Weak Solutions of the Navier-Stokes Equation and their Euler limits

Authors

• James Glimm, State University of New York at Stony BrookFollow

Document Type

Article

Publication Date

2026

Keywords

Euler Equation, Navier-Stokes Equation, entropy, spherical harmonics

Abstract

The existence of a weak solution of the incompressible isothermal Navier-Stokes equation with given initial conditions in the Sobolev space $\mathcal{H}_{-2}$ for energy fluctuations and in $\mathcal{H}_{-3}$ for enstrophy fluctuations is assumed. The existence is uniform with respect to the Euler limit of zero viscosity $\nu$. Thus, existence of weak solutions of the Euler equation with given initial conditions is established, and these Euler solutions are the zero viscosity limit of Navier-Stokes solutions, provided the Navier-Stokes solutions exist.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Recommended Citation

Glimm, James, "Weak Solutions of the Navier-Stokes Equation and their Euler limits" (2026). Department of Applied Mathematics & Statistics Faculty Publications. 11.
https://commons.library.stonybrook.edu/ams-articles/11