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## Kinetic theory: global solution to 3D Vlasov-Poisson

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)

\displaystyle \begin{aligned} \partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f & =0, \\ E = -\nabla \phi, \qquad -\Delta \phi &=\rho, \end{aligned} \ \ \ \ \ (1)

on ${\mathbb{R}^3 \times \mathbb{R}^3}$, with ${\rho = \int_{\mathbb{R}^3} f\; dv}$ 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 ${f_0 \in C^1(\mathbb{R}^3\times \mathbb{R}^3)}$, there is the unique classical solution ${f \in C^1(\mathbb{R}_+\times \mathbb{R}^3\times \mathbb{R}^3)}$ to the VP problem, with ${f_{\vert_{t=0}} = f_0}$. In addition, the velocity support grows at most ${t^{1+\epsilon}}$ in the large time, for any small positive ${\epsilon}$.

Remark 1 Over 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 ${t^{2/3}\log t}$ for large time ${t}$ 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

$\displaystyle \mathcal{E}(t) : = \iint_{\mathbb{R}^6} \frac{|v|^2}{2} f (x,v,t)\; dxdv + \frac{1}{2} \int_{\mathbb{R}^3} |E(x,t)|^2 \; dx$

is conserved in time. This in particular yields the a priori energy bound ${\mathcal{E}(t)\le \mathcal{E}_0}$. In addition, due to the transport structure, we have

$\displaystyle f(x,v,t) = f_0 (X(0;t,x,v), V(0;t,x,v))\ \ \ \ \ (2)$

for any time ${t}$ and for ${(X(s),V(s))}$ being the particle trajectory satisfying the ODEs

$\displaystyle \dot X = V, \qquad \dot V = E(X(s),s) \ \ \ \ \ (3)$

with initial data ${(X(t),V(t)) = (x,v)}$ at ${s=t}$. In particular, (2) yields the uniform bound: ${\| f(t)\|_{L^\infty} \le \|f_0\|_{L^\infty}}$, for all ${t\ge 0}$.

Lemma 2 There holds

$\displaystyle \|\rho(t)\|_{L^{5/3}} \lesssim \| f_0 \|_{L^\infty}^{2/5} \mathcal{E}_0^{3/5}, \qquad \|E\|_{L^\infty} \lesssim \|\rho\|_{L^\infty}^{4/9} \|\rho\|_{L^{5/3}}^{5/9} .$

Proof: For the first inequality, we write

\displaystyle \begin{aligned} \rho &= \int_{\mathbb{R}^3} f\; dv = \Big(\int_{\{|v|\ge R\}} + \int_{\{|v|\le R\}} \Big) f\; dv \\&\le R^{-2} \int_{\mathbb{R}^3} |v|^2 f\; dv + R^3 \| f\|_{L^\infty}. \end{aligned}

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

\displaystyle \begin{aligned} |E(x) | &\le \int_{\mathbb{R}^3} |x-y|^{-2} \rho(y)\; dy \\&= \Big(\int_{\{|x-y|\ge R\}} + \int_{\{|x-y|\le R\}} \Big) |x-y|^{-2} \rho(y)\; dy \\&\le \Big(\int_{\{|x-y|\ge R\}} |x-y|^{-5}\; dy\Big)^{2/5} \| \rho\|_{L^{5/3}} + \|\rho\|_{L^\infty}\int_{\{|x-y|\le R\}} |x-y|^{-2} \; dy \\&\lesssim \|\rho\|_{L^{5/3}} R^{-4/5} + R\|\rho\|_{L^\infty}. \end{aligned}

Again, by optimizing ${R}$, the lemma follows. $\Box$

Lemma 3 (Velocity support) For compactly support initial data ${f_0}$, the velocity support defined by

$\displaystyle Q(t) : = \sup\{ 1 + |v|~:~ f(x,v,t) \not = 0, \quad\mbox{for some } x\}$

satisfies

$\displaystyle Q(t) \le C_0 \Big[ 1 + \int_0^{t} Q(s)^{4/3} \; ds \Big]\ \ \ \ \ (5)$

for some constant ${C_0}$ depending only on the initial data.

Proof: Recalling (3), for bounded initial velocity ${v}$, we have

$\displaystyle |V(t)| \le |v| + \int_0^{t} \| E(s)\|_{L^\infty} \; ds \le C_0 \Big[ 1+ \int_0^t \| \rho(s)\|_{L^\infty}^{4/9} \; ds\Big].$

By definition, we have ${\|\rho(t)\|_{L^\infty} \le \|f_0\|_{L^\infty} Q(t)^{3}}$. The lemma follows. $\Box$

In particular, by the Gronwall’s lemma, there is a positive time ${T}$ so that

$\displaystyle Q(t) \le C_T, \qquad \forall~t\in [0,T].$

We stress that ${\|\rho(t)\|_{L^\infty}}$ and ${\|E(t)\|_{L^\infty}}$ are bounded, as long as the velocity support ${Q(t)}$ remains bounded.

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

1.2. Derivative estimates

Let us give bounds on derivatives of ${f}$ and the field ${E}$. We start with the following potential estimates.

Lemma 4 For ${-\Delta \phi = \rho}$, the field ${E = -\nabla \phi}$ satisfies

$\displaystyle \| \nabla E \|_{L^\infty} \lesssim \|\rho\|_{L^\infty} \log (2+ \|\rho\|_{L^1}^{1/3}+ \|\nabla \rho\|_{L^\infty}) .$

Proof: The lemma is classical. $\Box$

Lemma 5 As long as ${Q(t) \le c_T}$ for ${t\in [0,T]}$, there holds

$\displaystyle \| f(t)\|_{W^{1,\infty}} \le C_T, \qquad \forall~t\in [0,T],$

for some constant ${C_T}$ depending on ${T, c_T,}$ and ${ \| f_0\|_{W^{1,\infty}}}$.

Proof: We differentiate the Vlasov equation with respect to ${x}$ and ${v}$, yielding

\displaystyle \begin{aligned} \Big( \partial_t + v\cdot \nabla_x + E \cdot \nabla_v \Big)\partial_{x_j}f &= -\partial_{x_j}E \cdot \nabla_v f \\ \Big( \partial_t + v\cdot \nabla_x + E \cdot \nabla_v \Big)\partial_{v_j}f &= - \partial_{x_j} f . \end{aligned}

Using the method of characteristics and the fact that ${\|\nabla_x \rho\|_{L^\infty} \lesssim Q(t)^3 \|\nabla_x f\|_{L^\infty}}$, we obtain

\displaystyle \begin{aligned} \|\nabla_x f(t)\|_{L^\infty} &\le C_0 \Big[ 1 + Q(t)^3\int_0^t \|\nabla_v f(s)\|_{L^\infty} \log (2+ Q(t)^3 \|\nabla_x f(s)\|_{L^\infty})\; ds\Big] \\ \|\nabla_v f(t)\|_{L^\infty} &\le C_0 \Big[ 1 + \int_0^t \|\nabla_x f(s)\|_{L^\infty} \; ds\Big] . \end{aligned}

Setting ${\| f(t)\|_{W^{1,\infty}} = \| f(t)\|_{L^\infty} + \|\nabla_x f(t)\|_{L^\infty} + \|\nabla_v f(t)\|_{L^\infty}}$ and using the boundedness assumption on ${Q(t)}$, we have

$\displaystyle \| f(t)\|_{W^{1,\infty}} \le C_0 \Big[ 1 + c_T\int_0^t\| f(s)\|_{W^{1,\infty}} \log (2+ \| f(s)\|_{W^{1,\infty}} ) \; ds\Big] .$

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

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

$\displaystyle V(t) = V(s)+ \int_s^t E(X(\tau),\tau)\; d\tau \ \ \ \ \ (6)$

for any particle trajectory ${(X(t),V(t))}$. To improve the estimates in the last section, we need to estimate the time integral of ${E}$ along the particle trajectory.

Now for any ${T}$, we fix a ${(t,x_*,v_*)}$ in ${[0,T]\times \mathbb{R}^3\times \mathbb{R}^3}$, and the corresponding particle trajectory ${(X_*(s), V_*(s))}$ that starts from ${(x_*,v_*)}$ at ${s=t}$. For any ${\delta, we estimate

$\displaystyle I(t,\delta) = \int_{t-\delta}^t |E(X_*(s),s)|\; ds =\int_{t-\delta}^t \iint_{\mathbb{R}^3\times \mathbb{R}^3} \frac{f(x,v,s) \; dxdvds}{|x-X_*(s)|^2}.$

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

$\displaystyle I = I_G + I_B + I_U\ \ \ \ \ (7)$

in which

\displaystyle \begin{aligned} G & = \Big\{ (s,x,v)~:~ \min \{ |v|, |v-V_*(s)|\} \le P\Big\} \\ B & = \Big\{ (s,x,v)~:~ |x-X_*(s)|\le R\Lambda(s,v)\Big\} \setminus G \\ U & = [t-\delta, t]\times \mathbb{R}^3\times \mathbb{R}^3 \setminus (G \cup B) \end{aligned}

for ${P,R}$ to be determined later and for ${\Lambda(t,v) = |v|^{-2}|v-V_*(s)|^{-1}}$ (this choice will be clear in Lemma 7 below, eventually for the integral to be integrable and optimal). We shall use the notation ${\chi_\Omega(x,v,s)}$ for the characteristic function over ${\Omega}$.

Lemma 6 There holds ${ |I_G(t,\delta) |\lesssim \delta P^{4/3}.}$

Proof: In ${I_G}$, we shall first take the integration with respect to ${v}$, yielding

$\displaystyle I_G =\int_{t-\delta}^t \int_{\mathbb{R}^3} |x-X_*(s)|^{-2} \rho_G(x,s)\; dxds ,$

in which ${\rho_G = \int \chi_G f(x,v,s)\; dv}$, ${\chi_G}$ being the characteristic function over ${G}$. Since ${\|\rho_G \|_{L^\infty}\le P^3}$ and ${\|\rho_G \|_{L^{5/3}} \le \|\rho\|_{L^{5/3}}}$, the same computation as done in the proof of Lemma 2 gives the lemma. $\Box$

Lemma 7 There holds ${I_B(t,\delta) \lesssim \delta R \log(2+ Q(t)/P)}$.

Proof: In ${I_B}$, we first compute the integration with respect to ${x}$, yielding

\displaystyle \begin{aligned} I_B &\le R \|f\|_{L^\infty}\int_{t-\delta}^t \int |v|^{-2}|v-V_*(s)|^{-1}\; dvds \\ &\le R \|f\|_{L^\infty}\int_{t-\delta}^t \int \Big( |v|^{-3} + |v-V_*(s)|^{-3} \Big)\; dvds ,\end{aligned}

in which the ${v}$-integrals are taken over the set ${P\le |v|\le Q(t)}$ and ${P\le |v-V_*(s)|\le Q(t)}$. The lemma follows. $\Box$

It remains to give estimates on ${I_U(t,\delta)}$. For this, we need to make use of time integration. To this end, let us introduce ${(X(s), V(s))}$ the particle trajectory with initial value ${(x,v)}$ at ${s=t}$. Note that ${f(t,x,v) = f(s,X(s), V(s))}$ and the particle trajectory is an Hamiltonian flow (hence, incompressible in the phase space; in particular, the volume is preserved:$dxdv = dXdV$). It follows that

\displaystyle \begin{aligned} I_U &= \int_{t-\delta}^t \iint_{\mathbb{R}^3\times \mathbb{R}^3} \frac{\chi_U(x,v,s)f(x,v,s) \; dxdvds}{|x-X_*(s)|^2} \\ &= \int_{t-\delta}^t \iint_{\mathbb{R}^3\times \mathbb{R}^3} \frac{\chi_U(X(s),V(s),s)f(x,v,t) \; dxdvds}{|X(s)-X_*(s)|^2} \\ &= \iint_{\mathbb{R}^3\times \mathbb{R}^3} f(x,v,t) \Big( \int_{t-\delta}^t \frac{\chi_U(X(s),V(s),s) ds}{|X(s)-X_*(s)|^2} \Big) \; dvdx . \end{aligned}

We prove the following.

Lemma 8 As long as ${sup_{x,v}I(t,\delta) \le \frac1{12} P}$, there holds ${ I_U \lesssim R^{-1}.}$

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

$\displaystyle \int_{t-\delta}^t \frac{\chi_U(X(s),V(s),s) ds}{|X(s)-X_*(s)|^2} \lesssim R^{-1}(1 + |v|^2).\ \ \ \ \ (8)$

Let ${s \in [t-\delta,t]}$ be such that ${(s,X(s),V(s))\in U}$ and let ${s_0}$ be the argmin of ${|X(s)-X_*(s)|}$ over ${[t-\delta,t]}$. Introducing the distance ${Z(s) = X(s) - X_*(s)}$, we compute

\displaystyle \begin{aligned} |Z(s)| &\ge |Z(s_0) + (s-s_0)\dot Z(s_0)| - \int_{s_0}^s |s-u| |\ddot Z(u)|\; du \\ &\ge |Z(s_0) + (s-s_0)\dot Z(s_0)| - 2|s-s_0| \sup_{x,v} I(t,\delta). \end{aligned}

Since the minimum ${|Z(s)|}$ occurs at ${s=s_0}$, ${(s-s_0) Z(s_0)\cdot \dot Z(s_0) \ge 0}$ and

$\displaystyle |\dot Z(s_0)| \ge |\dot Z(t)| - \int_{s_0}^t |\ddot Z(u)\; du \ge |v-v_*| - 2 \sup_{x,v} I(t,\delta)$

upon recalling that ${\dot Z(t) = V(t) - V_*(t) = v - v_*}$. This yields

$\displaystyle |X(s) - X_*(s) | = |Z(s)| \ge |s-s_0| \Big( |v-v_*| - 4 \sup_{x,v} I(t,\delta)\Big).$

Recall that ${(s,X(s),V(s)) \in U}$ and in particular is not in ${G}$, that is, ${|V(s) - V_*(s)|\ge P}$. This implies that ${|v - v_*| \ge P - 2 \sup_{x,v} I(t,\delta)}$. Using the assumption that ${ \sup_{x,v} I(t,\delta) \le \frac1{12}P}$, the above yields

$\displaystyle |X(s) - X_*(s) | \ge \frac12 |s-s_0| |v-v_*| .\ \ \ \ \ (9)$

In addition, the assumption that ${(s,X(s),V(s)) \not\in B}$ implies that

$\displaystyle |X(s) - X_*(s) |\ge R |V(s)|^{-2} | V(s) - V_*(s)|^{-1} \ge \frac{R}{8}|v|^{-2}|v-v_*|^{-1},\ \ \ \ \ (10)$

upon using again the assumption that ${ \sup_{x,v} I(t,\delta) \le \frac1{12}P}$. We now take the integration over ${[t-\delta,t]}$, upon using (9) when ${|s-s_0| > R |v|^{-2}|v-v_*|^{-2}}$ and (10) otherwise, yielding at once (8), and hence the lemma. $\Box$

Remark 3 The proof of the above lemma shows that it suffices to assume that ${I(t,\delta) \le \frac{1}{12}|v-v_*|}$, 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 ${sup_{x,v}I(t,\delta) \le \frac1{12} P}$, we have

$\displaystyle \frac1\delta I(t,\delta) \le c_0 \Big[ P^{4/3} + R \log (2+ Q(t)/P) + (\delta R)^{-1}\Big],$

for some universal constant ${c_0}$. Fix an ${\alpha<1}$. We now choose ${\delta,P,R}$ so that the above is bounded by ${Q(t)^\alpha}$. Without optimizing them, we take ${P = Q(t)^{3\alpha/4} }$, ${R = Q(t)^\alpha}$, and ${\delta = c_0^{-1}Q(t)^{-2\alpha}}$. It follows that ${sup_{x,v}I(t,\delta) \le Q(t)^{-\alpha}}$, which is clearly smaller than ${\frac1{12} P}$, the condition used above. Hence this proves that

$\displaystyle \frac1\delta I(t,\delta) \lesssim Q(t)^\alpha \log (2 + Q(t)),\ \ \ \ \ (11)$

for all finite ${t \ge 0}$. We are now in the position to give estimates on ${Q(t)}$, starting from (6). Indeed, we partition the interval ${[0,t]}$ into roughly ${\delta^{-1}t}$ subintervals ${[t_j - \delta, t_j]}$ and apply the above bound. Setting ${\mathcal{Q}(t) = \sup_{s\in [0,t]} Q(s)}$, and repeatedly using(11), we have

\displaystyle \begin{aligned} \mathcal{Q}(t) &\lesssim 1 + \sum_j I(t_j, \delta) \\&\lesssim 1+ \delta \sum_j Q(t_j)^\alpha \log (2 + Q(t_j)) \\&\lesssim 1+ t \mathcal{Q}(t)^\alpha \log (2 + \mathcal{Q}(t)) . \end{aligned}

Since ${\alpha<1}$, this implies that ${\mathcal{Q}(t) \lesssim 1 + t^{1/(1-\alpha)} \log (2+t)}$. Taking ${\alpha}$ sufficiently small, we have ${Q(t) \lesssim 1 + t^{1+\epsilon}}$, for any small, but fixed, ${\epsilon}$. In particular, ${Q(t)}$ remains finite for any finite time. Thus, Lemma 5 yields the boundedness of ${\| f(t)\|_{W^{1,\infty}}}$ for all finite time.

Observe that the bound only depends on time ${t}$, initial energy ${\mathcal{E}_0}$, and ${W^{1,\infty}}$ 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 ${E_n}$, construct ${f_{n+1}}$ solving the linear Vlasov equation

$\displaystyle \partial_t f_{n+1} + v \cdot \nabla_x f_{n+1} + E_n \cdot \nabla_v f_{n+1} =0.$

The iterative field ${E_{n+1}}$ is then constructed by solving the Poisson equation ${E_{n+1} = -\nabla (-\Delta)^{-1} \rho_{n+1}}$. Performing the a priori estimates on these iterative system yields bounds on ${f_n,E_n}$ in ${C^1}$ norm, uniformly in ${n}$. Passing to the limit, one obtains a classical solution to the Vlasov-Poisson system. The uniqueness follows similarly.