[Paper Review] Lower bound of density for Lipschitz continuous solutions in the isentropic gas dynamics

Geng Chen, Ronghua Pan+1 more|arXiv (Cornell University)|Oct 13, 2014

Navier-Stokes equation solutions19 references7 citations

TL;DR

This paper establishes a sharp $O(1+t)^{-1}$ lower bound for density in Lipschitz continuous solutions of the isentropic Euler equations (p-system) in one dimension, under the condition $1 < \gamma < 3$. Using a polygonal scheme approximation and analysis of Riemann invariants, the authors prove that even with compression, density cannot decay faster than $O(1+t)^{-1}$, improving prior estimates and confirming the optimality of known examples.

ABSTRACT

For the Euler equations of isentropic gas dynamics in one space dimension, also knowns as p-system in Lagrangian coordinate, it is known that the density can be arbitrarily close to zero as time goes to infinity, even when initial density is uniformly away from zero. In this paper, for uniform positive initial density, we prove the density in any Lipschitz continuous solutions for Cauchy problem has a sharp positive lower bound in the order of O(1/(1+t)), which is identified by explicit examples in [9](Courant and Friedrichs, Supersonic Flow and Shock Waves, 1948.).

Motivation & Objective

• To resolve the sharp time decay rate of density lower bounds in Lipschitz continuous solutions of the isentropic Euler equations.
• To address the challenge of vacuum formation and loss of strict hyperbolicity in large data solutions.
• To improve upon prior $O((1+t)^{-4/(3- u)})$ lower bounds by establishing the optimal $O((1+t)^{-1})$ decay rate.
• To confirm the sharpness of the $O(1+t)^{-1}$ decay observed in classical rarefaction wave examples.

Proposed method

• Application of a polygonal scheme approximation to the p-system, inspired by Dafermos and Diperna, to model wave interactions.
• Classification of local solution behavior into rarefaction and compression waves using Riemann invariants $s$ and $r$, and their spatial derivatives.
• Analysis of wave propagation across blocks in the $(x,t)$-plane, tracking changes in specific volume $v$ along characteristic paths.
• Use of the Riemann invariant evolution equations $s_t + c s_x = 0$, $r_t - c r_x = 0$ to derive estimates on $v$-changes.
• Derivation of a priori estimate $v(x,t) \leq \max v(x,0) + L t$ via induction over time steps, ensuring uniform control.
• Leveraging weak-strong uniqueness to connect the polygonal scheme limit to the true Lipschitz solution.

Experimental results

Research questions

• RQ1 What is the optimal time decay rate for the lower bound of density in Lipschitz continuous solutions of the isentropic Euler equations with $1 < \gamma < 3$?
• RQ2 Can the $O((1+t)^{-1})$ lower bound be rigorously proven for solutions including compressive regions, not just rarefaction waves?
• RQ3 How does the polygonal scheme approximation preserve the structure of wave interactions and enable uniform bounds on $v$?
• RQ4 Is the $O((1+t)^{-1})$ decay rate sharp, and does it match known explicit examples from Courant?
• RQ5 Can this bound be used to improve the lifespan estimate for classical solutions with initial compression?

Key findings

• The density $v(x,t)$ in any Lipschitz continuous solution of the isentropic Euler equations satisfies $v(x,t) \geq c_0 (1+t)^{-1}$ for some $c_0 > 0$, uniformly in $x$, when $1 < \gamma < 3$.
• The $O((1+t)^{-1})$ lower bound is sharp, as confirmed by explicit examples in Courant’s work for $\gamma = \frac{2N+1}{2N-1}$.
• The bound is established via a polygonal scheme approximation that preserves wave classification and allows uniform control over time.
• The result improves upon the prior $O((1+t)^{-4/(3-\gamma)})$ lower bound for smooth solutions, now valid for all Lipschitz solutions.
• The lifespan of classical solutions with initial compression is bounded above by a time that depends on the initial minimum of $y = \sqrt{c} s_x$ and $q = \sqrt{c} r_x$, with singularity forming not later than $t = \frac{1}{L}\left\{\left(-\frac{4K_0}{\gamma+1} \frac{1}{G_0} + H_0^{\frac{\gamma+1}{4}}\right)^{\frac{4}{\gamma+1}} - H_0\right\}$.
• The proof confirms that even in compressive regions, density cannot vanish faster than $O((1+t)^{-1})$, resolving a key issue in large data theory.