Global solutions in the critical Sobolev space for the Landau equation

The Landau equation is studied for hard potential with −2 ≤ γ ≤ 1. Under a perturbation setting, a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space \(H_x^dL_v^2(d > {3 \over 2})\) $$H_x^d{L}_v^2(d>\frac{3}{2})$$ , which extends the results of [11] in the torus domain to the whole space \(\mathbb{R}_x^3\) $${ℝ}_x^3$$ . Here we utilize the pseudo-differential calculus to derive our desired result.