[Paper Review] Remarks on Blow-up of Smooth Solutions to the Compressible Fluid with Constant and Degenerate Viscosities

Quansen Jiu, Yuexun Wang+1 more|arXiv (Cornell University)|Oct 12, 2013

Navier-Stokes equation solutions34 references3 citations

TL;DR

This paper establishes the blow-up of smooth solutions to the compressible Navier-Stokes and Euler equations with constant and degenerate viscosities in arbitrary dimensions, under mild initial data conditions. Using a unified energy-type method and refined differential inequalities, it proves finite-time blow-up occurs without requiring compactly supported or vacuum-initial data, and provides precise blow-up time estimates for both full and isentropic models.

ABSTRACT

In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplified and unified proof on the blow-up results to the classical solutions of the full compressible Navier-Stokes equations without heat conduction by Xin \cite{Xin} and with heat conduction by Cho-Bin \cite{CJ} will be given.

Motivation & Objective

• To establish the blow-up of classical solutions to the compressible Navier-Stokes and Euler equations with constant and degenerate viscosities in arbitrary spatial dimensions.
• To remove restrictive assumptions such as compactly supported initial data or presence of vacuum in finite regions.
• To provide a simplified and unified proof for blow-up results in the full compressible Navier-Stokes equations with and without heat conduction.
• To derive precise estimates for the blow-up time in both isentropic and non-isentropic cases.
• To extend the analysis to models with density-dependent viscosities, including the shallow water (Saint-Venant) system.

Proposed method

• Derives a potential energy functional $ I(t) $ and its time evolution via energy estimates and differential inequalities.
• Applies a modified energy method using the function $ IJ(t) = I(t) \cdot J(t) $, where $ J(t) $ is a weighted $ L^2 $-norm of velocity.
• Implements a critical differential inequality of the form $ \frac{d}{dt}IJ(t) \leq C_1(t) IJ(t) + C_2(t) IJ(t)^{\frac{\alpha-1}{\gamma-1}} $, with time-dependent coefficients.
• Uses Gronwall-type inequalities with time-weighted exponential terms to control the growth of $ IJ(t) $, leading to blow-up when the coefficient becomes negative.
• Applies the condition $ \gamma < 1 + \frac{2}{n} $ and $ \frac{\gamma+1}{2} < \alpha \leq \gamma $ to ensure the decay of the coefficient in the differential inequality.
• Establishes blow-up via contradiction by showing $ I(t) \to 0 $ too rapidly unless the solution becomes singular.

Experimental results

Research questions

• RQ1 Under what conditions do smooth solutions to the compressible Navier-Stokes equations with constant and degenerate viscosities blow up in finite time?
• RQ2 Can blow-up be proven without assuming compactly supported initial data or the presence of vacuum?
• RQ3 How can the blow-up time be estimated more precisely in the isentropic and full compressible models?
• RQ4 Can a unified proof be constructed for blow-up results in the full compressible Navier-Stokes equations with and without heat conduction?
• RQ5 What role does the choice of viscosity law (constant vs. density-dependent) play in the blow-up behavior of smooth solutions?

Key findings

• Blow-up of smooth solutions occurs for the full compressible Navier-Stokes equations with constant viscosities and heat conduction, even without compactly supported or vacuum-initial data.
• For the isentropic case with $ \gamma < 1 + \frac{2}{n} $, the solution blows up in finite time under mild initial data conditions.
• The blow-up time is estimated via $ I(t) \leq C (1+t)^{1-\gamma} \exp\left(-C_{30}/(1+t)^{C_{28}}\right) $ in 1D and $ I(t) \leq C (1+t)^{-n(\gamma-1)} \exp\left(-C_{27}/(1+t)^{C_{28}}\right) $ in $ n \geq 2 $, showing rapid decay implying singularity.
• The method provides a simplified and unified proof for blow-up results in both the full compressible Navier-Stokes equations with and without heat conduction.
• The blow-up result extends to the compressible Euler equations and isentropic Euler equations, with explicit blow-up time estimates.
• The analysis covers the degenerate viscosity case with $ h(\rho) = \rho^\alpha $, $ g(\rho) = (\alpha-1)\rho^\alpha $, $ \alpha > 1 - \frac{1}{n} $, and includes the shallow water model as a special case.