Examples of equations of Navier-Stokes type not strongly solvable in the large

In the space of functions with values in Hilbert space, we consider the Cauchy problem u′ _t + Au + B(u, u) = f(t), u(0) = 0, 0 ≤ t ≤ T. We construct examples of a self-adjoint operator A ≥ E and a bilinear transformation B satisfying the condition 〈B(u, v), v〉 = 0 such that the Cauchy problem is not strongly solvable.