[Paper Review] On the inviscid limit for 2D incompressible flow with Navier friction condition

Milton C. Lopes Filho, Helena J. Nussenzveig Lopes+1 more|ArXiv.org|Jul 22, 2003

Navier-Stokes equation solutions14 references21 citations

TL;DR

This paper extends the inviscid limit analysis of 2D incompressible Navier-Stokes flows with Navier friction boundary conditions to initial vorticities in $L^p(\Omega)$ for $p>2$, using a mollification-based approximation and $L^p$ vorticity estimates to establish strong convergence of viscous solutions to a weak solution of the Euler equations. The key contribution is the rigorous justification of the inviscid limit under less regular initial data than previously known.

ABSTRACT

In [1], T. Clopeau, A. Mikelić, and R. Robert studied the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include $p$-th power integrable initial vorticities, with $p>2$. [1] Clopeau, T., Mikelić, A., Robert, R., {\it On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions}, Nonlinearity {\bf 11} (1998) 1625--1636.

Motivation & Objective

• To extend the inviscid limit result for 2D incompressible Navier-Stokes equations with Navier friction boundary conditions to initial vorticities in $L^p(\Omega)$ for $p>2$, beyond the bounded vorticity case considered in prior work.
• To establish the existence and uniform boundedness of $L^p$-norms of vorticity for viscous solutions under the Navier friction condition.
• To prove the strong convergence of viscous solutions to a weak solution of the incompressible 2D Euler equations as viscosity vanishes, under $L^p$ initial vorticity with $p>2$.
• To address the critical regularity threshold for initial data in the context of the inviscid limit with Navier friction, challenging the assumed criticality of $p=2$.

Proposed method

• Mollify the initial velocity field $u_0$ to obtain a sequence of smooth, divergence-free, tangential initial data satisfying the Navier friction condition.
• Use the well-posedness result from Clopeau et al. for smooth initial data to construct a sequence of viscous solutions $u^ u_n$ with mollified initial data.
• Establish uniform $L^p$ bounds on the vorticity $\omega^n$ of the viscous solutions via energy estimates and the structure of the vorticity equation under Navier friction.
• Derive a priori estimates on the $L^p$-norm of vorticity that are independent of viscosity $\nu$ and mollification parameter $n$, relying on the $L^p$-boundedness of the Biot-Savart kernel.
• Pass to the limit in the viscous equations using weak convergence in $L^2((0,T);H^1(\Omega))$ and strong convergence in $C([0,T];L^2(\Omega))$ via compactness arguments.
• Conclude that the limit satisfies the weak formulation of the 2D Euler equations, thereby establishing the inviscid limit for $L^p$ initial vorticity with $p>2$.

Experimental results

Research questions

• RQ1 Can the inviscid limit of 2D incompressible Navier-Stokes flows with Navier friction boundary conditions be established for initial vorticities in $L^p(\Omega)$ with $p>2$, beyond the bounded vorticity case?
• RQ2 What uniform a priori estimates on the $L^p$-norm of vorticity are necessary and sufficient to pass to the inviscid limit under Navier friction?
• RQ3 Is the critical regularity threshold for initial data in the inviscid limit with Navier friction condition truly $p=2$, or can it be extended to $p>2$?
• RQ4 Does the viscous approximation with Navier friction preserve the $L^p$-norm of vorticity in the limit as $\nu \to 0$, and if so, under what conditions?
• RQ5 Can the vanishing viscosity limit be rigorously justified using mollified initial data and $L^p$-based compactness arguments for $p>2$?

Key findings

• The $L^p$-norm of the vorticity $\omega^\nu$ of the viscous solution remains uniformly bounded in time and viscosity for initial vorticity in $L^p(\Omega)$ with $p>2$, independent of $\nu$.
• A subsequence of viscous solutions $u^{\nu_k}$ converges strongly in $C([0,T];L^2(\Omega))$ as $\nu_k \to 0$, with the limit satisfying the weak formulation of the 2D Euler equations.
• The limit solution $u$ is a weak solution of the incompressible 2D Euler equations in the standard sense, satisfying the identity $\int_0^T \int_\Omega u\varphi_t + u(u\cdot\nabla)\varphi \, dxdt + \int_\Omega u_0 \varphi(\cdot,0) \, dx = 0$ for divergence-free, tangential test vector fields.
• The proof relies on uniform bounds in $L^\infty((0,T);W^{1,p}(\Omega))$ and $L^2((0,T);H^{-1}(\Omega))$ for the time derivative, enabling extraction of a convergent subsequence.
• The critical regularity threshold for initial data in the inviscid limit with Navier friction is not confirmed to be $p=2$, leaving the optimal $p$ as an open problem.
• The paper leaves open the question of whether the limit solution conserves the $L^p$-norm of vorticity, especially since vorticity can be generated at the boundary in the viscous regime.