Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [6] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s<5/6$ for almost all time and the energy dissipates.
Moreover, it is proved that the unique equilibrium is an exponential global attractor.
We improve regolarity 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 k_n = 2^{bn}. Some regularity results are proved for positive solutions, namely \sup_n n^{-a} k_n^{1/3} X_n(t) < \infty for a.e. t and \sup_n k_n^{1/3-1/(3b)} X_n(t) \leq C t^{-1/3}$ for all $t$. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.
Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in $H^{5/6}$-norm. Onsager's conjecture is confirmed for the model system.
We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.
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.
We study the inviscid dyadic model of the Euler equations and prove some regularizing properties of the nonlinear term that occur due to forward energy cascade. We show every solution must have 3/5 L^2-based (or 1/10 L^3-based) regularity for all positive time. We conjecture this holds up to Onsager's scaling, where the L^2-based exponent is 5/6 and the L^3-based exponent is 1/3.
In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.
For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^2$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^2$-initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.
A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process.
A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov's theory for the actual Navier-S…
We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlović. They showed a finite time blow-up in the case where the dissipation degree $α$ is less than 1/4. In this paper we prove the existence of weak solutions for all $α$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for $α> 1/3$, and global regularity for $α\geq 1/2$. In addition, we prove a finite time blow-up in the case where $α<1/3$. It is remarkable that the model with $α=1/3$ enjoys the same estimates on the nonlinear term as the 4D Navi…
We study an infinite system of non-linear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3d Euler and Navier-Stokes equations in a rough approximation of a wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.
Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the $H^{3/2+ε}$ Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in $H^α$, for $α\ge 1/2$, but its dyadic restriction is even more singular, exhibiting blow-up for any $α> 0$. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in $H^{3/2+ε}$. Some inco…
Asymptotic-in-time feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [22] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of imposed energy dissipation the plate dynamics converge to a compact and finite dimensional set [16,17]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the stationary set in the natural energy topology. In doing so, we demonstrate the existence of an expon…
Dyadic models of the Euler equations were introduced as toy models to study the behaviour of an inviscid fluid in turbulence theory. In 1974 Novikov proposed a generalized mixed dyadic model that extends both Katz-Pavlovic and Obukhov models giving birth to a more complex structure: no results were found in literature until 2015 where blow up in finite time for smooth solutions and existence of self-similar solution for particular values of the model parameters were shown by Jeong I.J. We extend such partial results by giving a complete spectrum of existence and uniqueness results for two card…
We investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, we also show that the weak solutions of the Navier-Stokes-Voight equat…
A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced profile for the initial magnitude of vorticity along with a new algorithm for the initial vorticity direction. The new analysis uses a new adaptation to the Euler equations of a rescaling of the vorticity moments developed for Navier-Stokes analysis. All rescaled moments grow in time, with the lower-order moments bounding the higher-order moments from above,…
The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite volume limit for every fixed asymptotic slope; concentration inequalities for free energy implying a bound on its fluctuation exponent; straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given en…
