On the finite time blow-up of the Euler-Poisson equations in $\Bbb R^{2}$

Abstract

Abstract. We prove the finite time blow-up for C 1 solutions of the attractive Euler-Poisson equations in R n , n ≥ 1, with and without background state, for a large set of 'generic' initial data. We characterize this supercritical set by tracing the spectral dynamics of the deformation and vorticity tensors.Key words. Euler-Poisson equations, finite time blow-up AMS subject classifications. 35Q35, 35B30 The Euler-Poisson equationsWe are concerned with the pressureless Euler-Poisson equations in R n , n ≥ 1,The equations involve the unknown velocity field,, and the two constants, c and k. Here, c ≥ 0 is the constant "background" state; typical cases include the case of zero background, c = 0, or the case of a nonzero background given by the average mass, c =ρ, wherēFinally, k is a scaled physical constant which signifies whether the underlying forcing is attractive, when k < 0, or repulsive, when k > 0.The hyperbolic-elliptic system (1.1) appears in a variety of different applications, from small scale models in charge transport and plasma collision, e.g., [18,8], to large scale dynamics of (clusters of) stars in cosmological waves, and expansion of the cold ions, e.g., [1,7].For the questions of local regularity and global existence of weak solutions, we refer to [13,14,5] for local existence in the small H s -neighborhood of a steady state, and to [16,9] for the relaxation limit of the weak entropy solution in the isentropic and isothermal cases. Global existence with a "sufficient" amount of damping relaxation *

An improved local blow-up condition for Euler–Poisson equations with attractive forcing

“…They identify if-and-only-if, pointwise conditions for global existence of C 1 solutions to restricted Euler-Poisson systems. Chae and Tadmor [10] further extend the Critical Threshold argument to multi-D full Euler-Poisson systems (1.2a), (1.2b) with attractive forcing k < 0. Their result, however, offers a blow-up region When tracking other results on the well-posedness of Euler-Poisson equations, we find them commonly relying on (the vast family of) energy methods and thus fundamentally differ from our pointwise results obtained via the Lagrangian approach.…”

“…Both matrices are used to study the spectral dynamics of Euler systems (see e.g. [8] for M and [10] for S). The relation between the spectra of M and S is described in the following.…”

“…Phase plane of (4.1) with blow-up region Ω 1 ∪ Ω 2 which extends the Chae-Tadmor region[10] d ≤ d − .Here, ω is the n(n−1)2 vorticity vector which consists of the off-diagonal entries of A := 1 2 ∇u − (∇u) . Proof.…”

Blowup for the C1 solutions of the Euler–Poisson equations of gaseous stars in RN

“…The local existence for the systems can be found in [16,18,1,12]. The analysis of stabilities for the systems may be referred to [11,[19][20][21]9,10,23,3,7,25].…”

Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry

“…For the model without damping relaxation, global existence in the neighborhood of a steady state was obtained in [8]. On the other hand, finite time blowup results for attractive forces were obtained in [18], and for repulsive forces in [21,3,4]. Beyond the two scenarios of global existence of smooth solutions and finite-time breakdown, a third scenario of conditional regularity was promoted in [5,6,14,15], where it was shown that there exists a "large" set of O(1) initial configurations which lead to global smooth solutions, and the complementary "large" set of O(1) initial configurations which yield finite-time breakdown.…”