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

Introduction

The following system of differential equations

where $k_{n}=2^{n}$, has been introduced as a simplified model of 3D Euler evolution in order to investigate a number of properties which are out of reach at present for more realistic models of fluid dynamics. Let us mention in particular the works , , , , and references therein, devoted to this model and variants of it. This model differs from other dyadic or shell models, the analysis of which is more difficult and less explicit, see among many others , , , , ; system (1) has a basic monotonicity property that makes it more tractable.

Among the many interesting properties proved in the above mentioned works, let us recall: i) the dissipation of energy, in spite of the fact that formally the equation is conservative; ii) the blow-up of solutions in certain topologies, although the same solutions are global in larger spaces; iii) the relation with Kolmogorov (K41) scaling law. Our aim is to add some contribution to the understanding of these problems. The basic difference between the previous works dealing with energy dissipation and the present one is that a non-zero force was imposed until now, while we investigate the case without force, which contains a number of new phenomena.

About energy dissipation, the results known until now have the following form. A constant positive force $f>0$ is added to the first mode

and it is proved that there exists a unique fixed point, having Kolmogorov scaling, which attracts exponentially all positive solutions (solutions with non negative components), in the topology $l^{2}$. This implies that energy is dissipated. Positive solutions are better motivated in comparison with fluid dynamic equations (see ).

Although the case $f>0$ and the fixed point are very interesting, it is also of interest to analyze the free (unforced) dynamic, namely system (1) without any forcing term. Physically, if we accept that a dyadic model like (1) may describe something of turbulence, the unforced case would correspond to free decaying turbulence, a widely observed phenomenon, see and references therein. The results of energy dissipation from the previous papers do not extend to this case, they really require $f>0$ and make use of the non-trivial fixed point in the computations. For simplicity, one would conjecture the case $f=0$ to be similar (exponential decay to zero), maybe with a different proof, but it turns out this is not the case.

For the unforced case, namely system (1), we first prove the following result: all finite energy positive solutions have energy which decays to zero as $t\rightarrow\infty$. Nevertheless, for sufficiently regular initial conditions, energy is conserved for small times, as was shown for instance in , and .

Even a large class of solutions starting with infinite energy immediately enter $l^{2}$, namely they immediately get finite energy, and then continue their process of dissipation to zero.

However, the decay of energy to zero is no more exponential. By means of a scaling argument we prove an upper bound on the decay of the energy of the order $t^{-2}$. Then we prove a weak form of lower bound of the same order, which includes in particular the property $\int_{0}^{\infty}X_{n}\left(t\right)dt=\infty$ for all $n$ greater than some $n_{0}$. Thus exponential decay is ruled out.

Then we investigate positive self-similar solutions, of the form

We prove the existence of such solutions in the space $l^{2}$. The proof is highly non-trivial. The energy of these solutions decays exactly as $t^{-2}$ and we conjecture that this should be the case for all positive solutions.

The existence of such self-similar solutions relates also to the problem of blow-up and coalescence, but in a rather controversial way from the viewpoint of the physical interest. If we decide that only solutions with positive components have physical meaning, our self-similar solutions do not contribute to the problem of blow-up. If on the contrary we consider system (1) as a nonlinear model to be understood for any kind of initial condition, we have proved (by a simple inversion of time) that there exist solutions defined on the time interval $\left(-\infty,t_{0}\right)$ of the form

with $\left(a_{n}\right)\in l^{2}$. The components $X_{n}\left(t\right)$ are all negative. These solutions blow up in finite time, irrespective of the topology, in the sense that all components blow up in finite time. This result is much stronger than the blow-up results of the previous literature on dyadic models.

The existence of positive self-similar solutions also implies coalescence: we prove that at every point of a self-similar solution there is the coalescence of at least another solution (which cannot be positive).

Let us finally mention other properties of the self-similar solutions we construct and some open problems.

Given $t_{0}$, we prove that there is a unique solution of the form (2) in $l^{2}$, with strictly positive components. We also prove that the components $a_{n}$ decay as

so, in a sense, these are ‘Kolmogorov type decaying solutions’ . There are two degrees of freedom, however, in these solutions. One is the given value of $t_{0}$. The other is that we could take

and prove that there exists a unique (for given $t_{0}$) self-similar solution with these first components equal to zero and all the others strictly positive. In this case, closer inspection gives us

(numerical experiments give us $R\approx 0.885765931$).

Thus the picture is that in $l^{2}$ travel a family of self-similar solutions depending on the continuous parameter $t_{0}$ and the discrete parameter $n_{0}$.

What happens to all other solutions? Do they approach this set of special solutions? In this case, does a generic solution select one particular self-similar solution and get closer and closer to it, or does it slowly shift from one self-similar solution to the other? A sufficiently fast convergence to one self-similar solution would imply that Kolmogorov K41 scaling holds true for this simple dyadic model. But a slow convergence or a shift along different self-similar solutions could produce small deviations from Kolmogorov scaling. Further research is necessary to clarify these issues.

Concepts of solution, their existence, positivity and energy inequality

If $X$ is a componentwise solution, from system (1) we have

We say that a componentwise solution $X$ has finite energy for positive times if $X\left(t\right)\in H$ for all $t>0$. If also $X^{0}\in H$, we call $X$ a finite energy solution.

for every $n\geq 1$ and $t\geq 0$ we have

if $X_{n}^{0}>0$ for some $n\geq 1$, then $X_{m}(t)>0$ for all $m\geq n$ and all $t>0$.

For every $X^{0}\in H$, there exists at least one finite energy solution of system (1) with initial condition $X^{0}$, with the property

Energy dissipation

by a simple rearrangement of the series and the condition $X_{0}=0$. This rearrangement is rigorous if the solutions live in a sufficiently regular space (see for example ). Such kind of regularity may hold for small times if the initial condition is very regular itself (see ), but for sufficiently large times we prove that solutions dissipate energy. The intuitive mechanism is a very fast shift of energy from small to large $n$ components.

We give two results of energy dissipation for positive solutions: infinite initial energy becomes finite immediately; the energy of a finite energy solution tends to zero as $t\rightarrow\infty$. Although the degree of infinity of the energy of initial conditions can be generalized, for the simplicity of statements we restrict ourselves to $X^{0}$ of class $l^{\infty}$: the norm $||X^{0}||_{\infty}:=\sup_{n}\left|X_{n}^{0}\right|$ is finite.

If $X$ is a positive finite energy solution, then

Moreover, given $L>0$ and $\alpha>0$, there exists $\bar{t}>0$ depending only on $L$ and $\alpha$ such that for all positive finite energy solutions $X$ with $|X(0)|_{H}\leq L$ we have $\left|X\left(\bar{t}\right)\right|_{H}^{2}\leq\alpha$.

The proof of both statements is based on the following lemma

Then there exist two summable sequences of positive numbers $\{a_{n}\}_{n\geq 1}$ and $\{s_{n}\}_{n\geq 1}$ depending only on $L$, such that:

for all $n\geq 1$, for all $t>0$ and for all $\varepsilon\in(0,1]$ one has

Proof. For all $n$, let $m_{n}^{2}:=\phi_{n}(0)\vee L$, so that by equation (4), $X_{n}(t)\leq m_{n}$ for all $t\geq 0$.

We observe that it is possible to find two summable sequences $\{a_{n}\}_{n}$ and $\{s_{n}\}_{n}$ such that for all $n\geq 1$:

It is enough to set $s_{n}=2^{-n/4}$ and $a_{n}=C2^{-n/4}$ with $C$ a suitable constant, and recall that by hypothesis $m_{n}\leq L\sqrt{n}$. We observe that without changing the sequences, a fortiori for all $\varepsilon\leq 1$,

Part 1. Given the two sequences, we now show that the upper bounds (6) hold.

Let $h=\varepsilon^{-2}s_{n}$. Since $\phi$ is nonincreasing, for all $s\in[0,h]$,

hence, if for some $s$ inside the interval, $X_{n}^{2}(t+s)\leq\varepsilon a_{n}$ we are done.

On the other hand, let us suppose that $X_{n}^{2}(t+s)>\varepsilon a_{n}$ for all $0\leq s\leq h$. One has

For sake of notation simplicity, let $a=\varepsilon a_{n}k_{n}$ and let $\lambda=k_{n+1}m_{n+2}$. Then, by Theorem 3-i:

We need a lower bound for $X_{n+1}$. In the interval $[t;t+h]$ we know that

By substituting into the integral above one gets

Substituting next $h$, $a$ and $\lambda$ and recalling the condition (8), we finally get

Part 2. Let $M\geq 1$ and define the sequence $\{t_{n}\}_{n\geq M-1}$ by $t_{M-1}=0$ and $t_{n}=\varepsilon^{-2}\sum_{k=M}^{n}s_{k}$, for $n\geq M$. We substitute $t=t_{n-1}$ inside the inequality (6):

Adding the above inequalities for $n$ from $M$ to any number $N>M$, one has

hence letting $N$ go to infinity we get (7). The proof of the lemma is complete.

We are now ready to prove the above theorems. Consider the assumptions of Theorem 5. By letting $\varepsilon=1$ in the second part of Lemma 7, one has that $X(t)\in H$ for $t\geq\sum_{k=M}^{\infty}s_{k}$. Letting $M$ go to infinity we get the thesis of Theorem 5.

As to Theorem 6, let us prove that for all $\alpha>0$ there exists some $\bar{t}>0$ such that $\phi_{\infty}(\bar{t})\leq\alpha$. Since $||X(0)||_{\infty}\leq|X(0)|_{H}\leq L$, we let $M=1$ in the second part of Lemma 7: we only need to choose $\varepsilon$ in such a way that $\varepsilon\sum_{n=1}^{\infty}a_{n}\leq\alpha$. One gets $\bar{t}=\varepsilon^{-2}\sum_{k=1}^{\infty}s_{k}$. The proof is complete.

Bounds on the decay of energy as t→∞→𝑡t\rightarrow\infty

In this section we prove a bound from above and another from below, for the decay of energy as $t\rightarrow\infty$, which essentially say that solutions decay as $t^{-1}$. The results are restricted to positive componentwise solutions. The first result is due to a scaling argument based on the fact that the nonlinearity is homogeneous of degree two.

Proof. First, by Theorem 5, the solution has finite energy for positive times. Hence, by Theorem 6, there is a time $t_{0}$ such that $\left|X\left(t_{0}\right)\right|_{H}^{2}\leq 1$. It is thus not restrictive to prove the theorem in the case $\left|X^{0}\right|_{H}^{2}=1$.

First a general fact: from Theorem 6 we know that there exists $\bar{t}>0$ such that for all initial conditions $X^{0}$ with $\left|X^{0}\right|_{H}^{2}=1$ we have

This means that $Y$ is a finite energy solution of system (1). Moreover, $\left|Y\left(0\right)\right|_{H}^{2}=1$. Hence

for every $k\geq 1$. Since $\alpha_{i}\leq\frac{1}{2}$, we have

Recall that energy inequality holds for all positive finite energy solutions, see Theorem 3. Hence for all $t\geq\frac{2\bar{t}}{\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{k}}$ we have

If we restrict to $\frac{2\bar{t}}{\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{k}}\leq t\leq\frac{2\bar{t}}{\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{k}\cdot\alpha_{k+1}}$ we also have $\alpha_{1}^{2}\cdot\alpha_{2}^{2}\cdot\ldots\cdot\alpha_{k+1}^{2}\leq\frac{4\bar{t}^{2}}{t^{2}}$. Hence

for all $\frac{2\bar{t}}{\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{k}}\leq t\leq\frac{2\bar{t}}{\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{k}\cdot\alpha_{k+1}}$. This implies the claim of the theorem. The proof is complete.

Thus, in particular, for every $n>n_{0}$,

By the upper bound of the previous theorem, there exists a constant $C>0$ such that

This implies the claims of the theorem. The proof is complete.

Self-similar solutions and related facts

We call self-similar any solution $X$ of the form $X_{n}(t)=a_{n}\cdot\varphi(t)$. It is easy to verify that self-similar solutions satisfying the equations (1) have the form

with $t_{0}<0$. We are interested in finite energy self-similar solutions, hence we require also

In the next section we prove the following result.

Given $t_{0}<0$, there exists a unique finite energy self-similar solution with $a_{1}\neq 0$. In general, given $t_{0}<0$ and $n_{0}\geq 0$, there exists a unique finite energy self-similar solution with

(where the first conditions are meaningful only for $n_{0}>0$). In addition, the coefficients $a_{n}$ have the property

Thus we see that Kolmogorov scaling law (so called K41) appears in these special solutions, phenomenologically associated to decaying turbulence. But it is for us a very difficult open problem to understand whether all other solutions approach the self-similar ones and in which sense, so we cannot say how general this scaling property should be considered.

The existence of finite energy self-similar solutions is of conceptual interest in itself, in comparison with analogous investigations for Euler and Navier-Stokes equations. Also here, in this very simple context, the proof is highly non trivial. Apart from its intrinsic interest, the existence of such solutions has a number of implications.

A second implication is the existence of solutions that blow-up backward in time. This is of interest for two reasons. To explain them let us first clarify what happens to solutions when we reverse time.

We may consider system (1) for negative times, $t\leq 0$, and give a definition of componentwise solution exactly as for $t\geq 0$. All definitions and theorems can be rewritten for negative times. One also has the following correspondence between the forward and backward problem: let $X=\left(X\left(t\right)\right)_{t\geq 0}$ be a componentwise solution of system (1), for $t\geq 0$, as usual. Then $Y=\left(Y\left(t\right)\right)_{t\leq 0}$ defined as

is a componentwise solution for $t\leq 0$. Indeed, by the purely quadratic nature of the equation,

Thus, any positive solution over $[0,\infty)$ gives rise to a negative solution over $(-\infty,0]$, and vice versa.

Theorem 10 ensures that there exists a self-similar solution $X_{n}(t):=\frac{a_{n}}{t-t_{0}}$ with $t_{0}<0$ and $a_{n}>0$ for all $n>n_{0}$. It is easy to check that $X_{n}(t)$ is a componentwise solution on the open interval $(t_{0},+\infty)$. The energy is finite for all $t>t_{0}$ and $\lim_{t\to\infty}|X_{n}(t)|=0$ whereas $\lim_{t\to t_{0}^{+}}X_{n}(t)=+\infty$ for all $n>n_{0}$. With time inversion (10) it is possible to define $Y(t):=-X(-t)$ for all $n$ and $t\in(-\infty,-t_{0})$. $Y(t)$ is a componentwise solution of (1) on the open interval $(-\infty,-t_{0})$ and $\lim_{t\to-\infty}|Y_{n}(t)|=0$ whereas $\lim_{t\to-t_{0}^{-}}|Y_{n}(t)|=+\infty$. This means that $Y_{n}$ blows-up in finite time. Notice that every component blow-up, not only some norm of the solution.

If $X$ is a self-similar solution of the form (9), all its values are coalescence points, in the sense that for all $t_{1}\geq t_{0}$ there exists a finite energy solution $X^{t_{1}}$, defined for $t\in(-\infty,t_{1}]$, such that $X^{t_{1}}\left(t_{1}\right)=X\left(t_{1}\right)$ and $X^{t_{1}}(t)\neq X(t)$ on $(t_{0},t_{1})$.

Finally, we state the blow-up for negative self-similar finite energy solutions.

Given $t_{0}>0$ and $n_{0}\geq 0$, there exists a unique negative finite energy self-similar solution, defined on $[0,t_{0})$ with

Proof. Apply the inversion (10) to the solution given by Theorem 10.

Existence and uniqueness of self-similar solutions

In order to prove Theorem 10 it is best to restate the problem in terms of properties for sequences of positive numbers.

If a positive componentwise solution is of the form (9), then

for all $n$. It is indeed possible for the first terms $a_{1},a_{2},\dots,a_{n_{0}}$ to be zero, but by induction, if $a_{n_{0}+1}>0$ then the subsequent coefficients must satisfy

To simplify matters and take into account $n_{0}$, let

so that the condition becomes $\widetilde{a}_{0}=0$, $\widetilde{a}_{1}>0$ and

Since, given $a_{n_{0}+1}$ or $\widetilde{a}_{1}$, this recurrence uniquely defines the sequence, in order to prove Theorem 10 we have to show that there exists a unique positive number $\widetilde{a}_{1}$ such that the sequence $\{a_{n}\}_{n}\in H$. We will prove the following.

There exists a unique real number $\gamma$ such that the sequence $\{a_{n}\}_{n}$ is in $H$ iff $\widetilde{a}_{1}=\gamma$ (equivalently, iff $a_{0}=a_{1}=\dots=a_{n_{0}}=0$ and $a_{n_{0}+1}=2^{-n_{0}-1}\gamma$).

Moreover, let $\beta=2^{-1/3}$. One can find $R>0$ and a strictly decreasing bijective function $h:(0;R]\rightarrow[0;\infty)$ such that

if $\{a_{n}\}_{n}\in H$, then $a_{n}=2^{-n}h(\beta^{2(n-n_{0})}R)$ for all $n>n_{0}$. In particular $a_{n}\sim C_{n_{0}}\beta^{n}$, for $n\rightarrow\infty$, with $C_{n_{0}}=\beta^{2n_{0}}/R$.

Numerical computations give $\gamma\approx 0.917576296$.

It is clear that Theorem 10 follows immediately from Theorem 13. The proof of the latter requires some work and will follow from Theorem 17 below. Three technical lemmas will be needed.

We start by looking for another sequence $\{d_{k}\}_{k\geq-1}$ (not depending on $a_{n}$), satisfying the peculiar relation below:

Equations (13) uniquely define a sequence of real numbers $\{d_{k}\}_{k\geq-1}$ such that $d_{k}\geq 0$ for $k\geq 0$. The power series $\sum_{k=0}^{\infty}d_{k}x^{k}$ has a positive convergence radius $R>0$. If we let $h(x):=-\sum_{k=-1}^{\infty}d_{k}x^{k}$, the function $h$ is defined on the interval $(0;R)$ where it is analytic, nonnegative and strictly decreasing, with $h(0^{+})=+\infty$ and $h(R^{-})=0$ (so that it can be continuously extended on $(0;R]$).

Proof. The second one of (13) uniquely defines $d_{k+1}$ as a function of the previous terms. Truly, the coefficient of $d_{k+1}$ is $d_{-1}(8-\beta^{2k-2}-\beta^{4k+2})$ which is always nonzero.

With some algebraic manipulations, the above system of equations can be rewritten as:

All $\alpha_{k,i}$ are positive and so $d_{k}\geq 0$ for $k\geq 0$. We notice moreover that $\lim_{k\rightarrow\infty}\alpha_{k,i}=1$ uniformly in $i$ (exponentially in $k$, see the Appendix).

The recursion (14) is similar to the classical Catalan sequence, but since we have only some asymptotic control on the coefficients $\alpha_{k,i}$, and since the behaviour of $\{d_{k}\}_{k}$ strongly depends on the first values (even its convergence radius does), we need some explicit investigation of its properties.

For $k\geq 0$, let $d^{\prime}_{k+1}:=\frac{1}{2}\sum_{i=0}^{k}(\alpha_{k,i}-1)d_{i}d_{k-i}$ and let $d^{\prime}_{0}=d_{0}$.

Let $g(x):=\sum_{k=0}^{\infty}d_{k}x^{k}$ so that $h(x)=x^{-1}-g(x)$. Let $\hat{g}(x):=\sum_{k=0}^{\infty}d^{\prime}_{k}x^{k}$. By the third one of (14) it follows that

If $n$ is such that $|\alpha_{k,i}-1|<\epsilon$, we have

Writing also the corresponding lower bound and letting $\epsilon\rightarrow 0$ and $n\rightarrow\infty$ accordingly, yields

The above formula is true for all (complex) $x$ inside the radius of convergence of $g$ (in the Appendix we show that $\hat{g}$ has a convergence radius $\beta^{-2}$ times larger than $g$).

For all $x$ inside the convergence radius of $g$, $g(x)$ must be one of the roots of the above degree 2 polynomial, i.e.

(The other root does not satisfy $g(0)=d_{0}$.)

In the Appendix we show that the radius of convergence of $\hat{g}$ is greater than 1 and that inside the unit circle of the complex plane $1-2z\hat{g}(z)$ has only one zero, which is in fact some real point $R\in(4/5,1)$. It follows that $R$ is the radius of convergence of $g$, that $h$ is defined on $(0;R)$ and that $h(R^{-})=0$. The other properties follow from inspection of the first coefficients $d_{k}$.

From now on $h$ is intended to be extended up to $R$. Since $h$ is bijective and has image $[0,\infty)$ and recalling that $\widetilde{a}_{n}>0$ for all $n$, it makes sense to compute $h^{-1}(\widetilde{a}_{n})$.

Let $\{\widetilde{a}_{n}\}_{n\geq 0}$ be any positive sequence satisfying (12), not necessarily with $\widetilde{a}_{0}=0$. For $n\geq 0$ let $\lambda_{n}:=h^{-1}(\widetilde{a}_{n})\in(0,R]$. The numbers $\lambda_{n}$ satisfy:

Proof. Recall that $h(x)=-\sum_{k=-1}^{\infty}d_{k}x^{k}$,

Using relation (13) for $k\geq-1$ and since $\beta^{2k+2i}\equiv 4$ if $k=-2$ we get:

The ratio between $\lambda_{n}$ and $\beta^{2}\lambda_{n-1}$ plays a crucial rôle in what follows. Next lemma characterizes its logarithm.

Let $\{\widetilde{a}_{n}\}_{n\geq 0}$ be any positive sequence satisfying (12), not necessarily with $\widetilde{a}_{0}=0$. For $n\geq 1$ let $\lambda^{\prime}_{n}:=\log\lambda_{n}-\log\lambda_{n-1}-\log\beta^{2}$. The sequence $\lambda^{\prime}_{n}$ can present two behaviours only:

If for any $n\geq 1$, $\lambda^{\prime}_{n}=0$, then $\lambda^{\prime}_{m}=0$ for all $m\geq 1$.

If $\lambda^{\prime}_{n}\neq 0$ for some $n\geq 1$, then there exist two constants $c,d>0$ such that for all $k\geq 0$:

Proof. If $\lambda^{\prime}_{n}=0$, then of course $\lambda_{n}=\beta^{2}\lambda_{n-1}$, hence (16) reduces to:

Since $h$ is injective, $\lambda_{n+1}=\beta^{2}\lambda_{n}$, that is $\lambda^{\prime}_{n+1}=0$. This proves by induction the first part of the lemma, for $k\geq n$. If the same were not true for some $k<n$, then by the second part it would have been $\lambda^{\prime}_{n}\neq 0$.

In particular note that if $\widetilde{a}_{0}=0$ (yielding $\lambda_{0}=R$) and $\lambda_{1}=\beta^{2}R$ (yielding $\lambda^{\prime}_{1}=0$), then by the first part and recurrence (12), $h(\beta^{4}R)=\widetilde{a}_{2}=2$. This will be used in the second part.

For proving the second part, let $\psi(x):=\log h(e^{x})$ and let $\Delta_{x}(y):=h(xe^{y})/h(x)$.

$\Delta_{x}(y)$ on the other hand is defined for $x\in(0;R)$ and $y\in(-\infty;\log R-\log x)$ and this last interval always contains a neighbourhood of 0. $\Delta_{x}(\cdot)$ is analytic and decreasing on all its domain, moreover we have:

and $\psi^{\prime}(\xi)<-1$. By virtue of (16) we can write:

We need to prove that $K_{n}$ is always nonnegative, and bounded below by a positive constant for $n$ large enough.

Since $\lambda_{n-1}\leq R$, $h(\beta^{4}\lambda_{n-1})\geq h(\beta^{4}R)=2$. So $K_{n}\geq 0$ for all $n\geq 1$. For $n\geq 3$, $\widetilde{a}_{n-1}\geq 2$ by (12), hence $\lambda_{n-1}\leq\beta^{4}R$, so:

Going back to the previous expression, if $\lambda^{\prime}_{n}>0$, since both coefficients are positive,

A posteriori we get $\lambda^{\prime}_{n+1}<0$ and so, after letting $x:=e^{\lambda^{\prime}_{n}}$,

The above inequality holds for all $x\geq 1$ and follows by comparing the two functions, their derivatives and by the fact that $K_{n}<1$, in fact:

Finally $e^{-\lambda^{\prime}_{n+1}}>\widetilde{\theta}_{n}(x)=e^{(1+K_{n})\lambda^{\prime}_{n}}$ and hence

On the other hand, if $\lambda^{\prime}_{n}<0$, with analogous reasoning:

Summing up things, for any $\lambda^{\prime}_{n}\neq 0$, the sequence $\{\lambda^{\prime}_{n+k}\}_{k}$ has alternating signs, moreover:

The second part of the lemma is now proved by induction on $k$, recalling that $K_{n}\geq K>0$ for $n\geq 3$.

We are finally able to make the main statement.

A sequence $a_{n}$ satisfying (11) with $a_{n_{0}}=0$ and $a_{n_{0}+1}>0$ can present only two behaviours:

If for any $m\geq 1$, $\lambda^{\prime}_{m}=0$, then there exists $c>0$ such that for all $n>n_{0}$, $a_{n}=2^{-n}h(\beta^{2(n-n_{0})}R)$; in this case $a_{n}\rightarrow 0$ for $n\uparrow\infty$ and in the limit $a_{n}=C_{n_{0}}\beta^{n}+O(\beta^{3n})$, with $C_{n_{0}}=\beta^{2n_{0}}/R$.

If for any $m\geq 1$, $\lambda^{\prime}_{m}\neq 0$, then either the odd or the even terms of $a_{n}$ diverge more than exponentially and the power series $\sum_{n}a_{n}z^{n}$ has zero radius of convergence.

In the first case, for all $n\geq 1$, $\lambda^{\prime}_{n}\equiv 0$, hence $\lambda_{n}=\beta^{2n}\lambda_{0}=\beta^{2n}R$, by which $a_{n+n_{0}}=2^{-n_{0}-n}\widetilde{a}_{n}=2^{-n_{0}-n}h(\beta^{2n}R)$. The limit behaviour of $a_{n}$ follows from $h(x)=x^{-1}+O(1)$ for $x\downarrow 0$.

In the second case, notice that for all $k\geq 2$,

Let $n=m$ or $n=m+1$ so that $\lambda^{\prime}_{n}<0$ and hence $\lambda^{\prime}_{n+2k}<0$ for all $k$ and $\lambda^{\prime}_{n+2k}\rightarrow-\infty$ for $k\uparrow\infty$. By the properties of $\psi$, there exists $x_{0}$ such that $x\leq x_{0}\Rightarrow\psi(x)\geq-x-1$. We deduce that, for $k$ large enough:

so the power series $\sum_{k}\widetilde{a}_{k}z^{k}$ has zero radius of convergence and the same for $\sum_{k}a_{k}z^{k}$.

By Theorem 17, we are in the first case iff $\lambda^{\prime}_{1}=0$ that is if $\widetilde{a}_{1}=h(\beta^{2}R)$. In the first case $\{a_{n}\}\in H$, since $a_{n}=C_{n_{0}}\beta^{n}+O(\beta^{3n})$ and $\beta<1$. In the second case of course the sequence is not in $H$.

Appendix

The first claim is an obvious consequence of identity (3).

It has a unique solution: local existence and uniqueness comes from Cauchy theorem (the vector field on the right-hand-side is locally Lipschitz continuous), global existence is a consequence of the bound on maximal solutions derived from

which is easily proved by computing $\frac{d}{dt}\sum_{n=1}^{N}X_{n}^{2}(t)$. Denote by $X^{N}$ the unique solution.

Moreover, for the solution $X^{N}$, for every $n\in\left\{1,...,N\right\}$ we have

Hence, being positive, we get $\frac{d}{dt}\sum_{j=1}^{n}\left(X_{j}^{N}\right)^{2}\leq 0$, namely

On a bounded interval $[0,T]$, consider now the family $\left(X_{n}^{N}\right)_{N>n}$, for a given $n\geq 1$. The assumptions of Ascoli-Arzelà theorem are satisfied for this family. Indeed, equi-boundedness has been proved above in (18); equi-uniform-continuity (equi-Lipschitz continuity, in fact) comes from the identity

Thus the functions $X_{n}(\cdot)$ are continuously differentiable and satisfy system (1). Of course they are non negative, being the uniform limit of non negative functions. Continuation from an arbitrary bounded time interval to all $t\geq 0$ is classical.

Finally, we have to prove properties (i) and (ii) of the theorem, for any positive componentwise solution $X$. As to (i), we repeat the argument used above for $X^{N}$: for every $n\geq 1$, from system (1), we have

hence, $X$ being positive, we get $\frac{d}{dt}\sum_{j=1}^{n}\left(X_{j}^{N}\right)^{2}\leq 0$ and thus (i) is proved.

For $X_{n+1}(t)$ we use the inequality (again a consequence of (3))

and the fact that $X_{n}(t)>0$ for all $t>0$. By induction we get the result. The proof of the theorem is complete.

2 Proof of Theorem 4

We introduce the same finite dimensional system as above. We do not have anymore the positivity property, but we still have (17). Since the (global) initial energy is finite, we have

for all $t\geq 0$ and $N\geq 1$. This implies again a bound on single components:

for every $t\geq 0$, $N\geq 1$ and $n=1,...,N$. Having this bound, we proceed as above and prove, on a given $\left[0,T\right]$, the existence of a componentwise solution $X$, with $X_{n}^{N_{k}}\rightarrow X_{n}$ uniformly on $\left[0,T\right]$ as $k\rightarrow\infty$, along a diverging sequence

first by truncating the sum up to a given value $R$ and taking the limit in $k$, then sending $R$ to infinity. Hence in particular $X\left(t\right)\in H$ for all $t\geq 0$.

for every $n\geq 1$ and $t\geq 0$. This implies that $X$ is finite energy and satisfies (5). The proof is complete.

We study their behaviour for $k$ large and look for bounds. Notice that the denominators are all positive for $k\geq 0$.

There exists $C>0$ such that for all $k\geq 0$ and all $0\leq i\leq k$,

When $k\geq 1$ and $i=0$ or $i=k$, $\alpha_{k,i}<1$, in fact:

which is positive. The last factor is decreasing in $k$ and is equal to $4/3$ when $k=1$, so for $k\geq 1$

When $k\geq 2$, $0<i<k$, we have $\alpha_{k,i}>1$, since, recalling that $\cosh(x)\leq e^{|x|}$,

The right factor is decreasing in $k$, hence for $k\geq 2$

The right factor is increasing in $k$ for $k\geq 2$ and it tends to 1, so

The third case, namely $k=0$ is included by adjusting $C$.

4 Estimates on the convergence radius of g^^𝑔\hat{g} and g𝑔g

The proof is based on Rouché’s theorem for holomorphic functions.

(It follows from the fact that $\max_{i}\alpha_{k,i}=\alpha_{k,[k/2]}$ and that $\alpha_{k,[k/2]}$ is decreasing in $k$.)

We introduce another auxiliary sequence $\{D_{k}\}_{k}$.

Let $G(z):=\sum_{k=0}^{\infty}D_{k}z^{k}$. The radius of convergence of $g$ is larger than the radius of convergence of $G$ which is larger than $\beta^{2}$.

Proof. Thanks to (14), by induction $0\leq d_{k}\leq D_{k}$ for all $k$.

The right sign is ‘$-$’ for all $z$ since $G(0)=d_{0}$ and $G$ is continuous. After some algebraic manipulations we write

The radius of convergence of $G$ is the distance from the origin of the nearest zero of the radicand. The roots of the latter are both real and positive:

The smallest one is the convergence radius of $G$ and one can verify that

By the estimates on $\alpha_{k,i}$, we have $|1-\alpha_{k,i}|<C\beta^{2k}$; we also need some lower bound and, for $k\geq 1$, $\alpha_{k,i}\geq\alpha_{1,0}$. Then for all $k\geq 1$:

Thanks to (22), this proves that the radius of convergence of $\hat{g}$ is greater than 1.

Putting together the last two upper bounds we get

The value of $G(\beta^{2}|z|)$ can be computed thanks to (21), and the bounds one gets are those in the lemma.

The first factor is never 0 because $|c_{0}c_{1}+c_{1}c_{2}\rho^{2}|-|4c_{0}c_{2}\rho|>0$

Setting $\rho=1$ and $\rho=4/5$ and computing values we get the thesis.

Proof of Theorem 19.

References