scholarly journals Sharp critical thresholds in a hyperbolic system with relaxation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Manas Bhatnagar ◽  
Hailiang Liu
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Global Regularity ◽  
Critical Threshold ◽  
One Dimensional ◽  
Critical Thresholds ◽  
Threshold Phenomena ◽  
Local Relaxation ◽  
Eulerian System ◽  
Time Blowup

<p style='text-indent:20px;'>We propose and study a one-dimensional <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic <b>critical thresholds</b> for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.</p>

scholarly journals Critical thresholds in one-dimensional damped Euler–Poisson systems

2020 ◽  
Vol 30 (05) ◽  
pp. 891-916 ◽  
Author(s):  
Manas Bhatnagar ◽  
Hailiang Liu
Keyword(s):  
Initial Data ◽  
Local System ◽  
Critical Threshold ◽  
Phase Plane Analysis ◽  
Threshold Region ◽  
One Dimensional ◽  
Critical Thresholds ◽  
Critical Curves ◽  
Plane Analysis ◽  
Threshold Phenomenon

This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler–Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J. 50 (2001) 109–157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

scholarly journals On the Finite-Time Blowup of a One-Dimensional Model for the Three-Dimensional Axisymmetric Euler Equations

2017 ◽  
Vol 70 (11) ◽  
pp. 2218-2243 ◽  
Author(s):  
Kyudong Choi ◽  
Thomas Y. Hou ◽  
Alexander Kiselev ◽  
Guo Luo ◽  
Vladimir Sverak ◽  
...  
Keyword(s):  
Euler Equations ◽  
Finite Time ◽  
Three Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Finite Time Blowup ◽  
Time Blowup

scholarly journals Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Zaihong Jiang ◽  
Sevdzhan Hakkaev
Keyword(s):  
Initial Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Propagation Speed ◽  
One Dimensional ◽  
General Family ◽  
Infinite Propagation Speed

We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solutionu(t,x)with compactly supported initial datumu0(x)does not have compactx-support any longer in its lifespan.

Fractal-like singularities for an inviscid one-dimensional model of fluid jets

2000 ◽  
Vol 11 (1) ◽  
pp. 29-60 ◽  
Author(s):  
M. A. FONTELOS ◽  
J. J. L. VELÁZQUEZ
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Inviscid Fluid ◽  
Dimensional Model ◽  
Small Perturbations ◽  
One Dimensional ◽  
Fluid Jets ◽  
Time Singularities ◽  
Extreme Sensitivity

In this paper we describe solutions of a one-dimensional model of inviscid fluid jets that develop finite time singularities in a fractal-like manner. We also discuss the extreme sensitivity of the solutions of this problem with respect to small perturbations of the initial data.

scholarly journals Prioritizing conservation actions in urbanizing landscapes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
A. K. Ettinger ◽  
E. R. Buhle ◽  
B. E. Feist ◽  
E. Howe ◽  
J. A. Spromberg ◽  
...  
Keyword(s):  
Coho Salmon ◽  
Predictive Accuracy ◽  
Critical Threshold ◽  
Negative Effects ◽  
Conservation Action ◽  
Critical Thresholds ◽  
Species Response ◽  
Response Data ◽  
Conservation Actions ◽  
Preservation And Restoration

AbstractUrbanization-driven landscape changes are harmful to many species. Negative effects can be mitigated through habitat preservation and restoration, but it is often difficult to prioritize these conservation actions. This is due, in part, to the scarcity of species response data, which limit the predictive accuracy of modeling to estimate critical thresholds for biological decline and recovery. To address these challenges, we quantify effort required for restoration, in combination with a clear conservation objective and associated metric (e.g., habitat for focal organisms). We develop and apply this framework to coho salmon (Oncorhynchus kisutch), a highly migratory and culturally iconic species in western North America that is particularly sensitive to urbanization. We examine how uncertainty in biological parameters may alter locations prioritized for conservation action and compare this to the effect of shifting to a different conservation metric (e.g., a different focal salmon species). Our approach prioritized suburban areas (those with intermediate urbanization effects) for preservation and restoration action to benefit coho. We found that prioritization was most sensitive to the selected metric, rather than the level of uncertainty or critical threshold values. Our analyses highlight the importance of identifying metrics that are well-aligned with intended outcomes.

EXPLOSIVE INSTABILITY DUE TO 3-WAVE OR 4-WAVE MIXING, WITH OR WITHOUT DISSIPATION

2008 ◽  
Vol 06 (04) ◽  
pp. 413-428 ◽  
Author(s):  
HARVEY SEGUR
Keyword(s):  
Initial Data ◽  
Nonlinear Waves ◽  
Finite Time ◽  
Blow Up ◽  
Wave Mixing ◽  
Explosive Instability ◽  
Effective Threshold ◽  
Blowing Up ◽  
Gain Energy

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

scholarly journals Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations

Nonlinearity ◽  
2003 ◽  
Vol 16 (6) ◽  
pp. 1967-1989 ◽  
Author(s):  
Hamid Bellout ◽  
Said Benachour ◽  
Edriss S Titi
Keyword(s):  
Finite Time ◽  
Global Regularity ◽  
Time Singularity ◽  
Finite Time Singularity
Sign in / Sign up

Export Citation Format

Share Document