Regularity of the Vlasov-Poisson-Boltzmann System without angular cutoff

We consider the Vlasov-Poisson-Boltzmann system of two species in the whole space $\mathbb{R}^{3}$, cf. [22]:

	$\displaystyle\partial_{t}F_{+}+v\cdot\nabla_{x}F_{+}+E\cdot\nabla_{v}F_{+}=Q(F_{+},F_{+})+Q(F_{-},F_{+}),$		(1)
	$\displaystyle\partial_{t}F_{-}+v\cdot\nabla_{x}F_{-}-E\cdot\nabla_{v}F_{-}=Q(F_{-},F_{-})+Q(F_{+},F_{-}).$

The self-consistent electrostatic field is taken as $E(t,x)=-\nabla_{x}\phi$, with the electric potential $\phi$ given by

$\displaystyle-\Delta_{x}\phi=\int_{{\mathbb{R}^{3}}}(F_{+}-F_{-})\,dv,\quad\phi\to 0\text{ as }|x|\to\infty.$

(2)

The initial data of the system is

$\displaystyle F_{\pm}(0,x,v)=F_{\pm,0}(x,v).$

(3)

The unknown function $F_{\pm}(t,x,v)\geq 0$ represents the velocity distribution for the particle with position $x\in{\mathbb{R}^{3}}$ and velocity $v\in{\mathbb{R}^{3}}$ at time $t\geq 0$. The bilinear collision term $Q(F,G)$ on the right hand side of (1) is given by

$\displaystyle Q(F,G)(v)=\int_{{\mathbb{R}^{3}}}\int_{\mathbb{S}^{2}}B(v-v_{*},\sigma)\big{(}F^{\prime}_{*}G^{\prime}-F_{*}G\big{)}\,d\sigma dv_{*},$

(4)

where $F^{\prime}=F(x,v^{\prime},t)$, $G^{\prime}_{*}=G(x,v^{\prime}_{*},t)$, $F=F(x,v,t)$, $G_{*}=G(x,v_{*},t)$. The more rigorous definition in the form of Carleman representation can be found in [13]. $(v,v_{*})$ is the velocity before the collision and $(v^{\prime},v^{\prime}_{*})$ is the velocity after the collision. They are defined by

$\displaystyle v^{\prime}=\frac{v+v_{*}}{2}+\frac{|v-v_{*}|}{2}\sigma,\ \ v^{\prime}_{*}=\frac{v+v_{*}}{2}-\frac{|v-v_{*}|}{2}\sigma.$

This two pair of velocities satisfy the conservation law of momentum and energy:

$\displaystyle v+v_{*}=v^{\prime}+v^{\prime}_{*},\ \ |v|^{2}+|v_{*}|^{2}=|v^{\prime}|^{2}+|v^{\prime}_{*}|^{2}.$

The Boltzmann collision kernel $B$ is defined as

$\displaystyle B(v-v_{*},\sigma)=|v-v_{*}|^{\gamma}b(\cos\theta),$

for some function $b$ and $\gamma>-3$ determined by the intermolecular interactive mechanism with $\cos\theta=\frac{v-v_{*}}{|v-v_{*}|}\cdot\sigma$. Without loss of generality, we can assume $B(v-v_{*},\sigma)$ is supported on $(v-v_{*})\cdot\sigma\geq 0$, which corresponds to $\theta\in(0,\pi/2]$, since $B$ can be replaced by its symmetrized form $\overline{B}(v-v_{*},\sigma)=B(v-v_{*},\sigma)+B(v-v_{*},-\sigma)$ in $Q(f,f)$. The angular function $\sigma\mapsto b(\cos\theta)$ is not integrable on $\mathbb{S}^{2}$. Moreover, there exists $0<s<1$ such that

$\displaystyle\sin\theta b(\cos\theta)\approx\theta^{-1-2s}\ \text{ on }\theta\in(0,\pi/2],$

It’s convenient to call soft potential when $\gamma+2s<0$, and hard potential when $\gamma+2s\geq 0$. In this work, we always assume

$\displaystyle s\in(0,1),\quad\gamma\in(-3,\infty)\text{ and }\gamma+2s\geq 0.$

(5)

In this paper, we are going to establish the global existence as well as the smoothing effect of the solutions to Cauchy problem (1)-(3) of the Vlasov-Poisson-Boltzmann system near the global Maxwellian equilibrium. For global existence, Guo [19] firstly investigate hard-sphere model of the Vlasov-Poisson-Boltzmann system in a periodic box. Since then, the energy method was largely developed for Boltzmann equation with the self-consistent electric and magnetic fields; see [15, 21, 16]. For smoothing effect of Boltzmann equation, since the work [1] discover the entropy dissipation property for non-cutoff linearized Boltzmann operator, there’s been many discussion in different context. See [2, 5, 7, 18, 24] for the dissipation estimate of collision operator, and [3, 6, 8, 9, 10, 11, 14] for smoothing effect of the solution to Boltzmann equation in different aspect. These works show that the Boltzmann operator behaves locally like a fractional operator:

$\displaystyle Q(f,g)\sim(-\Delta_{v})^{s}g+\text{lower order terms}.$

More precisely, according to the symbolic calculus developed by [5], the linearized Boltzmann operator behaves essentially as

$\displaystyle L\sim\langle v\rangle^{\gamma}(-\Delta_{v}-|v\wedge\partial_{v}|^{2}+|v|^{2})^{s}+\text{lower order terms}.$

However, until now, the smoothing effect of the solutions to Vlasov-Poisson-Boltzmann system remains open and to the best of our knowledge, this is the first paper discussing such smoothing phenomenon.

We will reformulate the problem near Maxwellian as in [19]. For this we denote a normalized global Maxwellian $\mu$ by

$\displaystyle\mu(v)=(2\pi)^{-3/2}e^{-|v|^{2}/2}.$

(6)

Set $F_{\pm}(t,x,v)=\mu(v)+\mu^{1/2}f_{\pm}(t,x,v)$. Denote $f=(f_{+},f_{-})$ and $f_{0}=(f_{+,0},f_{-,0})$. Then the Cauchy problem (1)-(3) can be reformulated as

$$\partial_{t}f_{\pm}+v\cdot\nabla_{x}f_{\pm}\pm\frac{1}{2}\nabla_{x}\phi\cdot vf_{\pm}\mp\nabla_{x}\phi\cdot\nabla_{v}f_{\pm}\pm\nabla_{x}\phi\cdot v\mu^{1/2}-L_{\pm}f=\Gamma_{\pm}(f,f),$$

(7)

$$-\Delta_{x}\phi=\int_{{\mathbb{R}^{3}}}(f_{+}-f_{-})\mu^{1/2}\,dv,\quad\phi\to 0\text{ as }|x|\to\infty,$$

(8)

with initial data

$\displaystyle f_{\pm}(0,x,v)=f_{\pm,0}(x,v).$

(9)

The linear operator $L=(L_{+},L_{-})$ and $\Gamma=(\Gamma_{+},\Gamma_{-})$ are gives as

$$L_{\pm}f=\mu^{-1/2}\Big{(}2Q(\mu,\mu^{1/2}f_{\pm})+Q(\mu^{1/2}(f_{\pm}+f_{\mp}),\mu)\Big{)},$$

$$\Gamma_{\pm}(f,g)=\mu^{-1/2}\Big{(}Q(\mu^{1/2}f_{\pm},\mu^{1/2}g_{\pm})+Q(\mu^{1/2}f_{\mp},\mu^{1/2}g_{\pm})\Big{)}.$$

For later use, we introduce the bilinear operator $\mathcal{T}$ by

$\displaystyle\mathcal{T}_{\beta}(h_{1},h_{2})=\int_{{\mathbb{R}^{3}}}\int_{\mathbb{S}^{2}}B(v-v_{*},\sigma)\partial_{\beta}(\mu^{1/2}_{*})\big{(}h_{1}(v^{\prime}_{*})h_{2}(v^{\prime})-h_{1}(v_{*})h_{2}(v)\big{)}\,d\sigma dv_{*},$

for two scalar functions $h_{1},h_{2}$ and especially $\mathcal{T}=\mathcal{T}_{0}$. Thus,

$$L_{\pm}f=2\mathcal{T}(\mu^{1/2},f_{\pm})+\mathcal{T}(f_{\pm}+f_{\mp},\mu^{1/2}),\\$$

$$\Gamma_{\pm}(f,g)=\mathcal{T}(f_{\pm},g_{\pm})+\mathcal{T}(f_{\mp},g_{\pm}).$$

Through the paper, $C$ denotes some positive constant (generally large) and $\lambda$ denotes some positive constant (generally small), where both $C$ and $\lambda$ may take different values in different lines. $(\cdot|\cdot)$ is the inner product in $\mathbb{C}^{n}$. For any $v\in{\mathbb{R}^{3}}$, we denote $\langle v\rangle=(1+|v|^{2})^{1/2}$. For multi-indices $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ and $\beta=(\beta_{1},\beta_{2},\beta_{3})$, write

$\displaystyle\partial^{\alpha}_{\beta}=\partial^{\alpha_{1}}_{x_{1}}\partial^{\alpha_{2}}_{x_{2}}\partial^{\alpha_{3}}_{x_{3}}\partial^{\beta_{1}}_{v_{1}}\partial^{\beta_{2}}_{v_{2}}\partial^{\beta_{3}}_{v_{3}}.$

The length of $\alpha$ is $|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}$. The notation $a\approx b$ (resp. $a\gtrsim b$, $a\lesssim b$) for positive real function $a$, $b$ means there exists $C>0$ not depending on possible free parameters such that $C^{-1}a\leq b\leq Ca$ (resp. $a\geq C^{-1}b$, $a\leq Cb$) on their domain. $\mathscr{S}$ denotes the Schwartz space. $\text{Re}(a)$ means the real part of complex number $a$. $[a,b]=ab-ba$ is the commutator between operators. $\{a(v,\eta),b(v,\eta)\}=\partial_{\eta}a_{1}\partial_{v}a_{2}-\partial_{v}a_{1}\partial_{\eta}a_{2}$ is the Poisson bracket. $\Gamma=|dv|^{2}+|d\eta|^{2}$ is the admissible metric and $S(m)=S(m,\Gamma)$ is the symbol class. For pseudo-differential calculus, we write $(x,v)\in{\mathbb{R}^{3}}\times{\mathbb{R}^{3}}$ to be the space-velocity variable and $(y,\eta)\in{\mathbb{R}^{3}}\times{\mathbb{R}^{3}}$ to be the corresponding variable in frequency space (the variable after Fourier transform).

(i) As in [20], the null space of $L$ is given by

$$\ker L=\text{span}\Big{\{}(1,0)\mu^{1/2},(0,1)\mu^{1/2},(1,1)v_{i}\mu^{1/2}(1\leq i\leq 3),(1,1)|v|^{2}\mu^{1/2}\Big{\}}.$$

We denote $\mathbf{P}_{\pm}$ to be the orthogonal projection from $L^{2}_{v}\times L^{2}_{v}$ onto $\ker L$, which is defined by

$$\mathbf{P}f=\Big{(}a_{+}(t,x)(1,0)+a_{-}(t,x)(0,1)+v\cdot b(t,x)(1,1)+(|v|^{2}-3)c(t,x)(1,1)\Big{)}\mu^{1/2},$$

(10)

or equivalently by

$$\mathbf{P}_{\pm}f=\Big{(}a_{\pm}(t,x)+v\cdot b(t,x)+(|v|^{2}-3)c(t,x)\Big{)}\mu^{1/2}.$$

Then for given $f$, one can decompose $f$ uniquely as

$$f=\mathbf{P}f+(\mathbf{I}-\mathbf{P})f.$$

The function $a_{\pm},b,c$ are given by

	$\displaystyle a_{\pm}$	$\displaystyle=(\mu^{1/2},f_{\pm})_{L^{2}_{v}}=(\mu^{1/2},\mathbf{P}_{\pm}f)_{L^{2}_{v}},$
	$\displaystyle b_{j}$	$\displaystyle=\frac{1}{2}(v_{j}\mu^{1/2},f_{+}+f_{-})_{L^{2}_{v}}=(v_{j}\mu^{1/2},\mathbf{P}_{\pm}f)_{L^{2}_{v}},$
	$\displaystyle c$	$\displaystyle=\frac{1}{12}((|v|^{2}-3)\mu^{1/2},f_{+}+f_{-})_{L^{2}_{v}}=\frac{1}{6}((|v|^{2}-3)\mu^{1/2},\mathbf{P}_{\pm}f)_{L^{2}_{v}}.$

(ii) To describe the behavior of linearized Boltzmann collision operator, [4] introduce the norm ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$ while [17] introduce the norm $N^{s,\gamma}_{l}$. The previous work [5][13] give the pseudo-differential-type norm $\|(\tilde{a}^{1/2})^{w}f\|_{L^{2}_{v}}$. They are all equivalent and we list their results as follows.

Let $\mathscr{S}^{\prime}$ be the space of tempered distribution functions. $N^{s,\gamma}$ denotes the weighted geometric fractional Sobolev space

$\displaystyle N^{s,\gamma}=\{f\in\mathscr{S}^{\prime}:|f|_{N^{s,\gamma}}<\infty\},$

with the anisotropic norm

$\displaystyle|f|^{2}_{N^{s,\gamma}}:$

$\displaystyle=\|\langle v\rangle^{\gamma/2+s}f\|^{2}_{L^{2}}+\int(\langle v\rangle\langle v^{\prime}\rangle)^{\frac{\gamma+2s+1}{2}}\frac{(f^{\prime}-f)^{2}}{d(v,v^{\prime})^{d+2s}}\mathbf{1}_{d(v,v^{\prime})\leq 1},$

with $d(v,v^{\prime}):=\sqrt{|v-v^{\prime}|^{2}+\frac{1}{4}(|v|^{2}-|v^{\prime}|^{2})^{2}}$. In order to describe the velocity weight $\langle v\rangle$, [17] defined

$\displaystyle|f|^{2}_{N^{s,\gamma}_{l}}=|w^{l}\langle v\rangle^{\gamma/2+s}f|^{2}_{L^{2}_{v}}+\int_{{\mathbb{R}^{3}}}dv\,w^{l}\langle v\rangle^{\gamma+2s+1}\int_{{\mathbb{R}^{3}}}dv^{\prime}\,\frac{(f^{\prime}-f)^{2}}{d(v,v^{\prime})^{d+2s}}\mathbf{1}_{d(v,v^{\prime})\leq 1},$

which turns out to be equivalent with $|w^{l}f|_{N^{s,\gamma}}$. This follows from the proof of Proposition 5.1 in [17] since the $\psi$ therein has a nice support.

On the other hand, as in [4], we define

$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}:$

$\displaystyle=\int B(v-v_{*},\sigma)\Big{(}\mu_{*}(f^{\prime}-f)^{2}+f^{2}_{*}((\mu^{\prime})^{1/2}-\mu^{1/2})^{2}\Big{)}\,d\sigma dv_{*}dv,$

For pseudo-differential calculus as in [5, 13], one may refer to the appendix as well as [23] for more information. Let $\Gamma=|dv|^{2}+|d\eta|^{2}$ be an admissible metric. Define

$\displaystyle\tilde{a}(v,\eta):=\langle v\rangle^{\gamma}(1+|\eta|^{2}+|\eta\wedge v|^{2}+|v|^{2})^{s}+K_{0}\langle v\rangle^{\gamma+2s}$

(11)

to be a $\Gamma$-admissible weight, where $K_{0}>0$ is chosen as the following. Applying theorem 4.2 in [5] and Lemma 2.1 and 2.2 in [12], there exists $K_{0}>0$ such that the Weyl quantization $\tilde{a}^{w}:H(\tilde{a}c)\to H(c)$ and $(\tilde{a}^{1/2})^{w}:H(\tilde{a}^{1/2}c)\to H(c)$ are invertible, with $c$ being any $\Gamma$-admissible metric. The weighted Sobolev space $H(c)$ is defined by (123). The symbol $\tilde{a}$ is real and gives the formal self-adjointness of Weyl quantization $\tilde{a}^{w}$. By the invertibility of $(\tilde{a}^{1/2})^{w}$, we have equivalence

$\displaystyle\|(\tilde{a}^{1/2})^{w}(\cdot)\|_{L^{2}_{v}}\approx\|\cdot\|_{H(\tilde{a}^{1/2})_{v}},$

and hence we will equip $H(\tilde{a}^{1/2})_{v}$ with norm $\|(\tilde{a}^{1/2})^{w}(\cdot)\|_{L^{2}_{v}}$. Also $\|w^{l}(\tilde{a}^{1/2})^{w}(\cdot)\|_{L^{2}_{v}}\approx\|(\tilde{a}^{1/2})^{w}w^{l}(\cdot)\|_{L^{2}_{v}}$ due to Lemma 6.1. Notice that $\|\cdot\|_{L^{2}_{v}}\lesssim\|(\tilde{a}^{1/2})^{w}(\cdot)\|_{L^{2}_{v}}$ for hard potential $\gamma+2s\geq 0$ and we will use this property in our proof.

The three norms above are equivalent since for $l\in\mathbb{R}$,

$\displaystyle\|(\tilde{a}^{1/2})^{w}f\|^{2}_{L^{2}_{v}}\approx{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}\approx|f|^{2}_{N^{s,\gamma}}\approx(-Lf,f)_{L^{2}_{v}}+\|\langle v\rangle^{l}f\|_{L^{2}_{v}},$

which follows from (2.13)(2.15) in [17], Proposition 2.1 in [4] and Theorem 1.2 in [5]. An important result from [12] is that

$\displaystyle L\in S(\tilde{a}),$

where $S(\tilde{a})=S(\tilde{a},\Gamma)$ is the pseudo-differential symbol class; see [23]. This implies that

$\displaystyle|(Lf,f)_{L^{2}_{v}}|\lesssim\|(\tilde{a}^{1/2})^{w}f\|_{L^{2}}^{2}.$

The normal $L^{2}_{v,x}$ is defined as $L^{2}_{v,x}=L^{2}(\mathbb{R}^{3}_{v}\times\mathbb{R}^{3}_{x})$. $L^{2}(B_{C})$ is the $L^{2}_{v}$ space on Euclidean ball $B_{C}$ of radius $C$ at the origin. For usual Sobolev space, we will use notation

$\displaystyle\|f\|_{H^{k}_{v}H^{m}_{x}}=\sum_{|\beta|\leq k,|\alpha|\leq m}\|\partial^{\alpha}_{\beta}f\|_{L^{2}_{v,x}},$

for $k,m\geq 0$. We also define the standard velocity-space mixed Lebesgue space $Z_{1}=L^{2}(\mathbb{R}^{3}_{v};L^{1}(\mathbb{R}^{3}_{x}))$ with the norm

$$\|f\|_{Z_{1}}=\Big{\|}\|f\|_{L^{1}_{x}}\Big{\|}_{L^{2}_{v}}.$$

In this paper, we write Fourier transform on $x$ as

$\displaystyle\widehat{f}(y)=\int_{\mathbb{R}^{3}}f(x)e^{-ix\cdot y}\,dx.$

To state the result of the paper, we let $K\geq 0$ to be the total order of derivatives on $v,x$ and define the velocity weight function $w^{l}$ for any $l\geq 0$ by

$\displaystyle w^{l}=\langle v\rangle^{l}.$

In order to extract the smoothing effect, we define a useful coefficient

$$\psi_{k}=\left\{\begin{aligned} 1,\text{ if $k\leq 0$},\\ \psi^{k},\text{ if $k>0$},\end{aligned}\right.$$

where $\psi=1$ in Section 4 and Theorem 1.1 and $\psi=t^{N}$ with $N>0$ large in Section 5 and Theorem 1.2. When the second case $\psi=t^{N}$ arise, we assume $0\leq t\leq 1$, since the regularity is local property. We will carry $\psi$ in our calculation for brevity of proving the smoothing effect. Corresponding to given $f=f(t,x,v)$, we introduce the instant energy functional $\mathcal{E}_{K,l}(t)$ and the instant high-order energy functional $\mathcal{E}^{h}_{K,l}(t)$ to be functionals satisfying the equivalent relations

	$\displaystyle\mathcal{E}_{K,l}(t)$	$\displaystyle\approx\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}E(t)\|^{2}_{L^{2}_{x}}+\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}\mathbf{P}f\|^{2}_{L^{2}_{v,x}}+\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}$
		$\displaystyle\qquad+\sum_{\begin{subarray}{c}|\alpha|+|\beta|\leq K\end{subarray}}\|\psi_{|\alpha|+|\beta|-2}w^{l-|\alpha|-|\beta|}\partial^{\alpha}_{\beta}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}.$		(12)

	$\displaystyle\mathcal{E}^{h}_{K,l}(t)$	$\displaystyle\approx\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}E(t)\|^{2}_{L^{2}_{x}}+\sum_{1\leq|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}\mathbf{P}f\|^{2}_{L^{2}_{v,x}}+\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}$
		$\displaystyle\qquad+\sum_{\begin{subarray}{c}|\alpha|+|\beta|\leq K\end{subarray}}\|\psi_{|\alpha|+|\beta|-2}w^{l-|\alpha|-|\beta|}\partial^{\alpha}_{\beta}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}.$		(13)

Also, we define the dissipation rate functional $\mathcal{D}_{K,l}$ by

	$\displaystyle\mathcal{D}_{K,l}(t)$	$\displaystyle=\sum_{|\alpha|\leq K-1}\|\psi_{|\alpha|-2}\partial^{\alpha}E(t)\|^{2}_{L^{2}_{x}}+\sum_{1\leq|\alpha|\leq K}\|\psi_{|\alpha|-2}\partial^{\alpha}\mathbf{P}f\|^{2}_{L^{2}_{v,x}}+\sum_{|\alpha|\leq K}\|\psi_{|\alpha|-2}(\tilde{a}^{1/2})^{w}\partial^{\alpha}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}$
		$\displaystyle\qquad+\sum_{\begin{subarray}{c}|\alpha|+|\beta|\leq K\end{subarray}}\|\psi_{|\alpha|+|\beta|-2}(\tilde{a}^{1/2})^{w}w^{l-|\alpha|-|\beta|}\partial^{\alpha}_{\beta}(\mathbf{I}-\mathbf{P})f\|^{2}_{L^{2}_{v,x}}.$		(14)

Here $E=E(t,x)$ is determined by $f(t,x,v)$ in terms of $E=-\nabla_{x}\phi$ and (8). Notice that one can change the order of $(\tilde{a}^{1/2})^{w}$ and $w^{l-|\alpha|-|\beta|}$ due to Lemma 6.1. The main results of this paper are stated as follows.

This gives the global existence to the Vlasov-Poisson-Boltzmann system with the optimal large time decay as in [16], where Duan and Strain discover the optimal large time decay for Vlasov-Maxwell-Boltzmann system. Notice that we only require $K\geq i+1$, which improve the index $K\geq 8$ in [15]. In order to define the $a$ $priori$ assumption, for $0<T\leq\infty$ and $t\in[0,T]$, we define the time-weighted energy norm $X(t)$ by

$\displaystyle X(t)=\sup_{0\leq\tau\leq t}(1+\tau)^{3/2}\mathcal{E}_{K,l}(\tau)+\sup_{0\leq\tau\leq t}(1+\tau)^{5/2}\mathcal{E}^{h}_{K,l}(\tau).$

Here the high-order energy functional $\mathcal{E}^{h}_{K,l}$ has time decay rate $(1+t)^{-5/2}$ while $\mathcal{E}_{K,l}$ has time decay rate $(1+t)^{-3/2}$. They are all optimal as in the Boltzmann equation case [25] and the Vlasov-Maxwell-Boltzmann system case [16]. Let $\delta_{0}>0$ and the $a$ $priori$ assumption to be

$\displaystyle\sup_{0\leq t\leq T}X(t)\leq\delta_{0}.$

(16)

Then we will obtain the following closed $a$ $priori$ estimate

$\displaystyle X(t)\lesssim\epsilon^{2}_{0}+X^{3/2}(t)+X^{2}(t).$

In order to extract the smoothing effect on $x$, we define a symbol $\tilde{b}$ by

$\displaystyle\tilde{b}(v,y)=(1+|v|^{2}+|y|^{2}+|v\wedge y|^{2})^{\delta_{1}},$

(17)

where $\delta_{1}>0$ is defined by (110) and (111). Notice that we will require $\gamma+2s>0$ here and in the next main result.

This result is similar to the Boltzmann equation case; see [6]. That is, whenever the initial data has exponential decay, the solution $f$ is Schwartz in $v$ and smooth in $(t,x)$ for any positive time $t$.

In what follows let us point out several technical points in the proof of Theorem 1.1 and 1.2. For Theorem 1.1, firstly, we use $K\geq 2$ because of $H^{2}_{x}(\mathbb{R}^{3})$ is a Banach algebra when controlling (2.2) and it’s useful when dealing with the trilinear estimate. Secondly, the velocity weight $w^{l-|\alpha|-|\beta|}$ will help us deal with the term $v\cdot\nabla_{x}\phi f$ when bounding

$\displaystyle\big{(}\sum_{\alpha_{1}\leq\alpha}v\cdot\nabla_{x}\partial^{\alpha_{1}}\phi\partial^{\alpha-\alpha_{1}}f,e^{\pm\phi}w^{2l-2|\alpha|-2|\beta|}\partial^{\alpha}f\big{)}_{L^{2}_{v,x}},$

where $|\alpha|\leq K$. The case $\alpha_{1}=0$ will be eliminated by the similar term corresponding to $v\cdot\nabla_{x}f$ as in [21]. This is what $e^{\pm\phi}$ designed for. The case $\alpha_{1}\neq 0$ can be bounded due to the weight $w^{l-|\alpha|-|\beta|}$. For the term $\nabla_{x}\phi\cdot\nabla_{v}f$, one will need to bound

$\displaystyle\big{(}\sum_{\alpha_{1}\leq\alpha}\nabla_{x}\partial^{\alpha_{1}}\phi\partial^{\alpha-\alpha_{1}}\nabla_{v}f,e^{\pm\phi}w^{2l-2|\alpha|-2|\beta|}\partial^{\alpha}f\big{)}_{L^{2}_{v,x}}.$

This term will transfer one derivative from $x$ to one derivative on $v$ and so one should require $|\alpha|+|\beta|\leq K$. If $\alpha_{1}=0$, we can use integration by parts to move $\nabla_{v}$ to the weight $w^{2l-2|\alpha|-2|\beta|}$, while if $\alpha_{1}\neq 0$, the total order on the first $f$ is less or equal to $K$ and hence can be control by our energy functional $\mathcal{E}_{K,l}$ or $\mathcal{D}_{K,l}$. As in [15], one has to bound the term

$\displaystyle\|\partial_{t}\phi\|_{L^{\infty}_{x}}\mathcal{E}_{K,l},$

which cannot be absorbed by the energy dissipation norm. But observing that $\|\partial_{t}\phi\|_{L^{\infty}_{x}}$ is bounded by the high-order energy functional $\mathcal{E}^{h}_{K,l}$ and hence integrable as shown in [21], one can use the Gronwall’s inequality to close the $a$ $priori$ estimate.

The second technical point concerns the choice of $\psi=t^{N}$ in Theorem 1.2 and the usage of $\psi_{|\alpha|+|\beta|-2}$ is Section 5. Firstly, whenever $|\alpha|+|\beta|\geq 2$, $\psi_{|\alpha|+|\beta|-2}=t^{N(|\alpha|+|\beta|-2)}$ is equal to $0$ at $t=0$. Plugging this into energy estimate, one can easily deduce the smoothing effect locally in time, since the initial data becomes zero. By using the global energy control obtained in Theorem 1.1, the local regularity becomes global-in-time regularity. Notice that we use $-2$ to eliminate the index arising from Sobolev embedding $\|\cdot\|_{L^{\infty}_{x}}\lesssim\|\cdot\|_{H^{2}_{x}}$. However, after adding the term $\psi_{|\alpha|+|\beta|-2}$, one need to deal with the term

$\displaystyle\big{(}\partial_{t}(\psi_{|\alpha|+|\beta|-2})\partial^{\alpha}_{\beta}f,e^{\pm\phi}\partial^{\alpha}_{\beta}f\big{)}_{L^{2}_{v,x}}.$

Using the symbols (11) and (17), we can control this term by pseudo-differential norms with a little higher-order, where these pseudo-differential norms can be controlled by the functional $\mathcal{E}_{K,l}$ and $\mathcal{D}_{K,l}$. Hence, we can obtain a closed energy estimate locally. Together with the global energy control in Theorem 1.1, one can deduce the regularity for any positive time $t>0$.