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

Abstract

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 v 0. Using an inverse Laplace transform ofv(k, t) −v 0 in 1/t, we arrive at an integral equation forÛ (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 solutionÛ 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) l 1 (Z 3 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 ofv(k, t) −v 0 in 1/t n 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 solu-tionÛ is known on a finite interval [0, p 0 ]. We also show that for analytic v [0] and f , with finitely many nonzero Fourier-coefficients, the series forÛ (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.

Existence, Uniqueness, Analyticity, and Borel Summability of Boussinesq and Magnetic Benard Equations

“…In other words, given any solution in H 2 (R d ) to the Boussinesq equation for which the initial data and forcing satisfy the given assumption then the solution has the representation (7). iv) A sufficient condition for global existence of smooth solution to the Boussinesq equation is that e…”

“…where v, B : R d × R + → R d . This method has previously been applied to the Navier-Stokes equation in [5], [7], and [8]. We show that this approach can be used to show local existence for the Boussinesq and magnetic Bénard equation, either for d = 2…”

Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

“…Notice if (Ĥ,Ŝ) (a) (k, p) is known, thenĤ (s) (k, p),Ŝ (s) (k, p), G [1],(a) j (k, p), and G [2],(a) j (k, p) are also known functions. Also, recallû 1 andΘ 1 are quantities based on the initial condition and forcing given in (12).…”

“…A Borel based approach has also led to analysis of complex singularities for a specific PDE [9]. Recent developments include Navier-Stokes initial value problem (see [13], [12], [11]). Recently [20], numerical schemes have been suggested for nonlinear PDEs, based on a Borel plane reformulation.…”

“…ω is large enough so that (27) in the ensuing holds, where (û 1 ,Θ 1 ), defined in(12), depends on the initial data and forcing…”