A uniqueness theorem for the unbounded classical solution of the nonstationary Navier-Stokes equations in \(\mathbb{R}^ 3\) (Q1323178)

The author considers the Navier-Stokes equations in \(\mathbb{R}^ 3\), i.e. \[ u_ t - \nu \Delta u + (u \cdot \nabla) u + \nabla p = f, \quad \text{div} u = 0 \quad \text{in} \quad \mathbb{R}^ 3 \times [0,T], \quad u(0) = u_ 0 \quad \text{in} \quad \mathbb{R}^ 3 \tag{*} \] and proves the following uniqueness result: if \((u,p)\) and \((v,q)\) are classical solutions of \((*)\) having the same initial data and both satisfying the growth estimates \[ | u |, | v | = O \bigl( | x |^ \alpha \bigr), \quad | \nabla_ x u |, | \nabla_ x v | = O(1),\quad | p |, | q | = O \bigl( | x |^{-1/2} \bigr) \quad \text{as} \quad | x | \to \infty \] for some \(\alpha<1\) then \(u \equiv v\) in \(\mathbb{R}^ 3 \times [0,T]\). The same statement is true in two dimensions with the exception that the pressures \(p,q\) only have to be bounded.