Asymptotic Nonuniqueness of the Navier - Stokes Equations in Kinetic Theory

(1) dp/dt + £ • grad p = Qp/e, whose solution p = pe(t, x, £), t > 0, x G jR , % G R, e > 0. Q is the linearized collision operator corresponding to a spherically symmetric hard potential, and e is a parameter which represents the mean free path. In a series of basic papers, Grad [6], [7], [8] studied the existence and asymptotic behavior of the solution of the initial value problem for (1), where the initial data p e (0 + , x, £) = f(x, £) satisfies mild growth and smoothness conditions. Grad's method begins with the decomposition