[Paper Review] On the vanishing viscosity limit of the isentropic Navier-Stokes system

Eduard Feireisl, Martina Hofmanová|arXiv (Cornell University)|May 7, 2019

Navier-Stokes equation solutions4 citations

TL;DR

This paper establishes that in the vanishing viscosity limit for the isentropic Navier-Stokes system on $\mathbb{R}^d$ ($d=2,3$), any weakly converging sequence of solutions either converges strongly in the energy norm or fails to converge weakly to a solution of the Euler system. The key result is derived by analyzing turbulent defect measures via a system of differential equations that admit only trivial solutions, distinguishing the compressible from the incompressible case.

ABSTRACT

We show that any weakly converging sequence of solutions to the isentropic Navier-Stokes system on the full physical space $R^d$, $d=2,3$, in the vanishing viscosity limit either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. The same result holds for any sequence of approximate solutions in the spirit of DiPerna and Majda. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the associated turbulent defect measures and showing that it only has a trivial solution.

Motivation & Objective

• To resolve the behavior of solutions to the isentropic Navier-Stokes system in the vanishing viscosity limit on the full space $\mathbb{R}^d$ for $d=2,3$.
• To determine whether weak limits of solutions can yield weak solutions of the associated Euler system.
• To contrast the compressible case with the incompressible case, where oscillatory approximate solutions may converge weakly to Euler solutions.
• To establish that such weak limits are either strongly convergent or not valid Euler solutions, using defect measure analysis.

Proposed method

• Formulate a system of differential equations governing the turbulent defect measures associated with weakly converging sequences of solutions.
• Analyze the structure of these defect measures to show they satisfy a closed system of equations under the Navier-Stokes dynamics.
• Prove that the only solution to this system of equations is the trivial solution, implying no nontrivial defect measures can persist.
• Apply this result to both exact solutions and approximate solutions in the DiPerna-Majda framework.
• Use energy estimates and weak convergence arguments to control the behavior of solutions as viscosity vanishes.
• Leverage the structure of the isentropic system to rule out oscillatory or concentration effects that could lead to nontrivial weak limits.

Experimental results

Research questions

• RQ1 Can weak limits of solutions to the isentropic Navier-Stokes system in the vanishing viscosity limit yield weak solutions of the Euler system?
• RQ2 What conditions must be satisfied for a sequence of solutions to converge weakly to a solution of the Euler system in the compressible regime?
• RQ3 How do turbulent defect measures behave in the vanishing viscosity limit for the isentropic Navier-Stokes equations?
• RQ4 Why does the compressible case differ fundamentally from the incompressible case in terms of weak convergence to Euler solutions?
• RQ5 Can the system of equations for defect measures admit nontrivial solutions under the isentropic Navier-Stokes dynamics?

Key findings

• Any weakly converging sequence of solutions to the isentropic Navier-Stokes system on $\mathbb{R}^d$ ($d=2,3$) in the vanishing viscosity limit either converges strongly in the energy norm or does not converge weakly to a solution of the Euler system.
• The system of differential equations satisfied by the turbulent defect measures admits only the trivial solution, ruling out nontrivial concentration or oscillation effects.
• This result holds not only for exact solutions but also for approximate solutions in the DiPerna-Majda sense.
• The analysis reveals a fundamental difference between the compressible and incompressible Navier-Stokes systems in the context of weak limits.
• The absence of nontrivial defect measures implies that weak limits cannot yield weak solutions of the Euler system unless strong convergence occurs.
• The key mechanism is the structure of the defect measure equations, which are shown to be solvable only in the trivial case.