It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to the Euler equations which are compactly supported in space and time. Obviously these solutions do not respect the classical conservation law for the total kinetic energy and they are therefore very irregular. In recent joint works we have… (More)
In this paper, we establish the generalized Hyers–Ulam–Rassias stability of homomorphisms between ternary algebras associted to the generalized Jensen functional equation rf (sx+ty r) = sf (x) + tf (y).
For any ε > 0 we show the existence of continuous periodic weak solutions v of the Euler equations which do not conserve the kinetic energy and belong to the space L 1 t (C 1 /3−ε x), namely x → v(x, t) is (1 /3 − ε)-Hölder continuous in space at a.e. time t and the integraí [v(·, t)]1 /3−ε dt is finite. A well-known open conjecture of L. Onsager claims… (More)