The no-slip inviscid limit in two dimensions
Let $\Omega \subset \mathbb{R}^2$ be a bounded simply connected domain with smooth boundary and outer unit normal $n$, and let $u_0 \in C_c^\infty(\Omega; \mathbb{R}^2)$ be divergence-free. For $\nu > 0$ let $u^\nu$ solve the incompressible Navier–Stokes equations in $\Omega$ with viscosity $\nu$, no forcing, initial value $u_0$ and the no-slip condition $u^\nu = 0$ on $\partial\Omega$, and let $u^E$ solve the incompressible Euler equations with the same initial value and $u^E \cdot n = 0$ on $\partial\Omega$. Kato proved that $u^\nu \to u^E$ in $L^\infty(0, T; L^2(\Omega))$ if and only if the energy dissipation $\nu \int_0^T \int_{\Gamma_{c\nu}} |\nabla u^\nu|^2 \, dx \, dt$ in a boundary strip $\Gamma_{c\nu}$ of width proportional to $\nu$ tends to zero. Sammartino and Caflisch proved the inviscid limit for a short time for analytic data in a half-space, and Maekawa proved local-in-time convergence in the half-plane when the initial vorticity is located away from the boundary.
Problem. Is $$\lim_{\nu \downarrow 0} \sup_{0 \le t \le T} \|u^\nu(t) - u^E(t)\|_{L^2(\Omega)} = 0$$ for every $T < \infty$ and every such $\Omega$ and $u_0$? The datum does not depend on $\nu$, and a counterexample within these hypotheses also answers the question.
Source: A classical question of boundary-layer theory; see Kato (1984) and Maekawa and Mazzucato (2018). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, boundary layers, inviscid limits, Navier–Stokes equations
References
@incollection{kato1984,
author = {Kato, Tosio},
title = {Remarks on zero viscosity limit for nonstationary {N}avier--{S}tokes flows with boundary},
booktitle = {Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif., 1983)},
series = {Math. Sci. Res. Inst. Publ.},
volume = {2},
publisher = {Springer},
address = {New York},
year = {1984},
pages = {85--98},
doi = {10.1007/978-1-4612-1110-5_6},
}
@incollection{maekawa2018,
author = {Maekawa, Yasunori and Mazzucato, Anna},
title = {The inviscid limit and boundary layers for {N}avier--{S}tokes flows},
booktitle = {Handbook of Mathematical Analysis in Mechanics of Viscous Fluids},
editor = {Giga, Yoshikazu and Novotn{\'y}, Anton{\'i}n},
publisher = {Springer},
address = {Cham},
year = {2018},
pages = {781--828},
doi = {10.1007/978-3-319-13344-7_15},
eprint = {1610.05372},
archivePrefix = {arXiv},
}
@article{sammartino1998a,
author = {Sammartino, Marco and Caflisch, Russel E.},
title = {Zero viscosity limit for analytic solutions of the {N}avier--{S}tokes equation on a half-space. {I}. {E}xistence for {E}uler and {P}randtl equations},
journal = {Comm. Math. Phys.},
volume = {192},
number = {2},
year = {1998},
pages = {433--461},
doi = {10.1007/s002200050304},
}
@article{sammartino1998b,
author = {Sammartino, Marco and Caflisch, Russel E.},
title = {Zero viscosity limit for analytic solutions of the {N}avier--{S}tokes equation on a half-space. {II}. {C}onstruction of the {N}avier--{S}tokes solution},
journal = {Comm. Math. Phys.},
volume = {192},
number = {2},
year = {1998},
pages = {463--491},
doi = {10.1007/s002200050305},
}
@article{maekawa2014,
author = {Maekawa, Yasunori},
title = {On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane},
journal = {Comm. Pure Appl. Math.},
volume = {67},
number = {7},
year = {2014},
pages = {1045--1128},
doi = {10.1002/cpa.21516},
}
@misc{aim-no-slip-inviscid-limit,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The no-slip inviscid limit in two dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/075-no-slip-inviscid-limit.md},
note = {Entry 075 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00620, "The no-slip inviscid limit in two dimensions", clopen.solutions.