On local existence of the Vlasov-Fokker-Planck equation in a 2D anisotropic space
© Chen and He; licensee Springer. 2013
Received: 26 June 2013
Accepted: 14 October 2013
Published: 8 November 2013
We are concerned with local existence of the Vlasov-Fokker-Planck equation in a 2D anisotropic space in a bounded domain with respect to the space variable. The energy method is used to construct the result and the Hardy inequality is used to estimate the electric field.
The author required that has the same index of differentiability and integrability with respect to variables x and v, i.e., they are isotropic. Comparing with , we do not need , to be bounded, continuous to offset, we cannot obtain the global existence even in a 2D space because of the difficulty to get the Gronwall inequality which can be solved in . Roughly speaking, the essential difficult point lies in that is in proportion to , which is super-linear.
In this paper, we consider the initial boundary problem with 0 boundary value (see (1.1)). The problem about general boundary value [6, 7] is so tough that we are going to consider it in the next paper. Moreover, we focus on the case in which the initial value has different regularities with respect to the x and v variables, i.e., they have different integral and differential indices. An anisotropic space is natural since the variables x and v need not to have the same regularity. For instance, Strain  considered the anisotropic space when he studied the optimal decay rate of the solution to the hard and soft potential Boltzmann equation.
Before narrating the main theorem, we would like to introduce some notations.
, with , , , the distribution function of the particles, where Ω is a bound open set with a boundary (for Definition see p.626 of ).
is a space in which the elements are given the norm , , here is the Lebesgue space with the norm given by .
is a space given the norm , here is the Sobolev space with respect to v.
Now we are ready to state our main theorem.
In the theorem, we denote respectively by , the electric field and the electric charge. is a diffusion coefficient which is very small in physical situations.
admits a solution in the interval , i.e., . Here we mean the boundary value in the sense of trace (see p.259 of ). Moreover, we have that . The solution is unique.
The arrangement of this paper is the following. In Section 2, we cite or prove some lemmas which will be used in the proof of Theorem 1.1. In Section 3, we give the proof of the main theorem in several steps.
2 Some lemmas
Next we introduce some lemmas, which are modified versions of the lemmas in Appendix A in  and will be used in the proof of Theorem 1.1 in Section 3. We only point out the differences, the details can be found in the reference.
Proof Since , exists for any , which implies .
The other processes are similar. □
Remark 2.2 Similar to Lemma A.1 on p.535 of , we mean the initial condition that for any , u admits trace value for a.e. .
Taking the norm with respect to x yields the desired result. □
3 Proof of the main theorem
After the above preparation, we are in a position to prove our main theorem.
Proof To prove the theorem, we split the process into several steps. The first step is to construct the approximating solution sequence; in the second step, we prove the regularity of the solution we have obtained; and in the last step, we prove the uniqueness. To get the existence of a weak solution in the space , it is natural to require to be a Cauchy sequence in the strong topology of and to be bounded in , here . To estimate the norm of , we will use the Hardy inequality.
which implies converges to some f in .
Proof of the claim of (3.4).
here , .
Therefore, the claim of (3.4) holds.
According to the standard weak convergence process, we conclude that f is a solution of the Cauchy problem of equations of (1.1).
Step 2: Regularity of the solution.
Denote or by D. Since in , which deduces in , note that , we have .
By property (i) of Proposition A.3 in , we conclude that is nonnegative. Moreover, , a.e. , since in , which implies that f is nonnegative.
Step 3: Uniqueness of the solution.
which in turn is a result of a very similar process to (3.5).
Next we are going to deal with the boundary, i.e., to show that the solution satisfied the boundary condition.
The authors would like to thank the editors and the reviewers for providing us with valuable and detailed comments and suggestions which have brought great improvements to this manuscript. Also, they would like to thank Prof. C. J. Zhu (CCNU, School of Mathematics and Statistics) and Dr. H. Y. Wen (CCNU, School of Mathematics and Statistics) for helpful discussions and suggestions very much.
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