Upper-thresholds for shock formation in two-dimensional weakly restricted Euler–Poisson equations

Abstract

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 improved model of the restricted Euler-Poisson (REP) system introduced in [H. Liu and E. Tadmor, Comm. Math. Phys., 228:435-466, 2002]. We identify upper-thresholds for finite time blow up of solutions for WREP equations with attractive/repulsive forcing. It is shown that the thresholds depend on the size of the initial density relative to the initial velocity gradient through both trace and a nonlinear quantity.

Necessary conditions for blow-up solutions to the restricted Euler--Poisson equations

“…In this work, we attempt to advance our understanding of the critical threshold phenomenon by providing necessary conditions for the existence of finite-time blow-up solutions to the REP system (1.1). Our results thus provide a complement to the existing results in [15,16,13] for REP systems.…”

“…These results have been extended to multi-dimensional REP equations by Lee and Liu in [14]. While Lee identified upper-thresholds for finite time blow-up solutions to an improved REP equation in two dimensions ( [13]). It is worth mentioning that critical thresholds for restricted Euler equations were studied in [17] and [20].…”

On the Riccati dynamics of the Euler-Poisson equations with zero background state

“…However More discussions about the motivation of this study can be found in [8]. Furthermore, it turns out that many of so-called restricted/modified models [6,7,9,12] and the results under the vanishing initial vorticity assumption [2,3] can be reinterpreted using our proposed structure (2.6)-(2.7).…”

Wave breaking in a class of non-local conservation laws

“…That is, the global flux function F (u, ū) must be either concave up only or concave down only during a solution's life span. Essentially the same stipulation is inherent for the models in [4,25] and many blow-up analysis that use a Riccati-type equation e.g., [16,15,34]. This is because F uu serve as the leading coefficient function of the Riccati type equation and sign changes of F uu flip over the phase diagram of d := u x (setting aside other factors) which governed by…”