The Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and its Sharpness by Wojciech S. Ożański

Author:Wojciech S. Ożański
Language: eng
Format: epub, pdf
ISBN: 9783030266615
Publisher: Springer International Publishing

(3.60)

(We write “” to articulate that we restrict ourselves to points (x, t) of the form .) Note that by taking we obtain

Here, the small term will be used in the continuity argument to absorb the Laplacian term, (compare with (3.57)), see (3.81) and (3.84) for details. In other words, the time-dependent extensions , () will be chosen such that, by construction, we will obtain the NSI.

In particular, we will choose

where are certain oscillatory processes, which are discussed in detail in Sect. 3.3.3 below. The oscillatory process will have two remarkable features. The first is that

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