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Eulerian dynamics with a commutator forcing III. Fractional diffusion of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math>