Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

Abstract

Abstract. For the compressible Euler equations, even when initial data are uniformly away from vacuum, solutions can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this paper, for the initial value problems of isentropic and full Euler equations in one space dimension, assuming the initial density has positive lower bound, we prove that density functions in classical solutions have positive lower bounds in the order of O(1 + t) −1 and O(1 + t) −1−δ for any 0 < δ ≪ 1, respectively, where t is time. The orders of these bounds are optimal or almost optimal, respectively. Furthermore, for classical solutions in Eulerian coordinates (y, t) ∈ R × [0, T ), we show velocity u satisfies that u y (y, t) is uniformly bounded from above by a constant independent of T , although u y (y, t) tends to negative infinity when gradient blowup happens, i.e. when shock forms, in finite time.2010 Mathematical Subject Classification: 76N15, 35L65, 35L67.

Monokinetic solutions to a singular Vlasov equation from a semiclassical perspective

“…It remains to prove that the density is bounded from below to complete the proof of Theorem 1.4. The result from [10] Proof. In [10], the proof is presented for the pressure law p(ρ) = ρ in (1.2) replaced by the isentropic one, p(ρ) = ρ γ , with γ > 1.…”

“…The result from [10] Proof. In [10], the proof is presented for the pressure law p(ρ) = ρ in (1.2) replaced by the isentropic one, p(ρ) = ρ γ , with γ > 1. We simply perform the standard modification to adapt the proof to the isothermal case (see e.g.…”

Singularity formation for one dimensional full Euler equations

“…Remark 2.2. In [3], G. Chen improved the density lower bound to the order of (1 + t) −1+δ for any small δ > 0. In order to keep the presentation simple, we did not adopt this new estimate here.…”

Optimal density lower bound on nonisentropic gas dynamics

“…Now we briefly introduce the old ideas in [2] and then the new idea in this paper. First, we recall two gradient variables used in earlier papers [1,2,4]…”

“…Note in[1,2,4], variables α and β mean α and β in this paper, respectively. In this paper, we reserve α and β for other use.…”