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

Geng Chen|arXiv (Cornell University)|Aug 18, 2015

Navier-Stokes equation solutions13 references3 citations

TL;DR

This paper establishes optimal time-dependent lower bounds on density for classical solutions of the 1D compressible Euler equations under general initial data. Using characteristic analysis and energy-type estimates, it proves that for isentropic flows, density decays no faster than $O((1+t)^{-1})$, and for full Euler equations, it decays as $O((1+t)^{-1-\delta})$ for any $\delta > 0$, both bounds being sharp or nearly optimal, with uniform control on velocity gradients until shock formation.

ABSTRACT

For the compressible Euler equations, even when the initial data are uniformly away from vacuum, solution 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 initial density has positive lower bound, we prove that density functions in classical solutions have positive lower bounds in the order of $ extstyle O(1+t)^{-1}$ and $ extstyle O(1+t)^{-1-δ}$ for any $ extstyle 0

Motivation & Objective

• To establish sharp time-dependent lower bounds on density for classical solutions of the 1D compressible Euler equations.
• To resolve the long-standing challenge of quantifying how fast density can decay toward vacuum in the absence of initial vacuum.
• To extend known results from isentropic to full Euler equations, providing optimal decay rates.
• To prove that the decay rate of $O((1+t)^{-1})$ for isentropic flows and $O((1+t)^{-1-\delta})$ for full flows are optimal or nearly optimal.
• To show that velocity gradients remain uniformly bounded from above until shock formation, despite blowup in finite time.

Proposed method

• Adopting Lagrangian coordinates and analyzing the $p$-system for isentropic flows and the full Euler system with entropy conservation.
• Introducing modified characteristic variables $\alpha_\varepsilon$ and $\beta_\varepsilon$ to track wave interactions and control growth.
• Using a contradiction argument based on characteristic triangles to bound the growth of $\alpha_\varepsilon$ and $\beta_\varepsilon$.
• Applying energy-type estimates and Riccati-type differential inequalities along forward characteristics to control the evolution of $\alpha_\varepsilon$.
• Establishing uniform bounds on $\alpha_\varepsilon$ and $\beta_\varepsilon$ via careful selection of parameters $\varepsilon$ and $N$ to prevent blowup.
• Deriving the time decay of density from the boundedness of $\alpha_\varepsilon$ and $\beta_\varepsilon$, leveraging mass conservation and initial positive density bounds.

Experimental results

Research questions

• RQ1 What is the optimal time-dependent lower bound on density for classical solutions of the 1D compressible Euler equations with initial data uniformly bounded away from vacuum?
• RQ2 Can the decay rate of density be improved beyond previously known bounds, particularly for the full Euler system?
• RQ3 How does the velocity gradient behave as shock formation approaches in classical solutions?
• RQ4 Is the $O((1+t)^{-1})$ decay rate for isentropic flows optimal, and can it be extended to the full Euler system?
• RQ5 Can uniform upper bounds on velocity gradients be maintained up to shock formation, despite the eventual blowup of derivatives?

Key findings

• For isentropic Euler equations, the density in classical solutions has a lower bound of order $O((1+t)^{-1})$, which is optimal.
• For the full Euler equations, the density has a lower bound of order $O((1+t)^{-1-\delta})$ for any $\delta > 0$, which is nearly optimal.
• The time decay rates are sharp in the sense that no faster polynomial decay is possible under the given assumptions.
• Velocity gradients $u_y$ remain uniformly bounded from above by a constant independent of the life span $T$ before shock formation.
• The blowup of $u_y$ occurs only at the moment of shock formation, consistent with finite-time gradient blowup.
• The proof technique via characteristic analysis and boundedness of modified variables $\alpha_\varepsilon$, $\beta_\varepsilon$ provides a robust framework for handling general large data.