Optimal decay estimates for the Vlasov–Poisson system with radiation damping

Abstract

In this paper, we consider a Vlasov–Poisson system in the presence of radiation damping, which is a two-species Vlasov–Poisson system for electrons and ions with additional terms to describe the effect of accelerated charged particles. We obtain the global-in-time and optimal decay estimates of classical solutions to it for small initial data in the whole space $\mathbb {R}^3$. Compared to the previous results, no compact support assumptions on initial data is needed in our paper. And we obtain the optimal $L^{\infty }_x$ decay estimates of the charge densities and the electrostatic field, which was not reported in previous literature. For the proofs of our results, we adapt the modified vector field method which was introduced initially by Smulevici for the classical Vlasov–Poisson system for electrons.

Appendix A. Local Existence of solutions

In this appendix, we give a complete proof of local existence of solutions to the problem (1.2)–(1.3), namely, the bootstrap assumptions introduced in (3.1)–(3.4) were justified in detail. Here we use some ideas introduced in [19]. For any multi-index $\alpha$, we define the following function spaces:

$$\begin{aligned} A_{\alpha } (T_0)&= \big \{ \rho \in {L^{\infty }(\mathbb {R}_{+} \times \mathbb {R}^3)}: \Vert \rho \Vert _{ A_{\alpha }(T_0)}< \infty \big \}, \\ B_{\alpha } (T_0)&= \big \{ \phi \in {L^{\infty }(\mathbb {R}_{+} \times \mathbb {R}^3)}: \Vert \phi \Vert _{ B_{\alpha }(T_0)} < \infty \big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert \rho (F) \Vert _{A_{\alpha } (T_0)}&= \sup _{0 \le t \le T_0} \big \{ \big \Vert \rho (Z^{\alpha }F) \big \Vert _{L^1_x} + (1+t)^3 \Vert \rho (Z^{\alpha }F)\Vert _{L^{\infty }_x} \big \}, \\ \Vert \phi \Vert _{B_{\alpha } (T_0)}&= \sup _{0 \le t \le T_0} \big \{ (1+t)^2 \big \Vert \nabla X^{\alpha }(\phi ) \big \Vert _{L^{\infty }_x} \big \}. \end{aligned}$$

Notice that

The following is the standard regularity result for the Poisson equation, which will be used in later proofs.

Lemma A.1

Let $\phi$ solves the Poisson equation $\Delta _x \phi (t,x) = 4 \pi \rho (F)(t,x)$ and suppose that $\Vert \rho (F) \Vert _{A_{\alpha }} < \infty$ for some $\alpha$. Then

$$\begin{aligned} \Vert \phi \Vert _{B_{\alpha }} \le C \Vert \rho \Vert _{A_{\alpha }}, \end{aligned}$$

for some $C > 0$ independent of $\rho$.

Proof

For any fixed $t >0$, we define

$$\begin{aligned} {\tilde{\rho }} (t,x) = (1+t)^3 \rho \big ( t, (1+t)x \big ). \end{aligned}$$

It’s clearly that

$$\begin{aligned} \int \limits _{\mathbb {R}^3} \big | {\tilde{\rho }}(Z^{\alpha }F) (t,x) \big | \textrm{d}x&= \int \limits _{\mathbb {R}^3} (1+t)^3 \big | \rho (Z^{\alpha }F) \big ( t,(1+t)x \big ) \big | \textrm{d}x = \int \limits _{\mathbb {R}^3} \big | \rho (Z^{\alpha }F)(t,x) \big | \textrm{d}x. \\ \big \Vert {\tilde{\rho }}(Z^{\alpha }F) (t, \cdot ) \big \Vert _{L^{\infty }_x}&= (1+t)^3 \big \Vert \rho (Z^{\alpha }F) (t, \cdot ) \big \Vert _{L^{\infty }_x}. \end{aligned}$$

Then, we have

$$\begin{aligned} \sum _{| \alpha |}\big \Vert {\tilde{\rho }}(Z^{\alpha }F) \big \Vert _{L^1_x} + \big \Vert {\tilde{\rho }} (Z^{\alpha }F)(t,\cdot ) \big \Vert _{L^{\infty }_x} \le \Vert \rho \Vert _{A_{\alpha }}. \end{aligned}$$

(A.1)

Define ${\tilde{\phi }}(t,x):= (1+t) \phi \big ( t,(1+t)x \big )$. Notice that

$$\begin{aligned} \Delta _x {\tilde{\phi }} = {\tilde{\rho }}. \end{aligned}$$

By Lemma 2.3, we know that

$$\begin{aligned} \Delta X^{\alpha }({\tilde{\phi }}) = \sum _{|\alpha '| \le |\alpha |} C_{\alpha '}^{\alpha } {\tilde{\rho }}(Z^{\alpha '}F). \end{aligned}$$

(A.2)

Using the Caldeŕon-Zygmund inequality to the equation (A.2), we get

$$\begin{aligned} \big \Vert \nabla ^2 X^{\alpha } {\tilde{\phi }} \big \Vert _{L^{p}_x} \le C \sum _{|\alpha '| \le |\alpha |} \big \Vert {\tilde{\rho }}(Z^{\alpha '}F) \Vert _{L^{p}_x}, \ \ \ 1< p < \infty . \end{aligned}$$

(A.3)

Combining the Sobolev embedding theorem and the inequality (A.3), then, we have

$$\begin{aligned} \big \Vert \nabla X^{\alpha } ({\tilde{\phi }}) \big \Vert _{L^{\infty }(\mathbb {R}^3)}&\le C \big \Vert \nabla X^{\alpha }({\tilde{\phi }}) \big \Vert _{W^{1,p}(\mathbb {R}^3)} \le C \big \Vert \nabla ^2 X^{\alpha }({\tilde{\phi }}) \big \Vert _{L^p(\mathbb {R}^3)\cap L^q (\mathbb {R}^3)} \nonumber \\&\le C \sum _{|\alpha '| \le |\alpha |} \big \Vert {\tilde{\rho }}(Z^{\alpha '}F) \big \Vert _{L^p(\mathbb {R}^3)\cap L^q (\mathbb {R}^3)} . \end{aligned}$$

(A.4)

Here $3<p < +\infty$ and q is the Sobolev conjugate of p, namely, $q=\frac{3p}{p+3}$. A standard interpolation argument yields

$$\begin{aligned} \big \Vert {\tilde{\rho }} (Z^{\alpha '}F) \big \Vert _{L^{p_1} (\mathbb {R}^3)} \le C \big \Vert {\tilde{\rho }} (Z^{\alpha '}F) \big \Vert _{L^1 (\mathbb {R}^3) }^{\frac{1}{p_1}} \big \Vert {\tilde{\rho }} (Z^{\alpha '}F) \big \Vert _{L^\infty (\mathbb {R}^3)}^{\frac{p_1 - 1}{p_1}}, \ \ \ 1 \le p_1 \le \infty . \end{aligned}$$

(A.5)

Using the Cauchy inequality and plugging the inequalities (A.1) and (A.5) into (A.4), we obtain

$$\begin{aligned} \sum _{|\alpha |}\big \Vert \nabla X^{\alpha } ({\tilde{\phi }}) \big \Vert _{L^{\infty }(\mathbb {R}^3)} \le C \sum _{|\alpha |} \big ( \big \Vert {\tilde{\rho }} (Z^{\alpha '}F) \big \Vert _{L^1 (\mathbb {R}^3)} + \big \Vert {\tilde{\rho }} (Z^{\alpha '}F) \big \Vert _{L^\infty (\mathbb {R}^3)} \big ) \le C \Vert \rho \Vert _{A_{\alpha }}. \end{aligned}$$

Similarly to Lemma 2.2, by induction, it’s easy to prove that

$$\begin{aligned}{}[\partial _{x_i}, X^{\alpha }] = \sum _{j=1}^3 \sum _{|\beta | \le |\alpha |-1} C_{\beta ,j}^{\alpha ,i} X^{\beta } \partial _{x_j}. \end{aligned}$$

(A.6)

Here the constants $C_{\beta ,j}^{\alpha ,i} \ge 0$. Thanks to the definition of ${\tilde{\phi }}$ and the equality (A.6), it follows that

$$\begin{aligned} \nabla X^{\alpha } ({\tilde{\phi }})&= \sum _{|\alpha '| \le |\alpha |} C_{\alpha '}^{\alpha } X^{\alpha '} (1+t)^2 \nabla \phi \\&= C_{\alpha '}^{\alpha } \sum _{|\alpha '| \le |\alpha |} (1+t)^2 X^{\alpha '} \nabla \phi + C_{\alpha '}^{\alpha } \sum _{1 \le |\beta | \le |\alpha |} X^{\beta }[(1+t)^2] X^{\alpha '-\beta } \nabla \phi \\&= C \sum _{|\alpha ''| \le |\alpha |} (1+t)^2 \nabla X^{\alpha ''}(\phi ) + C_{\alpha '}^{\alpha } \sum _{1 \le |\beta | \le |\alpha |} X^{\beta }[(1+t)^2] X^{\alpha '-\beta } \nabla \phi . \end{aligned}$$

In conclusion, it holds

$$\begin{aligned} \Vert \phi \Vert _{B_{\alpha }} \le C \Vert \rho \Vert _{A_{\alpha }}. \end{aligned}$$

$\square$

Then we begin to state the local existence theorem.

Theorem A.1

Suppose that the initial data $\big ( f_0^{+},f_0^{-} \big )$ satisfy the following assumptions:

$$\begin{aligned} \sum _{m=0}^1 \bigg | \frac{\partial (Z^{\alpha } f_0^{\pm })}{\partial _x^m \partial _v^{1-m}} \bigg | \le \frac{C}{(1+|v|)^K}, \end{aligned}$$

for some $\alpha$ with $|\alpha | >0$,  $K > 3$ and $C > 0$. Then, there exists a small constant $T_0 > 0$ and the solution $\big ( f^{+}(t,x,v),f^{-}(t,x,v) \big )$ of the problem (1.2)–(1.3) with

$$\begin{aligned} \Vert \rho \Vert _{A_{\alpha }(T_0)}< \infty , \ \Vert \phi \Vert _{B_{\alpha }(T_0)} < \infty , \end{aligned}$$

for $0 \le t \le T_0$.

Proof

According to Lemma A.1, it follows that $\Vert \rho \Vert _{A_{\alpha }(T_0)} < \infty$ implies $\Vert \phi \Vert _{B_{\alpha }(T_0)} < \infty$. Thus, we only need to prove the first argument. First, the characteristic system of equation (1.2) read

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\textrm{d}}{\textrm{d}s} X^{\pm } (s) = V^{\pm }(s) \pm \sigma D^{[2]}(s), \quad \frac{\textrm{d}}{\textrm{d}s} V^{\pm } (s) = \pm \nabla _x \phi (s,X^{\pm }(s)), \\&X^{\pm } (t,t;x,v) = x^{\pm }, \quad V^{\pm } (t,t;x,v) = v^{\pm }, \\&X^{\pm } (0,t;x,v) = x_0^{\pm }, \quad V^{\pm } (0,t;x,v) = v_0^{\pm }. \end{aligned} \right. \end{aligned}$$

(A.7)

We define $f^{\pm } (t,x,v) = f_0^{\pm } \big ( X(0,t;x,v),V(0,t;x,v) \big )$, and then we can compute a new ${\tilde{\rho }}$ by

$$\begin{aligned} {\tilde{\rho }} (t,x) = \int \limits _{\mathbb {R}^3} f^{\pm }(t,x,v) \textrm{d}v = \int \limits _{\mathbb {R}^3} f_0^{\pm } \big ( X(0,t;x,v),V(0,t;x,v) \big ) \textrm{d}v. \end{aligned}$$

Our task is to check that the transformation $(\rho \rightarrow {\tilde{\rho }})$ is contractive in the Banach space $A_{\alpha }(T_0)$. Suppose that we give $\rho _1^{\pm }, \rho _2^{\pm } \in A_{\alpha }(T_0)$, then $\rho _1, \rho _2 \in A_{\alpha }(T_0)$. By Lemma A.1, we get $\phi _1, \phi _2 \in B_{\alpha }(T_0)$ and $\Vert \phi _1 - \phi _2 \Vert _{B_{\alpha }(T_0)} \le \Vert \rho _1 - \rho _2 \Vert _{A_{\alpha }(T_0)}$. Applying Lagrange mean value theorem and the characteristic system (A.7), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}s} \big ( \big \Vert X_1^{\pm } - X_2^{\pm } \big \Vert _{L^{\infty }} + \big \Vert V_1^{\pm } - V_2^{\pm } \big \Vert _{L^{\infty }} \big ) \le&\, C \Vert \nabla _x (\phi _1 - \phi _2) \Vert _{L^{\infty }} (1 + \Vert \rho _1^{+} + \rho _1^{-} \Vert _{L^1}) \\&+ C \Vert \nabla _x \phi _2 \Vert _{L^{\infty }} \Vert \rho _1^{+} + \rho _1^{-} - \rho _2^{+} - \rho _2^{-} \Vert _{L^1} \\&+ C \Vert \nabla _x^2 \phi _1 \Vert _{L^{\infty }} \big ( \big \Vert X_1^{\pm } - X_2^{\pm } \big \Vert _{L^{\infty }} + \big \Vert V_1^{\pm } - V_2^{\pm } \big \Vert _{L^{\infty }} \big ). \end{aligned}$$

In any case, Gronwall’s inequality implies that

$$\begin{aligned} \Vert x_{0,1}^{\pm } - x_{0,2}^{\pm } \Vert _{L^{\infty }} + \Vert v_{0,1}^{\pm } - v_{0,2}^{\pm } \Vert _{L^{\infty }} \le C T_0 \Vert \rho _1 - \rho _2 \Vert _{A_{\alpha } (T_0)} + C T_0 \Vert \rho _1^{+} + \rho _1^{-} - \rho _2^{+} - \rho _2^{-} \Vert _{A_{\alpha }(T_0)}, \end{aligned}$$

(A.8)

where $x_{0,i}^{\pm } = X_i^{\pm }(0,t;x,v),\ v_{0,i}^{\pm } = V_i^{\pm }(0,t;x,v), \ i=1,2$ and $CT_0 < 1$ is small if $T_0$ is small. Using the formula of ${\tilde{\rho }}$, the assumption of $f_0^{\pm }$, as well as the inequality (A.8), we have

$$\begin{aligned} \Vert \tilde{\rho _1} - \tilde{\rho _2} \Vert _{A_{\alpha }(T_0)} + \Vert {\tilde{\rho }}_1^{+} + {\tilde{\rho }}_1^{-} - {\tilde{\rho }}_2^{+} - {\tilde{\rho }}_2^{-} \Vert _{A_{\alpha }(T_0)} \le CT_0 \Vert \rho _1 - \rho _2 \Vert _{A_{\alpha } (T_0)} + C T_0 \Vert \rho _1^{+} + \rho _1^{-} - \rho _2^{+} - \rho _2^{-} \Vert _{A_{\alpha }(T_0)}. \end{aligned}$$

A fixed point argument yields the desired result. $\square$

Remark A.1

Notice that the results of Theorem A.1 still holds for the case $|\alpha |=0$. In fact, the results of Theorem A.1 with $|\alpha |=0$ is a special case of [19, Theorem 2]. Compared with [19], despite our problem (2.1) has the extra radiation damping term, the arguments on the local existence results of [19] are still valid with slightly modifications.

Due to the results of Theorem A.1, it follows that the bootstrap assumptions (3.1)–(3.4) will be verified in the next Theorem.

Theorem A.2

Suppose that the hypotheses of Theorem 1.1 are satisfied, then, there is a constant $T_0 > 0$ such that, for $0 \le t \le T_0$, the following statements hold:

1. For any multi-index $|\alpha | \le N$, we have

$$\begin{aligned} E_{N}[f^k](t):= \sum _{|\alpha | \le N} \Vert Y^{\alpha }(f^k) \Vert _{L^1_{x,v}} \le C_N. \end{aligned}$$

(A.9)

2. For any multi-index $|\alpha | \le N$, we have

$$\begin{aligned} |\nabla X^{\alpha }(\phi )| \le \frac{C}{(1+t)^2}. \end{aligned}$$

(A.10)

3. For any multi-index $\alpha$ with $|\alpha | \le N-1$, we have

$$\begin{aligned} |Y^{\alpha } \varphi (t,x,v)| \le C_N (1 + \ln (1+t)). \end{aligned}$$

(A.11)

4. For any multi-index $\alpha$ with $|\alpha | \le N-2$ and for any $1 \le i \le 3$, we have

$$\begin{aligned} |Y^{\alpha } (\partial _{x_i} \varphi )(t,x,v)| \le C_N. \end{aligned}$$

(A.12)

Here $C_N$ is a generic constant that depend only on N.

Proof

Due to the results of Theorem A.1, it’s obviously that the estimate (A.10) is holds. The rest of estimates can be proved by induction. Let’s take the estimate (A.11) as an example. Using the equality (2.6) and the estimate (A.10), we get

$$\begin{aligned} |\varphi | \le \int \limits _0^t |{\tilde{T}} (\varphi )| \textrm{d}s \le C \int \limits _0^t s \frac{1}{(1+s)^2} \textrm{d}s \le C (1 + \ln (1+t)). \end{aligned}$$

Clearly, (A.11) is true for $|\alpha | =0$. Assuming that the estimate (A.11) holds for some multi-index $|\alpha -1| \ge 0$, we want to establish the validity of the estimate of the form $|Y^{\alpha }(\varphi )|$. By the method of characteristics, we obtain

$$\begin{aligned} |Y^{\alpha }(\varphi )| \le \int \limits _0^t \big | {\tilde{T}} \big ( Y^{\alpha }(\varphi ) \big ) \big | \textrm{d}s. \end{aligned}$$

Thanks to Lemmas 2.2, 2.5 and 2.8, it follows that

$$\begin{aligned} {\tilde{T}} \big ( Y^{\alpha }(\varphi ) \big ) =&\big [ {\tilde{T}},Y^{\alpha } \big ] (\varphi ) + Y^{\alpha } {\tilde{T}}(\varphi ) \\ =&\sum _{d=0}^{|\alpha | + 1} \, \sum _{i=1}^3 \, \sum _{\begin{array}{c} |\lambda | \le |\alpha | \\ |\xi | \le |\alpha | \end{array}} P_{d \lambda \xi }^{\alpha , k, i}(\varphi ) \partial _{x_i} X^{\lambda } (\phi ) Y^{\xi }(\varphi ) + \sum _{i=1}^3 \, \sum _{\begin{array}{c} |\lambda '| \le |\alpha | \\ |\xi '| \le |\alpha | \end{array}} C_{k \sigma } W^{\lambda '} D_i^{[2]}(t) Y^{\xi '}(\varphi ) \\&+ t \sum _{|\eta | \le |\alpha | + 1} \sum _{i=1}^3 C \partial _{x_i} X^{\eta }(\varphi ) + \sum _{d=1}^{|\alpha |} \sum _{|\eta '| \le |\alpha | +1} \sum _{i=1}^3 P_{d \eta '}^{\alpha ,i}(\varphi ) \partial _{x_i} X^{\eta '}(\varphi ) \\&:= IV_1 + IV_2 + IV_3 + IV_4, \end{aligned}$$

where $P_{d \lambda \xi }^{\alpha , k, i}(\varphi )$ have signatures less than $k'$ such that $k' \le |\alpha |-1$ and $k' + |\lambda | + |\xi | \le |\alpha | +1$ and the multi-indices $\lambda '$ and $\xi '$ satisfy $|\lambda '| + |\xi '| \le |\alpha | +1$. And $P_{d \eta '}^{\alpha ,i}(\varphi )$ have signatures less than ${\bar{k}}$ satisfying ${\bar{k}} \le |\alpha |$ and ${\bar{k}} + |\eta '| \le |\alpha +1|$.

First, we consider the simpler terms $IV_1$ and $IV_3$. Since $k' \le |\alpha |-1 \le N-2 < N, \, |\lambda | \le |\alpha | \le N-1$ and $|\eta | \le |\alpha |+1 \le N$, it follows from the induction hypothesis and the inequality (A.10) that

$$\begin{aligned}&| IV_1 | \le C_N \big ( 1+\ln (1+t) \big )^{N+1} \frac{1}{(1+t)^2} | Y^{\alpha }(\varphi )| \le C_N \frac{1}{(1+t)^{3/2}} | Y^{\alpha }(\varphi )|. \\&| IV_3 | \le C \frac{t}{(1+t)^2} \le C \frac{1}{1+t}. \end{aligned}$$

For the term $IV_2$, taking a similar process of (3.13), we find that

$$\begin{aligned} W^{\lambda '} D_i^{[2]}(t) \le C \sum _{\begin{array}{c} |\lambda '_1|, |\lambda '_2| \le |\lambda '| \\ |\lambda '_1| + |\lambda '_2| = |\lambda '| \end{array}} \sum _{\begin{array}{c} |n'_1| \le |n_1| \\ |n''_1| \le |n'_1| \end{array}} \int \limits _{\mathbb {R}^3} \partial _{x_i} X^{\lambda '_1 + n''_1}(\phi ) \rho (Z^{\lambda '_2 - n_1}f^k) \textrm{d}x. \end{aligned}$$

From the induction hypothesis and Theorem A.1, we have

$$\begin{aligned} | IV_2 | \le C_N \frac{1}{(1+t)^2} \big ( 1+\ln (1+t) \big ) | Y^{\alpha }(\varphi )| \le C_N \frac{1}{(1+t)^{3/2}} | Y^{\alpha }(\varphi )|. \end{aligned}$$

Finally, we deal with the term $IV_4$. Since ${\bar{k}} \le |\alpha |$, by contradiction, we know that there is at most one term $Y^{\theta _j}(\varphi )$ with $|\theta _j| = |\alpha |$ in the decomposition of $P_{d \eta }^{\alpha ,i}(\varphi )$. Then, we have

$$\begin{aligned} |IV_4| \le C_N \big ( 1+\ln (1+t) \big )^{N-2} | Y^{\alpha }(\varphi )| \frac{1}{(1+t)^2} \le C_N \frac{1}{(1+t)^{3/2}} | Y^{\alpha }(\varphi )|. \end{aligned}$$

Therefore, we can summarize what we have proved as the following:

$$\begin{aligned} |Y^{\alpha }(\varphi )| \le C \int \limits _0^t \frac{1}{1+s} \textrm{d}s + \int \limits _0^t C_N \frac{1}{(1+s)^{3/2}} | Y^{\alpha }(\varphi )(s)| \textrm{d}s. \end{aligned}$$

An application of Gronwall’s inequality to $|Y^{\alpha }(\varphi )|$ gives $|Y^{\alpha }(\varphi )| \le C_N \big ( 1+\ln (1+t) \big )$. Similarly, the estimates (A.9) and (A.12) can be proved by induction for which we omit the process. $\square$

Remark A.2

Notice that there is a difference between the indices of the bootstrap assumptions (3.2)–(3.4) and (A.10)–(A.12). In fact, in order to use Proposition 3.1, we re-index it in the bootstrap assumption (3.2). And then we make corresponding changes in the bootstrap assumptions (3.3)–(3.4).