We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal Q-composition) for Q0<α<2−1(Rn≥2), and yet demonstrate that if f:R2→R2 is a homeomorphism then the boundedness of u↦u∘f on Q2−1<α<1(R2)⊂BMO(R2) yields the quasiconformality of f.

Using the convex integration technique for the three-dimensional Navier-Stokes equations introduced by T. Buckmaster and V. Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier-Stokes equations with fractional hyperviscosity $(-\Delta)^{\theta}$, whenever the exponent $\theta$ is less than J.-L. Lions' exponent $5/4$, i.e., when $\theta < 5/4$.

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of RN with prescribed measure m attains its maximum on the union of two disjoint balls of measure m/2. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.

This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. The starting point is the kinetic model considered in earlier work of Degond & Motsch with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in Degond & Motsch, the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.