The Existence of Solutions of a Space-Uniform Boltzmann Equation

Boltzmann equation is an equation which is related to the three variables of , , x v t . In this paper, we mainly study the space-uniform Boltzmann equation which unknown function F is not related to the position variable x. We mainly use the contraction mapping theorem to find the existence of the solution, so our mainly work is to prove the self-mapping, i.e. to prove its uniformly bounded, and then to prove the contraction mapping. There we can get the range of ( ) ( ) 1 B θ ∞ L L , next we can figure out the range of M and T from the conditions what we know. Finally, from these conditions, we can find the existence of the solution.


Introduction
Boltzmann equation is a dynamic model which describes the state of gases, and it is one of the important branches of mathematical and physical equations. It has a wide range of applications in scientific research, such as astronomy, semiconductors, aerospace technology and so on. Nowadays, many people have studied this equation, but due to its complexity and difficulty in dealing with the data, we simplify the problem most of time.
Boundary effects play an important role in the dynamics of Boltzmann solutions [1] of ( ) Despite extensive developments in the study of the Boltzmann equation, many basic questions regarding solutions in a physical bounded domain, such as their regularity [2], have remained largely open. This is partly due to the characteristic nature of boundary conditions in the kinetic theory. In [3], it is shown that in convex domains, Boltzmann solutions are continuous away from the grazing set.
On the other hand, in [4], it is shown that singularity (discontinuity) does occur for Boltzmann solutions in a non-convex domain, and such singularity propagates precisely along the characteristic emanating from the grazing set.
In the last years, the study of kinetic models for granular flows received a sig- where ( ) ( ) 3 , F t v C + ∈ ×   denotes the particle distribution at time t and velocity v. Throughout this paper, the collision operator takes the form Theorem Assume that

The Proof of the Result
For this equation, we need to integrate both sides of the equation with respect to t. Then it becomes that We denote Now we just need to prove that TF F = is a contractive mapping. Step 1. Prove uniformly bounded.
Before we prove that the Equation (6) is contraction mapping, we need to prove that it is uniformly bounded. So we prove TF is uniformly bounded first.
By the first mean value theorem of integral, it has In the above of Equation (9), let Then from Equation (8)  Step 2. Prove contraction mapping.
First, we prove TF is the contraction mapping, so it means we prove that Next, we just to prove the above formula For the above formula, we can simplify it as follows, There, we substituted Equation (15) into Equation (14), and we can get And then from Equation (13) and the first mean value theorem, we can get Equation (16) becomes as follows At last, to ensure TF is a contractive mapping, so we need to find the as long as the above formula holds, then the Equation (3) According to this inequality, the discriminant of this inequality is more than or equal to 0. So   (26)

Conclusion
In this paper, we mainly discussed the Boltzmann equation which is independent of the position relation x, and the equation independent of x can be regarded as a squeezed fixed point mapping. So our main work is to prove the compressed mapping, and before we prove the compressed mapping, we need to ensure that the TF we want to prove is self-mapping, that is, to prove that its uniformly bounded. In this process, we find out a relation to make the proof valid, and then in the process of proving the compressed mapping, we find out another condition to make the proof valid. Finally, we must ensure that the two conditions found earlier and the conditions required in the equation are satisfied at the same time. We mainly prove the existence of the equation by means of the compression mapping of the function.

Supported
The work is supported by National Natural Science Foundation of China (No.
11861078). The work is supported by National Natural Science Foundation of China (No. 11561076).