Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity

Nonlinear Differ. Equ. Appl. 20 (2013), 1105–1123 c 2012 Springer Basel AG  1021-9722/13/031105-19 Nonlinear Differential Equations published online October 9, 2012 and Applications NoDEA DOI 10.1007/s00030-012-0200-3 Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity David Barbato and Francesco Morandin Abstract. We improve regularity and uniqueness results from the literature for the inviscid dyadic model. We show that positive dyadic is globally well-posed for every rate of growth β of the scaling coefficients kn = 2βn . Some regularity results are proved for positive solutions, namely 1 1− 1 supn n−α kn3 Xn (t) < ∞ for a.e. t and supn kn3 3β Xn (t) ≤ Ct−1/3 for all t. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time. Mathematics Subject Classification. 76B03, 35Q35, 35Q31. 1. Introduction Well-posedness and regularity for Navier–Stokes and Euler equations represent a major point of interest in mathematics. The study of estimates of the nonlinear term (u · )u in particular is important, since this term is associated with the so-called cascade of energy from lower to higher modes. A very rough idea of this phenomenon is the following. Fix a time t and decompose the velocity u on the frequencies u = k uk ek , where ek are “wave ik,· functions” (for example ek = ) and uk are the corresin(k, ·) or ek = e sponding coefficients (with k |uk |2 = u2L2 ). Then the regularity of u can be associated to how fast the coefficients uk go  to zero as |k| tends to infinity. More precisely, u has N -th derivative in L2 if k (|k|N |uk |)2 < ∞. The bilinear term (u · )u acts on the dynamics of the coefficients uk by mixing different components, in that if h and k are two active frequencies (uh , uk = 0), then the term (u·)u will activate the component h+k and so on, activating higher and higher frequencies in a phenomenon called energy cascade [6]. In this paper we study the inviscid dyadic model, which is a shell-type model of Euler equations which was early introduced in [9] and then again in recent times in [11] and [13]. This model, represented in equations (1) below, exploits some of the properties of Euler equations (see among the others [6] 1106 D. Barbato and F. Morandin NoDEA and [10]) on a much simpler structure. Informal derivations of this model from Euler equations are given in [7,9,11]; from an intuitive point of view, one should imagine that the variable Xn in (1) represents a global coefficient for all components uk with |k| of order 2n . These equations, like Euler’s, are homogeneous of degree 2, they are formally conservative and moreover they show the energy cascade phenomenon, which in this setting is very clearly understood [4]. To collect previous results on the dyadic model, one should immediately distinguish whether the initial condition has all positive components or not, since the dynamics of energy cascade is strongly dependent on the sign of Xn . If all Xn ’s are positive, energy moves from lower modes to higher ones. If all Xn ’s are negative, energy moves from higher modes to lower ones. In [4] and [8] it is shown that in the case of positive components, energy moves to higher modes faster and faster, in such a way that a positive fraction of energy gets lost “at infinity” in finite time and one can show that in this case the energy X(t)2l2 goes to zero like t−2 . On the other hand in [8] it is proved that if a positive forcing term is included in the model (which puts energy into the first component), the energy converges exponentially fast to a fixed value (corresponding to the stationary solution). In both cases the model is not really conservative in the end, since energy moves to infinity and there it disappears: this phenomenon is called anomalous dissipation. On the other hand, if all components are negative, the opposite situation can occur: energy can enter from “infinity” into the system. In [4] explicit solutions are constructed in which there is an anomalous increase of energy, immediately yielding non-uniqueness of solutions for the negative dyadic. In this paper we prove that the positive dyadic is globally well-posed, extending the uniqueness result in [1] to arbitrary rate of growth of the coefficients kn in system (1). This means in particular that the escape of energy at infinity does never preclude well-posedness; on the contrary, in the negative dyadic, the input of energy from infinity immediately destroys well-posedness. It is interesting to confront this with the stochastic dyadic model introduced in [2], where the distinction between positive and negative solutions is meaningless, since noise causes infinite sign changes in every time interval. In [2,3] it is proved that there is escape of energy at infinity and (weak) uniqueness of solutions, so also in this case energy cannot enter from infinity and the problem is well-posed. Section 3 deals with the connection between negative and positive dyadic. It is shown that under minimal hypothesis all components become positive in finite time and stay positive forever. Based on this fact all the rest of the paper is restricted to positive initial conditions. Sections 4, 5 and 6 deal with uniqueness and regularity and are very much interlinked. Uniqueness was already proved in [1] for kn = 2βn and β ≤ 1 and in [5] for any β in a class of regular enough solutions. It is now extended by Theorem 13 to arbitrary β and l2 initial condition. As already stated we are interpreting the components of the solution as something similar to Fourier coefficients, so regularity means smallness of Vol. 20 (2013) Positive and non-positive 1107 −1/3 components Xn for n large. The vague idea is that Xn tends to zero as kn . The first results in the literature are of lack of regularity. In [4,7,8,11,12] it is shown that if the initial condition is in l2 , then all solutions are Leray–Hopf and nonetheless a blow-up occurs in finite time, in the sense that for all  > 0 the quantities  t  1 1 + kn3 Xn (t), sup kn(1+) Xn3 (s)ds, sup kn3 Xn (t), n n n 0 become infinite in finite time, even if they were finite for t = 0. On the other hand one first important regularity result can be found again in [8], where for β < 3, the authors prove that for all α > 1,  t 1 (n−α kn3 Xn (s))2 ds < +∞. 0 n Our main results on regularity are Theorem 10, Lemma 14, Theorem 15 and Theorem 17 which, through some corollaries, imply that 1 2. 3. supn n−α kn3 Xn (t) < ∞ for all α > 13 and for a.e. t > 0;  −α 13 4 n n kn Xn (t) < ∞ for all α > 3 and for a.e. t > 0; t supn 0 n−1 kn Xn3 (s)ds < ∞ for all t > 0; and moreover 4. supn kn3 1. 1 1 − 3β Xn (t) ≤ Ct−1/3 for all t > 0. 2. Model A very natural space for the dynamics of the dyadic is H := l2 (R), the Hilbert space of square-summable sequences with the usual norm which we will denote simply by  · . Let β > 0 and x = (xn )n≥1 ∈ H. Consider the following Cauchy p (...truncated)