This section is devoted to the derivation of a priori estimates of . We begin with the observation that the total mass is conserved. Moreover, if we multiply (1.3), (1.4), and (1.5) by , and , respectively, and sum up the resulting equations, we have by using (1.2) that
Integrating (1.2) and (2.1) over , we arrive at our first lemma.
For any , one has
where is the nonnegative function defined by
The next lemma gives us an upper bound of the density , which is crucial for the proof of Theorem 1.1.
For any , holds.
Notice that (1.3) can be rewritten as
from which and (2.4), we find that satisfies
In view of Lemma 2.1 and (2.6), we have by using Cauchy-Schwarz's inequality that
Letting denote the material derivative and choosing , we obtain after a straightforward calculation that
which, together with (2.8), yields Lemma 2.2 immediately.
To be continued, we need the following lemma because of the effect of magnetic .
The magnetic field satisfies the following estimates:
Multiplying (1.5) by and integrating over , we have
where we have used Cauchy-Schwarz's inequality, Lemma 2.1, and the following inequalities:
Since because of Lemma 2.1, we thus obtain the first inequality indicated in this lemma from (2.11) by applying Gronwall's lemma and then Sobolev's inequality.
To prove the second part, we multiply (1.5) by and integrate the resulting equation over to deduce that
where we have used Cauchy-Schwarz's inequality, Sobolev's inequality (2.12), Lemma 2.1, and the first part of the lemma. This completes the proof of Lemma 2.3.
The following estimates hold for the velocity :
Multiplying (1.3) by and then integrating over , by Young's inequality we obtain
It follows from (1.2) and (1.3) that
Thus, inserting (2.16) into (2.15), and integrating over , we see that
where the terms on the right-hand side can be bounded by using Lemmas 2.1–2.3 as follows:
Therefore, taking appropriately small, we conclude from (2.17)–(2.20) that
where, combined with the fact that due to Lemma 2.1, we obtain the first part of Lemma 2.4 by applying Gronwall's lemma and then Sobolev's inequality. Similarly, multiplying ((1.4) by and integrating the resulting equation over , we get that
where we have also used Cauchy-Schwarz's inequality. Integration of (2.22) over gives
where Lemmas 2.1–2.3 and the first conclusion of this lemma have been used. Therefore, from the above inequality we obtain the second part, and so Lemma 2.4 is proved.
Notice that (1.3), ((1.4) can be written as follows:
where and . Thus, by Lemmas 2.1–2.4, we see that
which immediately implies
Hence, we have the following lemma.
There exists a positive constant , such that
where and .
To prove the uniqueness of strong solutions, we still need the following estimates.
The pressure satisfies . Furthermore, if the compatibility conditions (1.9), (1.10) hold, then
It follows from the continuity equation (1.2) that satisfies
which, differentiated with respect to , leads to
Multiplying the above equation by and integrating over , we deduce that
where we have used the inequality
which follows from the definition of . Therefore, applying the previous Lemmas 2.1–2.5 and Gronwall's lemma, one has
which proves the first part of the lemma.
We are now in a position to prove the second part. We first derive the estimate for the longitudinal velocity . To this end, we firstly rewrite (1.3) as
Differentiation of (2.34) with respect to gives
which, multiplied by and integrated by parts over , yields
On the other hand, by virtue of (1.2) we have
from which and (2.36) we see that
Using the previous lemmas and Young's inequality, we can estimate each term on the right-hand side of (2.38) as follows with a small positive constant :
Putting the above estimates into (2.38) and taking sufficiently small, we arrive at
so that, using the relation between and again, one infers from (2.40) that
where the first term on the right-hand side of (2.41) is integrable on due to the previous lemmas. Thus, integrating (2.41) over , we deduce from (1.3) that
Letting and using the compatibility condition (1.9), we easily obtain from (2.42) that
which, together with and Gronwall's lemma, immediately yields
In a same manner as that in the derivation of (2.44), we can show the analogous estimate for the transverse velocity by using the previous lemmas, (2.44), and the compatibility condition (1.10) as well. Thus, we complete the proof of Lemma 2.6.
From the a priori estimates established above, one sees that the compatibility conditions are used to obtain the second part of Lemma 2.6 only. However, this is crucial in the proof of the uniqueness of strong solutions.