An integral equation approach to smooth 3D Navier-Stokes solution

We summarize a recently developed integral equation (IE) approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3D Navier-Stokes (NS) equation in the context of a periodic box problem with smooth time independent forcing and initial condition v₀. Using an inverse-Laplace transform of v̂(k,t)-v7circ;₀ in 1/t, we arrive at an IE for Û(k,p), where p is inverse-Laplace dual to 1/t and k is the Fourier variable dual to x. The advantage of this formulation is that the solution Û to the IE is known to exist a priori for ∈ ℝ⁺ and the solution is integrable and exponentially bounded at ∞. Global existence of NS solution in this formulation is reduced to an asymptotics question. If ∥Û(.,p)∥_l1(ℤ3) has subexponential bounds as p→∞, then global existence to NS follows. Moreover, if f=0, then the converse is also true in the following sense: if NS has global solution, then there exists n≥1 for which the inverse-Laplace transform of v̂(k,t)-v7circ;₀ in 1/tⁿ necessarily decays as q→∞, where q is the inverse-Laplace dual to 1/t ⁿ. We also present refined estimates of the exponential growth when the solution Û is known on a finite interval [0, p₀]. We also show that for analytic v^[0] and f, with finitely many nonzero Fourier-coefficients, the series for in powers of p has a radius of convergence independent of initial condition and forcing; indeed the radius gets bigger for smaller viscosity. We also show that the IE can be solved numerically with controlled errors. Preliminary numerical calculations for Kida (1985 J. Phys. Soc. Japan 54 2132) initial conditions, though far from being optimized, and performed on a modest interval in the accelerated variable q show decay in q. © 2008 The Royal Swedish Academy of Sciences.