above), is investigated. The condition, necessary and sufficient, either for the absence of subcritical instabilities or for the global non-linear stability is obtained by (1) discovering symmetries hidden in the Darcy–Boussinesq equations; (2) introducing “ symmetrizable double or triply diffusive convections”.

Global stability for thermal convection in a couple-stress fluid

We show that the global nonlinear stability threshold for convection in a couple-stress fluid is exactly the same as the linear instability boundary. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. It is also found that the couple-stress fluid is thermally more stable than the ordinary viscous fluid and then the effect of couple stress parameter on the onset of convection is also analyzed.

Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds number Re = 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higher Re. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes at Re ≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higher Re. For Re = 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds numberRe= 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higherRe. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes atRe≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higherRe. ForRe= 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Global nonlinear stability of convection in a heat generating fluid filled channel with a moving boundary

The energy method is employed to investigate the stability of a steady convective flow in a heat generating fluid arising due to the combined effect of buoyancy, shear and pressure gradient. By introducing a suitable generalized energy functional and using energy inequalities sufficient conditions for the existence of such a flow are found. An analysis through the variational principles is then made to find sharper limits for nonlinear stability. Comparisons are made with linear results in the literature and it is shown that the linear theory fails to capture the physics of the onset of secondary flow.

On a class of global characteristic problems for the Einstein vacuum equations with small initial data

We study a class of global characteristic problems for the Einstein vacuum equations with small initial data. In a previous work [Caciotta and Nicolò, “Global characteristic problem for Einstein vacuum equations with small initial data: (I) The initial data constraints,” Journal of Hyperbolic Differential Equations (JHDE) 2, 201 (2005)], denoted by Paper I, our attention was focused on prescribing the initial data satisfying the constraints imposed by the characteristic problem. In this paper we show how the global existence result can be achieved. This result is heavily based on the global results of Christodoulou and Klainerman [The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series Vol. 41 (Princeton University Press, Princeton, NJ, 1993)] and Klainerman and Nicolò [The Evolution Problem in General Relativity, Progress in Mathematical Physics Vol. 25 (Birkhauser, Boston, MA, 2002)].

An extension of the stability theorem of the Minkowski space in general relativity

In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in previous work by the present author. We discuss solutions of the Einstein vacuum (EV) equations. We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman or than in any other work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in $L^{\infty}(M)$. As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman.

Long-time behaviour of multi-component fluid mixtures in porous media

The long-time behaviour of a triply convective–diffusive fluid mixture saturating a porous horizontal layer in the Darcy–Oberbeck–Boussinesq scheme, is investigated. It is shown that the L 2- solutions are bounded, uniquely determined (by the initial and boundary data) and asymptotically converging toward an absorbing set of the phase-space. The stability analysis of the conduction solution is performed. The linear stability is reduced to the stability of ternary systems of O.D.Es and hence to algebraic inequalities. The existence of an instability area between stability areas of the thermal Rayleigh number (“ instability island”), is found analytically when the layer is heated and “salted” (at least by one “salt”) from below. The validity of the “linearization principle” and the global nonlinear asymptotic stability of the conduction solution when all three effects are either destabilizing or stabilizing, are obtained via a symmetrization.

Energy relaxation for transient convection in ferrofluids

The onset of transient instability driven by a coupling of thermal and magnetic effects in an initially quiescent ferrofluid layer is investigated using the energy method. Following the work of Kim [Phys. Lett. A 372, 4709 (2008)], an energy stability criterion is derived for the underlying dynamical system by taking into account the different boundary conditions and the Prandtl number effects. The critical onset time of the instability is determined as a function of the Rayleigh number, the Prandtl number, and the thermomagnetic parameter. For larger times, our analysis predicts that the energy stability theory and the linear theory yield essentially the same results irrespective of whether the fluid under consideration is a magnetically polarizable or a nonmagnetic fluid and subcritical instabilities are not possible. For the global nonlinear stability boundary in the impulsively heated ferrofluid layer, the minimum critical onset time is found to occur when the values of the Rayleigh number and the thermomagnetic parameter are same.

Penetrative convection via internal heating in anisotropic porous media

Penetrative convection in an anisotropic porous medium is simulated via internal heating. We consider both internal heat sources and sinks, where the porous medium has constant anisotropic thermal diffusivity and non-homogeneous permeability. Both linear instability and global nonlinear energy stability analyses are performed. By adopting a differential constraint approach, the nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds. In contrast, it is shown that adopting a standard energy analysis does not yield sharp results.

Rotating porous convection with prescribed heat flux

The problem of thermal convection in a rotating horizontal layer of porous medium is considered. The porous medium is described by the equations of Darcy. A novel aspect of this work is to consider boundary conditions for the temperature of Newton–Robin type with heat flux prescribed as a limiting case. The effect of rotation is found to be crucial. For the Taylor number small enough the critical wave number is zero but we find a threshold such that for Taylor numbers beyond this non-zero critical wave numbers are found. The threshold is verified via a weakly nonlinear analysis. Finally, a sharp global nonlinear stability analysis is given.

Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data

In this paper, we study the global structure stability of the Riemann solution u = U ( x t ) for general n × n quasilinear hyperbolic systems of conservation laws under a small BV perturbation of the Riemann initial data. We prove the global existence and uniqueness of piecewise C 1 solution containing only n contact discontinuities to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem, for general n × n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution u = U ( x t ) to the corresponding Riemann problem. Our result indicates that this kind of Riemann solution u = U ( x t ) mentioned above for general n × n quasilinear hyperbolic systems of conservation laws possesses a global nonlinear structure stability under a small BV perturbation of the Riemann initial data. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to the system describing the motion of the relativistic string in Minkowski space R 1 + n , are also given.

Sharp global nonlinear stability for a fluid overlying a highly porous material

The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the Navier–Stokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

A nonlinear stability analysis in a double-diffusive magnetized ferrofluid with magnetic-field-dependent viscosity saturating a porous medium

A rigorous nonlinear stability result is derived by introducing a suitable generalized energy functional for a magnetized ferrofluid layer heated and soluted from below with magnetic-field-dependent (MFD) viscosity saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic-body and inertia forces. For ferrofluids, we find that there is possibility of existence of subcritical instabilities, however, it is noted that, in case of a non-ferrofluid, the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of the magnetic parameter, M3; solute gradient, Sf; Darcy number, Da; and MFD viscosity parameter, δ; on the subcritical instability region has also been analyzed.

Power Law Inflation

The subject of this paper is Einstein’s equations coupled to a non-linear scalar field with an exponential potential. The problem we consider is that of proving future global non-linear stability of a class of spatially locally homogeneous solutions to the equations. There are solutions on \({\mathbb {R}_{+}\times \mathbb {R}^{n}}\) with accelerated expansion of power law type. We prove a result stating that if we have initial data that are close enough to those of such a solution on a ball of a certain radius, say \({B_{4R_{0}}(p)}\) , then all causal geodesics starting in \({B_{R_{0}}(p)}\) are complete to the future in the maximal globally hyperbolic development of the data we started with. In other words, we only make local assumptions in space and obtain global conclusions in time. We also obtain asymptotic expansions in the region over which we have control. As a consequence of this result and the fact that one can analyze the asymptotic behaviour in most of the spatially homogeneous cases, we obtain quite a general stability statement in the spatially locally homogeneous setting.