A Priori Bounds for the $$\Phi ^4$$ Equation in the Full Sub-critical Regime

Abstract

AbstractWe derive a priori bounds for the $$\Phi ^4$$ Φ 4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by "Equation missing"where the term $$+\infty \phi $$ + ∞ ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions $$d<4$$ d < 4 by adjusting the regularity of the noise term $$\xi $$ ξ , choosing $$\xi \in C^{-3+\delta }$$ ξ ∈ C - 3 + δ . Our main result states that if $$\phi $$ ϕ satisfies this equation on a space–time cylinder $$D= (0,1) \times \{ |x| \leqslant 1 \}$$ D = ( 0 , 1 ) × { | x | ⩽ 1 } , then away from the boundary $$\partial D$$ ∂ D the solution $$\phi $$ ϕ can be bounded in terms of a finite number of explicit polynomial expressions in $$\xi $$ ξ . The bound holds uniformly over all possible choices of boundary data for $$\phi $$ ϕ and thus relies crucially on the super-linear damping effect of the non-linear term $$-\phi ^3$$ - ϕ 3 . A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model $$(\Pi _x)_x$$ ( Π x ) x and the family of translation operators $$(\Gamma _{x,y})_{x,y}$$ ( Γ x , y ) x , y we work with just a single object $$(\mathbb {X}_{x, y})$$ ( X x , y ) which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.