Large-Amplitude Steady Downstream Water Waves

We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling gravity waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.