On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics

We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and Tolomeo [Invent. Math. 227 (2022), no. 3, 1323–1429]. More precisely, we will establish almost sure growth bounds for solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the equations of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript t element-of left-bracket negative upper T comma upper T right-bracket Endscripts double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar Subscript upper H Sub Superscript s Subscript left-parenthesis double-struck upper T right-parenthesis Baseline less-than-or-equivalent-to Subscript s comma u 0 Baseline log left-parenthesis 2 plus upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mo> − </mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:munder> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:msub> <mml:mo> ≲ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sup _{t \in [-T,T]} \|u(t)\|_{H^s(\mathbb {T})} \lesssim _{s, u_0} \log (2+T)</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> with initial data <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 0 element-of upper H Superscript s Baseline left-parenthesis double-struck upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u_0 \in H^s(\mathbb {T})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than one half"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">s&gt; \frac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The proof uses a generalisation of Bourgain’s invariant measure argument for measures in a suitable Orlicz space.