Critical Thresholds in 2D Restricted Euler-Poisson Equations

Upper-thresholds for Shock Formation in Two-dimensional Weakly Restricted 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 paper strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional weakly restricted Euler-Poisson (WREP) system. This system can be viewed as an im...

Thresholds in three-dimensional restricted Euler–Poisson equations

This work provides a description of the critical threshold phenomenon in multi-dimensional restricted Euler–Poisson (REP) equations, introduced in [H. Liu, E. Tadmor. Spectral dynamics of the velocity gradient field in restricted fluid flows, Comm. Math. Phys. 228 (2002) 435–466]. For three-dimensional REP equations, we identified both upper thresholds for the finite-time blow up of solutions a...

Spectral Dynamics of the Velocity Gradient Field in Restricted Flows

We study the velocity gradients of the fundamental Eulerian equation, ∂tu+ 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 λt + u · ∇λ+ λ2 = 〈l,∇Fr〉. We focus...

Critical Thresholds in Euler-Poisson Equations

Critical Thresholds in Multi-dimensional Restricted Euler Equations

Abstract. Using the spectral dynamics, we study the critical threshold phenomena in the multidimensional 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; i...