arXiv subjects
Publications and source records attributed to Blake Barker.
In this note, we are interested in the linear Rayleigh-Taylor instability problem for the incompressible fluid with an upper free surface. Using the spectral theory of self-adjoint and compact operator, we rigorously prove the existence of infinitely many normal mode solutions to the linearized equations. Some numerical computations are presented to support our theoretical study.
A habitat that is moving due to environmental change may result in tipping to extinction if the rate at which it moves is too great. We use a scalar reaction-diffusion equation with a non-autonomous reaction term, representing a spatially localized habitat moving from one asymptotic location to another, as a context for studying this phenomenon. The movement is characterized by displacement $d$ and rate parameter $r$. The system admits three steady states in both asymptotic habitat locations: a stable extinction state $u_0^*=0$, an unstable pulse (so-called edge state) $u_1^*(x)>0$, which gives rise to the Allee effect, and a stable pulse (populated base state) $u_2^*(x)>u_1^*(x)$, which corresponds to a thriving population at its carrying capacity. Numerical simulations for a specific model identify a critical displacement $d^*$ and, for $d > d^*$, demonstrate the existence of a \textit{critical rate} $r_c(d)$ at which rate-induced tipping occurs: for $r> r_c$ an initially thriving population becomes extinct due to habitat movement being too rapid. We provide analytical results for two limiting cases. For $r\ll 1$, solutions track the moving base state with error $O(r)$. For $r\gg 1$, solutions converge to the extinction state provided $d$ is sufficiently large. For $d$ too small, no tipping occurs regardless of $r$. Numerical simulations complement and extend these analytical results. At the critical rate $r=r_c(d)$, we identify a pulse-to-pulse heteroclinic connection between the base state at the past asymptotic location and the edge state at the future asymptotic location of the habitat. We also establish the uniqueness of this critical rate and non-degeneracy of the heteroclinic connection as $r$ varies.
For certain values of the wave speed parameter, evolution equations for the temperature of a region of fuel admit traveling wave solutions describing fire fronts. We consider such a system in the form of a nonlinear reaction-diffusion equation with a first-order forcing term capturing the combined effects of ambient and fire-induced wind. The fire-induced wind is introduced by way of a piecewise continuous function that ``switches'' in space. We demonstrate that, in the case of a spatially dependent wind, traveling wave solutions corresponding to fire fronts exist for a continuum of wave speeds rather than for a single unique speed. Using geometric methods, we determine the range of allowable speeds, refine this range to only those fronts which will persist in nature, and develop a selection mechanism to identify the specific wind configuration corresponding to the most stable solution. For this spectrally preferred front, we find that the wind switch occurs ahead of the fireline in a manner consistent with the physics of air entrainment. Even when the wind is not coupled to the temperature and is instead imposed as an external forcing, the conditions on the existence and stability of front solutions force the wind term in to a configuration reflective of physical reality.
Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the paper is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.
We investigate the stability of traveling front solutions to nonlinear diffusive-dispersive equations of Burgers type, with a primary focus on the Korteweg-de Vries-Burgers (KdVB) equation, although our analytical findings extend more broadly. Manipulating the temporal modulation of the translation parameter of the front and employing the energy method, we establish asymptotic, nonlinear, and orbital stability, provided that an auxiliary Schr\"odinger equation possesses precisely one bound state. Notably, our result is independent of the monotonicity of the profile and does not necessitate the initial condition to be close to the front. We identify a sufficient condition for stability based on a functional that characterizes the 'width' of the traveling wave profile. Analytical verification for the KdVB equation confirms that this sufficient condition holds for the relative dispersion parameter within an open interval $\nu \in [-0.25,0.25]$, encompassing all monotone profiles. Utilizing validated numerics or rigorous computation, we present acomputer-assisted proof demonstrating that the stability condition itself holds for parameter values within the interval [0.2533, 3.9].
We consider a free boundary problem on three-dimensional cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. Combining analysis and computer-assisted proof, we show that when c is less than 0.43, the free boundary may pass through the vertex of the cone.
We treat the 1D shock tube problem, establishing existence of steady solutions of full (nonisentropic) polytropic gas dynamics with arbitrary noncharacteristic data. We present also numerical experiments indicating uniqueness and time-asymptotic stability of such solutions. At the same time, we give an example of an (artificial) equation of state possessing a convex entropy for which there holds nonuniqueness of solutions. This is associated with instability and Hopf bifurcation to time-periodic solutions. In a second part, we study general systems of viscous conservation laws and we state results on existence, stability and spectral stability of steady states in restricted cases.
We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model and setting) an open question of Zumbrun and Serre.
The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.
The Evans function has become a standard tool in the mathematical study of nonlinear wave stability. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). We report on an unexpected complication that frustrates this computation for viscous shock profiles in gas dynamics. Although this phenomenon---related to the choice of Eulerian or Lagrangian coordinate system used to describe the gas---is present already in the one-dimensional setting, its implications are especially important in the multidimensional case where no computationally viable Lagrangian description of the gas is readily available. We introduce new "pseudo-Lagrangian" coordinates that allow us to overcome this difficulty, and we illustrate the utility of these coordinates in the setting of isentropic gas dynamics in two space dimensions.
The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman's integrated coordinates [G1]; this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.
In this note, we announce a complete classification of stability of periodic roll-wave solutions of the viscous shallow-water equations, from their onset at Froude number $F\approx 2$ up to the infinite-Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between $F$ and the boundaries of the region of stable periods, that appears potentially useful in hydraulic engineering applications. In the asymptotic regime $F\to 2$ (onset), we provide an analytic expression of the stability boundaries whereas in the limit $F\to\infty$, we show that roll-waves are always unstable.
We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $\approx 1.86 $, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $\approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.
We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general values of the Froude number $F$, including the limit $F\to +\infty$. In the former, $F\to 2^+$, limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $F\to 2^+$ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson--Noble--Rodrigues--Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around $F=2.3$ from weakly unstable to different, large-$F$ behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $2.5\leq F\leq 6.0$.
We document the MATLAB code used in the following study: Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow.
We present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the KdV-KS equation modeling weakly unstable flow of a thin fluid film on an incline in the small-amplitude KdV limit. The argument proceeds by verification of a stability condition derived by Bar-Nepomnyashchy and Johnson-Noble-Rodrigues-Zumbrun involving inner products of various elliptic functions arising through the KdV equation. One key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time. Another is the way of handling analytic interpolation in several variables by a two-step process carving up the parameter space into manageable pieces for rigorous evaluation. These and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability.
It has long been a standard practice to neglect diffusive effects in stability analyses of detonation waves. Here, with the principal aim of quantifying the impact of these oft-neglected effects on the stability characteristics of such waves, we use numerical Evans-function techniques to study the (spectral) stability of viscous strong detonation waves---particular traveling-wave solutions of the Navier-Stokes equations modeling a mixture of reacting gases. Remarkably, our results show a surprising synergy between the high-activation-energy limit typically studied in stability analyses of detonation waves and the presence of small but nonzero diffusive effects. While our calculations do show a modest delay in the onset of instability in agreement with recently reported calculations by direct numerical simulation of the physical equations, our approach, based on the Evans function, also provides additional spectral information. In particular, for each of the families of detonation waves in our computational domain, we find a completely unexpected kind of hysteresis in the limit of increasing activation energy; that is, our calculations suggest that, whenever diffusive effects are present, there is a return to stability as unstable eigenvalues return to the stable complex half plane.