[Paper Review] Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

Paolo Antonelli, Michele Dolce+1 more|arXiv (Cornell University)|Mar 3, 2020

Navier-Stokes equation solutions63 references4 citations

TL;DR

This paper establishes linear stability for 2D isentropic compressible Euler flows near Couette shear flow by analyzing the compressible and incompressible components separately. It proves superlinear-in-time growth for the compressible part and inviscid damping with slower rates for the incompressible part, using weighted energy estimates and a functional relation between vorticity and density in the absence of the standard conservation law found in the pure Couette case.

ABSTRACT

In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain $\mathbb{T} imes \mathbb{R}$. We begin by directly investigating the Couette shear flow, where we characterize the linear growth of the compressible part of the fluid while proving time decay for the incompressible part (inviscid damping with slower rates). Then we extend the analysis to monotone shear flows near Couette, where we are able to give an upper bound, superlinear in time, for the compressible part of the fluid. The incompressible part enjoys an inviscid damping property, analogous to the Couette case. In the pure Couette case, we exploit the presence of an additional conservation law (which connects the vorticity and the density on the moving frame) in order to reduce the number of degrees of freedom of the system. The result then follows by using weighted energy estimates. In the general case, unfortunately, this conservation law no longer holds. Therefore we define a suitable weighted energy functional for the whole system, which can be used to estimate the irrotational component of the velocity but does not provide sharp bounds on the solenoidal component. However, even in the absence of the aforementioned additional conservation law, we are still able to show the existence of a functional relation which allows us to recover somehow the vorticity from the density, on the moving frame. By combining the weighted energy estimates with the functional relation we also recover the inviscid damping for the solenoidal component of the velocity.

Motivation & Objective

• To analyze linear stability of 2D isentropic compressible Euler equations near monotone shear flows close to Couette flow.
• To understand the behavior of compressible and incompressible components under linearization around such shear flows.
• To overcome the absence of a conservation law for vorticity and density in general shear flows by constructing a functional relation.
• To establish weighted energy estimates that yield superlinear-in-time growth for the compressible part and inviscid damping for the incompressible part.
• To extend known results from the pure Couette case to nearby monotone shear flows with general velocity profiles.

Proposed method

• Linearize the isentropic compressible Euler equations around a monotone shear flow close to Couette, with constant density and velocity profile $ U(y) = y + b(y) $, where $ b $ is small.
• Decompose the velocity into irrotational and solenoidal components to separately analyze compressible (irrotational) and incompressible (solenoidal) parts.
• Use weighted energy estimates with time-dependent weights to control the growth of the compressible component, especially in the absence of the vorticity-density conservation law.
• Define a novel weighted energy functional that captures the dynamics of the full system and allows control of the irrotational velocity component.
• Establish a functional relation between vorticity and density on the moving frame to recover control over the solenoidal component despite the lack of conservation law.
• Apply Neumann series expansions and operator norm estimates to control the resolvent operators arising in the linearized system, particularly $ \Delta_t^{-1} $ and related operators.

Experimental results

Research questions

• RQ1 What is the long-time behavior of the compressible part of the fluid in linearized 2D compressible Euler equations near Couette flow?
• RQ2 How does the absence of the vorticity-density conservation law in general shear flows affect the stability analysis compared to the pure Couette case?
• RQ3 Can inviscid damping still be established for the incompressible (solenoidal) component in shear flows near Couette without the standard conservation law?
• RQ4 What functional relation between vorticity and density on the moving frame enables recovery of control over the solenoidal component in general shear flows?
• RQ5 Can weighted energy estimates be adapted to yield superlinear-in-time growth bounds for the compressible part in non-Couette shear flows?

Key findings

• The compressible part of the fluid exhibits superlinear-in-time growth, bounded by $ O(t^{1+\epsilon}) $ for any $ \epsilon > 0 $, in shear flows near Couette.
• The incompressible (solenoidal) part displays inviscid damping with slower decay rates than in the Couette case, due to the absence of the standard conservation law.
• A functional relation between vorticity and density on the moving frame is derived, enabling control of the solenoidal component even without the conservation law.
• Weighted energy estimates are successfully applied to the full system, yielding control over the irrotational velocity component and allowing the recovery of inviscid damping for the solenoidal part.
• The analysis extends the classical Couette stability results to a broader class of monotone shear flows by constructing a new energy functional and using Neumann series expansions for resolvent operators.
• The proof relies on operator norm estimates and weighted $ L^2 $ bounds, with $ \|B\Phi_b f\|_{H^s} \leq \frac{1}{1 - C\epsilon} \|Bf\|_{H^s} $, ensuring stability under small perturbations.