An Integral Equation Approach to Smooth 3-d Navier-stokes Solution

We summarize a recently developed integral equation approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3-D Navier-Stokes 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) − v̂0 in 1/t, we arrive at an integral equation 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 integral equation is known to exist a priori for p ∈ 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 the inverse Laplace transform of v̂(k, t)− v̂0 in 1/t necessarily decays as q → ∞, where q is the inverse-Laplace dual to 1/t. 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 Û(k, p) 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 the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations for Kida ([14]) initial conditions, though far from being optimized, and performed on a modest interval in the accelerated variable q show decay in q.