One of the beautiful results in kinetic theory is to construct the global classical solution to the 3D Vlasov-Poisson system. The result is now classical; see, for instance, chapter 4 of Glassey‘s book. However, I feel the result is a bit non-trivial to convey to students and beginners. Would you agree? Anyway, this post is to try to present this classical result, aiming to be as pedagogical as possible, with the original the good, the bad, and the ugly proof of J. Schaeffer ’91.

Precisely, we consider the Vlasov-Poisson system (considering the plasma case, only)

on , with denoting the charge density.

The global classical solution to the Cauchy problem for general compactly supported data was constructed by Pfaffelmoser ’91, Horst ’91, and Schaeffer ’91, and in fact even earlier, by Batt ’77 and Horst ’82 for data with symmetry and by Bardos-Degond ’85 for small data. Then, around the same time in 1991, Lions-Perthame proved the propagation of finite moments. It’s also worth mentioning the averaging lemma was introduced around this time by Golse-Lions-Perthame-Sentis ’88, giving the extra regularity on the macroscopic density.

In this section, we present the proof of Schaeffer (see Glassey‘s book, chapter 4) to construct the classical global solution. Precisely, the theorem reads

Theorem 1For compactly supported initial data , there is the unique classical solution to the VP problem, with . In addition, the velocity support grows at most in the large time, for any small positive .

Remark 1Over the years, there have been efforts to improve upper bounds on the velocity support. I shall not attempt to give the best possible results, but refer the readers to, for instance, Schaeffer ’11, Pallard ’11 and ’12, where an upper bound essentially of order for large time is obtained. In addition, the compactly supported data can be relaxed to have finite moments; see, for instance, Lions-Perthame ’91 and Pallard’ 12.

**1.1. A priori estimates**

We shall derive various uniform a priori estimates for smooth solutions to the VP problem (1). As seen in the last chapter, the Hamiltonian or total energy

is conserved in time. This in particular yields the a priori energy bound . In addition, due to the transport structure, we have

for any time and for being the particle trajectory satisfying the ODEs

with initial data at . In particular, (2) yields the uniform bound: , for all .

*Proof:* For the first inequality, we write

Optimizing and recalling the conservation of energy give the first inequality. Similarly, by definition, we write

Again, by optimizing , the lemma follows.

Lemma 3 (Velocity support)For compactly support initial data , the velocity support defined by

*Proof:* Recalling (3), for bounded initial velocity , we have

By definition, we have . The lemma follows.

In particular, by the Gronwall’s lemma, there is a positive time so that

We stress that and are bounded, as long as the velocity support remains bounded.

Remark 2In the two dimensional case, a similar analysis as in Lemma 3 yields the boundedness of velocity support for all (finite) time .

**1.2. Derivative estimates**

Let us give bounds on derivatives of and the field . We start with the following potential estimates.

Lemma 4For , the field satisfies

*Proof:* The lemma is classical.

Lemma 5As long as for , there holds

for some constant depending on and .

*Proof:* We differentiate the Vlasov equation with respect to and , yielding

Using the method of characteristics and the fact that , we obtain

Setting and using the boundedness assumption on , we have

The lemma follows from applying the Gronwall’s lemma to the above inequality.

**1.3. Velocity support**

As seen in the previous subsection, it suffices for the global classical solution to bound the velocity support. This turns out to be tricky and we shall follow the proof of Schaeffer. Recalling (3), we have

for any particle trajectory . To improve the estimates in the last section, we need to estimate the time integral of along the particle trajectory.

Now for any , we fix a in , and the corresponding particle trajectory that starts from at . For any , we estimate

The classical analysis is to divide the integral over three parts: namely,

for to be determined later and for (this choice will be clear in Lemma 7 below, eventually for the integral to be integrable and optimal). We shall use the notation for the characteristic function over .

Lemma 6There holds

*Proof:* In , we shall first take the integration with respect to , yielding

in which , being the characteristic function over . Since and , the same computation as done in the proof of Lemma 2 gives the lemma.

Lemma 7There holds .

*Proof:* In , we first compute the integration with respect to , yielding

in which the -integrals are taken over the set and . The lemma follows.

It remains to give estimates on . For this, we need to make use of time integration. To this end, let us introduce the particle trajectory with initial value at . Note that and the particle trajectory is an Hamiltonian flow (hence, incompressible in the phase space; in particular, the volume is preserved:). It follows that

We prove the following.

Lemma 8As long as , there holds

*Proof:* Due the the energy conservation, it suffices to prove that

Let be such that and let be the argmin of over . Introducing the distance , we compute

Since the minimum occurs at , and

upon recalling that . This yields

Recall that and in particular is not in , that is, . This implies that . Using the assumption that , the above yields

In addition, the assumption that implies that

upon using again the assumption that . We now take the integration over , upon using (9) when and (10) otherwise, yielding at once (8), and hence the lemma.

Remark 3The proof of the above lemma shows that it suffices to assume that , which plays a role in the improvements of the growth of the velocity support in large time. See, for instance, Schaeffer ’11 and Pallard ’11 and ’12.

Combining, as long as , we have

for some universal constant . Fix an . We now choose so that the above is bounded by . Without optimizing them, we take , , and . It follows that , which is clearly smaller than , the condition used above. Hence this proves that

for all finite . We are now in the position to give estimates on , starting from (6). Indeed, we partition the interval into roughly subintervals and apply the above bound. Setting , and repeatedly using(11), we have

Since , this implies that . Taking sufficiently small, we have , for any small, but fixed, . In particular, remains finite for any finite time. Thus, Lemma 5 yields the boundedness of for all finite time.

Observe that the bound only depends on time , initial energy , and norm of the initial data. One can now construct a solution to the Vlasov-Poisson problem following the standard iterative scheme. For instance, for fixed field , construct solving the linear Vlasov equation

The iterative field is then constructed by solving the Poisson equation . Performing the a priori estimates on these iterative system yields bounds on in norm, uniformly in . Passing to the limit, one obtains a classical solution to the Vlasov-Poisson system. The uniqueness follows similarly.

on March 1, 2018 at 7:05 pm |Bardos-Degond’s solutions to Vlasov-Poisson | Snapshots in Mathematics ![…] to the Vlasov-Poisson system. Of course, the global smooth solutions are already constructed in the previous lecture, without any restriction on size of initial data (e.g., Pfaffelmoser, Schaeffer ’91), however […]