Regularity and Conserved Quantities: Some Recent Developments

Evolution equations with a physical background always come with one or several conserved quantities, like the energy. However, when the solution and/or the coefficients are sufficiently irregular, the conservation law may break down, and the question arises precisely what regularity is needed to guarantee conservation. We will recall the classical theories of DiPerna-Lions and Constantin-E-Titi, and then discuss some recent results concerning bounded domains, optimal Besov regularity, and nonlocal effects.