Global-in-time inviscid limit of the Navier-Stokes equations for vortex-wave data

Problem Statement
This paper addresses the inviscid limit problem for the two-dimensional Navier–Stokes equations on a prescribed finite time interval $[0, T]$. Specifically, it investigates the convergence of solutions when the initial vorticity consists of a singular point vortex (a Dirac mass) interacting with a smooth, compactly supported background vorticity that vanishes in a neighborhood of the vortex. The goal is to rigorously justify the vortex-wave system, introduced by Marchioro and Pulvirenti, as the vanishing-viscosity limit of the Navier–Stokes equations for this class of "vortex-wave" initial data.

Previous results on the inviscid limit typically required regular initial data or imposed restrictive time intervals dependent on viscosity (e.g., short-time existence or conditions avoiding the support of the ambient wave). This work aims to provide a global-in-time justification (where the time interval $T$ is arbitrary but fixed, and constants may depend on $T$) without such stopping-time restrictions.

Methodology
The author employs a decomposition of the viscous vorticity $\omega_\nu$ into a regular component $\omega_{E,\nu}$ and a concentrated component $\omega_{B,\nu}$. The proof strategy involves constructing precise approximate solutions for both components and deriving quantitative error estimates that close a coupled bootstrap argument.

1. Regular Component Approximation:

   • The regular component is approximated by the solution to the vortex-wave system plus a corrector term.
   • The author constructs a "regular corrector" $\nu w_{1,a}$ by solving a linear transport equation driven by the difference between the viscous and inviscid velocity fields.
   • A key technical step involves localizing the velocity fields near the support of the background vorticity to handle the singularity of the point vortex. This allows for the construction of the corrector before proving that its support remains separated from the vortex trajectory.
   • The author proves that the support of the regular component remains separated from the point vortex trajectory $Z_\nu(t)$ throughout the interval $[0, T]$, provided the initial background vanishes near the vortex.

2. Concentrated Component Approximation:

   • The concentrated component is approximated by a Lamb–Oseen vortex centered at a modulated trajectory $Z_\nu(t)$, rather than the limiting trajectory $z(t)$.
   • The analysis utilizes self-similar variables $\xi = (x - Z_\nu(t))/\sqrt{\nu t}$ to resolve the core structure.
   • The author performs a multipole expansion of the ambient velocity (generated by the regular vorticity) around the vortex center. This expansion isolates the symmetric strain tensor $\Sigma(t)$ and higher-order terms.
   • Profile Construction: They construct a hierarchy of correction profiles ($F_{\neq 0}$, $F_{rad}$, $H$, $J$) to cancel the leading-order errors in the expansion. These profiles solve linearized equations involving the Oseen operator and the Biot–Savart law.
   • Strain Cancellation: A critical innovation is the use of weighted energy estimates with a specific radial weight $p_0$ and an angular correction $q$. The angular correction is designed to satisfy a cell equation that cancels the leading-order contribution of the ambient strain $\Sigma(t)$ in the energy identity. This technique, inspired by Gallay and Donati, removes the need for smallness conditions on the strain or short-time restrictions.

3. Coupled Error Estimates:

   • The errors in the regular and concentrated components are coupled: the regular error acts as a forcing term for the concentrated core, and the concentrated core's deformation affects the regular transport.
   • The author defines a combined energy functional $Y_*(t)$ involving the core energy and the accumulated dissipation.
   • By deriving differential inequalities for the regular remainder (in $L^4 \cap L^{4/3}$) and the concentrated remainder (in a weighted $L^2$ space), they close the estimates using a continuity argument. This yields bounds of order $O(\nu t)$ for the regular component and $O(\sqrt{\nu t})$ for the concentrated component.

Key Contributions and Results

• Global-in-Time Justification: The paper proves that for any finite time $T > 0$, the Navier–Stokes solution with initial data $\delta_{z_0} + \omega_0^E$ converges to the vortex-wave system solution as viscosity $\nu \to 0$. The constants in the estimates depend on $T$ but not on the viscosity in a way that restricts the time interval.
• Quantitative Estimates:
  • The regular vorticity $\omega_{E,\nu}$ converges strongly to the regular wave $\omega_E$ in $L^4 \cap L^{4/3}$ with an error bound of $O_T(\nu)$.
  • The concentrated vorticity $\omega_{B,\nu}$ converges to a Lamb–Oseen vortex $\Gamma_{\nu t, Z_\nu(t)}$ centered at a modulated trajectory $Z_\nu(t)$ with an $L^1$ error of $O_T(\nu t)$.
  • The modulated trajectory $Z_\nu(t)$ stays close to the limiting trajectory $z(t)$, with $|Z_\nu(t) - z(t)| \lesssim_T \nu t$.
  • Consequently, the concentrated vorticity converges to a Dirac mass $\delta_{z(t)}$ with an $L^1$ error of $O_T(\sqrt{\nu t})$.
• Removal of Stopping-Time Restrictions: Unlike previous works (e.g., Nguyen and Nguyen [29]) that required the vortex to avoid regions swept by earlier wave supports or imposed smallness conditions on the ambient strain, this result holds for the entire interval $[0, T]$ by controlling the cumulative deformation of the core via the strain-cancelling weighted estimates.
• Separation Persistence: The paper rigorously establishes that if the initial background vorticity vanishes near the vortex, the support of the regular component remains separated from the vortex trajectory for all $t \in [0, T]$, ensuring the smoothness of the ambient velocity near the core.

Significance
The paper provides a rigorous mathematical foundation for the vortex-wave system as the correct inviscid limit model for flows containing both concentrated vortices and smooth backgrounds. By extending the validity of this approximation to arbitrary finite time intervals and removing restrictive smallness assumptions on the strain, the work resolves a long-standing technical barrier in the theory of concentrated vortices. The methodology, particularly the strain-cancelling weighted energy estimates, offers a robust framework for analyzing the stability and deformation of viscous vortices in non-trivial external flows, bridging the gap between the singular point-vortex dynamics and the diffusive Lamb–Oseen profiles.