Space-times paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

Abstract

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a 3-dimensional Riemannian setting, in either bounded or unbounded domains. Aiming that, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity. This constitutes to a first step in building a higher order paracontrolled calculus via semigroup methods.Contents 1 I.Bailleul thanks the U.B.O. for their hospitality, part of this work was written there.

Untitled

“…Hence F is a contraction mapping, provided that |τ − τ 0 | ≤ δ := (2C) −1 . Then, F gives a unique fixed point f ∈ Y, which is the unique bounded solution to the Cauchy problem (3)(4)(5)(6)(7)(8)(9) in Y. By dividing the interval [0, 1] into subintervals of length less than δ, we conclude that 1 ∈ I.…”

“…((τ, ξ, η) −1 • (t, x, v))s(τ, ξ, η) dτ dξ dη (3)(4)(5)(6)(7) is the unique bounded solution in C 2+α l ( ) to (3-1) with L 1 replaced by L 0 and f in = 0.…”

“…Let the set I be the collection of τ ∈ [0, 1] such that the Cauchy problem (3)(4)(5)(6)(7)(8)(9) is solvable for any s ∈ C α l ( ) with ∥s∥ (2−σ ) α < ∞: there is a unique bounded solution f ∈ Y satisfying…”

“…Thus, we can define the mapping F : Y → Y by setting F(w) = f . Armed with (3)(4)(5)(6)(7)(8)(9)(10) and (3)(4)(5)(6)(7)(8), there exists a universal constant C > 0 such that, for any u, w ∈ Y, ∥F(u) − F(w)∥ (−σ ) 2+α ≤ C|τ − τ 0 |∥(L 0 − L 1 )(u − w)∥ (2−σ ) α ≤ C|τ − τ 0 |∥u − w∥ (−σ ) 2+α .…”

“…Remark 1.2. If the general measurable initial data f in satisfies f in ≤ µ and an extra locally uniform lower bound assumption (see (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) below for a precise description), the existence of solutions still holds in some weak sense, as pointed out in Remark 4.9 below. In order to describe the diffusion asymptotics of (1-2), we introduce the (Bregman) distance characterized by the relative phi-entropy functional H β .…”