Josef Demmel | Searcharxiv

arXiv2026

Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equations

In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.

math.AP↗

arXiv2025

Strong Solutions and Applications to the Inviscid Limit for 2D Navier-Stokes with Navier Slip

We analyze the two-dimensional incompressible Navier-Stokes equations on a smooth, bounded, simply connected domain with Navier boundary conditions with friction coefficient $\alpha\in C^2$. For initial vorticity in $L^2$, we show that the unique weak solution for velocity is, in fact, strong and satisfies the Navier slip conditions for any positive time. The key idea is to consider a shifted vorticity, which vanishes on the boundary, and to study the Laplacian subject to Navier boundary conditions. We prove that this boundary-value problem is elliptic in the sense of Agmon-Douglis-Nirenberg. As an application of this scheme, we establish uniform-in-time strong convergence of the vorticity in the vanishing viscosity limit for initial vorticity in $L^p$ with $p>2$. We utilize a purely interior framework from Seis, Wiedemann, and Wo\'{z}nicki, originally derived for no-slip, and upgrade local to global convergence. Moreover, we show that the total bulk viscous enstrophy dissipation vanishes.

math.AP↗