Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

Abstract

Abstract. A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations

“…Their approach did not involve a limiting process, and this led to a new existence theorem. This was later generalized [2] to a physical space analogue.…”

Some Recent Developments

“…Decay assumptions are implicit to most of the theory to be described throughout this article, from orthogonality conditions (6) to Fourier transform formulations. Such boundary conditions at infinity on Navier-Stokes are the subject of extensive investigation for both free-space and exterior domain problems since Finn's conception of the notion of "physically reasonable" solution; e.g.…”

“…They then observe that the resulting equation is precisely the form for a branching random walk recursion for χ(ξ, t) := ν|ξ| 2û (ξ, t), for which the kernel |ξ−η| −2 |η| −2 is naturally constrained by integrability to dimensions n ≥ 3 for normalization to a transition probability. Alternatively, Bhattacharya et al [6] introduce a Fourier multiplier 1/h, where h(ξ), ξ ∈ W h := {ξ ∈ R 3 : ξ = 0} is a positive function such that 0 < h * h(ξ) < ∞, which is used to re-scale the Fourier transformed equation (FNS) by factors 1/h(ξ) in compensation for the temporal normalization of the exponential factor to a probability. Namely, we consider the equation (FNS h ) obtained from (FNS) as…”

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

“…In Bhattacharya et al [3] the branching trees are pruned after n generations. This gives a stochastic representation of a Picard iteration scheme converging to the original PDE, but, as stated in [3], the existence of the expectation is equivalent to the convergence of the Picard iteration scheme. In another approach, Morandin [12] suggested a clever re-summation of the expectation in order to improve the convergence for large times, but he was only able to rigorously verify the global convergence of his method in a simple example where (1.1) is a one-dimensional ODE.…”