Large Initial Perturbation Research Articles

The conduit equation: Hyperbolic approximation and generalized Riemann problem

The conduit equation is a dispersive non-integrable scalar equation modeling the flow of a low-viscous buoyant fluid embedded in a highly viscous fluid matrix. This equation can be written in a particular form reminiscent of the famous Godunov form proposed in 1961 for the Euler equations of compressible fluids. We propose a hyperbolic approximation of the conduit equation by retaining the Godunov-type structure. The comparison of solutions to the conduit equation and those to the approximate hyperbolic system is performed: the wave fission of a large initial perturbation of a rectangular or Gaussian form. The results are in good agreement. New generalized solutions to the conduit equation composed of a finite set of waves of the same period and linked with a constant solution by generalized Rankine-Hugoniot relations are discovered. Such multi-hump structures interact with each other almost as solitary waves: they collide, merge, and reconstruct after the interaction. This partly indicates the stability of such multi-hump solutions under small perturbations. The exact and approximate hyperbolic system describes such an interaction with good accuracy.

Global stability of viscous contact wave for compressible Navier–Stokes equations with temperature‐dependent transport coefficients and large data

We are concerned with the large‐time behavior of the viscous contact wave for a one‐dimensional compressible Navier–Stokes equations whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent satisfies a suitable inequality and strength is small enough, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the viscous contact wave under large initial perturbation for any . Subsequently, we also prove that the large‐time asymptotic stability of the contact wave has a uniform convergence rate under initial data. Our results improve ‐rate in [F. Huang, Z. Xin, and T. Yang, Adv. Math. 219 (2008), 1246–1297] which studied the constant coefficients Navier–Stokes system. The proofs are given by the elementary energy estimate and anti‐derivative method.

Effective statistical control strategies for complex turbulent dynamical systems

Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties.

Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier–Stokes–Poisson system under large initial perturbation

Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier–Stokes–Poisson system under large initial perturbation

Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties

The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle method for collisional kinetic models of plasmas under the effect of uncertainties. This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles' position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG projection and to obtain the corresponding system for the gPC coefficients. We show that the sG particle method preserves the main physical properties of the problem, such as conservations and positivity of the solution, while achieving spectral accuracy for smooth solutions in the random space. Furthermore, in the fluid limit the sG particle solver is designed to possess the asymptotic-preserving property necessary to obtain a sG particle scheme for the limiting Euler-Poisson system, thus avoiding the loss of hyperbolicity typical of conventional sG methods based on finite differences or finite volumes. We tested the schemes considering the classical Landau damping problem in the presence of both small and large initial uncertain perturbations, the two stream instability and the Sod shock tube problems under uncertainties. The results show that the proposed method is able to capture the correct behavior of the system in all test cases, even when the relaxation time scale is very small.

Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data

Abstract In this article, we study the large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].

Large-amplitude effects on interface perturbation growth in Richtmyer–Meshkov flows with reshock

Experimental and theoretical studies on the Richtmyer–Meshkov (RM) instability of heavy/light gaseous interfaces with reshock are performed. Both small and large initial perturbation amplitudes of single- and quasi-single-mode interfaces are considered, highlighting the effects of interface amplitude and shape on the linear and nonlinear growths of the RM instability. The results indicate that for small-amplitude interfaces distorted before and after the first reshock arrival, the perturbation growths at linear stages can be well predicted by the impulsive model. For large-amplitude interfaces, however, the reshock acceleration on the evolving interface promotes the mode interaction and enhances the nonlinear effects, making the perturbation growth rates reduced in comparison with those in the singly shocked cases. The complete evolution, especially the bubble evolution, has a strong memory of initial shapes, while for large-amplitude cases, the spike evolution is nearly independent of them owing to the destruction of large-scale vortices and multiple-shock-induced small-scale structures. Compared with that of the single-mode case, the normalized perturbation growths after reshock for the quasi-single-mode cases are more sensitive to initial amplitudes. To better describe the linear growth rates of the RM instability induced by the incident shock and reshock, the reduction factor models for large-amplitude cases are developed, which successfully predict the non-monotonic dependence of linear growth rates on initial perturbation amplitudes. For small-amplitude cases, the nonlinear model proposed for the singly shocked case can predict the reshocked nonlinear growth, while for large-amplitude cases, it is invalid because the perturbation growth shows a linear characteristic after reshock.

Least-current maneuver sequence for electromagnetic suspension control subjected to large perturbation found by direct collocation

ABSTRACT Conventional approaches in handling the electromagnetic suspension (E.M.S.) problems usually start with building nonlinear models and linearizing them around the equilibrium point. For this sake, the applied perturbation must remain considerably small, and this is often inapplicable to the industry. This study applies optimal control to determine the least-current maneuver sequences for an E.M.S. system. The optimal sequences determined by simulations successfully remove the large undesirable initial perturbation and bring the system back to the equilibrium states. Taking advantage of direct collocation and nonlinear programming (DCNLP), this study challenges the two-point boundary-value problem (TPBVP) without linearizing the nonlinear system. The results of simulations indicate that the DCNLP provides a rigorous approach in solving the TPBVP. The optimal controls successfully subdue the large initial perturbations of displacement, as large as 0.8 m, and bring the mass back to the desired states. The accuracy of simulation is also exploited in order to achieve a reasonable balance between numerical solution quality and iteration load of computations. The study suggests an optimal control approach for handling nonlinear E.M.S. problems using the DCNLP method, which guarantees strong convergence.

Evolution mechanism of double-layer heavy gas column interface with sinusoidal disturbance induced by convergent shock wave

Based on Navier-Stokes equations, combining the fifth-order weighted essentially non-oscillatory scheme with the adaptive structured grid refinement technique, the interactions between converging shock and annular SF<sub>6</sub> layers with different initial perturbation amplitudes and thickness are numerically investigated. The evolution mechanism of shock structure and interface are revealed in detail, and the variations of the circulation, mixing rate and turbulent kinetic energy are quantitatively analyzed. The dynamic mode decomposition method is used to analyze the dynamic characteristics of the vorticity. The results show that in the case with large initial perturbation amplitude, the transmitted shock wave forms Mach reflection structures both inside and outside of the inner interface of SF<sub>6</sub> layer, and multiple shock focusing phenomena occur in the center. After the transmitted shock wave penetrates the outer interface, the circulation increases faster, and the “spike” and “bubble” structure on inner interface develop faster, so that the amplitude of the inner and outer interfaces and the gas mixing rate increase. As for the case with larger thickness of the gas layer, the phase of the transmitted shock wave changes inside the layer, which forms “bubble” at the crest of the inner interface and “spike” at the trough. When the thickness of the gas layer decreases, the crest of the inner interface does not move inside after being impacted, and “spike” and “bubble” are generated in the late stage. The dynamic modes show that the main structure of vorticity and the exchange of positive and negative vorticity on the main structure are determined by the modes with weak growth and low frequency, but the modes with weak growth and high frequency determine rapid exchange of positive and negative vorticity at the interface in the cases with weak coupling effect.

Nonlinear stability of rarefaction waves for micropolar fluid model with large initial perturbation

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature.

Asymptotic behaviors of global solutions to the two-dimensional non-resistive MHD equations with large initial perturbations

This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in Pan et al. (2018) [24], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the H2-norm of the initial perturbation value of both the velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that H2-norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution converges in a rate of the field strength to a solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented.

Stability of stationary solutions for the unipolar isentropic compressible Navier–Stokes–Poisson system

The stationary solutions to the outflow problem for unipolar isentropic Navier–Stokes–Poisson system in a half line (0, ∞) have recently been shown to be asymptotically stable in13 and26, provided that all the L2 norms of initial perturbations and their derivatives are small. The main purpose of this paper is to study the asymptotic stability of the stationary solutions under large initial perturbations. First, for the outflow problem, we show that the stationary solutions are asymptotically stable, provided that only the L2 norm of initial perturbation is small. Next, for the inflow problem, we show asymptotic stability under a large initial perturbation in the H1 norm. The main ingredient in the proofs is a careful analysis in order to control the growth induced by nonlinearities of the system in a priori estimates.

Long-time behaviors for the Navier–Stokes equations under large initial perturbation

Long-time behaviors for the Navier–Stokes equations under large initial perturbation

On the problem of stability of a motion of essentially nonlinear systems

This article discusses essentially nonlinear systems. Following the approach of applying the pseudolinear inequalitiesdeveloped in a number of works, new estimates for the variation of Lyapunov functions along solutionsof the considered systems of equations are obtained. Based on these estimates, we obtain sufficient conditionsfor the equiboundedness of solutions of second-order systems and sufficient conditions for the stability of anessentially nonlinear system under large initial perturbations. Conditions for the stability of affine systems arealso obtained.

Linear and nonlinear hydromagnetic stability in laminar and turbulent flows.

We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

Dynamics of Screening in Modified Gravity.

Gravitational theories differing from general relativity may explain the accelerated expansion of the Universe without a cosmological constant. However, to pass local gravitational tests, a "screening mechanism" is needed to suppress, on small scales, the fifth force driving the cosmological acceleration. We consider the simplest of these theories, i.e., a scalar-tensor theory with first-order derivative self-interactions, and study isolated (static and spherically symmetric) nonrelativistic and relativistic stars. We produce screened solutions and use them as initial data for nonlinear numerical evolutions in spherical symmetry. We find that these solutions are stable under large initial perturbations, as long as they do not cause gravitational collapse. When gravitational collapse is triggered, the characteristic speeds of the scalar evolution equation diverge, even before apparent black-hole or sound horizons form. This casts doubts on whether the dynamical evolution of screened stars may be predicted in these effective field theories.

Contact discontinuity for a viscous radiative and reactive gas with large initial perturbation

Contact discontinuity for a viscous radiative and reactive gas with large initial perturbation

Nonlinear bubble competition of the multimode ablative Rayleigh–Taylor instability and applications to inertial confinement fusion

The self-similar nonlinear evolution of the multimode ablative Rayleigh–Taylor instability (RTI) and the ablation-generated vorticity effect are studied for a range of initial conditions. We show that, unlike classical RTI, the nonlinear multimode bubble-front evolution remains in the bubble competition regime due to ablation-generated vorticity, which accelerates the bubbles, thereby preventing a transition into the bubble-merger regime. We develop an analytical bubble competition model to describe the linear and nonlinear stages of ablative RTI. We show that vorticity inside the multimode bubbles is most significant at small scales with large initial perturbation. Since these small scales persist in the bubble competition regime, the self-similar growth coefficient αb can be enhanced by up to 30% relative to ablative bubble competition without vorticity effects. We use the ablative bubble competition model to explain the hydrodynamic stability boundary observed in OMEGA low-adiabat implosion experiments.

Global stability of rarefaction waves for the 1D compressible micropolar fluid model with density-dependent viscosity and microviscosity coefficients

This paper is concerned with the global existence and large time behavior of strong solutions to the Cauchy problem of one-dimensional compressible isentropic micropolar fluid model with density-dependent viscosity and microviscosity coefficients, where the far-fields of the initial data are prescribed to be different. The pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R are considered. For the case when the corresponding Riemann problem of the resulting Euler equations admits two rarefaction waves solutions, it is shown that if the parameters α and γ satisfy some conditions and the initial data is sufficiently regular, without vacuum and mass concentrations, then the Cauchy problem of the one-dimensional compressible micropolar fluid model has a unique global strong nonvacuum solution, which tends to a superposition of these two rarefaction waves as time goes to infinity. This result holds for arbitrarily large initial perturbation and large-amplitudes rarefaction waves. Moreover, the exponential time decay rate of the microrotation velocity ω(t,x) under large initial perturbation is also derived. The proof is given by an elaborate energy method and the key ingredient is to deduce the uniform-in-time lower and upper bounds on the specific volume.

Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation

We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of an one-dimensional compressible Navier-Stokes type system for a viscous, compressible, radiative and reactive gas, where the constitutive relations for the pressure $p$, the specific internal energy $e$, the specific volume $v$, the absolute temperature $\theta$, and the specific entropy $s$ are given by $p=R\theta/v +a\theta^4/3$, $e=C_v\theta+av\theta^4$, and $s=C_v\ln \theta+ 4av\theta^3/3+R\ln v$ with $R>0$, $C_{v}>0$, and $a>0$ being the perfect gas constant, the specific heat and the radiation constant, respectively. For such a specific gas motion, a somewhat surprising fact is that, general speaking, the pressure $\widetilde{p}(v,s)$ is not a convex function of the specific volume $v$ and the specific entropy $s$. Even so, we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant $a$ and the strength of the rarefaction waves are sufficiently small. The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature, which are uniform with respect to the space and the time variables, but are independent of the radiation constant $a$.