Abstract
The Cauchy problem for the linearization around one of its equilibria of a non linear system of equations, arising in the kinetic theory of a condensed gas of bosons near the critical temperature, is solved for radially symmetric initial data. As time tends to infinity, the solutions are proved to converge to an equilibrium of the same linear system, determined by the conservation of total mass and energy. The asymptotic limit of the condensate’s density is proved to be larger or smaller than its initial value under a simple and explicit criteria on the initial data. For a large set of initial data, and for values of the momentum variable near the origin, the linear approximation n(t) of the density of the normal fluid behaves instantaneously as the equilibria of the non linear system.
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1 Introduction
Correlations between the superfluid component and the normal fluid in a uniform condensed Bose gas, at temperature below but close to the condensation temperature, and for a small number density of condensed atoms, may be described by the equation
where n(t, p) represents the density of particles in the normal gas that at time \(t>0\) have momentum p and \(n_c(t)\) is the density of the condensate at time t, that satisfies
The collision integral \(I_3\) in (1.1) was first derived in [9] and [20] and their treatment was afterwards extended to a trapped Bose gas. By including Hartree–Fock corrections to the energy of the excitations the so called ZNG system was obtained (cf. [18]). On the interest of system (1.1), (1.4) for the description of condensed Bose gases see also [19, 24, 26].
Of course, even in presence of the superfluid component, interactions between particles in the normal gas continue to take place. In order to take them into account the Nordheim–Boltzmann collision operator should be added at the right hand side of the equation (1.1) as it is done in the references above. Such an operator is generally written as
It may be of some interest to understand the respective effects of each of the two terms \(I_3\) and \(I_4\) at the different stages of the evolution of the condensate and its thermal cloud. It is known that the term \(I_4\) provokes by itself the finite time blow up of some solutions of Nordheim equation (cf. [15]). The finite time blow up phenomena does not seem to be expected in presence of \(I_3\) alone. On the contrary, as proved below, the Cauchy problem for the linearization of (1.1, 1.4) around an equilibria is globally well posed in a suitable functional space; its solutions conserve the natural physical quantities and converge to a suitable equilibrium. Some regularizing effects are also observed as shown in Theorem 1.9 and Theorem 3.2, below. Such effects are not observed for \(I_4\) (cf. [12]). However, a detailed understanding of the respective mathematical properties of \(I_3\) and \(I_4\) is still missing, and that makes difficult to precisely foresee what may be expected in presence of both terms. Several of these questions are considered in [3] where in particular some regularizing property of the term \(I_3\) is proved in a different functional setting.
Other theoretical models do exist to describe Bose gases in presence of a condensate (cf. [23]) but ZNG system, and (1.1), (1.4) in particular, are very appealing by their simplicity and are well suited for analytical PDE methods.
The two functions of time,
give, respectively, the total number of particles and the total energy of the normal fluid part in the gas, with density function n(t, p). The total number of particles at time t in the system condensate-normal fluid is \(n_c(t)+{\mathcal {N}}(t)\) and its total energy is \({\mathcal {E}}(t)\). It formally follows from (1.1), (1.4) that these two quantities are constant in time: \(\mathcal E(t)={\mathcal {E}}(0)\) and \(n_c(t)+{\mathcal {N}}(t)=n_c(0)+{\mathcal {N}}(0)\) for all \(t>0\). This corresponds to the conservation of the total mass and energy property that is satisfied by the particle system in the physical description (cf. for example [9]). It is also well known that equation (1.1) has a family of non trivial equilibria,
where the mass of the particles is taken to be \(m=1/2\) and \(\beta \) is a positive constant related to the temperature of the gas whose particle’s density is at the equilibrium \(n_0\). It is easily checked that \(R(p, p_k, p_\ell )\equiv 0\) in (1.3) for \(n=n_0\).
Our purpose is to prove the existence of classical solutions to the Cauchy problem for the “radially symmetric linearization” of (1.1)–(1.4) around an equilibrium \(n_0\), and describe some of their properties. Such linearization is deduced through the change of variables (cf. [10, 13])
and keeping only linear terms with respect to u in (1.1). Since equation (1.4) is linear with respect to \(n_c\) its linearization (1.15) follows by just keeping the terms of \(I_3(n_0(p)+n_0(p)(1+n_0(p))|p|^2u (t, |p|))\) that are linear with respect to u. This finally reads, for dimensionless variables in units which minimize the number of prefactors, as
where, for all \(x>0\), \(y>0\), \(x\not =y\),
With some abuse of notation the function \(\nu _0(x)=n_0(p)\) is still denoted \(n_0(x)\).
For uniform condensed Bose gases at very low temperatures and large number density of condensed atoms, the limit of the ZNG system deduced in references [9] and [20] is slightly different. In that limit, the interactions involving only particles in the thermal cloud are neglected, ZNG system is reduced to (1.1), (1.4), up to lower order terms but with a different expression for \(R(p, p_1, p_2)\) in (1.3) since the dispersion relation and scattering amplitude are different. Related works in the mathematical literature for the isotropic case may be found in [1, 2, 27]. The non isotropic linearized system around an equilibrium is treated in [11].
1.1 The Isotropic Linearization of (1.1), (1.4).
The linearization of (1.1), detailed in [13] and recalled in [10], is briefly presented here for the sake of completeness. When \(R(p, p_1, p_2)\!-\!R(p_1, p, p_2)\!-\!R(p_2, p_1, p)\) is written in terms of the function \(\Omega \) defined in (1.11) and only linear terms in \(\Omega \) are kept, the result is
where a is the s-wave scattering length, \(k=|p|\) and \(k'=|p'|\). The functions \({\mathscr {U}}(k, k')\) and \({\mathscr {V}}(k, k')\) have a non integrable singularity along the diagonal \(k=k'\). However, these singularities cancel each other when the two terms are combined as in (1.20) as far as it is assumed that, for all \(t>0\), \(\Omega (t)\in C^\alpha (0, \infty )\) for some \(\alpha >0\). However, the integrand \(\left( {\mathscr {U}}(k, k')\Omega (t, k')-\mathscr {V}(k, k')\Omega (t, k)\right) \) cannot be split as for example in the linearization of Boltzmann equations for classical particles. However an explicit calculation shows that, for all \(k>0\),
from which we deduce, for all \(k>0\),
We may then write,
Since equation (1.4) is linear with respect to \(n_c\), its linearization (1.15) follows by just keeping the terms of \(I_3(n_0(p)+n_0(p)(1+n_0(p))|p|^2u (t, |p|))\) that are linear with respect to u. The linearized system then reads as
or, in terms of \({\tilde{\Omega }} (t, k)=\Omega (t, k)/k^2\),
Since \(\left( k^2n_0(k)(1+n_0(k))\right) ^{-1}=4k^{-2}\sinh ^2\left( \frac{\beta k^2}{2}\right) \),
Use of the change of variables (1.11–1.12) in (1.24) yields system (1.13), (1.15) for \((u, p_c)\), after scaling the time variable to get rid of some positive numerical constants.
1.2 A Nonlinear Approximation
Another approximation of the system (1.1), (1.4) is possible where, in the equation (1.4), the function n is replaced by \(n_0+n_0(1+n_0)x^2 u\) in the nonlinear collision term \(I_3\) given by (1.2) to obtain the system
instead of (1.13), (1.15). In that way the non linearity of \(I_3\) in the equation for \({\tilde{p}}_c\) is kept, but the conservation in time of \({\tilde{p}}_c(t)+N(t)\) does not hold, and so an important global property of the original system (1.1)–(1.4) is lost. As a consequence the time existence of the solutions to system (1.26), (1.27) cannot be proved to be \((0, \infty )\). Then, system (1.26), (1.27) is not too satisfactory to describe global properties of the particle’s system. But it may be a better approximation of the local properties of the solutions to the nonlinear system of equations (1.1), (1.4). In order to avoid any confusion, system (1.26), (1.27) is considered in the Appendix.
1.3 Further Motivation
It is known that for all non negative measure \(n _{ in }\) with a finite first moment, and for every constant \(\rho >0\), system (1.1)–(1.4) has a weak solution \((n(t), n_c(t))\) with initial data \((n _{ in }, \rho )\) that satisfies the conservation of mass and energy (cf. [7]). For all \(t>0\), n(t) is a non negative measure that does not charge the origin, with finite first moment, and \(n_c(t)>0\). However, one basic aspect of the non equilibrium behavior of the system condensate–normal fluid is the growth of the condensate after its formation (cf. [4, 18, 23] and references therein). In the kinetic formulation (1.1)–(1.4), this behavior is driven by the integral of \(I_3(n)\) in the right hand side of equation (1.4). A was shown in [25], the behaviour of that term crucially depends on the behavior of n(t, p) as \(|p|\rightarrow 0\) (this was discussed also in [7, 21, 26]). If, for example, the measure n(t) is a radially symmetric, bounded function near the origin then, from a simple use of Fubini’s Theorem,
for some constant \(C>0\) independent of n, and this would give a monotone decreasing behavior of \(n_c(t)\). On the contrary, as it is shown in [25], if the measure n(t) is a function such that
for some \(a(t)>0\), and satisfies some Hölder regularity property with respect to p in a neighborhood of the origin, then for some other constant \(C_1>0\) independent of n,
On the other hand, it was proved in [7] that if the measure \(|p|^2 n(t, p)\) has no atomic part and has an algebraic behavior as \(|p|\rightarrow 0\) then it satisfies (1.28). Both results in [25] and [7] assume some regularity of the solution n with respect to p, although no regular solutions to (1.1) are known yet. The existence of regular classical solutions to (1.1)–(1.4) satisfying (1.28) is one of the motivations of our present work.
Since (1.28) is the behavior of the equilibrium \(n_0\) (with \(a(t)\equiv \beta \)), it is natural to first consider the existence of such regular solutions for the linearization of (1.1) around \(n_0\). Because of the singular behavior (1.28) of \(n_0\) near the origin, the linear operator \({\mathcal {L}}\) in (1.14) has regularizing effects. Similar regularizing effects may be expected also in the non linear equation (1.1).
1.4 Basic Arguments and Main Results
The function \(p_c(t)\) in the right hand side of (1.13) may be absorbed by the change of variables,
to obtain
This equation may be written as
where, from (1.14), (1.33) and (1.35), the operator F is given by,
The equation (1.32) is then solved as a perturbation of
with a forcing term in (1.35).
Equation (1.37) is of interest by itself and has been considered in [10]. For example, one may consider what is seen somehow as the “classical approximation” of the nonlinear equation (1.1)–(1.3), where the factor of the Dirac’s measure in (1.3) is replaced by \(n_1n_2-n(n_2+n_1)\). The resulting equation also appears in the theory of wave turbulence for nonlinear optic waves (cf. for example [8, 28]), related to the Schrödinger equation. The function \(|p|^{-2}\) is an equilibrium solution of that equation, and (1.37) is its isotropic linearization, around that equilibrium.
It is proved in [10] that equation (1.37) has a fundamental solution \(\Lambda \in C((0,\infty ); L^1(0, \infty ))\) that satisfies (1.37) in \({\mathscr {D}}'((0, \infty )\times (0, \infty ))\) and almost every \(t>0\), \(x>0\). For all initial data \(f_0\in L^1\), there exists a weak solution of (1.37), denoted \(S(t)f_0\),
such that \(S(\cdot )f_0\in C([0, \infty ); L^1(0, \infty ))\), \(S(t)f_0\in C([0, \infty ))\) for all \(t>0\) and (1.37) is satisfied in \({\mathscr {D}}'((0, \infty )\times (0, \infty ))\). It was also proved that if \(f_0\in L^1(0, \infty )\cap L^\infty _{ loc }(0, \infty )\) then \(L(u)\in L^\infty ((0, \infty )\times (0, \infty ))\), \(u_t\in L^\infty ((0, \infty )\times (0, \infty ))\) and (1.37) is satisfied pointwise, for \(t>0\) and \(x>0\). (cf. Appendix for more detailed statements).
Once the Cauchy problem for equation (1.32) is solved using the semigroup S(t), the change of time variable in (1.30) is inverted to obtain the function u(t), and deduce \(p_c(t)\) using the conservation of mass of system and equation (1.15). Our first result, then, is as follows:
Theorem 1.1
Suppose that \(u_0\in L^1(0, \infty )\) satisfies
for some \(\theta \in (0, 1)\). Then, there exists a pair \((u, p_c)\),
such that, for each \(t>0\), u(t) is locally Lipschitz on \((0, \infty )\) and, for all almost every \(t>0\) and \(x>0\),
Moreover,
and there exists a function \(H\in L^\infty ((\delta , 0)\times (0, \delta ))\) for all \(\delta >0\), defined in (5.12), such that
For all \(\varphi \in C^1_0(0, \infty )\), the map \( t\mapsto \displaystyle {\int _0^\infty \varphi (x)u(t, x)\textrm{d}x} \) belongs to \(W _{ loc }^{1,1}(0, \infty )\) and for almost every \(t>0\),
Theorem 1.1, shows that the possible singular behavior as \(x^{-\theta }\) for some \(\theta \in (0, 1)\) of the initial data \(u_0\) at the origin is instantaneously regularized to \(u(t)\in L^\infty (0, \infty )\) for all \(t>0\). This is a direct consequence of the same property of the equation (1.37), where L may be seen as a pseudo differential operator with a symbol whose logarithmic growth at infinity induces a logarithmic regularizing effect (cf. Remark 6.6 below). The equation (1.37) enjoys a “local \(C^1\)” regularizing property (cf. Lemma (6.4) below), that cannot be extended to the equation (1.42), due precisely to its local character. However, the equation (1.42) regularizes the initial data \(u_0=\delta _1\) and gives a weak solution \(u \in C((0, \infty ); L^1(0, \infty ))\) (cf. Theorem 3.2). No regularizing effects have been observed for the isotropic linearization of the “classical approximation” of the Boltzmann Nordheim collision integral (1.5) around its stationary solutions ( [8, 16]).
In view of (1.11), (1.12) and (1.30), if u is a solution of (1.32) given by Theorem 1.1, the pair of functions
may be seen as an approximated solution of (1.1), (1.4), as far as \(n_0(p)(1+n_0(p))|p|^2 u (t, |p|)\) remains small compared to \(n_0\). In view of (1.9) it is natural to look at the quantities
These represent, respectively, the variation of the total number of particles and of energy caused by the initial perturbation \(n_0(p)(1+n_0(p))|p|^2 u(0)\) of the equlibrium \(n_0\). Let us also define,
The two following properties then hold true:
Corollary 1.2
Let \(u_0\) and u be as in Theorem 1.1 and n defined by (1.49). Then,
Corollary 1.3
Let \(u_0\) and u be as in Theorem 1.1. Then,
It follows from Corollary 1.3 that the mass and the energy variations due to the perturbation \(n_0(1+n_0)|p|^2u (t)\) tend to the mass and energy of \( C_*n_0(1+n_0)|p|^2\), and this however small the perturbation is at infinity, even if, for example, \(u_0\) is compactly supported. This kind of energy flux towards infinity could be expected, since it is well known to happen in the nonlinear homogeneous version of wave turbulence type of the system (1.1), (1.4) and is called direct energy cascade ( [8, 28] and [16, 27]).
Corollary 1.4
Let \(u_0\) and u be as in Theorem 1.1. Then, the function \(p_c\in C[0, \infty )\) is bounded on \([0, \infty )\) and
1.5 Some Remarks
Several remarks follow from the previous results.
1.5.1 On the Formal Approximation
The approximation of (1.1), (1.4) by (1.13), (1.15) may be expected to be reasonable only as long as the perturbation remains small with respect to \(n_0\),
and this requires \(x^2|f(t, x)|\) small for \(x\rightarrow \infty \). However, although it could be proved that (1.63) holds for small values of time if it holds at \(t=0\), it follows from (1.58) that it can not be true for all \(t>0\). Notice indeed that, for all \(R\ge R_0>0\),
and the right hand side tends to zero as \(t\rightarrow \infty \). If, on the other hand, we had \(x^2|u(t, x)|\le C\), for some \(C>0\), \(R_0>0\) and \(t_0>0\) for all \(x>R_0\) and all \(t>t_0\),
and then, for \(R>C/|C_*|\) and all \(t>t_0\),
and this would contradict (1.64). System (1.13), (1.15) may then be considered “close to” (1.1), (1.4) only for small values of t. Of course, u could be such that, for some C(t) that tends to \(\infty \) with t, \(x^2|u(t, x)|\le C(t)\) for all \(x>0\).
1.5.2 The Behavior of the Perturbation as \(\varvec{|p|\rightarrow 0}\)
For all \(t>0\), the perturbation \(n_0(1+n_0)\Omega (t, p)\) of \(n_0\) satisfies (1.28), for any \(f_0\) as in the hypothesis of Theorem 1.1, where a(t) is given in Proposition 4.16. The behavior \(|p|^{-2}\) at the origin (that of the equilibria of (1.1), (1.4)) is then instantaneously fixed, whatever the behavior at the origin of \(f_0\) may be, as far as the hypothesis of Theorem 1.1 are satisfied.
1.5.3 The Function \(p_c(t)\)
In view of Corollary 1.4, if the initial data \(u_0\) is such that
or equivalently,
then \(\lim _{ t\rightarrow \infty }p_c(t)>p_c(0)\), and conversely.
Condition (1.65) and its converse are both compatibles with \(n_0(1+n_0)x^2u_0\) being a small perturbation of \(n_0\). For example
and
1.6 Very Low Temperature and Large \(n_c\)
The linearization of system (1.1), (1.4) for large number density of condensed atoms and very low temperature may be performed following similar arguments as to those above (cf. [6] and [11]). No regularizing effects have been observed and the existence of a first positive eigenvalue and spectral gap for a suitable integrable operator ( [6] and [17]) provide a convergence rate to the equilibrium for a large set of initial data (cf. [11], Theorem 2.2) A necessary and sufficient condition on \(p_c(0)\) to have a global solution.
2 The Operator F
Equation (1.32) may be treated as a perturbation of (1.37) only whenever the term F(f) in (1.35) is bounded in spaces where the properties of the solutions of (1.37) may be used. The purpose of this section is to establish that this is the case.
Proposition 1.5
(i) For all \(g\in L^\infty (0, \infty )\), \(F(g)\in L^\infty (0, \infty )\) and
(ii) For all \(g\in L^1(0, \infty )\),
where
(iii) For all \(\theta \in [0, 1)\) there exists a positive constant \(C(\theta )\) depending on \(\theta \), such that, if \(g\in L^\infty _{ loc }(0, \infty )\) satisfies \( |||g|||_\theta <\infty \) then \(|||F(g)|||_\theta \le C(\theta )|||g|||_\theta .\)
(iv) For all \(g\in L^1(0, \infty )\cap L^\infty _{ loc }(0, \infty )\), \(F(g)\in L^1(0, \infty )\cap L^\infty _{ loc }(0, \infty )\).
(v) For all \(R>0\) there exists a constant \(C=C(R)>0\) such that:
Proposition 2.1 follows from estimates of the kernel T defined in (1.36), that we split as follows:
The kernels \(T_1\) and \(T_2\) are estimated in the two next Propositions.
Proposition 1.6
Proof
0.- Proof of (2.1). When \(y\in (0, R)\) and \(x\in (0, R)\). We may use the series expansion of the function \(1/\sinh x\) to obtain
For \(y\in (0, R)\) and \(x\in (0, R)\),
and
1.- Proof of (2.2). Consider first the set where \(y <1/2\) and \(x <1/2\), and use the Taylor’s expansion of \(1/\sinh z \) around \(z =0\),
Then
and this proves (2.2).
2.- Proof of (2.4). When \(x\in (0, 1)\) and \(y>x\), in the identity,
we use the mean value Theorem to obtain
and then,
We deduce,
On the other hand, since \(y>\min (2, 3x/2)\), \(y^2-x^2>Cy^2\) and
it follows that, \( |T_1(x, y)|\le C\frac{x^2}{y^4}\frac{y}{x}\le C\frac{x}{y^3} \) and that proves (2.4).
3.- Proof of (2.3) Suppose now that \(x+y>1\), and \(|x-y|\le 1/8\). We may still use the Taylor’s expansion of \(1/\sinh (|x^2-y^2|) \) around \(|x^2-y^2|=0\),
Then, if \(x+y>1\), \(|y-x|\le 1/8\),
and this proves (2.3).
4.- Proof of (2.5). Consider now the cases where \(x+y>1\), \(|y-x|>1/8\) and \(|x-y|<x/2\). Since
if \(x+y>1\), \(|y-x|>1/8\) and \(|x-y|<x/2\),
The estimate (2.8) is nothing but (2.5).
5.- Proof of (2.6) On the other hand, if \(x>1\) and \(|y-x|>x/2\), we may still use hat,
If \(y>x\),
Then, if \(y>3x/2\),
If \(x>2y\),
which proves (2.6). However, we also have, in the case \(x>2y\), \(x^2-y^2>3x^2/4\), and so
which proves (2.7). \( \quad \square \)
Proposition 1.7
If \(x+y>1\) and \(|x-y|>x/2\),
Proof
1.- Proof of (2.10). Let us write first,
and
Using Taylor’s expansion we have, for some \(\xi (x^2, y^2)\) between \(x^2 \) and \(y^2\),
and
We deduce that, for all \(x\in (0, R)\) and \(y\in (0, R)\),
Similarly, using the change \(x\leftrightarrow y\),
It follows that, for all \(x\in (0, R)\), \(y\in (0, R)\),
and
On the other hand, since \(\sinh \) is locally Lipschitz,
for some constant \(C_R>0\). We deduce that
and
This proves (2.10).
2.- Proof of (2.11). Suppose now that \(x\in (0, 1)\) and \(y>x+\delta \) for any \(\delta >0\). Then, as we have ween in the proof of Proposition 2.2,
We have also, since \(\sinh x^2 \le \sinh 1\),
Then,
and this proves (2.11).
3.- Proof of (2.12) We consider now the case where \(x+y>1\) and \(|x-y|<1/8\). Suppose again that \(0<y^2<x \). We may still use the Taylor’s expansion around \(x^2 =y^2\), and write that
and
If \(|x-y|<1/8\) and \(x+y>1\) then, for some \(x_0>0\), \(y\ge x_0\) and \(x\ge x_0\). Then, there exists \(C>0\) such that
and,
On the other hand, if \(|x+y|>1\) and \(|x-y|<1/8\),
for some positive C. It follows that
and
and this shows (2.12).
4.- Proof of (2.13) Suppose now that \(x+y>1\), \(|x-y|>1/8\) and \(|x-y|\le x/2\). Then
and
and this shows (2.13).
5.- Proof of (2.14) and (2.15). Suppose now \(x+y>1\), \(|x-y|>x/2\). As we have seen in the proof of Proposition 2.2, if \(y>x\),
and
Then,
If, on the other hand, \(y<x/2\), \(x^2-y^2>3x^2/4\), we then have that
We deduce that
and this proves (2.14) and (2.15). \( \quad \square \)
Two Corollaries follow from Proposition (2.2) and Proposition (2.3). We first have
Corollary 1.8
Proof
By definition, for all \(x>0\),
Suppose first that \(x\in (0, 1)\). Then, by (2.2) and (2.4),
With a similar argument, using (2.10) and (2.11) instead,
Suppose now that \(x>1\) and write, for \(\ell =1, 2\), that
When \(y\in (0, x/2)\) we may use (2.7) to obtain that
If we use (2.14) in the same region we obtain
When \(y\in (x/2, x-1/8)\), by (2.9),
We now use (2.13) in the same region to get that
Suppose now that \(y\in (x-1/8, x+1/8)\), by (2.3) and (2.12) \(T_1(x, y)\) and \(T_2(x, y)\) are both bounded on \([x-1/8, x+1/8]\), and so
In the region \(y\in (x+1/8, 3x/2)\), we may use (2.9) again to obtain, as in (2.21), that
By (2.13) we have, in the same region,
If \(y>3x/2\), by (2.6),
and we may write that
from which
By (2.15),
\( \square \)
Similar arguments show the second Corollary,
Corollary 1.9
Proof
As before,
Suppose first that \(y\in (0, 2)\). We have then,
We use (2.1) in the first integral of the right hand side of (2.29). Since \(x>5>2y\) in the second integral of the right hand side of (2.29), we may use (2.7) and deduce that \(\sup _{ y\in (0, 2) }\mu _1 (y)<\infty \). Similarly,
where (2.10) and (2.14) yield \(\sup _{ y\in (0, 2) }\mu _2(y)<\infty \).
Suppose now that \(y\in (2, 15/2)\). We then write
The first integral in the right hand side of (2.30) is estimated using (2.1):
The second integral in the right hand side of (2.30) may be estimated using (2.7) to obtain that
and it follows that \(\sup _{ y\in (2, 15/2 )}\mu _1(y)<\infty \). A similar argument using (2.10) and (2.14) gives \(\sup _{ y\in (2, 15/2 )}\mu _2(y)<\infty \).
Suppose now that \(y>15/2\). Then,
In the first and third integrals of the righthand side of (2.31) we use (2.6) to get that
Since \(|x-y|<x/2\) in the second integral at the right hand side of (2.31) we write,
Using (2.3)
We use now (2.5) in the second integral on the right hand side of (2.32):
from where \(\sup _{ y>15/2 }\mu _1(y)<\infty \). A similar argument shows that \(\sup _{ y>15/2 }\mu _2(y)<\infty \), using and (2.13), (2.12) instead of (2.3) and (2.15), (2.14) instead of (2.5). \(\quad \square \)
Proof
The proofs of (i) and (ii) are now straightforward:
Proof of (iii). Consider again the right hand side of (2.33) and notice that, by (2.1) and (2.10), for \(x\in (0, 1)\),
By Corollary 2.4,
and then,
By (2.7) and (2.14), for \(x>2\),
and by, Corollary 2.4,
and \( |||F(g)||| _{ \theta }\le C|||g|||_\theta \).
Proof of (iv). By (ii) only \(F(g)\in L^\infty _{ loc }(0, \infty )\) remains to be proved. For \(K=[a, b] \subset (0, \infty )\), \([a, b] \subset (A, B )\) and all \(x\in [a, b]\),
where,
and
A simple inspection of the expression of T(x, y) given by (1.36) shows that \(C(K)=\sup \{|T(x, y)|;x\in [a, b],\,y>B \}<\infty \). Then,
and
Proof of (v). For all \(x>0\),
If \(x\le 2R\), by (2.1) and (2.10),
For \(x>2R\), we use the original expression of T(x, y) in (1.36):
Then
and (iv) follow, since,
\( \square \)
3 Existence of Global Solution f
Using the properties of the operator L, Proposition 2.1 and a fixed point argument, classical solutions \(f\in C([0, \infty ); L^1(0, \infty ))\) of the Cauchy problem for (1.32) with initial data \(f_0\in L^1(0, \infty )\) are obtained. If, moreover, \(f_0\in L^1(0, \infty )\cap L^\infty (0, \infty )\) then \(f\in C([0, \infty ); L^1(0, \infty ))\cap L^\infty ((0, \infty ; L^\infty )\). However it is interesting to consider initial data slightly more general than in \(f_0\in L^1(0, \infty )\cap L^\infty (0, \infty )\) but whose solutions are more regular than just integrable with respect to x in \((0, \infty )\).
Theorem 1.10
Suppose that \(f_0\in L^1(0, \infty )\) satisfies
for some \(\theta \in (0, 1)\). Then, there exists a function
satisfying that
The function f also satisfies that
Proof
Given \(f_0\) fixed and satisfying the hypothesis, consider the operator
on the space
By (ii) in Proposition (2.1),
On the other hand,
By (iv) in Proposition (2.1) and Proposition 6.3 in the Appendix, for all \(t>0\), \(s\in (0, t)\) and \(x\in (0, 2)\),
Since we also have, for \(x\in (0, 2)\), that
we deduce, for \(t>0\), \(s\in (0, t)\) and \(x\in (0, 2)\),
and then
If \(x>2\),
from which,
Adding (3.8), (3.10) and (3.11),
and we deduce
If we denote that
we have then proved that
Let \(\rho >0\) and \(T>0\) be such that
Then, for all \(f\in Z_T\) such that \(||f|| _{ Z_T }\le \rho \),
and then
On the other hand,
and arguing as before,
The map \({\mathscr {L}}\) is then a contraction form \(B _{ Z_T }(0, \rho )\) into itself if T is small enough, and has a fixed point \(u\in B _{ Z_T }(0, \rho )\) that satisfies
in \(Z_T\). Property (3.5) follows from and Gönwall’s Lemma on (0, T) and by (3.10) and (3.11):
Then, there exists a constant \(C=C(T)>0\) such that, (3.31) holds true.
On the other hand, since \(f_0\in L^1(0, \infty )\) and \(|||f_0|||_\theta <\infty \), by Proposition 6.3 in the Appendix, \(S(t)f_0\in L^\infty (0, \infty )\). Moreover, by Proposition (6.3) and Proposition (2.1), for \(t\in (0, T)\) and \(x\in (0, 2)\),
It immediately follows that, for all \(t\in (0, T)\) and \(x\in (0, 2)\),
and then \(f(t)\in L^\infty (0, \infty )\) for all \(t\in (0, T)\). We wish to extend now this function f for all \(t>0\). We notice, to this end that, for all \(x>1\),
Since, by Proposition 2.1,
we obtain
It follows by Gönwall’s Lemma, that, for some constant C depending on T and \(\theta \),
On the other hand, for \(x\in (0, 2)\), using (3.23),
and by (3.25), for all \(x\in (0, 2)\),
We deduce from (3.24), (3.25) and (3.26) for all \(t\in (0, T)\) that
By a classical argument it follows that the function f may be extended to a function, still denoted f, for all \(t>0\) such that \(f\in Z_t\) for all \(t>0\) and satisfies (3.20) for all \(t>0\).
The same arguments used to prove the estimates (3.5), (3.31) and (3.7) on the interval of time given by (3.15) may now be applied to obtain (3.5), (3.31), (3.7) on all finite interval (0, T) for all \(T>0\).
Since \(f_0\in L^1(0, \infty )\) and \(|||f_0|||_\theta <\infty \) it follows from Proposition 6.2 that \(S(t)f_0\in C(0, \infty )\) for every \(t>0\). Since \(f(s)\in L^\infty (0, \infty )\) for all \(s>0\) it follows by Proposition 2.1 that \(F(f(s))\in L^\infty (0, \infty )\) too and therefore, again by 6.2, \(S(t-s)F(f(s))\in C(0, \infty )\) for all \(t>0\) and \(s\in (0, t)\). This shows that \(f(t)\in C(0, \infty )\) for all \(t>0\). \(\quad \square \)
The \(L^1-L^\infty \) regularizing effect of the equation (1.30) observed in Theorem 3.1 follows from a similar property as to that of the equation (1.37) proved in [10]. Our next result shows how equation (1.30) regularizes the Dirac’s delta given as initial data. This is a consequence of the same property of (1.37) proved Theorem 1.2 of [10] (cf. Section 6.1.1 in Appendix below). It also holds for example for the linearized coagulation equation around its equilibria ([14]), but not for the linearization of the classical approximation (or wave turbulence “version”) of the Nordheim equation [12].
Theorem 1.11
There exists a function
satisfying
and such that
where the notation \( \langle L(f(t)), \varphi \rangle \) is defined in Section 6.1.1.
Proof
By definition, \(S(t)\delta _1\) with \(t>0\) is nothing but the fundamental solution \(\Lambda (t)\) of the equation (1.37). Consider the following operator T, defined on functions \(f\in C((0, T); L^1(0, \infty )\):
By (6.5) in the Appendix below, there exists a constant \(C_0>0\) such that,
and then, by (6.13) in Section 6.1.2 and Proposition (2.1), for all \(f\in C((0, T); L^1(0, \infty ))\) such that \(||f(t)||_1\le R\) for all \(t\in [0, T]\),
Therefore, if
the operator T has a fixed point \(f\in C((0, T);L^1(0, \infty ))\) such that \(||f(t)||_1\le R\) for all \(t\in (0, T)\). It easilly follows using Gönwal’s Lemma that the solution f may be extended for all \(t>0\) and satisfies, for all \(t>0\),
The proof of (3.32) follows now by classical arguments. Multiplication of (3.29) by \(\varphi \in {\mathscr {D}}(0, \infty )\) yields
where in the first term (cf. Section 6.1.1 in the Appendix below),
The integral in the second term is denoted as
and, in order to obtain the derivative of v, consider the quotient,
Since the function \(t\rightarrow \left\langle [S(t+h-s)F(f(s))], \varphi \right\rangle \) is continuous on \((0, \infty )\),
On the other hand, since, for all \(g\in L^1(0, \infty )\),
integration from zero to \(h>0\) gives
Use of (3.33) with \(g=S(t-s)F(f(s))\) yields, for all \(s\in (0, t)\), \(h>0\),
Then,
and by the continuity of the function \(\sigma \rightarrow \langle L [S(\sigma )S(t-s)F(f(s))], \varphi \rangle \),
It follows that
\( \square \)
A stronger regularizing effect of the equation (1.37) takes place for \(t>0\) and \(x\in (0, t)\), and is given below in Lemma 6.4 of the Appendix. However, because of its “local” feature, this property does not extends to the equation (1.30).
In the next Theorem are presented some additional properties of the function f obtained in Theorem 3.1.
Theorem 1.12
Suppose that \(f_0\in L^1(0, \infty )\) satisfies (3.1) and f is the function given by Theorem 3.1. Then,
and, for all almost every \(t>0,\, x>0\),
There exists a function \({\tilde{H}}\in L^\infty ((\delta , \infty )\times (\delta , \infty ))\) for all \(\delta >0\) such that
(\({\tilde{H}}\) is defined in (3.45) below). For all \(\varphi \in C^1_0([0, \infty ))\), the map \( t\mapsto \displaystyle {\int _0^\infty \varphi (x)f(t, x)}\textrm{d}x \) belongs to \(W _{ loc }^{1,1}(0, \infty )\) and, for almost every \(t>0\),
Proof
We begin proving (3.34), (3.35), and (3.36). Since \(f_0\in L^1(0, \infty )\) and \(|||f_0|||_\theta <\infty \), by Proposition 6.3 in the Appendix \(S(t)u_0\in L^\infty (0, \infty )\) and Theorem 1.2 in [10],
Since for all \(s>0\), \(F(f(s))\in L^\infty (0, \infty )\cap L^1(0, \infty )\), for almost every \(x>0\), \(t\in (0, T)\) and \(s\in (0, t)\), by the same argument,
where both terms belong to \(L^\infty (0, \infty )\). Let us define
By (6.27), and (iv) of Proposition (2.1), for all \(t\in (0, T)\), \(s\in (0, t)\) and \(x>0\),
The right hand side of (3.41) may now be estimated for all \(s\in (0, t)\) using
We deduce, for all \(t>0\), \(x>0\),
and
Then,
Then, for all \(T>0\) and \(t\in (0, T)\), there exists \(C=C(T, \theta )\) such that
Using Proposition 6.8 in the Appendix it follows that
We deduce that
with
Estimates (3.34) and (3.35) for \(\frac{\partial f}{\partial t}\) follow using again Proposition 6.8. Moreover, by (3.42), (3.43) and (3.38),
and for almost every \(t>0\) and \(x>0\),
By Proposition 2.1 this shows \({\mathcal {L}}(f)\in L^\infty _{ loc }((0,\infty ); L^\infty (\delta , \infty )\cap L^1(0, \infty ))\) for all \(\delta >0\). This ends the proof of (3.34) and (3.35), and proves (3.36).
On the other hand, if we multiply both sides of (3.20) by \(\varphi \in C^1_0([0, \infty ))\) and integrate,
In order to derive this expression with respect to t, we use (3.38) and (3.43) again to obtain
and, for all \(t\in (0, T)\),
which shows that
Identity (3.37) follows now, since
\( \square \)
4 Further Properties of the Solution \(\varvec{f}\)
We describe in this section some further properties of the solutions f given by Theorem 3.1. We first consider what are the variations of mass and energy induced by the initial perturbation \(n_0(1+n_0)x^2f(\tau )\) of the equilibrium \(n_0\) introduced in (1.11), (1.12). Then we prove that for all \(\delta >0\), \(f\in L^\infty ((\delta , \infty )\times (0, \infty ))\) and that for every \(t>0\) the function f(t) has a limit as \(x\rightarrow 0\).
4.1 Mass and Energy
It will be sometimes denoted in what follows that
With some abuse of notation the function \(\nu _0(x)=n_0(p)\) is still denoted \(n_0(x)\). A first basic property is the following:
Proposition 1.13
Let \(f_0\) and f be as in Theorem 3.3. Then, for all \(p>1\),
Proof
Since f satisfies (3.34 ), (3.36), and \(x^6n_0(1+n_0)|f|^{p-2}f \in L^1(0, \infty )\), multiplication of both sides of (3.36) and integration over \((0, \infty )\) gives, using (1.31) and the symmetry of W(x, y) as follows:
Since \(\hbox {d}\mu \) is a non negative finite measure on \((0, \infty )\),
\( \square \)
Lemma 1.14
For all \(\theta \in [0, 3)\),
Proof
(i) For all \(C>0\) denote,
Then, if \(C>||f|| _{ L^\infty (1, \infty ) }\), the Lebesgue measure of \(A_c\cap (1, \infty )\) is zero and
On the other hand,
Then, since \(x^\theta f\in L^1(0, 1)\), if \(C>\sup _{ x\in (0, 1) }x^\theta |f(x)|\),
It follows that, for all \(C>\max (||f|| _{ L\infty (1, \infty ) }, \sup _{ x\in (0, 1) }x^\theta |f(x)|)\),
and this proves (4.4).
(ii) Let us denote now that
and suppose that \(C>||f|| _{ L^\infty (\hbox {d}\mu ) }\). Then
\(\square \)
We how easily have the following:
Corollary 1.15
Let \(f_0\) and f be as in Theorem 3.3. Then,
Proof
Since \(n_0(1+n_0)x^6\in C_0^1([0, \infty ))\), identity (3.37) with \(\varphi =n_0(1+n_0)x^6\) gives, by the definition of \({\mathcal {L}}\) (cf. (1.31)),
The result follows from the symmetry of the kernel W(x, y). \( \quad \square \)
The following property will show the boundedness of the variation of the mass:
Proposition 1.16
Let \(f_0\) and f be as in Theorem 3.3. Then for all \(t>0\) and all \(p>3\),
Proof
By Hölder’s inequality,
If \(p>3\),
It follows by Proposition 4.1, for all \(t> 0\),
Since,
it follows from Lemma 4.2 that
and the Proposition follows. \(\quad \square \)
From Proposition 4.4 we immediately have
Corollary 1.17
Let \(f_0\) and f be as in Theorem 3.3. Then, for \(N(\tau )\) defined in (1.54), and all \(p>3\),
where \(C_p\) is given in Proposition 4.4.
The following property also follows from similar arguments:
Proposition 1.18
Suppose \(f_0\) and f are as in Theorem 3.3, \(C>0\) is a constant and \(f_0\le C\). Then \(f(t)\le C\) for all \(t>0\).
Proof
Since \({\mathcal {L}} (C)\equiv 0\), it follows that
If we multiply the equation by \(n_0(1+n_0)x^6f_C(t)^+\), with \(f_C(t)=(f(t)-C)\),
If \(f_C(t, y)>f_C(t, x)\) and \(f_C^+(t, y)\le f_C^+(t, x)\) then \(f_C(t, y)\le 0\) because if \(f_C(t, y)>0\) we would have \(f_C^+(t, y)=f_C(t, y)>f(t, x)=f_C^+(t, x)\), which is a contradiction. However this implies \(f_C^+(t, x)=f_C^+(t, y)=0\). On the other hand, if \(f_C(t, y)<f_C(t, x)\) and \(f_C^+(t, y)\ge f_C^+(t, x)\) then, \(f_C(t, x)<0\) because if \(f_C(t, x)>0\) then \(f_C^+(t, x)=f_C(t, x)>0\) then, as before we would have, \(f_C^+(t, y)\ge f_C^+(t, x)>0\) and so \(f_C(t, y)=f_C^+(t, y)\) but this would give \(_Cf^+(t, y)=f_C(t, y)<f_C(t, x)=f_C^+(t, x)\), which is a contradiction. Thus, in this case, \(f_C^+(t, x)=f_C^+(t, y)=0\). We deduce
from which \(f_C^+(t)(x)=(f(t, x)-C)^+=0\,\, a. e.\) and the Proposition follows. \( \quad \square \)
We have the following Corollary:
Corollary 1.19
Suppose that f and g are two solutions of (3.35) given by Theorem 3.3 with initial data \(f_0\) and \(g_0\) satisfying the hypothesis of Theorem 3.3 and such that \(f_0\le g_0\). Then, \(f(t)\le g(t)\) for all \(t>0\). In particular, there exists a unique function f satisfying (3.2), (3.34) and (3.35).
Let let us also deduce the following:
Corollary 1.20
Suppose that \(f_0\) and f are as in Theorem 3.1. Then, for all \(\delta >0\) and all \(t\ge \delta \),
Proof
By Theorem 3.1, for all\(\delta >0\), \(f(\delta )\in L^\infty (0, \infty )\cap L^1(0, \infty )\) and
Since the constant \(||f(\delta )||_\infty \) is a solution of (3.36), the result follows, by Corollary 4.7. \( \quad \square \)
Our next result concerns the long time behavior of the solution f. The \(L^2(\hbox {d}\mu )\) norm and \(D(f(\tau ))\) play here the usual roles of entropy and entropy’s dissipation through the identity (cf. (4.3)),
For every \(n\in \mathbb {N}\setminus \{0\}\), define the regularized kernel
so that, \(\hbox {d}\sigma _1(x, y)=W_1(x, y) x^4y^4\hbox {d}y\hbox {d}x\) is now a bounded measure on \(\mathbb {R}^2\).
Now let j be the function
consider, for any pair of functions f, g defined on \((0, \infty )\) the function U defined on \((0, \infty )^2\) as
and denote that
Lemma 1.21
The function \(J_1\) is weakly l.s.c. on \(L^2(\textrm{d}\mu )\).
Proof
Let us show that \(J_1\) is convex and continuous on \(L^2(\hbox {d}\mu )\times L^2(\hbox {d}\mu )\). Since the Hessian of j is positive semi definite on \(\mathbb {R}^2\), the function j is convex on \(\mathbb {R}^2\) and then so is \(J_1\). On the other hand, the estimate of \(W_1(x, y)x^4y^4\) for \(x>0\) and \(y>0\) easily follows from this definition:
Then, for \(f\in L^2(\hbox {d}\mu )\),
and the terms \(I_i\), \(i=1, 2\) are bounded as
and
Therefore, if \(\{f_n\} _{ n\in \mathbb {N}}\) and \(\{g_n\} _{ n\in \mathbb {N}}\) are two sequences in \(L^2(\hbox {d}\mu )\) converging, respectively, to \(f\in L^2(\hbox {d}\mu )\) and \(g\in L^2(\hbox {d}\mu )\),
Using the previous estimate, we deduce in the second integral as
In the first,
Since the sequences \(\{f_n\} _{ n\in \mathbb {N}}\) and \(\{g_n\} _{ n\in \mathbb {N}}\) are bounded in \(L^2(\hbox {d}\mu )\) it follows that \(J_1(f_n, g_n)\rightarrow J_1(f, g)\) as \(n\rightarrow \infty \). The function \(J_1\) is then convex and continuous on \(L^2(\hbox {d}\mu )\times L^2(\hbox {d}\mu )\) and the weakly l.s.c. on \(L^2(\hbox {d}\mu )\times L^2(\hbox {d}\mu )\), (cf. for example [5], Corollary 3.9). \( \quad \square \)
Proposition 1.22
Let \(f_0\) and f be as in Theorem 3.1. Then, for all \(\varphi \in L^2(\textrm{d}\mu )\),
Remark 4.11
Notice that, since \(\hbox {d}\mu \) is a non negative bounded measure on \((0, \infty )\), \(L^2(\hbox {d}\mu )\subset L^1(\hbox {d}\mu )\). Corollary 4.10 shows the weak convergence,
Proof
Consider any sequence \(\{\tau _k\} _{ k\in \mathbb {N}}\) where \(\tau _k\rightarrow \infty \) as \(k\rightarrow \infty \) and define
By (4.6), for all \(T>0\),
from which we deduce that \(D(f(s))\in L^1(0, \infty )\) and
Since \(0\le D_1(f_k(t))\le D(f_k(t))|\) for all \(k\ge 1\),
and since \(D_1(f_k(t))\ge 0\) for all \(t>0\),
On the other hand, by (4.6) again, the sequence \(\{f_k\} _{ k\in \mathbb {N}}\) is bounded in \(L^\infty ((0, \infty ); L^2(\hbox {d}\mu ))\) and there exists a subsequence still denoted \(\{t_k\} _{ k\in \mathbb {N}}\), such that \(t_k\rightarrow \infty \) as \(k\rightarrow \infty \) and \(g\in L^\infty ((0, \infty ); L^2(\hbox {d}\mu ))\) satisfying
It then follows by the weak lower semicontinuity of \(D_1\) in \(L^2(\hbox {d}\mu )\) that
Then,
and, for almost every \(s \in (0, T)\) the measure \(|g(s, y)-g(s, x)|^2y^4x^4\hbox {d}y\hbox {d}x\) is concentrated on the diagonal \(\{(x, y)\in (0, \infty )^2;\,y=x\}\). Since \(g(s )\in L^2(\hbox {d}\mu )\) for almost every \(s\in (0, T)\), it follows that \(g=C_*\) for some constant \(C_*\in \mathbb {R}\). Since \(\mathbb {1}_{ (0, 1) }\in L^1(0, \infty ; L^2(\hbox {d}\mu ))\), it follows that
and then
For all \(\varphi \in C_c^1(0, \infty )\), the function
and,
There exists then a sequence of \(\{t_j\} _{ k\in \mathbb {N}}\) and \(C(\varphi )\) such that
Since \(\mathbb {1}_{ (0, 1) }(t)\varphi (x) \in L^1(0, \infty ; L^2(\hbox {d}\mu ))\), by (4.11),
and
\(\square \)
The following auxiliary result is used in the proof of the next Proposition:
Lemma 1.24
For all \(\varepsilon >0\), there exists a constant \(C_\varepsilon >0\) such that
Proof
There certainly exists \(C _{1, \varepsilon } >0\) such that
Then, by continuity,
and the result follows. \( \quad \square \)
Proposition 1.25
Let \(f_0\) and f be as in Theorem 3.1. Then,
Proof
For all \(\varepsilon >0\),
and similarly,
Then, for \(\delta >0\) given, let \(\varepsilon \) be fixed such that
On the other hand, it follows from Lemma 4.12, for all \(\varepsilon >0\), for all \(x\in (\varepsilon /2, 2/\varepsilon )\) and \(y\in (\varepsilon /2, 2/\varepsilon )\),
Let then be \(\theta _\varepsilon \in C_0(0, \infty )\), \(\theta _\varepsilon (x)=1\) for \(x\in (\varepsilon , 1/\varepsilon )\) and \(\theta _\varepsilon (x)=0\) if \(x\in (0, \varepsilon /2)\) or \(x>2/\varepsilon )\) and consider \(\{f_k\} _{ k\in \mathbb {N}}\) the sequence constructed in (4.7).
It was been proven in (4.8) that the first term in the right hand side of (4.15) tends to zero as \(k\rightarrow \infty \). In order to prove that the last term in the right hand side tends to zero the following Lemma is needed, whose proof is delayed after the end of the proof of Proposition 4.13.
Lemma 1.26
For all \(\varphi \in C_c(0, \infty )\) and all \(T>0\),
The second term in the right hand side of (4.15) may be split as follows:
The first term may be written as
Since,
we deduce that
by Lemma 4.14 with \(\varphi (x)=\theta _\varepsilon (y)\frac{\sinh y^2}{y^3}\). The same argument shows that \(J_2\) tends to zero too as k goes to \(\infty \) and this shows that, for all \(\varepsilon >0\) fixed,
We deduce from (4.14) and (4.16) that
since, by Proposition 4.1,
Proposition 4.13 follows. \( \quad \square \)
Proof
For all \(\varphi \in C_0(0, \infty )\),
and
If \(\rho >0\) and \(R>0\) are such that \(\textrm{supp } \varphi \subset (\rho , R)\), then \(\left| \varphi (y)-\varphi (x) \right| =0\) for \(x\in (0, \rho )\) and \(y\in (0, \rho )\). If, on the other hand, \(x\ge \rho \) or \(y\ge \rho \), that
If \(x>R\) and \(y>R\), then \(\left| \varphi (y)-\varphi (x) \right| =0\) again, and therefore,
The term \(I _{ k, 1 }\) is easily estimated as
Using Hölder’s inequality and Proposition 4.1,
The integral \(I _{ k, 2 }\) may be split again as
where,
and \(I _{ k, 2, 1}\) is then estimated as \(I _{ k,1 }\). The estimate of the term \(I _{ k, 2, 2}\) uses that when \(y\le R<x/2\) then \(\sinh |x^2-y^2|\ge \sinh (3x^2/4)\) as
and using Hölder’s inequality again,
We deduce that
We have then, for all \(T>0\), that
and by the compactness of the injection \(W^{1, 1}(0, T)\subset L^1(0, T)\), there exists a sequence \(t_j \underset{j\rightarrow \infty }{\rightarrow }~\infty \) and \(h\in L^1(0, T)\) satisfying
However, since, by Corollary 4.10,
we deduce that
and by the fundamental Theorem of Calculus,
It follows that h is the constant given by \(C_*\int _0^\infty \varphi (x)\hbox {d}\mu (x)\) and this ends the proof of Lemma 4.14. \( \quad \square \)
Corollary 1.27
Let \(f_0\) and f be as in Theorem 3.1. Then,
Proof
Arguing as in Proposition 4.4, for all \(\varepsilon >0\) and \(t>1\),
For \(p>3\), \(n_0(1+n_0) x^{\left( \frac{4p}{p-1}-\frac{6}{p-1} \right) }\in L^\infty (0, \infty )\), and then
On the other hand,
and the Corollary 4.15 follows from Corollary 4.13. \(\quad \square \)
4.2 The Limit of f(t, x) as \(x\rightarrow 0\) for \(t>0\)
We show now the existence of the limit
for all \(f_0\) and f as in Theorem 1.1, and describe its time evolution. We use the property, proven in Proposition 1.3 of [10],
for some constants \(A_1\), \(A_2\) and a function \(b_1\) such that \(b _1(t)={\mathcal {O}}( t^{-8})\) for \(t>1\). Slightly more precise information may be obtained and is shown here, since it is of further interest.
Proposition 1.28
Suppose that \(f_0\) and f are as in Theorem 3.1. Then for all \(t>0\), and \(\delta >0\) as small as desired,
where,
and there exists a constant \(C>0\) such that
Proof
By construction, for all \(t>0\) and \(x>0\),
By Proposition 1.5 in [10],
By (iv) in Proposition (2.1),
and, by (3.25) for \(T>0\),
On the other hand,
and then,
This shows (4.20), (4.21), (4.22). On the other hand, by property (iv) in Proposition 2.1, there exists a constant C that depends on \(\theta \) such that
Then, for \(t\in (0, 1)\) and \(s\in (0, t)\),
For \(t>1\),
with,
and
It follows, for \(t>1\), that
and
The same arguments show that \(|\ell (f_0; t)|\le C|||f_0|||_\theta (1+t^{-\theta })\) for all \(t>0\) and (4.23) follows. \( \quad \square \)
5 The Functions \(\varvec{u(t)}\) and \(\varvec{p_c(t)}\)
We now return to the notation of the time variable as in sub Section 1.4. Then, given the function \(f(\tau , x)\) obtained in Theorem 3.1, \(t=t(\tau )\) and \(p_c(t)\) must be determined in order to define
The functions \(t, \tau \) and \(p_c(t)\) are related by the change of time variable (1.30), i.e.,
Proposition 1.29
For all \(\tau >0\),
and, if
Proof
The proposition is a direct and straightforward consequence of the integrability property (3.34) of \({\mathcal {L}}(f)\). \( \quad \square \)
Let us denote that
Proposition 1.30
For all \(t>0\) there exists a unique \(\tau >0\) such that
Proof
By Proposition 5.1, \(|{\mathcal {M}}(\tau )|<\infty \) for all \(\tau >0\) and then \(q_c(\tau )\in (0, \infty )\) for all \(\tau >0\) and the integral in the right hand side of (5.7) is well defined and convergent. Since \(q_c(t)>0\) this integral is a monotone increasing function of \(\tau \). It only remains to check that its range is \([0, \infty )\).
By Corollary 4.15, for \(\varepsilon >0\) and \(\tau _\varepsilon \) large enough,
Since, on the other hand,
it follows, for \(\tau >\tau _\varepsilon \), that
Therefore, the function \(e^{{\mathcal {M}}(\sigma )}\) is not integrable at infinity, and
and, for all \(t>0\), there exists a unique \(\tau >0\) satisfying (5.7). \( \quad \square \)
Proof
For all \(t>0\), let \(\tau >0\) be given by Proposition 5.2 and define that
where f is obtained by Theorem 3.1 with initial data \(f_0=u_0\). From the definition of \(p_c\),
and (5.1) is satisfied. On the other hand,
It follows from (5.6) that
and then \(p_c\in C^1(0, \infty )\) and satisfies (1.15). On the other hand, by (5.1) and Theorem 3.3,
Then, Theorem 1.1 follows from Theorems 3.1 and 3.3, where the function H in (1.45) is given by
with \({\tilde{H}}\) given in (3.45). \( \quad \square \)
Proof
It follows from (1.42), and Corollary 4.3 that
and this yields property (1.56) in Corollary 1.2. Property (1.57) of Corollary 1.2 follows from (1.13) and (1.15), since by (1.44), integration of (1.13) on \((0, \infty )\) yields
From Proposition 4.13 property (1.58) in Corollary 1.3 follows, and property (1.59) is deduced from Corollary 4.15.
Since \(f\in C([0, \infty ); L^1(0, \infty ))\), by the identity (5.8) and (5.6), \(q_c\in C([0, \infty ))\). It follows that \(p_c\in C([0, \infty ))\) and it is bounded on any compact subset of \([0, \infty )\). Passing to the limit in (5.8) as \(t\rightarrow \infty \) and using Corollary 4.15 property (1.61) is obtained. Then, it also follows that \(p_c\) is bounded on \((0, \infty )\). \( \quad \square \)
Proposition 1.31
Suppose that \(u_0\) and u are as in Theorem 1.1. Then for all \(t>0\), and \(\delta >0\) as small as desired,
satisfies, for some constant \(C>0\),
Proof
By construction, \(u(t, x)=f(\tau , x)\) were \(\tau \) is given in terms of t by (5.1) and therefore, (5.13), (5.14) follow from (4.20) and (5.15) follows from (4.23). \( \quad \square \)
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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Acknowledgements
The research of the author is supported by grants PID2020-112617GB-C21 of MINECO and IT1247-19 of the Basque Government. The hospitality of IAM of the University of Bonn, and its support through SFB 1060 are gratefully acknowledged. The author thanks Pr. M. A. Valle at the Universidad del País Vasco (UPV/EHU) for enlightening discussions. The author is indebted to Pr. J. Bandyopadhyay and Pr. J. Lukkarinen for pointing to him some mistake in a previous version of this article. The author also thanks the referee for his careful reading of the manuscript, his comments and recommendations that greatly improved the final result.
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Appendix
Appendix
1.1 Some Results from [10]
1.1.1 The Fundamental Solution \(\Lambda \)
The equation (1.37) has a weak formulation, obtained as follows: if u is a regular function,
Then, for \(u\in {\mathscr {D}}'(0, \infty )\), a possible definition of L(u) in the weak sense is
Such a definition makes sense if the distribution u belongs to one of the spaces \(E' _{ p, q }\), with \(p<q\), of Mellin transformable distributions whose Mellin transform is analytic on the strip of the complex plane defined by \(\{s\in \mathbb {C};\,\,\Re e(s)\in (p, q)\}\) (cf. [22]).
The function \(\Lambda \), fundamental solution (1.37), is defined in Corollary 2.11 of [10] for all \(t>0\), as the inverse Mellin transform of a function \(U(t, \cdot )\). For each \(t>0\), the function \(U(t, \cdot )\) is holomorphic on \( \mathbb {C}\), analytic for \(\Re e(s)\in (\sigma _0^*, 2)\) for some real negative constant \(\sigma _0^*\in (-2, -1)\) and for all \(T>0\) there exists a constant \(C>0\) for which
It follows that \(\Lambda \in E' _{ 0, 2 }\) and satisfies the equation (1.37) in the weak sense described just above. For \(t>1/2\), and every \(\alpha \in (0, 2)\),
As shown in Corollary 3.15 fof [10], \(\Lambda \in C((0, \infty ); L^1(0, \infty ))\), and
We recall in the next Proposition the results of Proposition 3.1, Proposition 3.2, Proposition 3.4 and Proposition 3.5 from [10] about the function \(\Lambda \), the fundamental solution of (1.37), that are frequently needed along the present work.
Proposition 5.32
For all \(\delta >0\) as small as desired there exists positive constants, denoted as \(C_\delta \) but whose numerical value may change from line to line, such that
(i) For \(t>1\),
(ii) For \(t>1/2\),
(iii) For \(t\in (0, 1/2)\),
Properties (i) directly follow from Proposition 3.1, Proposition 3.2 and Proposition 3.3 of [10]. Properties (ii) come from Proposition 3.4 and those of (iii) from Proposition 3.5.
1.1.2 The Cauchy Problem Associated to Equation (1.37).
As proved in Theorem 1.4 of [10], for all \(f_0\in L^1(0, \infty )\), the function \(S(t)f_0\) defined by (1.38) is such that \(S(t)f_0 \in C((0, \infty ); L^1(0, \infty ))\), there exists a constant \(C>0\) such that
\(S(t)f_0\) satisfies (1.37) in \(\mathscr {D}'((0,\infty )\times (0,\infty ))\) and \(u(t) \rightharpoonup f_0,\,\, \hbox {in}\,\, {\mathscr {D}}'(0, \infty )\), as \(t\rightarrow 0\).
If \(f_0\in L^1(0, \infty )\cap L^\infty _{ \text {loc} }(0, \infty )\) then \(|\partial _t S(t)f_0|+ |L[S(t)f_0])|\in L^\infty _{loc } ((0, \infty ); L^\infty (0, \infty ))\), and \(S(t)f_0\) satisfies (1.37) for \(x>0\) and \(t>0\).
1.2 Some Further Properties of S(t)
We prove in this Appendix two properties of the solution \(S(t)f_0\) of (1.37) with initial data \(f_0\), that are not given in [10]. The first is just an elementary continuity result. We look for the continuity of \(S(t)f_0\) with respect to x in order to deduce the same property for the solution u of (1.13) and to be able later to speak of \(\ell (u(t))\).
We first briefly recall, from [10], the function W:
where \(\gamma _e\) is the Euler constant and \(\psi (z)=\Gamma '(z)/\Gamma (z)\) is the Digamma function.
For any \(\beta \in (0, 2)\) fixed, the function
is analytic in the domain \(\{s\in \mathbb {C};\,\mathscr {R}e(s)\in (\beta , \beta +1)\}\) and satisfies
Proposition 5.33
If \(f_0\in L^1(0, \infty )\cap L^\infty _{ loc }(0, \infty )\), then \(S(t)f_0\in L^\infty _{ loc }\cap C((0, \infty ))\) for every \(t>0\) and \(S(\cdot )f_0(x)\in C(0, \infty )\) for all \(x>0\).
Proof
By definition,
and then
We choose \(\delta _1\) small enough and R large enough, both depending on t, to get that
using that \(f_0\in L^1\) and the asymptotics of \(\Lambda \) for large and small arguments. Consider for example, the integral for \(y<\delta \), with \(\delta < t\). Then \(t/y>1\) and
and, if we use that \(y>x/2\),
A similar argument yields the limit of the integral for \(y>R\) as \(R\rightarrow \infty \).
On the other hand, if we suppose \(y\in (\delta _1, R),\,\,\,|x-y|<\delta \), then
Then,
since \(t/y\ge t/R\) for all \(y\in (\delta _1, R)\). If \(t/R>1/2\), then the function
is continuous on
and by Lebesgue’s convergence Theorem,
On the other hand, for \(|x-y|<\delta \), by (6.17),
and
If \(t/R<1/2\), we must divide the domain D in two sub domains:
With the previous argument,
In the domain \(D_-\), with \(r=t/R>0\) and \(a =r/2\),
and then,
A arguing as before,
By (6.18), if \(|x-y|<\delta \) and \(|1-x/y|>\rho \), then \(|1-y/y|>\rho -\frac{\delta }{\delta _1 } \) and
It follows that, for all \(y\in (\delta _1, R)\) such that \(|1-x/y|>\rho \), the function
is continuous at x. Then, for all \(t>0\) fixed and \(y\in (\delta _1, R)\) such that \(|1-x/y|>\rho \),
We deduce from Lebesgue’s convergence Theorem,
and this gives the continuity of with respect to x. On the other hand, fix \(x>0\) and \(t>0\) and suppose that \(t_n\rightarrow t\). Then
In the first integral, for \(\rho \in (0, 1)\), we have
In that range of values of y, we have the estimates
since we may assume that \(t_n\ge t\) and \(1-\rho >1/2\). We deduce that
This fixes \(\rho \). We then have that
and by Lebesque’s convergence Theorem, \(J_2\rightarrow 0\) as \(n\rightarrow \infty \). \( \quad \square \)
The next result is useful to consider initial data \(f_0\) that is unbounded near the origin.
Proposition 5.34
Suppose that \(g\in L^1(0, \infty )\) is such that, for some \(\theta >0\),
Then, \(S(t)g\in L^\infty (0, \infty )\), and, more precisely,
Proof
By hypothesis, for all \(\delta >0\), \(g\in L^\infty _{ loc }(\delta , \infty )\) and for all \(x>0\),
On the other hand, for \(x>2\),
where \(x/y>2\) for \(y\in (0, 1)\). Several cases are now possible. If \(t>x\), then \(t>y\) and it follows from by (6.6) in Proposition 6.1,
The same argument yields, for \(t\in (2, x)\), that
and, for \(0<y<t<2\), that
By estimate (6.10) in Proposition 6.1, when \(t<1\), and \(y\in (t, 1)\),
and this ends the proof of (6.20).
When \(x\in (0, 2)\), we first write that
When \(t>y\) and \(x<t\), by (6.6) of Proposition 6.1,
and for \(x>t\), by (6.7) of Proposition 6.1,
In both cases,
Similar arguments show the same estimate for \(y\in (t, 1)\). Therefore,
and (6.19) follows. \(\quad \square \)
The next Lemma shows a strong regularizing effect of the equation (1.37), but only in the domain \(\{(t, x)\subset \mathbb {R}_+^2;\,\,t>0, x\in (0, t)\}\).
Lemma 5.35
For all \( \theta \in (0, 1)\), \(t>0\) and all closed interval \(I\subset (0, t)\) there exists a constant \(C>0\) such that, for all \(f_0\in L^1\), \(y^\theta f_0\in L^\infty (0, 1)\),
where \(|||f_0||| _{\theta , 1 }=\sup _{ 0\le y\le 1 }(y^\theta |f_0(y)|)+||f_0||_1\),
with \(\delta >0\) such that \(x (1+\delta )<t\) for all \(x\in I\), and
and \(\sigma _0^*\) is the constant that appears in (6.3).
Proof
Suppose first that \(x\in (0, t)\), and write that
In the term \(I_2\), \(x/y<t/y<1\) and \(x/y<x/t<1\). Then, by (6.12) in Proposition 6.1, there exists constant C such that
Then,
where the first term in the right hand side may estimated as follows. If \(t\in (0, 1)\),
and then,
Since \(x\in (0, t)\) and \(\sigma _0^*<0\), it follows that
and then
If, on the contrary, \(t>1\), than
Since \(y\in (0, t)\) and \(x\in (0, t)\) in the term \(I_1\), it must be decomposed as follows: for \(\delta >0\) small enough to have \(x(1+\delta )<t\),
Since \(x/t<1\), \(t/y>1\) and \( |(x/y)-1|>1/(1+\delta )\) in \(I _{ 1,1 }\) and \(I _{ 1, 3 }\), by (6.8) of Proposition 6.1 may be applied to obtain that
A slightly different argument must be used to estimate \(I _{ 1,2 }\). Since \(t/y>1\), by (6.3) and (6.4), for all \(\alpha \in (0, 2)\),
and then,
Because \(y\in ((1-\delta )x, (1+\delta )x)\), if \(s=\alpha +iv\), then
from which,
and the last integral converges, since \(\frac{2t}{(1+\delta )x}>2\), to
This ends the proof of Lemma 6.4. \(\quad \square \)
Remark 6.5
When \(x>t\) the arguments of the proof of Lemma 6.4 give similar estimates for \(I _{ 1,1 }, I _{ 1, 3 }\) and \(I_2\) using the estimates in Proposition 6.1. However, this is not true for \(I _{ 1, 2 }\) because when \(x>t\) the integral in the right hand side of (6.26) does not converge anymore. This is due of course to the fact that U(t, s) decreases slowly as \(|s|\rightarrow \infty \). All of that makes if impossible to extend the regularizing effect of Lemma 6.4 to the domain \(x>t\) nor to the solutions of equations (1.31) and (1.13).
Remark 6.6
Under the change of variables \(x=e^{\xi /2}\) and \(xu(t, x)=w(t, \xi )\),
Then, using
it follows that \(L(u(t))=P(w(t))\) with
and the operator P is then a pseudo differential operator with symbol \(p(\xi , k)\):
Here \( \Psi (s)\) is the Digamma function and \(\gamma _E\) denotes the Euler’s constant. Use of Mathematica yields that
This growth of \(\rho (k)\) at infinity is responsible for the slow decreasing of U(t, s) as \(|s|\rightarrow \infty \) in (6.3). Operator \(P(w(t))(\xi )\) may then be seen as a derivative of order logarithmic after multiplication of w(t) by \(e^\xi \).
We close this Appendix with the following Remark and some simple estimates:
Remark 6.7
If \(f_0\in L^\infty \cap L^1(0, \infty )\) it follows from the estimate (1.37) in [10] that
where the function \(\zeta _{ \theta }(t, x)\) is defined in (3.40).
Proposition 5.39
For all \(t>0\) fixed,
Proof
On the other hand, we easily obtain, if \(t<2x/3\), that
and, if \(2x/3<t<2x\),
For \(2x<t\),
On the other hand,
\( \square \)
1.3 The System (1.26), (1.27)
We briefly present in this section the results for the non linear system (1.26), (1.27):
A change of time variable, similar to (1.30),
leads again to the equation (1.31). All of the results of Section 4 are then available.
The argument goes now as in Section 5. First, define the auxiliary function
1.3.1 The Function \(I_3(n_0+n_0(1+n_0)|p|^2 f(t, |p|))\)
The first result is a simpler expression of the term \(I_3(n_0+n_0(1+n_0)|p|^2 f(t, x))\), when f is the solution obtained in Section 4.
Proposition 5.40
There exists two numerical constants, \(C_1>0\) and \(C_2>0\), such that, if u is the solution of (1.13)–(1.15) obtained in Theorem 1.1 for \(u_0\) satisfying (1.39) and \(\rho >0\), than
Proof
Following [25, 26], the argument is more clear and the calculations simpler in the energy variable \(\omega \). Then define the function \(F=F(\omega )\) as follows:
After suitable time rescaling to absorb a positive constant,
We arrive now at the point. The integral in the right hand side of (1.4) is obtained by integration over \((0, \infty )\) of (6.33) multiplied by \(\sqrt{\omega }\). The singular behavior of the function \(F(t, \omega )\) as \(\omega \rightarrow 0\) makes delicate the estimate of that integral. This was done in detail in [25] under some Hölder conditions on F that are not known to hold true for the function F. Following the notations of [25], let us define that
The right hand side of (6.33) is then, strictly speaking,
If we define, as in [25], for some \(d>0\) small fixed, that
then,
As in [25], by Lebesgue’s convergence,
and we are then left with \(A_\delta (g_<, g_<)\). We now define the function \(h(t, \omega )\) such that
Then
and
The last term is explicit and gives \(A_\delta (h_0, h_0)=-\pi ^2 a(t)^2/3\). On the other hand, by (4.20),
we deduce that
The function \(h(t,\cdot )\) is then integrable for all \(t>0\) fixed, and
We now slightly rearrange the term \(A_\delta (h_0, h)+A_\delta (h, h_0)\) as
with
and
The argument still follows as in [25], even if our function h satisfies slightly different conditions than (A.13) and (A.14). Indeed we claim that here also, the two functions under the integral signs in (6.35) and (6.36) are integrable on \((0, \infty )\). The only delicate region is where bot \(\omega \) and \(|\omega '-\omega |\) are arbitrarily small.
Consider, for example, the term \((h(t, \omega )-h(t, \omega '))h_0(t, \omega -\omega ' )\) for \(\omega '\in (0, \omega )\) in (6.35). Rewrite first, with \(u(t)=u(t, \sqrt{\omega })\) and \(u'(t)=u(t, \sqrt{\omega '})\),
It is now immediately apparent that the function \(\varphi \) is globally Lipschitz on \([0, \infty )\). On the other hand, if the difference \(\psi (\omega )-\psi (\omega ')\) is written as
Then the two terms on the right hand side of (6.40) are estimated as follows: in the first case, for \(\omega \) and \(\omega \) small,
By Lemma 6.4, for each \(t>0\) fixed, if x and y are small enough,
and then, if \(\omega \) is small enough and \(\omega '\in (0, \omega )\),
and
The second term on the right hand side of (6.40) is written as
The function \(\omega \mapsto n_0(1+n_0)\omega -\omega ^{-1}\) is Lipschitz and then, the factor of a(t) in the right hand side of (6.42) yields that
For the last term in the right hand side of (6.42) we notice that
from which, if we say that \(g(\omega )\equiv n_0(1+n_0)\omega \), for \(\omega \) and \(\omega '\) small, is
It follows from (6.37 ), (6.40)– (6.44) that for d small, \(\omega \in (0, d)\) and \(\omega '\in (0, \omega )\),
Moreover, if \(\omega '\in (0, \omega )\), were such that \(\omega >d+\omega '\) or \(\omega '>d\) it would follow that \((h(t, \omega )-h(t,\omega '))h_0(t, \omega -\omega ' )=0\). Therefore,
Arguing in the same way for all the other terms in (6.35) and (6.36), it follows that
Proposition follows when the initial time rescaling is inverted. \( \quad \square \)
By Proposition 6.9, \({\tilde{m}}\) is well defined and finite for all \(\tau >0\). Let us then define
Proposition 5.41
There exists \(T_*\in (0, \infty ]\), such that for all \(t\in (0, T_*)\) there exists a unique \(\tau >0\) such that
Proof
By Proposition 6.9, \(|\widetilde{{\mathcal {M}}}(\tau )|<\infty \) for all \(\tau >0\) and then \({\tilde{q}}_c(\tau )\in (0, \infty )\) for all \(\tau >0\) and the integral in the right hand side of (6.45) is well defined and convergent. Since \(\tilde{q}_c(t)>0\) this integral is a monotone increaing function of \(\tau \). The value of \(T^*\) is then given by
\( \square \)
Remark 6.11
The function \(\widetilde{{\mathcal {M}}}\) can not be estimated as \({\mathcal {M}}\) in Proposition 5.2, using the conservation of the total number of particles. By Proposition 6.9 and Proposition 4.16,
By Corollary 4.13 and Corollary 4.15,
from which, for some constant \(C>0\),
However, the first term on the right hand side of (6.46) may be estimated only by (4.23) to get,
and this is not enough to prove that \(T^*=\infty \), since it implies only that, for \(\tau \) large nough,
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Escobedo, M. On the Linearized System of Equations for the Condensate-Normal Fluid Interaction Near the Critical Temperature. Arch Rational Mech Anal 247, 92 (2023). https://doi.org/10.1007/s00205-023-01923-3
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DOI: https://doi.org/10.1007/s00205-023-01923-3