Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

Abstract

Abstract.A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved, and decay of energy like t −2 is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.

A dyadic model on a tree

“…The results are analogous to those provided for the dyadic model in [2] and [8], but the proofs require some new ideas to cope with the more general structure.…”

“…so we base instead our argument upon [2], where it is proven existence and some kind of uniqueness of self-similar solution. We obtain the following statement.…”

“…We use Theorem 10 in [2] which, translated in the notation of this paper, states that there exists a unique sequence of non-negative real numbers (b n ) n≥0 such that b 0 > 0 and Y n := bn t−t0 is a positive l 2 solution of the unforced inviscid classic dyadic (8). Thanks to Proposition 4.1 this solution may be lifted to a solution of the inviscid tree dyadic (7) with the required features.…”

“…All these models are formally conservative: the global kinetic energy E(t) = j X 2 j (t), or E(t) = n Y 2 n (t) depending on the case, is formally constant in time; it can be easily seen in both cases, using the telescoping structure of the series dE(t) dt . However, in previous papers ([6], [2]) it has been shown that the dyadic model (3) is not rigorously conservative: anomalous dissipation occurs. The flux of energy to high values of n becomes so fast after some time of evolution that, in finite time, part of the energy escapes to infinity in n.…”

“…To be precise, we have dissipation for a class of coefficients c j which covers (2). The proof is similar to the one in [2] but requires new ideas and ingredients.…”