For the proof of our main result, firstly we give some properties about the gradient of velocity. Recall that the full gradient of the velocity, ∇u, can be decomposed into symmetric and antisymmetric parts,
is called the deformation tensor.
In the following lemmas, we recall some important properties of and without proof [7, 8].
Lemma 3.1 For both the symmetric and the antisymmetric parts , of ∇u, the bound
The antisymmetric part
for any vector . The vorticity ω satisfies the identity
(‘’ denotes principal value) where , with . Notably,
where denotes the standard measure on the sphere .
The matrix components of the symmetric part have the form
where are the vector components of ω, and where the integral kernels have the properties
Thus, in particular, is a Calderon-Zygmund operator, for every .
We can also give the following useful lemma to provide an upper bound of singular integral operator for the incompressible Euler equations in .
Lemma 3.2 For fixed, and , let be defined as above. Moreover, let () denote the components of the vorticity vector . Then any singular integral operator
for and the constant C independent of u and t.
Now we are ready to give a proof of Theorem 2.1, which is based on combining an energy estimate for ideal MHD equations with the estimate of .
For , we recall the definitions of the homogeneous and inhomogeneous Besov norms for ,
where is the Paley-Littlewood projection of f of scale j. We take the Besov norm of and ; then
However, applying the results of Lemma 3.1 and Lemma 3.2 to u and h, and by the definition of , we obtain
Therefore, we get
for . Thus we complete the proof of Theorem 2.1.