Smoothing effects and decay estimate of the solution of the linearized two species Landau equation

TL;DR

This paper investigates the linearized two-species Landau equation, demonstrating instant smoothing effects and analyzing the long-term decay of solutions without requiring initial smoothness.

Contribution

It establishes the instant regularization of solutions and provides decay estimates for the linearized two-species Landau equation with minimal initial smoothness assumptions.

Findings

Solutions become smooth instantly in space and momentum variables.

Decay estimates for solutions are obtained over large times.

No initial smoothness assumptions are needed for the results.

Abstract

We study the Landau equation for a mixture of two species in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state. For the linearized level, without any smoothness assumption on the initial data, it is shown that the solution becomes smooth instantaneously in both the space and momentum variables. Moreover, the large time behavior of the solution is also obtained.

Equations

$$\left\{\begin{array}[c]{l}\displaystyle\partial_{t}F_{A}+\frac{1}{m_{A}}p\cdot\nabla_{x}F_{A}=Q^{AA}(F_{A},F_{A})+Q^{AB}(F_{A},F_{B})\,,\\[11.38109pt] \displaystyle\partial_{t}F_{B}+\frac{1}{m_{B}}p\cdot\nabla_{x}F_{B}=Q^{BB}(F_{B},F_{B})+Q^{BA}(F_{B},F_{A})\,.\end{array}\right.$$

$$Q^{XY}(F_{X},F_{Y})=\nabla_{p}\cdot\Big{\{}\int_{{\mathbb{R}}^{3}}\Phi^{X,Y}\left(\frac{p}{m_{X}}-\frac{p_{\ast}}{m_{Y}}\right)\big{[}F_{Y}(p_{\ast})\nabla_{p}F_{X}(p)-F_{X}(p)\nabla_{p_{\ast}}F_{Y}(p_{\ast})\big{]}dp_{\ast}\Big{\}}\,,$$

$${\Phi }^{X,Y}(z)=\frac{m_X{m}_Y}{m_X+m_Y}\left[I_3-\frac{z\otimes z}{|z{|}^2}\right]\phi (|z|),$$

$$\left\{\begin{array}[c]{l}\displaystyle\int_{{\mathbb{R}}^{3}}\big{\{}1,p,|p|^{2}\big{\}}Q^{XX}(F_{X},F_{X})dp=0\quad\hbox{for}\quad X\in\{A,B\}\,,\\[8.53581pt] \displaystyle\int_{{\mathbb{R}}^{3}}Q^{AB}(F_{A},F_{B})dp=\int_{{\mathbb{R}}^{3}}Q^{BA}(F_{B},F_{A})dp=0\,,\\[8.53581pt] \displaystyle\int_{{\mathbb{R}}^{3}}p\left[Q^{AB}(F_{A},F_{B})+Q^{BA}(F_{B},F_{A})\right]dp=0\,,\\[8.53581pt] \displaystyle\int_{{\mathbb{R}}^{3}}\frac{|p|^{2}}{2}\left[\frac{1}{m_{A}}Q^{AB}(F_{A},F_{B})+\frac{1}{m_{B}}Q^{BA}(F_{B},F_{A})\right]dp=0\,.\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}F_{X}dpdx=0\quad\hbox{for}\quad X\in\{A,B\}\,,\\[8.53581pt] \displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}p\left[F_{A}+F_{B}\right]dpdx=0\,,\\[8.53581pt] \displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}\frac{|p|^{2}}{2}\left[\frac{1}{m_{A}}F_{A}+\frac{1}{m_{B}}F_{B}\right]dpdx=0\,.\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle F_{A}=\sqrt{M_{A}}f_{A}\,,\\[8.53581pt] \displaystyle F_{B}=M_{B}+\sqrt{M_{B}}f_{B}\,,\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle F_{A}(0,x,p)=\sqrt{M_{A}}g_{in}(x,p)\,,\\[8.53581pt] \displaystyle F_{B}(0,x,p)=M_{B}+\sqrt{M_{B}}h_{in}(x,p),\end{array}\right.$$

$$M_{B}(p)=\frac{1}{(2\pi)^{3/2}}\exp\Big{(}\frac{-|p|^{2}}{2}\Big{)}\,,$$

$$M_{A}(p)=\frac{1}{(2\pi m_{A}/m_{B})^{3/2}}\exp\Big{(}-\frac{m_{A}}{m_{B}}\frac{|p|^{2}}{2}\Big{)}\,,$$

$$Q^{AB}(M_A,M_B)=Q^{BA}(M_B,M_A)=0,$$

$$\left\{\begin{array}[c]{l}\displaystyle Q^{AA}(M_{A},M_{A})=0,\quad Q^{BB}(M_{B},M_{B})=0\,,\\[8.53581pt] \displaystyle Q^{AB}(M_{A},M_{B})=0,\quad Q^{BA}(M_{B},M_{A})=0\,.\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle\partial_{t}f_{A}+\frac{1}{m_{A}}p\cdot\nabla_{x}f_{A}=L_{AB}f_{A}+\Gamma^{AA}(f_{A},f_{A})+\Gamma^{AB}(f_{A},f_{B})\,,\\[8.53581pt] \displaystyle\partial_{t}f_{B}+\frac{1}{m_{B}}p\cdot\nabla_{x}f_{B}=L_{BB}f_{B}+\Gamma^{BB}(f_{B},f_{B})+L_{BA}f_{A}+\Gamma^{BA}(f_{B},f_{A})\,,\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle L_{AB}f_{A}=\frac{1}{\sqrt{M_{A}}}Q^{AB}(f_{A}\sqrt{M_{A}},M_{B})\,,\\[8.53581pt] \displaystyle L_{BA}f_{A}=\frac{1}{\sqrt{M_{B}}}Q^{BA}(M_{B},f_{A}\sqrt{M_{A}})\,,\\[8.53581pt] \displaystyle L_{BB}f_{B}=\frac{1}{\sqrt{M_{B}}}\Big{(}Q^{BB}(M_{B},f_{B}\sqrt{M_{B}})+Q^{BB}(f_{B}\sqrt{M_{B}},M_{B})\Big{)}\,,\\[8.53581pt] \displaystyle\Gamma^{XY}(f_{X},f_{Y})=\frac{1}{\sqrt{M_{X}}}Q^{XY}(f_{X}\sqrt{M_{X}},f_{Y}\sqrt{M_{Y}})\,.\end{array}\right.$$

$$\left\{\begin{array}[c]{l}\displaystyle\partial_{t}g+\frac{1}{m_{A}}p\cdot\nabla_{x}g=L_{AB}g\,,\\[8.53581pt] \displaystyle\partial_{t}h+\frac{1}{m_{B}}p\cdot\nabla_{x}h=L_{BB}h+L_{BA}g\,.\end{array}\right.$$

$$|f|_{L_{p}^{2}}=\Big{(}\int_{{\mathbb{R}}^{3}}|f|^{2}dp\Big{)}^{1/2}\,,$$

$$|f|_{L_{p}^{2}(m)}=\Big{(}\int_{{\mathbb{R}}^{3}}|f|^{2}mdp\Big{)}^{1/2}\,,\quad|f|_{L_{\theta}^{2}}=\Big{(}\int_{{\mathbb{R}}^{3}}|f|^{2}\left\langle p\right\rangle^{2\theta}dp\Big{)}^{1/2}\,.$$

$$|f|_{L_{x}^{2}}=\Big{(}\int_{{\mathbb{R}}^{3}}|f|^{2}dx\Big{)}^{1/2}\,.$$

$$|f{|}_{L_x^{\infty }}=\underset{x\in {\mathbb{R}}^3}{\sup }|f(x)|.$$

$$\mathbb{P}(p)u=\frac{u\cdot p}{|p{|}^2}p.$$

$$|f|_{L_{\sigma}^{2}}^{2}=|\left\langle p\right\rangle^{\frac{\gamma+2}{2}}f|_{L_{p}^{2}}^{2}+|\left\langle p\right\rangle^{\frac{\gamma}{2}}\mathbb{P}(p)\nabla_{p}f|_{L_{p}^{2}}^{2}+\big{|}\left\langle p\right\rangle^{\frac{\gamma+2}{2}}\big{[}I_{3}-\mathbb{P}(p)\big{]}\nabla_{p}f\big{|}_{L_{p}^{2}}^{2}\,,$$

$$|f|_{L_{\sigma,\theta}^{2}}^{2}=|\left\langle p\right\rangle^{\frac{\gamma+2}{2}+\theta}f|_{L_{p}^{2}}^{2}+|\left\langle p\right\rangle^{\frac{\gamma}{2}+\theta}\mathbb{P}(p)\nabla_{p}f|_{L_{p}^{2}}^{2}+\big{|}\left\langle p\right\rangle^{\frac{\gamma+2}{2}+\theta}\big{[}I_{3}-\mathbb{P}(p)\big{]}\nabla_{p}f\big{|}_{L_{p}^{2}}^{2}\,.$$

$$\Vert f{\Vert }_{L^2}^2={\int }_{{\mathbb{R}}^3}|f{|}_{L_p^2}^{2}dx,\Vert f{\Vert }_{L^2(m)}^2={\int }_{{\mathbb{R}}^3}|f{|}_{L_p^2(m)}^{2}dx,\Vert f{\Vert }_{L_{\theta }^2}^2={\int }_{{\mathbb{R}}^3}|f{|}_{L_{\theta }^2}^{2}dx,$$

$$\Vert f{\Vert }_{L_{\sigma }^2}^2={\int }_{{\mathbb{R}}^3}|f{|}_{L_{\sigma }^2}^{2}dx,\Vert f{\Vert }_{L_{\sigma ,\theta }^2}^2={\int }_{{\mathbb{R}}^3}|f{|}_{L_{\sigma ,\theta }^2}^{2}dx,$$

$$\Vert f{\Vert }_{L_x^{\infty }{L}_p^2}=\underset{x\in {\mathbb{R}}^3}{\sup }|f{|}_{L_p^2},\Vert f{\Vert }_{L_x^1{L}_p^2}={\int }_{{\mathbb{R}}^3}|f{|}_{L_p^2}dx.$$

$${\Vert f\Vert }_{H^k(m)}=\sum _{k_1+k_2=k}{\Vert {\partial }_x^{k_1}{\partial }_p^{k_2}f\Vert }_{L^2(m)}.$$

$$\Vert {\nabla }_p^{\ell }{\nabla }_x^{k}g{\Vert }_{L^2}\lesssim (1+t{)}^{-(3+k)/2}\left({\Vert g_{in}\Vert }_{L_x^1{L}_p^2}+{\Vert g_{in}\Vert }_{L^2\left(w_{k+\ell +2}\right)}\right),$$

$$\Vert {\nabla }_p^{\ell }{\nabla }_x^{k}h{\Vert }_{L^2}\lesssim (1+t{)}^{-(3+k)/2}\left({\Vert (g_{in},h_{in})\Vert }_{L_x^1{L}_p^2}+{\Vert (g_{in},h_{in})\Vert }_{L^2\left(w_{k+\ell +2}\right)}\right).$$

$$w_n\equiv \{\begin{array}{ll}1,& \gamma \in \left[0,1\right],\\ {⟨p⟩}^{|\gamma |n},& \gamma \in \left[-2,0\right).\end{array}$$

$$\{\chi_{0},\chi_{j},\chi_{4}\}=\big{\{}\sqrt{M_{B}},p_{j}\sqrt{M_{B}},\frac{1}{\sqrt{6}}(|p|^{2}-3)\sqrt{M_{B}}\big{\}}\,,\quad j=1,2,3\,.$$

$$L_{XY}f=-{\tilde{\Lambda }}^{XY}f+{\tilde{K}}^{XY}f,$$

Full text

The first author is supported by the Ministry of Science and Technology under the grant MOST 105-2115-M-006-002-. Part of the project was done while the second author was in Institute of Mathematics, Academia Sinica, Taiwan. The third author is supported by the Ministry of Science and Technology under the grant 104-2628-M-006-003-MY4 and National Center for Theoretical Sciences.

1. Introduction

1.1. The Model

This paper is concerned with the Cauchy problem for a system of Landau equations describing collisions in an ideal plasma mixture. The mixture is constituted by two species with mass $(m_{A},m_{B})$ and is described by density functions $(F_{A}(t,x,p),F_{B}(t,x,p))$ defined in the phase-space of position and momentum. It takes the form of the following system:

[TABLE]

The right hand side consists of the usual collision terms which for $X,Y\in\{A,B\}$ are given by:

[TABLE]

here

[TABLE]

the potential $\varphi(|z|)=|z|^{\gamma+2}$. We assume throughout this paper that $\gamma\in[-2,1]$.

We can find the following invariant properties of the collision operator (see [10]):

[TABLE]

This means that the system $\left(\ref{mix.1.a}\right)$ is possessed of the conservation of mass, total momentum and total energy:

[TABLE]

1.2. The Linearized Problem

Based on the idea introduced by [17], we consider the following perturbation:

[TABLE]

with the initial conditions of $F_{A}$ and $F_{B}$ satisfying

[TABLE]

where $M_{A}$ and $M_{B}$ are the Maxwellian states. Here, we fix the Maxwellian state of species $B$ as the standard Gaussian, i.e.,

[TABLE]

and then $M_{A}$, the Maxwellian state of species $A$, is chosen uniquely as

[TABLE]

for which the condition

[TABLE]

is valid. Therefore, under this choice, the two Maxwellian states satisfy

[TABLE]

It is easy to check that the system for $(f_{A},f_{B})$ satisfies

[TABLE]

where

[TABLE]

If we assume the total mass of species $A$ and the perturbations in species $B$ are sufficiently small, in general, the basic time asymptotic behavior of the nonlinear solution is governed primarily by the linearized equation (up to a large time scale), since the nonlinear term is quadratic. In this regard, we will consider the linearized problem as follows:

[TABLE]

In this paper, we shall study the regularization effect and large time behavior of the system (1.5).

1.3. Review of Previous Works

Let us review the mixtures of the Boltzmann equation first. The studies of gas mixtures in terms of the Boltzmann equations were firstly introduced by [20]. Many interesting physical problems such as “ghost effects” and “Knudsen layer” for gas mixtures have been investigated in [2, 16]. From a mathematical point of view, if we assume that all species are close to equilibrium, the explicit spectrum gap estimate and the hypocoercivity for a linearized multi-species Boltzmann system in the torus can be found in [7], and exponential decay towards equilibrium in general function spaces was done by Briant [3] and Braint and Daus [4]. Moreover, the global existence and stability of mild solutions to the Boltzmann system were completed by Ha, Noh and Yun in [12]. This research is also interested in phenomena related to vapor-vapor mixtures. This corresponds to the mathematical formulation that one species is near a vacuum and the other is near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis in the whole space case was introduced by Sotirov and Yu [17], and the torus case can be found in [22].

For the mono species Landau equation, we refer to Alexandre and Villani [1] for existence of renormalized solutions, to Desvillette and Villani [9] for conditional almost exponential convergence towards equilibrium and to a recent work by Carrapatoso, Tristani and Wu [5] for exponential decay towards equilibrium when initial data are close enough to equilibrium. Moreover, Guo [11] and Strain and Guo [18, 19] developed an existence and convergence towards equilibrium theory based on energy methods for initial data close to the equilibrium state in some Sobolev norms. Recently the set of initial data for which this theory is valid has been enlarged by Carrapatoso and Mischler [6] via a linearization method.

For the multi species Landau equation, we refer to Chapter 4 in [13] for a physical derivation and discussion. However, on the mathematical side, there are very few results. One can only find the work done by Gualdani and Zamponi [10], in which all species are assumed to be close to equilibrium; the explicit spectrum gap estimate and the hypocoercivity for a linearized multi-species Landau system in the torus were obtained.

In this paper, we are concerned with the Landau equation in the whole space, with initial condition of one species near a vacuum and the other near a Maxwellian equilibrium state, and study the regularization effect and large time behavior of solutions of this Cauchy problem.

1.4. Main theorem

Before the presentation of the main theorem, let us define some notations in this paper. We denote $\left\langle p\right\rangle^{s}=(1+|p|^{2})^{s/2}$, $s\in{\mathbb{R}}$. For the microscopic variable $p$, we denote

[TABLE]

and the weighted norms $|f|_{L_{p}^{2}(m)}$ and $|f|_{L_{\theta}^{2}}$ respectively by

[TABLE]

The $L_{p}^{2}$ inner product in ${\mathbb{R}}^{3}$ will be denoted by \big{<}\cdot,\cdot\big{>}_{p}. For the space variable $x$, we have similar notations. In fact, $L_{x}^{2}$ is the classical Hilbert space with norm

[TABLE]

We denote the sup norm as

[TABLE]

The standard vector product will be denoted by $(\cdot,\cdot)$. For any vector function $u\in L_{p}^{2}$, $\mathbb{P}(p)u$ denotes the orthogonal projection to the direction of vector $p$, i.e.,

[TABLE]

For the Landau equation, the natural norm in $p$ is $|\cdot|_{L_{\sigma}^{2}}$, which is defined by

[TABLE]

and the weighted norm by

[TABLE]

Moreover, we define

[TABLE]

[TABLE]

and

[TABLE]

Finally, we define the high order Sobolev norm: Let $k\in{\mathbb{N}}$ and $k_{1},k_{2}\geq 0$,

[TABLE]

The domain decomposition plays an important role for the coercivity of the collision operators, hence we define a cut-off function $\chi:{\mathbb{R}}\rightarrow{\mathbb{R}}$, which is a smooth non-increasing function, $\chi(s)=1$ for $s\leq 1$, $\chi(s)=0$ for $s\geq 2$ and $0\leq\chi\leq 1$. Moreover, we define $\chi_{R}(s)=\chi(s/R)$.

For simplicity of notations, hereafter, we abbreviate “ $\leq C$ ” to “ $\lesssim$ ”, where $C$ is a positive constant depending only on fixed numbers.

In the following, we state our main result.

Theorem 1**.**

Let k\and\ \ell\be nonnegative integers, and assume that the initial conditions $(g_{in},h_{in})\in L_{x}^{1}L_{p}^{2}\cap L^{2}(w_{k+\ell+2})$. Then the solutions of $(\ref{mix.2.a})$ have the following estimates:

[TABLE]

and

[TABLE]

Here

[TABLE]

Let us remark that this theorem is a simplified version, and it is covered in more details in Theorems 8 and 16.

1.5. Method of proof and plan of the paper

The basic principles used to develop the theory are the “separation of scales” and “regularization estimate”. The separation of scales is a concept created in [14] for a 1D Boltzmann equation with hard sphere in the whole space. The regularization estimate for the Landau equation was constructed in [5] initially.

In this paper, a long wave-short wave decomposition is introduced. Under this decomposition, one can follow the spectrum analysis of the Landau collision operator in [23] to decompose the solution to the linearized Landau equation as the long wave-fluid part, the lone wave-nonfluid part and the short wave part, which separate the scales, and hence get different decay rates for each part.

On the other hand, the regularization estimate plays an important role in this paper, which leads us to remove the regularity assumption on the initial condition. Based on the basic regularization estimate established in [5], we are able to construct new functionals for high order derivatives, and then by using bootstrap process to get the smoothing estimates in both the space variable $x$ and momentum variable $p$.

The separation of scales helps us get different decay rate in each part, but this idea can not improve regularity. Contrarily, the regularization estimate helps us improve regularity, but this can not get time decay estimates. Fortunately, if putting these two ideas together, by interpolation argument, one can get a smoothing effect and time decay simultaneously. To the best of our knowledge, this thinking is itself new for Landau kinetic equations.

In the following we point out some significant points of our method:

• •

The problem we consider here is the whole space case, and hence we expect the algebraic decay estimate of the solutions rather than the exponential decay in the torus case [3, 4, 5, 7, 10, 18, 19]. Under our linearization (one species near a vacuum and the other near equilibrium), the system is decoupled, so that one can estimate each equation $(g,h)$ separately. For solution $g$, as we mentioned above, one can get smoothing effect and time decay simultaneously by interpolation argument.

For solution $h$, we solve it by using the Duhamel principle and treating $g$ as a source term. This method usually results in that the decay rate of $h$ is slower than that of $g$. Indeed, the characteristics for the macroscopic equations for $h$ coincide with the direction of mass diffusion for $g$. Thus, there is a possible wave resonance, which leads to the slower decay of solution $h$. On the microscopic level, the resonance is from the small divisor caused by the closeness between different wave frequencies (i.e., $\lambda(\eta)$ in Lemma 6 and $\sigma_{j}(\eta),\leavevmode\nobreak\ j=2,3,4$ in Lemma 7). It was shown in [17] that for a 1D Boltzmann gas mixture there is only one characteristic for $h$ coinciding with $g$, which is the diffusion sound wave carrying mass and energy. The resonance problem was resolved by considering a microscopic cancellation representing the conservation of mass and total energy. For our 3D case, there are other two characteristics for $h$ having the same direction as mass diffusion for $g$ in addition to the diffusion sound wave. Their resonances are resolved by other microscopic cancellations from the conservation of total momentum and orthogonality between propagating directions (see the proof of Proposition 17 for details). These crucial cancellation properties lead $h$ to have the same decay rate as $g$. It would also be interesting to understand the detailed wave propagation of $(g,h)$, and they are postponed in our future studies.

• •

The method to prove high order regularization estimates in this paper is interesting and itself new. In the paper [5], they get a first order regularization estimate for the semigroup $e^{t\mathcal{L}_{AB}}$ in both the $x$ and $p$ variables (see Lemma 9), where $\mathcal{L}_{AB}$ contains the transport part $p\cdot\nabla_{x}$ and the ellipticity part $\Lambda_{AB}$, and the regularization mechanism is their combined effect. In order to improve the high order $x$ and $p$ regularities based on this result, we employ different arguments respectively. For the space variable $x$, we design a Picard type iteration, which treats $\mathcal{L}_{AB}$ as an anstaz and the operator $K_{AB}$ as a source term. The first several terms in the iteration contain the most singular part of the solution. Thereupon, after finite iterations (depending on the differentiation order), we can extract the singular waves and then the remainder part will become smooth in $x$ variable for all time (see Lemma 10). For the momentum $p$, we construct a new functional including high order $p$ derivatives and all lower order $x$, $p$ and mixed derivatives, where the lower order derivatives are used to take care of the nonzero commutator $[p\cdot\nabla_{x},\nabla_{p}]$. By making use of $x$ regularity obtained previously, and choosing suitable combinations of derivatives, one can get high order regularization estimates in $p$ variable for the semigroup $e^{t\mathcal{L}_{AB}}$. Then the smoothing effect of the solution in the $p$ variable will be obtained inductively (see subsection 3.2.1).

The paper is organized as follows: we list some properties of the linearized collision operator in section 2; then we prove the estimates of $g$ and $h$ in sections 3 and 4 respectively.

2. Preliminaries

The basic structures of the linearized Landau collision operator are well known [8]. Here we take a closer look at these structures with more modifications (see Lemma 2-Lemma 4). After that, Lemma 5 is done by Guo [11], which would be of great help to us regarding the regularization estimate. On the other hand, the separation of scales relies on the spectrum analysis of the operator $L_{XY}^{\eta}=-\frac{i\eta\cdot p}{m_{X}}+L_{XY}$ in the classical Hilbert space $L_{p}^{2}$, where $(X,Y)=(A,B)$ or $(B,B)$; this is why our argument can not include the case of very soft potentials and Coulomb potential. Following the systematic procedure established by Yang and Yu [23], we can easily derive the spectrum of the operator $L_{XY}^{\eta}.$ Moreover, we demonstrate that when $\left|\eta\right|$ is small, the corresponding eigenfunctions are not only smooth but also decays faster than any polynomial in $p.$

Lemma 2**.**

(The null spaces of $L_{BB}$ and $L_{AB}$ ) The null space of $L_{BB}$ is a five-dimensional vector space with an orthonormal basis $\{\chi_{i}\}_{i=0}^{4}$, where

[TABLE]

On the other hand, the null space of $L_{AB}$ is a one-dimensional vector space $\{E_{D}\}$, where $E_{D}=M_{A}^{1/2}$.

Lemma 3**.**

*(Decomposition) Let $(X,Y)=(B,B)$ or $(A,B)$. Then

(i) the collision operator $L_{XY}$ consists of an elliptic type operator $\widetilde{\Lambda}^{XY}$ and an integral operator $\widetilde{K}^{XY}$:*

[TABLE]

where

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

(ii) The spectrum of $\sigma^{XY}\left(p\right)$ consists of a simple eigenvalue $\lambda_{1}\left(p\right)$ associated with the eigenvector $p$, and a double eigenvalue $\lambda_{2}\left(p\right)$ associated with the eigenspace $p^{\bot}.$ Furthermore

[TABLE]

[TABLE]

with

[TABLE]

and as$\ \left|p\right|\rightarrow\infty,$

[TABLE]

Immediately, we have

[TABLE]

(iii) For any $k\in{\mathbb{N}}$, we have

[TABLE]

and as $|p|\rightarrow\infty$,

[TABLE]

(iv) (Coercivity) Rewrite

[TABLE]

Let

[TABLE]

where $\varpi,R$ are positive constants large enough. Then

[TABLE]

for some c_{0}>0,\and

[TABLE]

The behavior of the operator $L_{BA}$ is different from $L_{BB}$ and $L_{AB}$, but similar to $K^{BB}.$

Lemma 4**.**

The operator $L_{BA}$ is a bounded integral operator:

[TABLE]

where

[TABLE]

Lemma 5** (Lemma 6, [11]).**

Let $(X,Y)=(B,B)$ or $(A,B)$, assume $\left|\alpha\right|\geq 0$, for small $\delta>0$, we have

[TABLE]

and

[TABLE]

In particular,

[TABLE]

Taking the Fourier transform in the space variable $x,$ we consider the operator $L_{XY}^{\eta}=-\frac{i\eta\cdot p}{m_{X}}+L_{XY}$ , where $(X,Y)=(A,B)$ or $(B,B)$. Then the corresponding spectrum $\mathrm{Spec}_{XY}(\eta)$ in the classical Hilbert space $L_{p}^{2}$ are derived as follows.

Lemma 6**.**

*(Spectrum of $L_{AB}^{\eta}$ [23]) Given $\delta>0$

(i) there exists $\tau_{2}=\tau_{2}(\delta)>0$ such that if $|\eta|>\delta$,*

[TABLE]

(ii) If $\eta=|\eta|\omega$ and $|\eta|<\delta$, the spectrum within the region $\{z\in\mathbb{C}:Re(z)>-\tau_{2}\}$ is consisting of exactly one eigenvalue $\{\lambda(\eta)\}$,

[TABLE]

associated with its eigenfunction $\{e_{D}(\eta)\}$. They have the expansions

[TABLE]

with $a_{2}=-\left<L_{AB}^{-1}(p\cdot\omega/m_{A})E_{D},(p\cdot\omega/m_{A})E_{D}\right>_{p}>0$ and $E_{D,1}=L_{AB}^{-1}(p\cdot\omega/m_{A})E_{D}.$ Here $\{e_{D}(\eta)\}$ can be normalized by \big{<}e_{D}(-\eta),e_{D}(\eta)\big{>}_{p}=1.

More precisely, the semigroup $e^{(-i\eta\cdot p+L_{AB})t}$ can be decomposed as

[TABLE]

and there exist $a(\tau_{2})>0$, $\overline{a}_{2}>0$ such that $|e^{(-i\eta\cdot p+L_{AB})t}\Pi_{\eta}^{D\perp}|_{L_{p}^{2}}\lesssim e^{-a(\tau_{2})t}$ and $|e^{\lambda(\eta)t}|\leq e^{-\overline{a}_{2}|\eta|^{2}t}$.

Moreover, the eigenfunction $e_{D}(p,\eta)$ is smooth and decays faster than any polynomial in $p$, i.e., for any $\theta\geq 0$, $|\alpha|\geq 0$, $|\left\langle p\right\rangle^{\theta}\partial_{p}^{\alpha}e_{D}(p,\eta)|_{L_{p}^{2}}\leq C_{\theta,\alpha}|e_{D}(p,\eta)|_{L_{p}^{2}}$.

Proof.

The spectrum analysis can be found in [23]. Here it is suffices to show that $e_{D}(p,\eta)$ as a function of $p$ is in the Schwartz space. Note that

[TABLE]

for any $\theta\geq 0$ and $|\alpha|>0$. Taking the inner product, we have

[TABLE]

and

[TABLE]

In view of (2.1),

[TABLE]

∎

Lemma 7**.**

*(Spectrum of $L_{BB}^{\eta}$ [23] ) Given $\delta>0$,

(i) there exists $\tau_{1}=\tau_{1}(\delta)>0$ such that if $|\eta|>\delta$,*

[TABLE]

(ii) If $\eta=|\eta|\omega$ and $|\eta|<\delta$, the spectrum within the region $\{z\in\mathbb{C}:Re(z)>-\tau_{1}\}$ consists of exactly five eigenvalues $\{\sigma_{j}(\eta)\}_{j=0}^{4}$,

[TABLE]

associated with the corresponding eigenfunctions $\{e_{j}(\eta)\}_{j=0}^{4}$. They have the expansions

[TABLE]

with $a_{j,2}=\left<L_{BB}^{-1}\mathbb{P}_{1}(p\cdot\omega/m_{B})E_{j},\mathbb{P}_{1}(p\cdot\omega/m_{B})E_{j}\right>_{p}>0$ and

[TABLE]

where $\Psi=(\chi_{1},\chi_{2},\chi_{3})$, $\{\omega,\omega_{1}^{\perp},\omega_{2}^{\perp}\}$ is an orthonormal basis of ${\mathbb{R}}^{3}$, $\{e_{j}(\eta)\}_{j=0}^{4}$ can be normalized by \big{<}e_{j}(-\eta),e_{l}(\eta)\big{>}_{p}=\delta_{jl}, $0\leq j,l\leq 4$.

More precisely, the semigroup $e^{(-i\eta\cdot p+L_{BB})t}$ can be decomposed as

[TABLE]

and there exist $a(\tau_{1})>0$, $\overline{a}_{1}>0$ such that $|e^{(-i\eta\cdot p+L_{BB})t}\Pi_{\eta}^{\perp}|_{L_{p}^{2}}\lesssim e^{-a(\tau_{1})t}$ and $|e^{\sigma_{j}(\eta)t}|\leq e^{-\overline{a}_{1}|\eta|^{2}t}$ for all $0\leq j\leq 4$.

Moreover, the eigenfunction $e_{j}(p,\eta)$, $0\leq j\leq 4$ are smooth and decay faster than any polynomial in $p$, i.e., for any $\theta\geq 0$, $|\alpha|\geq 0$, $|\left\langle p\right\rangle^{\theta}\partial_{p}^{\alpha}e_{j}(p,\eta)|_{L_{p}^{2}}\leq C_{\theta,\alpha}|e_{j}(p,\eta)|_{L_{p}^{2}}$.

3. The result for $g$

In accordance with the theory of the semi-groups, the solution of the linearized Landau equation

[TABLE]

can be represented by

[TABLE]

where $\mathbb{G_{AB}}^{t}$ is the solution operator and $\hat{g}$ means the Fourier transform in the space variable $x$. Furthermore, we introduce a long wave-short wave decomposition for the solution $g$:

[TABLE]

where

[TABLE]

[TABLE]

In order to study the long wave part $g_{L},$ we need further to decompose it into the fluid part and non-fluid part, i.e., $g_{L}=g_{L,0}+g_{L,\bot},$ where

[TABLE]

[TABLE]

The main result for the solution $g$ is stated as below.

Theorem 8**.**

Let k\and\ \ell\be nonnegative integers. Then for $t\geq 1,$

[TABLE]

[TABLE]

and

[TABLE]

the constant $C>0$ depending only upon $k$ and $\ell$. Here

[TABLE]

In what follows, we prove this theorem by several stages. To begin with, we establish the $x$-regularity of $g$ in subsection 3.1. After that, we take advantage of it to improve the $p$-regularity via certain energy functionals and Duhamel’s principle in the next subsection. On the basis of these regularization estimates, we finally obtain the decay rate by invoking the Sobolev inequality and interpolation inequality.

3.1. Improvement of the $x$-regularity

It immediately follows from the Fourier transform in $x$ and the $p$-regularity of the eigenfunction $e_{D}(\eta)$ that

[TABLE]

On the other hand, from the spectral gap it follows that

[TABLE]

and

[TABLE]

Noticing that $g_{S}=g-g_{L},$ we need the regularization estimate of $g$ in $x$ further. To that end, we introduce a new operator $\mathcal{L}_{AB}$:

[TABLE]

Let $n$ be a nonnegative integer and let $\theta\geq 0$. Hereafter, we define m_{0}=m_{\theta}=\left\langle p\right\rangle^{\theta}\and

[TABLE]

Recall the energy estimate and the regularization property of the operator $e^{t\mathcal{L}_{AB}}$ in [5]:

Lemma 9**.**

If $u$ solves the equation

[TABLE]

then there exists $C>0$ such that

[TABLE]

Moreover, for $0<t\leq 1$, we have

[TABLE]

and

[TABLE]

Utilizing this lemma, we establish the regularity in $x$ as below. Worthy to be mentioned here, the proof for the case $k=1$ is crucial. For the $\mathbf{x}$-derivatives, from Lemma 9, $t^{-3/2}$ is not integrable as $t$ is small. It appears to be harmful whenever we use the Duhamel principle. In effect, one can see that the $L_{x}^{2}L_{p}^{2}\left(m_{\theta}\right)$ norms of the integrands in $\left(\ref{aa.9}\right)$ and $\left(\ref{aa.10}\right)$ are integrable in $t$ if we appropriately couple the operator $t\nabla_{x}$ with $e^{t\mathcal{L}_{AB}}$.

Lemma 10**.**

Let $f$ be a solution of the linearized Landau equation

[TABLE]

Let $k\in\mathbb{N\cup\{}0\mathbb{\}}$. Then

*(i)  for *$0<t\leq 1$

[TABLE]

*(ii) For *$t\geq 1,$

[TABLE]

Proof.

For $k=0,$ from Lemma 5 and the fact that $\left\|f\right\|_{L^{2}}\leq\left\|f_{in}\right\|_{L^{2}}$, it follows that

[TABLE]

which implies that there exists a universal constant $C>0$ such that

[TABLE]

Moreover, since the operator $\partial_{x}^{\alpha}$ is commutative with the equation $\left(\ref{Equ-AB}\right)$,

[TABLE]

as well.

To prove this lemma for $k\geq 1$, we design a Picard type iteration, treating $K^{AB}$ as a source term, as below: The zero order approximation $f^{(0)}$ is defined by

[TABLE]

and thus the difference $f-f^{(0)}$ satisfies

[TABLE]

Therefore, we define the first order approximation $f^{(1)}$ by

[TABLE]

Inductively, we can define the $j^{\mathrm{th}}$ order approximation $f^{(j)}$, $j\geq 1$, as

[TABLE]

Now, rewrite $f$ as

[TABLE]

here the remainder $\mathcal{R}^{(k)}$ solving the equation

[TABLE]

Under this decomposition,** we shall tackle the $x$-regularity for each component in the cases of $k=1$** and $k=2.$ The case of $k\geq 3$ follows the same argument as $k=2.$

Case 1: $k=1$, $f=\sum_{j=0}^{2}f^{(j)}+\mathcal{R}^{(1)}$.

Step 1: The first $x$-derivative in small time. We claim that for $0<t\leq 1$ and $0\leq j\leq 2$,

[TABLE]

and

[TABLE]

The estimate of $f^{(0)}$ is obviously from Lemma 9. Notice that

[TABLE]

Hence,

[TABLE]

By Lemma 9,

[TABLE]

Likewise,

[TABLE]

and

[TABLE]

By Lemma 9 again,

[TABLE]

Here we remark that $\nabla_{x}f^{(2)}$ is regular when $t$ is close to [math].

For the remainder $\mathcal{R}^{(1)}$, since

[TABLE]

we deduce

[TABLE]

by $\left(\ref{f-weighted-energy}\right)$ and $\left(\ref{aa.1}\right)$. In conclusion, for $0<t\leq 1$,

[TABLE]

Step 2: The first $x$-derivative in large time. It readily follows from $\left(\ref{fx-weighted-energy}\right)$ and $\left(\ref{f-x-stime}\right)$ that for $t>1$,

[TABLE]

Case 2: $k=2$, $f=\sum_{j=0}^{4}f^{(j)}+\mathcal{R}^{(2)}$.

Step 1: The second $x$-derivative in small time. Let $0<t\leq 1$. We shall show that

[TABLE]

and

[TABLE]

For the first term, we only give the estimates of $f^{(0)}$ and $f^{(1)}$ and the others are similar.

Let $0<t_{0}\leq 1$ and $t_{0}/2<t\leq t_{0}$. Since

[TABLE]

we have

[TABLE]

by Lemma 9. Taking $t=t_{0}$ gives

[TABLE]

Since $t_{0}\in(0,1]$ is arbitrary, this completes the proof of $f^{(0)}$. Similarly, we have

[TABLE]

and thus it follows from Lemma 9, $\left(\ref{f1-x}\right)$ and $\left(\ref{f0-xx}\right)$ that

[TABLE]

Plugging $t=t_{0}$ into the above inequality yields

[TABLE]

as desired.

For the remainder $\mathcal{R}^{(2)}$, since

[TABLE]

we deduce

[TABLE]

by $\left(\ref{f-weighted-energy}\right)$ and $\left(\ref{fj-xx}\right).$ Consequently, for $0<t\leq 1,$

[TABLE]

Step 2: The second $x$-derivative in large time. For $t>1$, we have

[TABLE]

due to $\left(\ref{fx-weighted-energy}\right).$ ∎

We now return to the $x$-regularity problem for $g_{S}.$ By Lemma 10 $\left(ii\right)$ with initial condition $f_{in}(x,p)=g_{in}(x,p)$ and taking $m_{\theta}=1,$

[TABLE]

for $t\geq 1.$ Therefore,

[TABLE]

for each $k\in\mathbb{N\cup}\left\{0\right\}$. On the other hand, the interpolation inequality says that

[TABLE]

Hence,

[TABLE]

Combining this with the Sobolev inequality [21, Proposition 3.8], yields

[TABLE]

Summing up, we conclude the following proposition.

Proposition 11**.**

Let $k,$ $\ell\in\mathbb{N\cup\{}0\}.$ Then for $t\geq 1,$

[TABLE]

[TABLE]

and

[TABLE]

where $C>0,\ C_{k}>0.$ Here

[TABLE]

3.2. Improvement of the $p$-regularity (general discussion)

In this subsection, we will study the $p$-regularity of the linearized Landau equation

[TABLE]

3.2.1. Improvement of the $p$-regularity in finite time

Assume that $0<t\leq 1$. In order to improve the $p$-regularity of $f$ in finite time, we first establish the $p$-regularity of the operator $e^{t\mathcal{L}_{AB}}.$ Specifically, let $u$ be a solution of the equation $\partial_{t}u=\mathcal{L}_{AB}u,$ $u(0,x,p)=u_{0}(x,p),$ and then we prove that for each $\ell\in\mathbb{N}$, there exists a functional $\mathcal{F}_{\ell}$ such that

[TABLE]

where $\mathcal{F}_{\ell}\left(0,u\right)$ involves only the other derivatives of the initial data $u_{0}$ with differentiation order less or equal to $\ell.$ With the aid of the operator $e^{t\mathcal{L}_{AB}}$ and the Duhamel principle, we use a bootstrap process to get the regularization estimate of $f$ in Both the space variable $x$ and momentum variable $p$ simultaneously.

Before proceeding, we derive some useful inequalities to simplify tedious computations regarding the evolutions we will consider later on. Besides, these inequalities clue us in on the form of the desired functional $\mathcal{F}_{\ell}$.

Lemma 12**.**

Let $u$ be a solution to the equation $\partial_{t}u=\mathcal{L}_{AB}u.$ Then for $k,\ell\in\mathbb{N\cup\{}0\mathbb{\}},$ there exist $c_{0}>0$ independent of $k$ and $\ell$, and $C>0$ such that

[TABLE]

[TABLE]

and

[TABLE]

Proof.

Let \alpha\and $\beta$ be any multi-indices. Direct computation shows

[TABLE]

where $\partial_{j}=\partial_{p_{j}}$, $\partial_{ij}^{2}$ $=\partial_{p_{i}p_{j}}^{2}$ and

[TABLE]

By Lemma 3 $(iii)$,

[TABLE]

which implies that

[TABLE]

provided $\varpi$ and $R$ are sufficiently large. Owing to Lemma 3 $(ii)$, there exists $c_{2}>0$ independent of $\alpha$ such that

[TABLE]

as required.

For $\left|\alpha\right|\geq 1,$ compute the evolution

[TABLE]

here we denote \binom{\alpha}{\beta}\by $C_{\beta}^{\alpha}.$ By choosing $0<\delta\ll 1$ and the Cauchy inequality,

[TABLE]

and

[TABLE]

From Lemma 5,

[TABLE]

Together with $\left\|\partial_{p}^{\alpha}u\right\|_{L^{2}\left(m_{\theta}\right)}^{2}\leq\left\|\partial_{p}^{\alpha}u\right\|_{L_{\sigma}^{2}\left(m_{\theta}\right)}^{2},$ we therefore have

[TABLE]

for $\ell\in\mathbb{N}$. Since $\partial_{x}^{\alpha}$ is commutative with $\mathcal{L}_{AB},$ $\left(\ref{mixed-evolu}\right)$ is a consequence of $\left(\ref{p-evolu}\right)$. ∎

Now, we embark on the $p$-regularity estimate for $e^{t\mathcal{L}_{AB}}$ and $f$ in turn. For clarification, we first elaborate our procedure in the cases of $\ell=1$ and $\ell=2.$ For general $\ell,$ we provide the explicit form of the desired functional $\mathcal{F}_{\ell}$ and complete the proof inductively on $\ell.$

Step 1: The estimate of $\nabla_{p}\nabla_{x}^{k}f$ (i.e., $\ell=1$). Define the functional

[TABLE]

In view of Lemma 12,

[TABLE]

[TABLE]

Collecting terms gives

[TABLE]

Let $\kappa=\varepsilon^{2}$. Choosing \varepsilon>0\sufficiently small, we thereby obtain

[TABLE]

It implies that for $0\leq t\leq 1,$

[TABLE]

Next, we return to the estimate for $f$. Let $0<t_{0}\leq 1$ be given. By Duhamel’s principle,

[TABLE]

for $0<t_{0}/2<t\leq 1.$ Hence, we have

[TABLE]

due to Lemma 10, \left(\ref{f-weighted-energy}\right)\and $(\ref{aa.2}).$ Consequently,

[TABLE]

On the other hand, since $\partial_{x}^{\alpha}$ commutes with the equation $\left(\ref{Landau-f}\right)$, we can improve the mixed regularity simultaneously. Precisely, replacing $f$ by $\partial_{x}^{\alpha}f$ in $\left(\ref{fp}\right)$ gives

[TABLE]

By Lemma 10 and $\left(\ref{f-weighted-energy}\right),$

[TABLE]

Step 2: The estimate of \nabla_{p}^{2}\nabla_{x}^{k}f\(i.e., $\ell=2$). Define the functional

[TABLE]

By Lemma 12,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence,

[TABLE]

Set $\kappa_{i}=\varepsilon^{i}$. Choosing $\varepsilon>0$ sufficiently small , we find

[TABLE]

so that

[TABLE]

for $0\leq t\leq 1.$

Let $0<t_{0}\leq 1.$ By Duhamel’s principle,

[TABLE]

for $0<t_{0}/2<t\leq 1.$ Using Lemma 10, $\left(\ref{f-weighted-energy}\right),(\ref{aa.5})$ and $(\ref{aa.4})$, we obtain

[TABLE]

Namely,

[TABLE]

Again, since $\partial_{x}^{\alpha}$ is commutative with the equation $\left(\ref{Landau-f}\right)$, replacing $f$ by $\partial_{x}^{\alpha}f$ in $\left(\ref{fpp}\right)$ gives

[TABLE]

That is,

[TABLE]

Step 3: The estimate of $\nabla_{p}^{\ell}\nabla_{x}^{k}f$. For general $\ell\in\mathbb{N}$, consider the functional

[TABLE]

for $\varepsilon>0$ sufficiently small. Applying the above argument inductively on $\ell$, we obtain high order $p$-regularity in small time and conclude our result as below.

Proposition 13**.**

Let $k,$ $\ell\in\mathbb{N}\cup\{0\}.$ Then for $0<t\leq 1,$

[TABLE]

3.2.2. Improvement of the $p$ regularity in large time

In what follows, we establish the $p$-regularity in large time through the Gronwall type inequalities.

Proposition 14**.**

Let $f$ be a solution to equation $\left(\ref{Landau-f}\right)$ and let $k,\ \ell\in\mathbb{N}\cup\{0\}.$ Then for $t\geq 1,$

[TABLE]

the constant $C$ depending only upon $k$ and $\ell.$

Proof.

Let $n\in\mathbb{N}$. Define

[TABLE]

We first claim that

[TABLE]

for some constants $c>0$ small and $C>0$ large. To confirm this, in view of Lemma 5, we find

[TABLE]

so that

[TABLE]

whenever we choose $0<\varepsilon\ll 1$ with $1-nC\varepsilon\geq 1/2$. Hence,

[TABLE]

as required. Now, for $t\geq 1$, we have

[TABLE]

the last inequality being true due to Proposition 13.

Next, we shall claim that

[TABLE]

for some constant $C>0$ depending only upon $n.$ When $n=1,$ it is from Lemma 10 $\left(ii\right)$. When $n\geq 2,$ consider the functional

[TABLE]

for $\varepsilon>0$ sufficiently small. By Lemma 5 again, we have

[TABLE]

and

[TABLE]

Hence, following the same argument as $H^{n}[f],$ we deduce

[TABLE]

for some constants $c>0$ small and $C>0$ large as well. Therefore, for $t\geq 1$

[TABLE]

due to Proposition 13. It implies that for $t\geq 1,$

[TABLE]

As a consequence, we have

[TABLE]

Therefore, for $\ell\in\mathbb{N},$

[TABLE]

In effect, it holds that for any $k,\ \ell\in\mathbb{N}$

[TABLE]

Together with Lemma 10 $\left(ii\right),$ the proof is completed. ∎

3.3. Proof of Theorem 8

Regarding the regularity of $g_{L,0},$ it has been done in Lemma 6. Therefore, it remains to demonstrate that for any $k,$ $\ell\in\mathbb{N\cup\{}0\}$,

[TABLE]

and

[TABLE]

whenever $t\geq 1.$ Here $C_{k,\ell}>0$ depends only upon $k$ and $\ell.$ Among them, the case in which $k\in\mathbb{N\cup\{}0\}$ and $\ell=0$ is a consequence of Proposition 11 and thus we may assume that $k\in\mathbb{N\cup\{}0\}$ and $\ell\in\mathbb{N}$ in the following discussion.

Firstly, we prove that for $t\geq 1,$ the $L^{2}$-norms of any derivatives of $g_{L}$ and $g_{S}$ are bounded above uniformly in $t$ by multiples of certain weighted $L^{2}$-norms of the initial data $g_{in}$. To see this, note that the long wave part $g_{L}$ satisfies

[TABLE]

and the short wave part $g_{S}$ satisfies

[TABLE]

respectively, where

[TABLE]

Hence, from Proposition 14 (with $m_{\theta}=1$), it readily follows that

[TABLE]

However,

[TABLE]

for all $k\in\mathbb{N\cup\{}0\}$. During the process of the proof of Propositions 13 and 14, we can improve the regularity of $g_{L}$ to obtain

[TABLE]

In sum, we summarize the $L^{2}$-norm bounds for derivatives as the following:

Proposition 15**.**

Let $k,\ell\in\mathbb{N}\cup\{0\}$. Then for $t\geq 1$

[TABLE]

and

[TABLE]

Thereupon, write $g_{L,\bot}=g_{L}-g_{L,0}$ and then we find

[TABLE]

for $t\geq 1,$ $k,$ $\ell\in\mathbb{N\cup\{}0\}.$

Further, to obtain the decay rate, we employ the Sobolev inequality and the interpolation inequality to attain this end. Specifically, by the Sobolev inequality [21, Proposition 3.8], we find

[TABLE]

On the other hand, the interpolation inequality says that

[TABLE]

and

[TABLE]

for all $t\geq 1.$ Accordingly, combining these with $\left(\ref{g_kl-norm}\right)$ yields

[TABLE]

for all $t\geq 1.$ Meanwhile, with the same argument, we also obtain

[TABLE]

by $\left(\ref{aa.8}\right)$ and Proposition 15. This completes the proof of Theorem 8.

4. The result for $h$

Recall the equation

[TABLE]

In previous sections, the regularity of the solution $g$ has been investigated well. Now, regarding $g$ as a source term, we shall devote to establish the regularization estimate for the solution $h$ in the rest of the article. Especially, we demonstrate that $h$ has the same decay rate as $g$ (see Theorem 1).

Let $\mathbb{G_{BB}}^{t}$ be the solution operator of

[TABLE]

and then the homogeneous part of $h$ is $\widetilde{h}=\mathbb{G_{BB}}^{t}h_{in}$. Concerning estimates on the homogeneous part, it is the same as those on $g_{L}$ and $g_{S}$ in previous sections and hence we only state the results without proofs. As before, we decompose $\widetilde{h}$ into the long wave-fluid part $\widetilde{h}_{L,0}$, long wave-nonfluid part $\widetilde{h}_{L,\bot}$ and short wave part $\widetilde{h}_{S}$. Then they respectively satisfy

[TABLE]

[TABLE]

and

[TABLE]

with the constant $C>0$ depending on $k$ and $\ell$, and

[TABLE]

Thereupon, it remains to control the inhomogeneous part. Hereafter, we may assume $h_{in}=0$.

We now decompose the solution $h$ into the long wave part $h_{L}$ and the short wave part $h_{S}$, which respectively satisfy

[TABLE]

and

[TABLE]

It is easy to see that

[TABLE]

where

[TABLE]

Here the eigenfunctions \{e_{j}\left(\eta\right)\}_{j=0}^{4}\ and $e_{D}\left(\eta\right)$ are defined as in $\left(\ref{pre.bb.h}\ \right)$ and $\left(\ref{pre.ab.h}\right)$ with $\left\langle e_{j}(-\eta),e_{l}(\eta)\right\rangle_{p}=$ $\delta_{jl}$ and $\left\langle e_{D}(-\eta),e_{D}(\eta)\right\rangle_{p}=1.$ The main result for the solution $h$ is stated as below.

Theorem 16**.**

(Main result of $h$) Let k\and\ \ell\be nonnegative integers. Then for $t\geq 1$

[TABLE]

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

the constant $C>0$ depending on $k$ and $\ell$. Here

[TABLE]

4.1. Improvement of the $x$-regularity

In this subsection, we deal with the $x$-regularization estimate for the long wave part $h_{L}$. Owing to the fact that the system $\left(\ref{mix.2.a}\right)$ is decoupled, we instinctively use the Duhamel principle to solve $h$ through treating $g$ as a source term. However, there are some possible wave resonances between $g$ and $h$ as mentioned in the Introduction, and thus this method usually leads the solution $h$ to lower decay rate than $g.$ To remedy this, further physical properties of the collision operators, namely the microscopic cancellations representing the conservation of mass, total momentum and total energy, are employed in the following discussion.

Proposition 17**.**

(Long wave $h_{L}$, improve of the $x$ regularity) For $k,\ell\in\mathbb{N\cup}\{0\}$

[TABLE]

[TABLE]

and

[TABLE]

Proof.

We first split $h_{00}^{L}=I_{0}+I_{1}+I_{2}+I_{3}+I_{4}$, where each $I_{j}$ is the orthogonal projection of $h_{00}^{L}$ along $e_{j},$ i.e.,

[TABLE]

Notice that $\lambda\left(\eta\right)-\sigma_{j}(\eta)\neq 0$ for j=0\and $1.$ It is natural to evaluate $I_{j}$ ($j=0,1$) by the fundamental theorem of calculus, that is,

[TABLE]

On the other hand, we have such a “microscopic cancellation” $L_{BA}E_{D}=0$ from Q_{BA}\left(M_{B},M_{A}\right)=0\that

[TABLE]

Therefore,

[TABLE]

for all $k\in\mathbb{N\cup\{}0\}.$ Further, since

[TABLE]

we deduce

[TABLE]

In contrast with $I_{0}$ and $I_{1},$ we lose $\left(\ref{1st-FTOC}\right)$ or $\left(\ref{eigen-difference}\right)$ in the case of $I_{j},\ j=1,2$ and 3.\To maintain the decay rate of $h,$ we come up with further microscopic cancellations as well as $L_{BA}E_{D}=0.$ In the light of the conservation of mass and total energy, the operator

[TABLE]

is orthogonal to the collision invariants $\sqrt{M_{A}}$ and $|p|^{2}\sqrt{M_{A}}$. With this applied to $J=E_{D,1}=L_{AB}^{-1}\left(p\cdot\omega/m_{A}\right)E_{D}$ (where we recall that $E_{D}=\sqrt{M_{A}}$) it follows

[TABLE]

for any linear combination $\Phi_{1}$ of $1$ and $|p|^{2},$ so that

[TABLE]

Choosing $\Phi_{1}=E_{2}/\sqrt{M_{B}}=\sqrt{\frac{1}{10}}(-5+|p|^{2})$, we have

[TABLE]

since the integrand is odd. It turns out that \big{<}e_{2}(-\eta),L_{BA}e_{D}(\eta)\big{>}_{p}=O(|\eta|^{2}). Hence

[TABLE]

here $\overline{a}=1/2\min\{a_{2},a_{2,2}\}>0$ whenever $\delta>0$ is small.

Furthermore, owing to the conservation of momentum, the operator

[TABLE]

is orthogonal to all collision invariants $p_{1}\sqrt{M_{A}},\ p_{2}\sqrt{M_{A}}$ and $p_{3}\sqrt{M_{A}}$. With this applied to $J=E_{D,1}=L_{AB}^{-1}\left(p\cdot\omega/m_{A}\right)E_{D}$, it follows

[TABLE]

for any linear combination $\Phi_{2}$ of $p_{1},\ p_{2}$ and $p_{3}$, so that

[TABLE]

Choosing $\Phi_{2}=E_{3}/\sqrt{M_{B}}=\omega_{1}^{\bot}\cdot p,$ we find

[TABLE]

due to the fact that $\int p_{i}p_{j}\frac{M_{A}}{m_{A}}dp=\frac{1}{3m_{B}}\delta_{ij}$ and $\omega_{1}^{\bot}\bot\omega.$ It turns out that \big{<}e_{3}(-\eta),L_{BA}e_{D}(\eta)\big{>}_{p}=O(|\eta|^{2}). Likewise, \big{<}e_{4}(-\eta),L_{BA}e_{D}(\eta)\big{>}_{p}=O(|\eta|^{2}). Therefore,

[TABLE]

This completes the estimate of $h_{00}^{L}$.

For $h_{0\perp}^{L}$, direct computation gives

[TABLE]

In fact, one can improve $p$ regularity immediately; that is, for any $\ell,\ k\in\mathbb{N\cup\{}0\mathbb{\}}$,

[TABLE]

For $h_{\perp 0}^{L}$, we have

[TABLE]

Finally,

[TABLE]

here $a(\tau)=\min\{a(\tau_{1}),a(\tau_{2})\}.$ This completes the proof of the proposition. ∎

4.2. Improvement of the $x$ and $p$ regularities (general discussion)

Let $f$ be a solution of the linearized Landau equation

[TABLE]

where $g$ is known and satisfies the equation $\partial_{t}g+\frac{1}{m_{A}}p\cdot\nabla_{x}g=L_{AB}g.$ In the following, we first apply similar arguments in section $3$ to establish the regularization estimates for $f$ in short time and in large time. After that, based on these regularization estimates, one can readily obtain the decay rate of $h_{S}$ through the interpolation trick as before. Contrary to $h_{S}$, the $L_{x}^{\infty}L_{p}^{2}$ norm of $\nabla_{x}h^{L}_{\bot 0}$ decays only algebraically, rather than exponentially. To maintain the decay rate of $\nabla_{p}^{\ell}\nabla_{x}^{k}h^{L}_{\bot 0}$ the same as that of homogeneous problem, we interpolate the $L_{x}^{\infty}L_{p}^{2}$ norm of it from higher $p$-derivatives. However, the method used before for improving $p$-regularity results in the weights imposed on initial data grow drastically (In fact, exponentially grows with respect to $\ell$). We refine the $p$-regularization estimate by making use of equation (4.15) and the $p$-regularity of source term $g_{L;0}$. In such a way, the $p$-regularity of $h^{L}_{\bot 0}$ has been improved without imposing any weight on initial data.

Let the operator $\mathcal{L}_{BB}=-\frac{1}{m_{B}}p\cdot\nabla_{x}-\Lambda^{BB}.$ As shown in [5], the operator $e^{t\mathcal{L}_{BB}}$ has the same energy estimate and regularization estimate as $e^{t\mathcal{L}_{AB}}$ in Lemma 9. Based on this estimate, we deduce the $x$-regularity of $f$ in small time as below.

Lemma 18**.**

Let $f$ be a solution of the linearized Landau equation $\left(\ref{fBB}\right).$ Then for $k\in\mathbb{N\cup\{}0\},$

*(i) there is $C_{k}>0$ such that for *$0<t\leq 1$

[TABLE]

*(ii) For *$t\geq 1,$

[TABLE]

Proof.

Firstly, similar to $\left(\ref{f-weighted-energy}\right)$ and $\left(\ref{fx-weighted-energy}\right)$, there exists a universal constant $C>0$ such that

[TABLE]

and

[TABLE]

for any $0<s_{1}<s_{2}.$ By the Duhamel principle,

[TABLE]

Notice that $L_{BA}$ is an integral operator like $K^{BB}.$ Hence, for $k=0,$ we find

[TABLE]

from $\left(\ref{f-weighted-energy}\right)$ and $\left(\ref{f-energy-BB1}\right)$. In addition, since $\partial_{x}^{\alpha}$ is commutative with equation $\left(\ref{fBB}\right),$ we as well have

[TABLE]

for any $0<s_{1}<s_{2}.$

For $k\geq 1,$ using a Picard type iteration introduced in Lemma 10, we rewrite

[TABLE]

where f^{\left(0\right)}\satisfies

[TABLE]

$f^{\left(j\right)}$ , $1\leq j\leq 2k,$ satisfies

[TABLE]

and $\mathcal{R}^{\left(k\right)}$ solves the equation

[TABLE]

Under this decomposition, following the same procedure as in Lemma 10, we finish the proof. ∎

Proposition 19**.**

Let $f$ be a solution to equation $\left(\ref{fBB}\right)$ and let $k,$ $\ell\in\mathbb{N}\cup\{0\}.$ Then for $0<t\leq 1$

[TABLE]

Proof.

Let $u$ be a solution of equation

[TABLE]

Then for any $\ell\in\mathbb{N}$

[TABLE]

where $\mathcal{F}_{\ell}\left(t,u\right)$ is defined as in the proof of Proposition 13.

Let $0<t_{0}\leq 1.$ By Duhamel’s principle,

[TABLE]

for $0<t_{0}/2\leq t\leq 1,$ so that

[TABLE]

due to Lemma 18, $\left(\ref{fx-energy-BB2}\right)$ and $\left(\ref{L-estimate}\right)$. It follows that

[TABLE]

Since $\partial_{x}^{\alpha}$ commutes with the equation $\left(\ref{fBB}\right)$, we also have

[TABLE]

For general $k,\ell,$ we apply the induction argument combined with $\left(\ref{L-estimate}\right)$ to conclude that

[TABLE]

for $0<t\leq 1.$ ∎

Proposition 20**.**

Let $f$ be a solution to equation $\left(\ref{fBB}\right)$ and let k,\ \ell\in\mathbb{N}\cup\{0\}.\Then for $t\geq 1,$

[TABLE]

the constant $C$ depending only upon $k$ and $\ell.$

Proof.

Define

[TABLE]

In view of Lemmas 4 and 5,

[TABLE]

Hence, following a similar argument in Proposition 14, we deduce

[TABLE]

for some constants $c>0$ small and $C>0$ large. It implies that for $t\geq 1$

[TABLE]

due to $\left(\ref{f-energy-BB2}\right)$ and Proposition 19. Further, applying a similar argument to the same functional $\mathcal{A}_{N}\left(t,f\right)$ defined in Proposition 14 gives

[TABLE]

so that

[TABLE]

It implies that for $t\geq 1$

[TABLE]

As a result, we have

[TABLE]

and

[TABLE]

Therefore, for $\ell\in\mathbb{N},$

[TABLE]

On the other hand, for any $k,\ \ell\in\mathbb{N}$ we also have

[TABLE]

Together with Lemma 18 $\left(ii\right),$ the proof is completed. ∎

4.3. Proof of Theorem 16

Let $k,$ $\ell\in\mathbb{N}\cup\{0\}.$ It follows from Lemma 18, Propositions 19 and 20 that for $0<t\leq 1$

[TABLE]

and for $t\geq 1$

[TABLE]

Owing to the fact that

[TABLE]

it is easy to derive the regularity estimate on $h_{S}$ via the Sobolev inequality and the interpolation inequality, i.e., for any $k,\ell\in\mathbb{N\cup\{}0\}$,

[TABLE]

for all $t\geq 1.$ We complete the estimate for $h_{S}.$

Recall the regularization estimate of the homogeneous part of $h.$ To maintain the decay rate of $h_{L}$ without imposing higher weights on the initial data, we further refine the regularization estimate on $h_{\perp 0}^{L}$ and $h_{\perp\perp}^{L}$ before employing the Sobolev inequality and interpolation trick. Specifically, we prove that for any $k,\ell\in\mathbb{N}\cup\{0\},$

[TABLE]

and

[TABLE]

for $t\geq 1.$ Now, let $h_{\ast,0}^{L}=h_{00}^{L}+h_{\perp 0}^{L}$ and then it satisfies the equation:

[TABLE]

Following the energy estimate in Proposition 20, together with $m_{\theta}=1,$ gives

[TABLE]

which implies that

[TABLE]

Combined

[TABLE]

with

[TABLE]

it follows that

[TABLE]

which implies that

[TABLE]

and

[TABLE]

for all $t\geq 1.$ Hence,

[TABLE]

By the Sobolev inequality, we obtain

[TABLE]

Consequently, employing the interpolation inequality in $p$ (NOT in $x$) yields

[TABLE]

whenever $q\geq\left\lceil\log_{2}\left(5+k\right)\right\rceil.$

Finally, let $h_{\ast,\perp}^{L}=h_{0\perp}^{L}+h_{\perp\perp}^{L}$ and then it satisfies the equation:

[TABLE]

Going through the proofs of Propositions 19 and 20 with the fact that $\left\|\nabla_{p}^{\ell}\nabla_{x}^{k}g_{L,\perp}(t)\right\|_{L^{2}}$ $\leq C\|g_{in}\|_{L^{2}\left(w_{\ell}\right)}$, we find

[TABLE]

Since

[TABLE]

we improve

[TABLE]

Therefore,

[TABLE]

This completes the proof of Theorem 16.