This section is devoted to obtaining the proof of Theorem 1.1. According to Lemma 2.1, a local strong solution of the Cauchy problem (1)-(6) exists. Suppose is the first blowup time of the strong solution to the Cauchy problem, it suffices to prove there actually exists a generic positive constant M (), depending only on the initial data and , such that
(15)
where . Then due to the local existence theorem (Lemma 2.1), it can easily be shown that the strong solution can be extended beyond . This conclusion contradicts the assumption on . Thus, the strong solution exists globally on for any . Hence the proof of Theorem 1.1 is therefore completed.
The proof of (15) is based on a series of lemmas. For simplicity, throughout the remainder of this paper, we denote by C a generic constant which depends only on the initial data and and may change from line to line.
First, the -norm of the density can be obtained easily by using the method of characteristics, we list the following lemma without proof.
Lemma 3.1 For every , we have
(16)
Next, the basic energy inequalities are used.
Lemma 3.2 For every , we have
(17)
The following estimates are the key estimates in the proof of Theorem 1.1, which depends on the critical Sobolev inequality of logarithmic type (see Lemma 2.6).
Lemma 3.3 For every , we have
(18)
where is the material derivative of f.
Proof First, multiplying (2) by and integrating the resultant equation by parts over on x, one deduces that
(19)
For the first term on the right-hand side of (19), using Young’s inequality and (16), one shows that
Next, the second term can be deduced as follows:
Then, substituting the above two estimates into (19), one obtains
(20)
Multiplying (3) by and integrating over by parts, one deduces that
(21)
The term on the left-hand side of (20) cannot be determined positive or negative, thus we have to control it by some appropriate positive terms. Note that it follows from Gagliardo-Nirenberg inequality that we may deduce
Then multiplying (21) by , adding it to (20), and integrating the resulting equation over on time, we finally deduce that
(22)
To proceed, we have to estimate and . First, due to (11), we obtain
(23)
For convenience, we denote
Then, combining (17), (22), and (23), we conclude that
(24)
To proceed, we have to get the appropriate bound on and . Thus, due to (13), we obtain
(25)
where we have used (11) and (24). Similarly, we conclude from (12), (11), and (24) that
(26)
Hence, combining (25) and (26), we obtain
(27)
Thus, keeping the definition of in mind, we conclude from (27) that
Substituting the above estimate into (25), we conclude that
(28)
It follows from the basic energy estimate that one can choose the interval small enough, such that
Substituting the above estimate into (25), we conclude that
which implies that
from which we complete the proof of this lemma. □
Remark 3.1 Due to (18) and the definition of the material derivative , we show that
(29)
by the following simple fact, i.e.:
where we have used (9), (16), (17), and (18).
The following lemma is devoted to improving the time regularity of u and B.
Lemma 3.4 For every , we have
(30)
Proof Differentiating (2) with respect to t, we obtain
Multiplying the above equation by , then integrating the resulting equation over on x, we deduce that
(31)
Now, we estimate each term on the right-hand side of (31). First, due to (1), we have
Next, it follows from (1), (18), Hölder’s inequality, and Young’s inequality that
Then one obtains
where we have used (9), (11), and (24). Finally, as for and , we see that
Hence, substituting all the above estimates into (31), we conclude that
(32)
From now on, we focus on the estimate for B. Differentiating equation (3) with respect to t, multiplying the resulting equation by , and then integrating by parts over on x, we finally obtain
(33)
We estimate each term on the right-hand side of (33). First, for , ones deduce from (11) and (9) that
Similarly, for , we show that
For , one deduces
Then, substituting the above estimates on , , , one deduces
(34)
Thus, combining (32) and (34), together with Gronwall’s inequality, one easily completes the proof of (30). This completes the proof of Lemma 3.4. □
Next, we will apply (12) and (13) to improve the higher regularity on the velocity u and magnetic fields B, respectively.
Lemma 3.5 For every , we have
(35)
Proof Let us rewrite (2) in the following form:
Then, using Lemma 2.4, we conclude that
(36)
Similarly, due to Lemma 2.5, we obtain
(37)
Thus, combining the above two inequalities and Young’s inequality, we arrive at
(38)
Then, by Lemmas 2.4 and 2.5, we have
(39)
and
(40)
Then, combining all the above estimates (36)-(40) together we show that (35). This completes the proof of Lemma 3.5. □
Lemma 3.6 For every , we have
(41)
Proof Differentiating (1) with respect to (), multiplying the resultant equation by , then integrating the resulting equation by parts over with respect to x, we finally deduce after summing them up that
which, combined with (35) and Gronwall’s inequality, yields
(42)
Similarly, we can also obtain from (1) that
which combined with (42), together with (35) and Gronwall’s inequality, yields
It follows from (12) that
which implies . Similarly, we can obtain . Thus, we obtain (41), and thus complete the proof of Lemma 3.6. □
The proof of Theorem 1.1 is based on all the estimates that we deduced in Lemmas 3.1-3.6. From all the estimates obtained, we arrive at (15), and, finally, the proof of Theorem 1.1 is therefore completed.