OAI.AlternatingNS.alternating

The theorem states that for every positive real viscosity ν computable by rational approximations with error at most 2⁻ⁿ, there exist computable compilers for a force f and velocity U, and one compact set K⊂ℝ³, with the following property for every well-formed finite deterministic Turing machine M and valid finite input w. The compiled programs describe smooth fields f,U on [0,∞)×ℝ³, both spatially supported in K for all nonnegative times, such that every mixed space-time derivative D satisfies sup_{t≥0,x}(1+t+‖x‖)ᴶ‖Df(t,x)‖<∞ and the analogous bound for U, for every nonnegative integer J. Each program supplies rational approximations to all mixed derivatives at rational space-time points with requested error 2⁻ᵏ, effective moduli of continuity on bounded regions, and integer bounds for all these weighted derivative norms. The velocity starts from zero, is divergence-free, and solves the classical forced Navier–Stokes equation ∂ₜU+(U·∇)U=νΔU+f with identically zero pressure; specifically f=∂ₜU+(U·∇)U−νΔU. On every interval [0,T], U is continuous in H² and continuously differentiable in L², and U and its spatial derivative are uniformly bounded in space and time. Moreover, any classical solution (v,p) with the same force and zero initial velocity equals (U,0) pointwise for all nonnegative times, provided on every [0,T] it has v continuous in H², continuously differentiable in L², and uniformly bounded, and p continuous in H¹. Here these Sobolev continuity conditions mean continuous L² representatives of every spatial derivative through the indicated order; time derivatives at zero are taken within [0,∞). Every initial point a has a unique trajectory X(a,t) for t≥0 satisfying X(a,0)=a and ∂ₜX(a,t)=U(t,X(a,t)). Finally, the trajectory starting at (−1,0,0) has strictly positive first coordinate at some nonnegative time if and only if M halts on w, where encountering a missing instruction also counts as halting.