Critical Threshold Phenomena Research Articles

Critical thresholds in 1D pressureless Euler–Poisson systems with variable background

The Euler–Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. This paper is to advance our understanding of critical threshold phenomena in such systems in the presence of different forces. We identify critical thresholds in two damped Euler–Poisson systems, with and without alignment, both with attractive potential and spatially variable background state. For both systems, we give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critical threshold for each scenario. Key tools include comparison with auxiliary systems and the phase space analysis of the transformed system.

Wave Breaking, Global Existence and Persistent Decay for the Gurevich–Zybin System

In this paper, the blow-up phenomenon, global existence and persistent decay of the solutions to the Gurevich–Zybin system are studied. We show that the system possesses a so called critical threshold phenomena, that is, global smoothness versus finite time breakdown depends on whether the initial configuration crosses an intrinsic critical threshold. We prove that a finite maximal life span for a solution necessarily implies wave breaking for this solution, and show some conditions which are local-in-space on the initial data to ensure wave-breaking for this system by making use of the characteristics method, otherwise, the system has a global smooth solution. Furthermore, we establish the persistence properties for the system in weighted $$L^p$$ spaces.

Blow-up conditions for two dimensional modified Euler–Poisson equations

The multi-dimensional Euler–Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow-up for some initial configurations. This article strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional modified Euler–Poisson system with a modified Riesz transform where the singularity at the origin is removed. We identify upper-thresholds for finite time blow-up of solutions for the modified Euler–Poisson equations with attractive/repulsive forcing.

Critical thresholds in multi-dimensional restricted Euler equations

Using the spectral dynamics, we study the critical threshold phenomena in the multi- dimensional restricted Euler (RE) equations. We identify sub-critical and sup-critical initial data for all space dimensions, which extends the previous result for the 3D and 4D restricted Euler equations. Our result suggests that: if the number of dimensions is odd, the finite time blowup is generic; in contrast, if the number of dimensions is even, there is a rich set of initial data which yields global smooth solutions.

Global regularity of the 4D restricted Euler equations

We are concerned with the critical threshold phenomena in the restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for the 3D and 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that the 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.

Critical thresholds in hyperbolic relaxation systems

Critical threshold phenomena in one-dimensional 2 × 2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.

Critical thresholds in a quasilinear hyperbolic model of blood flow

Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.

Critical thresholds in a relaxation system with resonance of characteristic speeds

We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.

Critical thresholds in a relaxation model for traffic flows

In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traYc flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual sub-characteristic con- dition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formation of solutions si- multaneously by showing the critical threshold phenomena associated with the underlying relaxation system. We identify five upper thresh- olds for finite time singularity in solutions and three lower thresholds for global existence of smooth solutions. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope. Our results show that the shorter the drivers' respond- ing time to the traYc, the larger the set of initial conditions leading to global smooth solutions which correctly predicts the empirical findings for traY cfl ows.

Semiclassical Limit of the Nonlinear Schrödinger-Poisson Equation with Subcritical Initial Data

We study the semi-classical limit of the nonlinear Schrodinger-Poisson (NLSP) equa- tion for initial data of the WKB type. The semi-classical limit in this case is realized in terms of a density-velocity pair governed by the Euler-Poisson equations. Recently we have shown in (ELT, Indiana Univ. Math. J., 50 (2001), 109-157), that the isotropic Euler-Poisson equations admit a critical threshold phenomena, where initial data in the sub-critical regime give rise to globally smooth solutions. Consequently, we justify the semi-classical limit for sub-critical NLSP initial data and confirm the validity of the WKB method.

Spectral Dynamics of the Velocity Gradient Field¶in Restricted Flows

We study the velocity gradients of the fundamental Eulerian equation, ∂ t u+u·∇u=F, which shows up in different contexts dictated by the different modeling of F's. To this end we utilize a basic description for the spectral dynamics of ∇u, expressed in terms of the (possibly complex) eigenvalues, λ=λ (∇u), which are governed by the Ricatti-like equation λ t +u·∇λ+λ2=<a, ∇F r>. We focus our investigation on four prototype models associated with different forcing F, ranging from simple linear damping and viscous dusty medium models to the main thrust of the paper – the restricted models of Euler/Navier–Stokes equations and Euler–Poisson equations. In particular, we address the question of the time regularity for these models, that is, whether they admit a finite time breakdown, a global smooth solution, or an intermediate scenario of critical threshold phenomena where global regularity depends on initial configurations. Using the spectral dynamics as our essential tool in these investigations, we obtain a simple form of a critical threshold for the linear damping model and we identify the 2D vanishing viscosity limit for the viscous irrotational dusty medium model. Moreover, for the n-dimensional restricted Euler equations we obtain [n/2]+1 global invariants, interesting for their own sake, which enable us to precisely characterize the local topology at breakdown time, extending previous studies in the n= 3-dimensional case. Finally, as a fourth model we introduce the n-dimensional restricted Euler–Poisson (REP)system, identifying a set of [n/2] global invariants, which in turn yield (i) sufficient conditions for finite time breakdown, and (ii) characterization of a large class of 2-dimensional initial configurations leading to global smooth solutions. Consequently, the 2D restricted Euler–Poisson equations are shown to admit a critical threshold.

Critical thresholds in Euler-Poisson equations

We present a preliminary study of a new phenom- ena associated with the Euler-Poisson equations — the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic,ON1O critical thresh- old. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems with or without various forcing mech- anisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical thresh- old which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock dis- continuity, except for the non-generic case of increasing initial profile,u 0 0. In contrast, we show that the corresponding one- dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initialN0;u 0O-configuration satisfiesu 0 p 2k0 - see (2.11) below, allowing a finite, crit- ical negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenom- ena.