Abstract
Assuming that T is a potential blow up time for the Navier–Stokes system in mathbb {R}^{3}_{+}, we show that the norm of the velocity field in the Lorenz space L^{3,q} with q<infty goes to infty as time t approaches T.
Highlights
The question that is addressed in the paper is as follows
It is assumed that the initial velocity field v0 is smooth, compactly supported, and divergence free in, i.e., v0 belongs to the space C0∞,0( ), and that is a domain in R3 with sufficiently smooth boundary
Though we focus on the half space, this provides an alternative to the proof given for the whole space in [21]
Summary
The question that is addressed in the paper is as follows. Let us consider the initial boundary value problem for the Navier–Stokes system in the space-time domain Q+ =. We are able to work without Lemarie–Rieusset type solutions in half space to get a local energy ancient solution to which a Liouville type theorem based on backward uniqueness is applicable. In comparison to [21], our approach doesn’t require a notion of local energy solution, since we get all desired properties for the ancient solution directly from a priori estimates associated to the blow up procedure Those ideas can be used for the construction of a global in time solution, for non energy initial data that is contained in the sum of certain Lesbesgue spaces, that applies in R3 and is extendible to other unbounded domains with boundaries. Further developments in this direction will be published elsewhere.
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