Introduction
Iteration Estimate
Now we give a significant iteration to verify the
The estimate below is necessary for the following iteration.
Before the contradiction, first we take a scaling by following formulation:
: Notice is suitable weak solution of following system on :
Moreover, the uniform bound implies
Now we attempt to derive a strong -convergence for by Aubin-Lion lemma: The local energy inequality implies2
and following calculation implies :
Since , together with and , we see is precompact in and thus . Moreover, since , so the Holder interpolation implies3 .
Consequently, solves4 following system in :
Following we show that is Holder continuous:
It comes for . Then the Campanato characterization in parabolic version gives out following bound since :
which implies . notice on , so we split where
We estimate by Riesz potential, and by properties of harmonic function:5
And then the estimate follows:
if . Consequently, we have .
Combine together, we get
criterion
:
6 Since , then we decompose , where cutoff on . Then
and since is sub-harmonic\footnote{Sub-harmonicity is preserved under convex function.}, it comes
: Apply Holder interpolation:
The second condition is similar: For ,
Integrating in time and apply Holder interpolation :
We take cutoff on , then the local energy inequality gives:
Take at both hand, we get that
7
Singularity
Appendix
Attachments List
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Reference
. Similar for .
↩\qst is the initial data for each
determined? ↩Particularly, it is a suitable weak solution which is preseved by weak limit. Details see Lin's Paper, Theorem 2.2. ↩
this is reached by following estimate:
Here we will encounter critical case for if no amplification by cutoff function. ↩Here
↩Notice
. ↩
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