Existence and uniqueness for arbitrary finite-energy initial data

Establish existence and uniqueness of weak solutions to the two-dimensional anisotropic incompressible Navier–Stokes equations with arbitrary solenoidal initial data in $L^2(\mathbb{R}^2)$.

Background

The paper studies the anisotropic incompressible Navier–Stokes system on $\mathbb{R}^2$, in which diffusion acts only in the $x_1$ direction. Its natural weak-solution class consists of velocities in $L^\infty(0,T;L^2(\mathbb{R}^2))$ with $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$.

The authors explain that, for arbitrary solenoidal initial data $u_0\in L^2(\mathbb{R}^2)$, neither existence nor uniqueness is known. Existing well-posedness results cited in the paper require additional vertical regularity, such as $\partial_{x_2}u_0\in L^2(\mathbb{R}^2)$ or the stronger condition $u_0\in H^1(\mathbb{R}^2)$. The paper proves energy equality for every weak solution and a weak–strong uniqueness result when one solution has additional regularity, but it does not resolve existence or unconditional uniqueness for arbitrary $L^2$ data.