We study anomalous dissipation in hydrodynamic turbulence the context of passive scalars. Our main result produces an incompressible $$C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))$$ velocity field which explicitly exhibits dissipation. As a consequence, this example also shows non-uniqueness solutions to transport equation with $$L^1([0,T]; drift, is smooth exc...

We summarize recent advances in the application of equilibrium partition function formalism for study transport coefficients relativistic fluids induced by quantum anomalies, at first and second order hydrodynamic expansion. provide results theories with Abelian non-Abelian chiral fermions, discuss some features corresponding constitutive relations.