Ill-posedness of the Maxwell–Dirac system below charge in space dimension three and lower

The Maxwell–Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension $$d=3$$ d = 3 the system is charge-critical, that is, $$L^2$$ L 2 -critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions $$d=1,2$$ d = 1 , 2 , global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if $$d=3$$ d = 3 ), by demonstrating ill-posedness below the charge regularity. In fact, for $$d \le 3$$ d ≤ 3 we exhibit a spinor datum belonging to $$H^s(\mathbb {R}^d)$$ H s ( R d ) for $$s<0$$ s < 0 , and to $$L^p(\mathbb {R}^d)$$ L p ( R d ) for $$1 \le p < 2$$ 1 ≤ p < 2 , but not to $$L^2(\mathbb {R}^d)$$ L 2 ( R d ) , which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.

The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

The Zakharov–Kuznetsov equation in spatial dimension $$d\ge 5$$ d ≥ 5 is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces, and it is proved that solutions scatter to free solutions as $$t \rightarrow \pm \infty $$ t → ± ∞ . The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $$(d-1)$$ ( d - 1 ) -dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $$d=4$$ d = 4 .

Approximation of critical regularity functions on stratified homogeneous groups

Let [Formula: see text] be a stratified homogeneous group with homogeneous dimension [Formula: see text] and whose Lie algebra is generated by the left-invariant vector fields [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text]. We prove that for any function [Formula: see text] there exists a function [Formula: see text] such that [Formula: see text] [Formula: see text] where [Formula: see text] is the largest integer smaller than [Formula: see text] and [Formula: see text] is a positive constant depending only on [Formula: see text]. Here, [Formula: see text] is a homogeneous Triebel–Lizorkin type space adapted to [Formula: see text]. This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9(2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal. 276(5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.