We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution u has maximal existence time T * < ∞, then the non-endpoint critical Besov norms must become infinite at the blow-up time:In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spacesḂ −1+ 3 p p,∞ , p > 3. These solutions satisfy a certain stability property with respect to the weak- * convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.

This paper concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. These results in particular yield that the Lions exponent is the sharp viscosity threshold for the uniqueness/non-uniqueness in law of Leray solutions. Our proof utilizes the self-similarity and instability programme developed by 43] and Albritton-Brué-Colombo [1], together with the theory of martingale solutions including stability for non-metric spaces and gluing procedure.

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [31]. Consider a passive scalar in a shear flow, whose evolution is described by the advection-diffusion equationwith periodic, Dirichlet, or Neumann conditions in y. We demonstrate that decay is enhanced on the timescale T ∼ ν −(N+1)/(N+3) , where N −1 is the maximal order of vanishing of the derivative b ′ (y) of the shear profile and N = 0 for monotone shear flows. In the periodic setting, we recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are new in the presence of boundaries.

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