Uniqueness for a stochastic inviscid dyadic model

Abstract

For the deterministic dyadic model of turbulence, there are examples of initial conditions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l squared"> <mml:semantics> <mml:msup> <mml:mi>l</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">l^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which have more than one solution. The aim of this paper is to prove that uniqueness, for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l squared"> <mml:semantics> <mml:msup> <mml:mi>l</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">l^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.