The attractor ?~t and dynamical system {~;Vt,-~<~< ~ of the initial-boundary problem for the two-dimensional equations of motion of an Oldroyd fluid of order are constructed. I. O. A. Ladyzhenskaya [1-3] gave life to a new direction in the theory of initialboundary problems and the theory of asymptotic methods for partial differential equations, finding an attractor ~ of initial-boundary problems for nonlinear evolutions with dissipation and the construction and study of dynamical systems on ~ , generated by these initial-boundary problems (cf. [4, 5]). In [6, 7], using the methods of [1-3] the attractor and dynamical system I D~;V~, -~<~ < ~ I of the basic initial-boundary problem for twodimensional equations of motion of an Oldroyd fluid of order 1 are constructed and studied. In the present note we extend these results to the case of an Oldroyd fluid of order L= 1,z,,., 2. By an Oldroyd fluid of order L = LZ .... is meant a linear viscoelastic fluid, whose defining equation, connecting the deviator of the stress tensor g and the rate of deformations tensor ~ has the form [8, 9]: ~:~ ~~ -~, v~.,~>0. (1) Let us assume that the times of relaxation tAel satisfy the following conditions: the roots {~t} of the polynomial (~(p)=-~* ~" %tp ~ r ~=~ are different, i.e., ~(~t)r , ~=~,...,h , real, and negative ~<0 , ~=~,...,L, and moreover, the times of relaxation I,~r , the coefficient of viscosity v , and the times of retardation ~I satisfy the conditions: b-4 [ -I -4 -= " ~c~,~,)%l>o, ~=~ .... ,L . (2) The conditions (2) generalize the familiar Oldroyd condition [8] for an Oldroyd fluid of order i: ~-~?~>0 . For an Oldroyd fluid of order 2, the conditions on v,[~l, lmr described above reduce to the following [10-12]:It is shown in [11][12] that under the conditions (2), the motion of an Oldroyd fluid of order L is described either by a system of integro-differential equations
