C 0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules*

In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwel-lian molecules. We first show that the global existence in time of the mild solution of the viscosity equation (,) t v f Q f f f         . We then study the asymptotic behaviour of the mild solution as the coefficients 0    , and an estimate on 0 f f   is derived.


Introduction
In this paper we shall investigate the asymptotic properties of the solution of the viscosity Boltzmann equation for Maxwellian molecules where ( , )   denotes the scalar product.
The problem of viscosity approximation of the spatially homogeneous Boltzmann equation, namely whether the solution of (1) converges to the solution of the equation , is very interested for mathematical theory of Boltzmann equation as well as practical applications.We know that the energy of the solution of (1) is increasing with the time t due to the diffusion effect.We cannot expect that the solution of (1) approaches to the Maxwellian equilibrium in large time.This observation has recently been shown by Li-Matsumura [1].In early work of the authors an explicit estimate of was derived which indicates also the dependence of time [2].It must be stressed this result excludes the case of Maxwellian molecules.Actually, the produce of moments for cutoff potential is not valid for Maxwellian molecules.In this paper we shall study the viscosity approximation for Maxwellian molecules.Let us mention some works about the spatially homogeneous Boltzmann equation with cutoff potential, see [3][4][5][6][7][8][9][10][11] for example.For the Maxwellian molecules Morgenstern first deduced the existence and uniqueness of the solution in 1  L space [12].We also remark that the approximation with diffusion term in velocity variable was present in the work of DiPerna-Lions [13].
Now we complement the equation ( 1) and ( 3) with the same initial condition: In the sequel we always assume that It must be emphasized that the nonnegative hypothesis of ( ) v  is not necessary in present paper.
In the following we denote the m C norm by | | m  , and Here  is the multi-index.
This paper is organized as follows.We introduce a mild solution to the Cauchy problem (1) and (4) in Section 2. We prove local existence of the mild solution by the contracted mapping principle.In Section 3, we propose the global existence of the mild solution.Our main tool is the interpolation inequalities.Finally, we study the

The Local Existence
In this section we shall study the local existence of the solution of the Cauchy problem (1), ( 4).Definition 1.Given 0   .We call f  is the mild solution to the Cauchy problem ( 1) and ( 4), if where The following is the local existence theorem.Theorem 1.Given 0    and satisfy (5).Then there exists 0 T  such that the Cauchy problem ( 1) and ( 4) has a unique mild solution , 0 f t T     .In order to prove Theorem 1, let us recall a well-known result which is often called convolution property.
Proposition 2 ([10,14]).For any 1 p    , if The proof of Theorem 1 Consider the following space where * v denotes the convolution in variable v .By and the definition of the mild solution (6), we have Making use of the Prop. 2 and Gronwall's lemma, we obtain the estimate of 1 L f  .In terms of (7) Here the nonnegative constant C depends on b only.In what following, we denote C for various nonnegative constants independent of  unless special statements.On the other hand, , , So, one deduces that the mapping  is locally Lipschitz continuous.By choosing 0 T  suitably, such that 0 t T   , the Cauchy problem ( 1) and ( 4) exists a unique mild solution.

The Global Existence of the Wild Solution
In order to prove the global existence of the mild solution above, it suffices to show that 1, , 0 First, let us recall the N dimensional Gagliardo-Nirenberg's inequality: let 1 , q r  , j, m are integers and 0 j m   .Suppose that is a nonnegative integer).Then there exists a constant C dependent on q, r, j, m, a, N such that for any and satisfy (5).Then Obviously 1 0 I  .By Prop. 2 and Holder's inequality By these estimates above and Gronwall's lemma, we derive This finishes the proof of the lemma.Lemma 4. Given 0 and satisfy (5).Then , the solution of Cauchy problem (1) and ( 4) satisfies Proof In terms of ( 6), for any By Young's inequality where Next we estimate ) By Gagliardo-Nirenberg's inequality, where By the translation invariance of Q, it is easily to show that So making us of Prop. 2 again , .
Next using the basic theory of parabolic equation and the a priori estimate above, we have the following theorem.
Theorem 5. Given and satisfy (5).Then for any 0 T    the Cauchy problem (1) and ( 4) exists a unique mild solution

2, p W
Estimate and 0 C Approximation In this section we shall make Proof By the equation of ( 1), one has Next we estimate 1 I and 2 I respectively. and Employing Young's inequality again, the second term of the above formulation can be estimated by (30) By (20) and Gronwall's lemma, and the Schauder theory, we conclude the desired result.Now, we consider whether the mild solution of the Cauchy problem (1) and ( 4) converges to the solution of ( 3) and ( 4 and f is the solution of where A and k are constants independent of  .
Furthermore, for any 0 Proof By the theorem above and the result of spatially homogenous Boltzmann equation, we know that Noting that Noting that   Together these estimate with (35) we have , Q satisfies the symmetrization and translation invariance.For Maxwellian potential Q can be split into Q  and _ Q :

W
estimate on f  and deduce the explicit estimate on the viscosity approximation. the mild solution f  of the Cauchy problem (1) and (4) belongs to

s inequality we can deduce the bound of 2 f
 and the bound is independent of  .


This finishes the proof of Theorem 7.