Elliptic problems in Besov and Sobolev—Triebel—Lizorkin spaces of low regularity

Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev–Triebel–Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.

Global Existence and Temporal Decay of Large Solutions for the Poisson--Nernst--Planck Equations in Low Regularity Spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.

Corrigendum: Local well-posedness of the coupled Yang–Mills and Dirac system for low regularity data (2022 Nonlinearity 35 1)

An error in the proof of the main theorem is fixed.

KdV on an incoming tide

Given smooth step-like initial data V(0, x) on the real line, we show that the Korteweg–de Vries equation is globally well-posed for initial data u ( 0 , x ) ∈ V ( 0 , x ) + H − 1 ( R ) . The proof uses our general well-posedness result (2021 arXiv:2104.11346). As a prerequisite, we show that KdV is globally well-posed for H 3 ( R ) perturbations of step-like initial data. In the case V ≡ 0, we obtain a new proof of the Bona–Smith theorem (Bona and Smith 1975 Trans. R. Soc. A 278 555–601) using the low-regularity methods that established the sharp well-posedness of KdV in H −1 (Killip and Vişan 2019 Ann. Math. 190 249–305).

Local well-posedness of the coupled Yang–Mills and Dirac system for low regularity data

We consider the classical Yang–Mills system coupled with a Dirac equation in 3 + 1 dimensions. Using that most of the nonlinear terms fulfill a null condition we prove local well-posedness for data with minimal regularity assumptions. This problem for smooth data was solved forty years ago by Choquet-Bruhat and Christodoulou. Our result generalises a similar result for the Yang–Mills equation by Selberg and Tesfahun.