Xiaojin Bai | Searcharxiv

arXiv2026

On well-posedness theory of very weak solutions to Navier-Stokes equations on irregular domains with nonhomogeneous Dirichlet boundary data

The well-posedness theory of very weak solutions is a central topic in mathematical hydrodynamics, especially in the regularity theory for Navier-Stokes equations. It has been fully developed for incompressible fluid flows on bounded domains in R^3 of C^{2,1}-regularity. In this paper, based on the analytic theories in [D. Breit and A. Gaudin, ArXiv Preprint: 2511.19091 (2025)] and [V.G. Maz'ya and T.O. Shaposhnikova, Vol.337, Grundlehren der mathematischen Wissenschaften (2009)], we establish the well-posedness theory of very weak solutions to the Navier-Stokes equations on bounded Lipschitz domains whose boundary has local graphing functions with sufficiently small Sobolev multiplier norm, which contain the bounded Lipschitz domains with sufficiently small Lipschitz constants as a special case.

math.AP↗

arXiv2026

A wedge product theorem of compensated compactness theory with critical exponents on Riemannian manifolds

We formulate and prove compensated compactness theorems concerning the limiting behaviour of wedge products of weakly convergent differential forms on closed Riemannian manifolds à la Robbin--Rogers--Temple [Trans. Amer. Math. Soc. 303 (1987), 609--618]. The case of critical regularity exponents is considered, which generalises the div-curl lemma in Briane--Casado-Díaz--Murat [J. Math. Pures Appl. 91 (2009), 476--494] for vectorfields, thus going beyond the regularity regime entailed by Hölder's inequality. Implications on the weak continuity of Gauss--Codazz--Ricci equations and $L^p$-extrinsic geometry of isometric immersions of Riemannian manifolds are discussed.

math.DG↗