The regularity of bidimensional solutions of the third grade fluids equations

It was recently proved in [1,2] that the third grade fluids equations have a unique global bidimensional solution provided that the initial velocity belongs to the Sobolev space H2. Here, we complete this result by proving that this solution preserves the Sobolev regularity of the initial data, i.e., if the initial velocity belongs to Hm, m ≥ 2, then the evolved velocity v(t, ·) also belongs to Hm for every time t.