Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

Abstract

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in R N. For time t ≥ 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = Kρ γ, where ρ is the density function, K is a constant, and γ > 1, we can show that the nontrivial C 1 solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include r N−1ln(r + 1), r N−1 e r, r N−1(r 3 − 3r 2 + 3r + ε), r N−1sin((π/2)(r/R)), and r N−1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

Blowup for the compressible isothermal Euler equations with non-vacuum initial data

“…In 2011, Yuen obtained the initial functional conditions for the blowup of the Euler-Poisson equations for testing functions f (r) = r n (with n = 1 in [4] and an arbitrary positive constant n in [5]). Subsequently, the authors in [9] designed general testing functions to obtain the initial functional conditions for showing the blowup phenomena of the Euler and Euler-Poisson equations using the integration method under the nonslip boundary condition [3]. Recently, the authors in [7] obtained improved blowup results for the Euler and Euler-Poisson equations with repulsive forces based on [4].…”

Blowup of regular solutions for the relativistic Euler–Poisson equations

“…Recently, Geng [1] used the integration method described in [18] to obtain a blowup result for the regular solutions of relativistic Euler and Euler-Poisson equations. Note that in [16,17], the authors generalized the testing function in the integration method to any strictly increasing function. By combining the method in [16,17] with the results of Geng [1], we obtain the following theorem, which is our main contribution.…”

“…Note that in [16,17], the authors generalized the testing function in the integration method to any strictly increasing function. By combining the method in [16,17] with the results of Geng [1], we obtain the following theorem, which is our main contribution. Moreover, we remove the condition that |v| ≥ c/2 in [1].…”

Stabilities for Nonisentropic Euler‐Poisson Equations

“…When δ = 1, the system is self-attractive. The system ( 1 ) is the Newtonian description of gaseous stars [ 1 ]. When δ = −1, the system comprises the Euler-Poisson equations with repulsive forces and can be used as a semiconductor model [ 2 , 3 ].…”

“…When δ = 0, the system comprises the compressible Euler equations and can be applied as a classical model in fluid mechanics [ 3 ]. For more classical and recent results in these systems, readers can refer to [ 1 , 4 – 10 ].…”