Time-dependent rescalings and Lyapunov functionals for some kinetic and fluid models

Abstract We apply the method of time-dependent rescalings which has been developed by G. Rein and the author to a model kinetic equation, to the Euler equations for a perfect polytropic gas and to a model with friction and heat transfer. We build Lyapunov functionals which are in the case of the fluid models improved versions of the estimates which have been found by J.-Y. Chemin and D. Serre (see [2], [14]).

into asymptotically stationary solutions of the rescaled problem, even if selfsimilar solutions are not well defined, for at least the corresponding initial data. The preservation of the L'-norm by this rescaling introduces a friction term which decreases the energy of the rescaled system and is therefore a Lyapunov functional of the original system. This Lyapunov functional may be used directly to measure the dispersion effects, or by the means of an interpolation between a moment and an Lp-norm (with p > l), or even the entropy (see [6] in the case of the Boltzmann equation and other models with a collision term).
Actually, one can reinterpret the Lyapunov functional (or at least the term which comes out from the rescaled kinetic energy) as a measure of the dispersion around an average velocity $x which decays to 0 for any fixed x E IRd as t --f foo, and a direct computation allows us to find the expression of R(t) and the corresponding decay. The fact that the average velocity decays to 0 might be surprising at first sight but it essentially means that the velocity of the particles remaining in a fixed bounded region for large times is small.
The time dependent rescaling considered in [7] and in this paper has the interesting property that it does not introduce any singularity at t = 0: the initial data for the rescaled problem can be choosen to be the same as for the unscaled system. The rescaled equation therefore connects the initial data to a stationary solutionwhen the solution convergeswhich is the asymptotic dispersion profile or, in the language of parabolic equations, the "intermediate asymptotics" (see [3] for the use of entropy methods in the context of porous media and fast diffusion equations). The purpose of this paper is to present a general methodwhich seems very efficient in the context of kinetic equations and fluid dynamicsfor getting estimates, more than to give new results, which were already known (up to minor improvements), especially in the context of Euler or Navier Stokes equations (see for instance the papers  [13] for instance). Since we are only looking for a priori estimates, the computations will be done in the context of classical solutions, which are supposed to exist globally in time, and we will not give further justifications to the integrations by parts. The method may be applied to other equations, for instance to nonlinear Schrodinger equations or even parabolic equations, but we will not insist here on these aspects and refer to [7] or [3] for further comments on this aspect of the question.

Nonlinear Vlasov equation
Consider the nonlinear Vlasov equation which has been studied for instance by I. Gasser, P. Markowich and B. Perthame in [8]. This nonlocal equation provides a very simple example to study the dispersion effects in kinetic theory. As in [7], we may consider the change of variables If F and v are respectively the rescaled distribution function and the spatial density in the rescaled variables:

R2 AZd
(it is not restrictive to choose the constants to be equal to 1). This can be solved into An easy computation shows that the energy L ( t ) given by satisfies which provides the following decay estimates

Euler Equations for a perfect compressible fluid
In [7], it has been noticed that estimates for fluid equations can be obtained in a similar way using Remark 2.3 and the case of an Euler equation with a pressure given by p(t,x) = p'(t,z) has been analyzed, providing exactly the same kind of results as the nonlinear Vlasov equation] at least for IIp(t,.)IIL7cRd, bounded.
One has to mention that the close analogy of the dispersion estimates in kinetic theory and in fluid dynamics has been noticed for instance in [13]. This analogy is still true if one interprets the Lyapunov functional as the energy after a time-dependent rescaling, and that is what is investigated here.
We will focus our attention on more realistic models than the nonlinear Vlasov equation and begin with the Euler equations for a perfect polytropic gas dtp+div,(pu) =O, (3.1) where the pressure is given by the law p=(y--l)pe. (3.4) The system has been studied by D. Serre (see [14]) and one of the main tools was the computation of dispersion estimates. We shall give here a slightly improved version of these dispersion estimates. Note that these estimates are sufficient to prove the global existence of a solution corresponding to a small initial data, even if larger solutions are known to have a finite existence time interval only (see [14], [15]). Exactly as in Remark 2.3, we may consider a new velocity variable ~( t , z ) = u ( t , x ) --Act) According to Equation If we write L ( t ) = Rq(t)E(t) and if we express p(t,z) in terms of p ( t , z ) e ( t , z ) according to Equation (3.4), then The two first coefficients are nonpositive provided q 5 min(2, (y -1)d).
If we make the ansatz the coefficient of the last term can be written as

R=RP,
and t H L(t) is decreasing if p = -(q+ 1 ) . For any y > 1, we recover the results due to J.-Y. Chemin and D. Serre (see [2], [14]) with constants which are explicit in terms of the initial data, and an additional term f & p ( t , z ) 1~1~& (with a coefficient vanishing as t + +w).  If h= RP, with p < -( q + l ) , then -(4-1)RQ-2R2 + --oo ast 3 +oo -(RQ-l&)=(q-d 1)RQ-2&2+RP+q-l dt provided q < 1. As a consequence, we may recover the dispersion results given by D. Serre in [14] (for the special case g(p) =Cp6 with 5> l), extend them to the general case and provide an additional momentum JRdp(t,z)1z12dx. (ii) If y 2 1 + i, then for any t , to E R, Note that in the case y 2 1 + $, R(t) = t is exactly the function which is used by D. Serre in [14] (for g ( p ) = Cp6 with 6 > 1) and that all the above estimates are equivalent t o the ones of D. Serre and J.-Y. Chemin as t -+ +m.

Conclusion
Extracting the right scale and finding the rate of dispersion or even the Lyapunov functional that governs this rate is probably the easy part of the study of the asymptotic dispersion profile or "intermediate asymptotics" , and the next (open) question is to understand in which cases the difference of the rescaled solution with a stationary solution of the rescaled equation has a faster, eventually exponential, decay in this framework of a priori estimates. The Lyapunov functional can indeed be used to study the dynamical stability, and in some cases is a relative to the stationary states entropy. This question is not easy, and has been partially answered in the context of systems where the field is coupled to the spatial density by the (coulombic) Poisson equation: the rescaled Vlasov-Poisson system converges to its unique stationary state if d = 1 (see [l]) but counter-examples to the convergence of the solution to the rescaled Euler-Poisson system to its unique stationary state were found in [7] when d= 3.