Theorem 3.1 Let θ be a Leray-Hopf weak solution of (1), namely
(5)
Let and let . If
(6)
then for .
Proof First we notice that (5) and (6) imply that
for any and . In fact, for any ,
Since , we have when
Next, we show that
implies
for some to be specified. Let j be an integer. Applying to the first equation of (1), we get
(7)
By Bony’s notion of paraproduct,
Multiplying (7) by , integrating with respect to x and applying the lower bound
of Proposition 2.4, we obtain
(8)
We now estimate . The standard idea is to decompose it into three terms: one with commutator, one that becomes zero due to the divergence-free condition and the rest. That is, we rewrite as
where we have used the simple fact that , and the brackets represent the commutator, namely
Since u is divergence free, becomes zero. We now bound and . By Hölder’s inequality,
To bound the commutator, we have, by the definition of ,
Using the fact that and thus
we obtain
Therefore,
The estimate for is straightforward. By Hölder’s inequality,
We then bound . By Hölder’s inequality, Bernstein’s inequality and the fact , we obtain
Last, we bound . By Hölder’s inequality and Bernstein’s inequality,
Inserting the estimates for , and in (8) and eliminating from both sides, we get
Integrating with time t, we have
Multiplying both sides by and taking the supremum with respect to j, we get
Here we have used the fact that
Therefore, we conclude that if
then
Since , we have and thus gain regularity. In addition, according to the Besov embedding of Proposition 2.2,
where
We have when
Noticing that
we conclude that, for ,
for some . The above process can then be iterated with replaced by . A finite number of iterations allow us to obtain that
for some . The regularity in the spatial variable can then be converted into regularity in time. We have thus established that θ is a classical solution to the supercritical porous media equation. Higher regularity can be proved by well-known methods. □