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To reduce equation (2) to the form (1) one has to add a new variable p = 8,~). We assume that the functions ,f (71, s) and ,g(:z. s) satisfy some general conditions providing the solvability of the Cauchy problem for (2) (see [ 191). But, generally speaking, under these conditions, the corresponding solution IL(~) = ,(/,(.I:; t) need not be unique. A trajectory attractor A is constructed for the family of equations (1). We start from the fact that the attractor A may not change when the initial symbol (TO(S) is replaced by any shifted symbol CT"(S + h.). h > 0. This is why, together with the initial equation (1) having the symbol (T,~(s)~ we consider the family of equations (1) with shifted symbols "a(.~ + /L); h > 0. This family contains also any symbol 'T(S) that is a limit of some sequence {cru(s + h,) ] h

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The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R > 0 are considered and the\ud existence of a strong trajectory attractor in the space L∞\ud loc(R+,H1) is established under the assumption that the external forces have\ud bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded\ud vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the\ud weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method