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Abstract We prove well‐posedness of a class ofkinetic‐typemean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non‐separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitablevector field methodto control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift‐diffusion operators and non‐linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non‐smoothing, that is, also depending locally on the final measure. Our well‐posedness results hold under an appropriate smallness condition, assumed jointly on the data. 

Abstract Periodic traveling waves at the interface of two incompressible, inviscid fluids subject to gravity and surface tension are studied. We focus on the case in which the linearization about the quiescent state has a two-dimensional kernel. We prove the existence of sheets of traveling waves in this circumstance. We also compute Wilton ripples in which the leading term has a (1:2) harmonic resonance, the triad ripple configuration. Global branches of waves are computed, terminating in three types of self-intersecting waves. 

Abstract We consider the three-dimensional Navier–Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier–Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity. 

We prove the existence of solutions to the Kuramoto–Sivashinsky equation with low regularity data in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data that are in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data that are in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.