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

Abstract

Abstract We derive a priori bounds for the $$\Phi ^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Φ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> 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 $$+\infty \phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:math> represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions $$d&lt;4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> by adjusting the regularity of the noise term $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> , choosing $$\xi \in C^{-3+\delta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Our main result states that if $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> satisfies this equation on a space–time cylinder $$D= (0,1) \times \{ |x| \leqslant 1 \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mo>{</mml:mo> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> <mml:mo>⩽</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , then away from the boundary $$\partial D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> the solution $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> can be bounded in terms of a finite number of explicit polynomial expressions in $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> . The bound holds uniformly over all possible choices of boundary data for $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> and thus relies crucially on the super-linear damping effect of the non-linear term $$-\phi ^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> . 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 $$(\Pi _x)_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Π</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> </mml:math> and the family of translation operators $$(\Gamma _{x,y})_{x,y}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:math> we work with just a single object $$(\mathbb {X}_{x, y})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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”.

Topics & Concepts