Variable exponent Besov-Morrey spaces

In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work.publishe

Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions

Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Suppose that u(t) is a solution of the three-dimensional Navier–Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev space s must be bounded below by cst−(2s−1)/4 for 1/2 5/2. (The result for 1/2 < s < 3/2 follows from well-known lower bounds on blowup in Lp spaces.) We show in particular that the local existence time in s(3) depends only on the s-norm for 1/2 < s < 5/2, s ≠ 3/2