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Bounded minimisers of the functional w ↦ ∫ ( | D w | p + a ( x ) | D w | q ) d x , where 0 ≦ a ( · ) ∈ C 0 , α and 1 < p < q , are C 1 , β -regular provided the sharp bound q ≦ p + α holds.

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

The power of any kind of network approach lies in the ability to simplify a complex system so that one can better understand its function as a whole. Sometimes it is beneficial, however, to include more information than in a simple graph of only nodes and links. Adding information about times of interactions can make predictions and mechanistic understanding more accurate. The drawback, however, is that there are not so many methods available, partly because temporal networks is a relatively young field, partly because it is more difficult to develop such methods compared to for static networks. In this colloquium, we review the methods to analyze and model temporal networks and processes taking place on them, focusing mainly on the last three years. This includes the spreading of infectious disease, opinions, rumors, in social networks; information packets in computer networks; various types of signaling in biology, and more. We also discuss future directions.

The seminal work by Sadi Carnot in the early nineteenth century provided the blueprint of a reversible heat engine and the celebrated second law of thermodynamics eventually followed. Almost two centuries later, the quest to formulate a quantum theory of the thermodynamic laws has thus unsurprisingly motivated physicists to visualize what are known as ‘quantum thermal machines’ (QTMs). In this article, we review the prominent developments achieved in the theoretical construction as well as understanding of QTMs, beginning from the formulation of their earliest prototypes to recent models. We also present a detailed introduction and highlight recent progress in the rapidly developing field of ‘quantum batteries’. Graphic Abstract

In the past two and a half decades machine learning potentials have evolved from a special purpose solution to a broadly applicable tool for large-scale atomistic simulations. By combining the efficiency of empirical potentials and force fields with an accuracy close to first-principles calculations they now enable computer simulations of a wide range of molecules and materials. In this perspective, we summarize the present status of these new types of models for extended systems, which are increasingly used for materials modelling. There are several approaches, but they all have in common that they exploit the locality of atomic properties in some form. Long-range interactions, most prominently electrostatic interactions, can also be included even for systems in which non-local charge transfer leads to an electronic structure that depends globally on all atomic positions. Remaining challenges and limitations of current approaches are discussed. Graphic Abstract

We study the McKean–Vlasov equation ∂tϱ=β-1Δϱ+κ∇·(ϱ∇(W⋆ϱ)),with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Krausse model for opinion dynamics.

Recently, during the investigations on planetary oceans, Hirota-Satsuma-Ito-type models have been developed. In this paper, for a (2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito system describing the fluid dynamics of shallow-water waves in an open ocean, non-characteristic movable singular manifold and symbolic computation enable an oceanic auto-Bäcklund transformation with three sets of the oceanic solitonic solutions. The results rely on the oceanic variable coefficients in that system. Future oceanic observations might detect some nonlinear features predicted in this paper, and relevant oceanographic insights might be expected.

We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient a ( · ) . The model case is given by the functional w ↦ ∫ ( | D w | p + a ( x ) | D w | q ) d x , where q >  p and a ( · ) ≧ 0 .

With respect to oceanic fluid dynamics, certain models have appeared, e.g., an extended time-dependent (3+1)-dimensional shallow water wave equation in an ocean or a river, which we investigate in this paper. Using symbolic computation, we find out, on one hand, a set of bilinear auto-Bäcklund transformations, which could connect certain solutions of that equation with other solutions of that equation itself, and on the other hand, a set of similarity reductions, which could go from that equation to a known ordinary differential equation. The results in this paper depend on all the oceanic variable coefficients in that equation.

Minima of functionals of the type w ↦ ∫ Ω | D w | log ( 1 + | D w | ) + a ( x ) | D w | q d x , 0 ≤ a ( · ) ∈ C 0 , α , with Ω ⊂ R n , have locally Hölder continuous gradient provided 1 < q < 1 + α / n .