[Paper Review] Development of singularities for the compressible Euler equations with external force in several dimensions

Olga Rozanova|ArXiv.org|Nov 30, 2004

Navier-Stokes equation solutions21 references3 citations

TL;DR

This paper establishes sufficient conditions for finite-time singularity formation in solutions to the compressible Euler equations with external forces in multiple spatial dimensions, using energy and momentum estimates. It proves that even smooth initial data in a class with finite mass and energy develop singularities in finite time under these conditions, with applications to viscous flows, rotation, and damping, and identifies the 'best' sufficient condition in a precise quantitative sense.

ABSTRACT

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.

Motivation & Objective

• To establish sufficient conditions for finite-time singularity formation in smooth solutions of the compressible Euler equations with external forces in n-dimensional space.
• To generalize prior results on singularity formation for compactly supported perturbations of constant states to broader classes of initial data with finite mass and energy.
• To analyze the influence of external forces such as Coriolis force, frictional damping, and viscous terms on singularity development.
• To identify the 'best' sufficient condition for singularity formation, meaning that there exist globally smooth solutions violating this condition arbitrarily little.
• To extend the analysis to viscous compressible flows under decay conditions at infinity, showing singularity formation under the same sufficient conditions.

Proposed method

• Define a function class $\mathfrak{K}$ of solutions with finite mass, energy, and momentum, and derive evolution equations for key quantities like $G(t)$, $F(t)$, and $\Theta_2(t)$.
• Use energy and momentum estimates to derive differential inequalities involving $G(t)$, $F(t)$, and $\Theta_2(t)$, particularly through the auxiliary function $\mathcal{K}$.
• Introduce a modified energy functional $\mathcal{K} = \mathcal{M}^2 - l^2 G_+^2 + \delta$, where $\delta$ depends on $\gamma$ and $n$, to capture pressure and vorticity effects.
• Apply comparison theorems and integration of differential inequalities to show that $F(t)$ becomes unbounded in finite time if $\mathcal{K} > 0$ or $\mathcal{K} \leq 0$ and $F(0) > \sqrt{-\mathcal{K}}$.
• Use the fact that $F(t)$ represents a measure of gradient growth to conclude that unboundedness implies loss of smoothness.
• Construct explicit counterexamples to show that the sufficient condition is sharp in the sense that solutions exist globally if the condition is violated arbitrarily little.

Experimental results

Research questions

• RQ1 Under what conditions on initial data do solutions to the compressible Euler equations in $\mathbb{R}^n$ develop finite-time singularities, even when initially smooth?
• RQ2 How do external forces such as Coriolis force, friction, and viscous terms affect the formation of singularities in compressible flows?
• RQ3 Can the sufficient condition for singularity formation be considered 'best' in the sense that there exist smooth global solutions violating it arbitrarily little?
• RQ4 What role does the initial vorticity (interpreted via $F_{\perp}(0)$) play in singularity formation, particularly in meteorological contexts?
• RQ5 To what extent can the results be extended to viscous compressible flows with decay at infinity?

Key findings

• Solutions in the class $\mathfrak{K}$ (finite mass, energy, momentum) develop singularities in finite time if $\mathcal{K} > 0$ or $\mathcal{K} \leq 0$ and $F(0) > \sqrt{-\mathcal{K}}$, where $\mathcal{K}$ combines mass, momentum, and pressure terms.
• The condition $\mathcal{K} > 0$ is 'best' in the sense that for any $\epsilon > 0$, one can construct a globally smooth solution violating the condition by less than $\epsilon$.
• Finite-time singularity formation occurs even in viscous compressible flows under decay conditions at infinity, provided the sufficient condition on $\mathcal{K}$ holds.
• The presence of Coriolis force or rotation modifies the singularity condition, and the condition remains sharp even for non-constant Coriolis parameters close to constant.
• The initial vorticity $F_{\perp}(0)$ must be negative (cyclonic) for the condition to be satisfied in the limit of small $l$, suggesting a link to atmospheric frontogenesis in extratropical cyclones.
• The time of singularity formation can be estimated from above using the differential inequality $F'(t) \geq \frac{F^2(t) + \mathcal{K}}{2G_+}$, which leads to blow-up in finite time when $\mathcal{K} > 0$.