High order approximation for the Boltzmann equation without angular cutoff

TL;DR

This paper introduces a new model to approximate the Boltzmann equation without angular cutoff, combining the Boltzmann and Landau operators, and demonstrates higher accuracy and well-posedness.

Contribution

It proposes a novel approximation model that integrates Boltzmann and Landau operators, achieving higher order accuracy without angular cutoff.

Findings

Proved well-posedness of the approximate equation

Established error estimates between approximate and original solutions

Achieved higher order accuracy compared to standard methods

Abstract

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cut-off and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cut-off approximation method, our method results in higher order of accuracy.

Equations

$${\partial }_{t}f=Q(f,f),$$

$$\frac{Q(g,h)(v)\text{buildrel}def}{={\int }_{R^3}{\int }_{{\text{SS}}^2}B(v-v_{\ast },\sigma )(g_{\ast }^{\prime }{h}^{\prime }-g_{\ast }h)d\sigma d{v}_{\ast }.}$$

$$v^{\prime }=\frac{v+v_{\ast }}{2}+\frac{|v-v_{\ast }|}{2}\sigma ,v_{\ast }^{\prime }=\frac{v+v_{\ast }}{2}-\frac{|v-v_{\ast }|}{2}\sigma ,\sigma \in {\text{SS}}^2.$$

$$\bar{B}(v-v_{\ast },\sigma )=[B(v-v_{\ast },\sigma )+B(v-v_{\ast },-\sigma )]1_{⟨\frac{v-v_{\ast }}{|v-v_{\ast }|},\sigma ⟩\ge 0}.$$

$$B(v-v_{\ast },\sigma )=\Phi (|v-v_{\ast }|)b(\cos \theta ),$$

$$K^{-1}{\theta }^{-1-2s}\le \sin \theta b(\cos \theta )\le K{\theta }^{-1-2s},\text{mbox}with0<s<1,K\ge 1.$$

$$\Phi (|v-v_{\ast }|)=|v-v_{\ast }{|}^{\gamma }.$$

$${\int }_{R^3}f(t,v)\varphi (v)dv={\int }_{R^3}f(0,v)\varphi (v)dv,\varphi (v)=1,v,|v{|}^2.$$

$$-\frac{d}{dt}{\int }_{R^3}f\log fdv=-{\int }_{R^3}Q(f,f)\log fdv\ge 0.$$

$$\frac{Q^{ϵ}(g,h)(v)\text{buildrel}def}{={\int }_{R^3}{\int }_{{\text{SS}}^2}{B}^{ϵ}(v-v_{\ast },\sigma )(g_{\ast }^{\prime }{h}^{\prime }-g_{\ast }h)d\sigma d{v}_{\ast }.}$$

$${\partial }_{t}f=Q^{ϵ}(f,f)$$

$$f=f^{ϵ}+O({ϵ}^{2-2s}).$$

$$||{ϵ}^{2-2s}{Q}_L(f,f)-Q_{ϵ}(f,f)|{|}_{L^1}\lesssim {ϵ}^{3-2s}||f|{|}_{H_{\gamma +12}^5}^2,$$

$$\frac{Q_L(g,h)(v)\text{buildrel}def}{={\nabla }_v\cdot \{{\int }_{R^3}a(v-v_{\ast })[g(v_{\ast }){\nabla }_{v}h(v)-{\nabla }_{v}g(v_{\ast })h(v)]d{v}_{\ast }\}.}$$

$$a(v)=\Lambda |v{|}^{\gamma +2}(I-\frac{v\otimes v}{|v{|}^2}),$$

$$\frac{M^{ϵ}(g,h)\text{buildrel}def}{=Q^{ϵ}(g,h)+{ϵ}^{2-2s}{Q}_L(g,h),}$$

$${\partial }_{t}f=M^{ϵ}(f,f).$$

$$f={\tilde{f}}^{ϵ}+O({ϵ}^{3-2s}).$$

$$H^{N}=\bigg{\{}f(v):\sum_{|\alpha|\leq N}\|\partial_{v}^{\alpha}f\|_{L^{2}}<+\infty\bigg{\}},$$

$$H^{m}_{l}=\bigg{\{}f(v):\|\langle D\rangle^{m}\langle\cdot\rangle^{l}f\|_{L^{2}}<+\infty\bigg{\}},$$

$$\frac{(a(D)f)(v)\text{buildrel}def}{=\frac{1}{(2\pi {)}^3}{\int }_{R^3}{\int }_{R^3}{e}^{i(v-u)\cdot \xi }a(\xi )f(u)dud\xi .}$$

$$\displaystyle\|f\|_{L^{p}_{q}}=\big{(}\int_{\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}}|f(v)|^{p}\langle v\rangle^{qp}dv\big{)}^{\frac{1}{p}},\quad\|f\|_{L\log L}=\int_{\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}}|f|\log(1+|f|)dv.$$

$$\frac{||f|{|}_{ϵ,m,l}^2\text{buildrel}def}{=||f|{|}_{H_l^{m+s}}^2+{ϵ}^{2-2s}||f|{|}_{H_l^{m+1}}^2,}$$

$$W^{ϵ}(\xi )=⟨\xi {⟩}^s{\text{1}}_{|\xi |\le \frac{1}{ϵ}}+{ϵ}^{-s}{\text{1}}_{|\xi |>\frac{1}{ϵ}},$$

$$ab\le \frac{a^p}{p}+\frac{b^q}{q}.$$

$$ab\le \eta a^p+(p\eta {)}^{-\frac{q}{p}}\frac{b^q}{q}.$$

$$\frac{dy}{dt}\le a+by(t),$$

$$y(t)\le y(0)e^{bt}+\frac{a}{b}(e^{bt}-1).$$

$$y(t)\le \alpha (t)+{\int }_a^t\beta (r)y(r)dr,$$

$$y(t)\le \alpha (t)+{\int }_a^t\alpha (r)\beta (r)\exp ({\int }_r^t\beta (u)du)dr.$$

Full text

Keywords: homogeneous Boltzmann equation, long-range interactions, hard potentials, high order approximation.

AMS Subject Classification (2010): 35Q20, 35R11, 75P05.

1. Introduction

1.1. The Boltzmann equation

Our interest is to consider the numerical method for the spatially homogeneous Boltzmann equation with long-range interaction in the case of hard potentials. Here, the spatial homogeneity means the unknown function is assumed to be independent of the position variables. In this case, the Boltzmann equation reads:

[TABLE]

where $f(t,v)\geq 0$ is the distribution function of collision particles which at time $t\geq 0$ move with velocity $v\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$. The Boltzmann collision operator $Q$ is a bilinear operator which acts only on the velocity variables $v$, that is,

[TABLE]

Here we use the standard shorthand $h=h(v)$, $g_{*}=g(v_{*})$, $h^{\prime}=h(v^{\prime})$, $g^{\prime}_{*}=g(v^{\prime}_{*})$ where $v^{\prime}$, $v_{*}^{\prime}$ are given by

[TABLE]

The nonnegative function $B(v-v_{*},\sigma)$ in the collision operator is called the Boltzmann collision kernel. It is always assumed to depend only on $|v-v_{*}|$ and $\langle\frac{v-v_{*}}{|v-v_{*}|},\sigma\rangle$. We introduce the angle variable $\theta$ through $\cos\theta=\langle\frac{v-v_{*}}{|v-v_{*}|},\sigma\rangle$. Without loss of generality, we may assume that $B(v-v_{*},\sigma)$ is supported in the set $0\leq\theta\leq\frac{\pi}{2}$ , i.e, $\langle\frac{v-v_{*}}{|v-v_{*}|},\sigma\rangle\geq 0$, for otherwise $B$ can be replaced by its symmetrized form:

[TABLE]

Here, $\mathbf{1}_{E}$ is the characteristic function of the set $E$.

1.2. Assumptions on the collision kernel

We consider the collision kernel satisfying the following assumptions:

• •

(A-1) The cross-section $B(v-v_{*},\sigma)$ takes a product form of

[TABLE]

where both $\Phi$ and $b$ are nonnegative functions.

• •

(A-2) The angular function $b(t)$ is not locally integrable and it satisfies

[TABLE]

• •

(A-3) The kinetic factor $\Phi$ takes the form of

[TABLE]

• •

(A-4) The parameter $\gamma$ verifies that $0<\gamma\leq 2$.

We remark that under assumption (A-2), we have $A_{2}\buildrel\hbox{\footnotesize def}\over{=}\int_{\SS^{2}}b(\cos\theta)\sin^{2}\theta d\sigma<\infty$.

The solutions of the Boltzmann equation (1.1) have the fundamental physical properties of conserving the total mass, momentum and kinetic energy, that is, for all $t\geq 0$,

[TABLE]

Moreover, there exists a quantity called entropy satisfying the Boltzmann’s $H$ theorem, which formally is

[TABLE]

1.3. Existing results, motivations and difficulties

The well-posedness of the spatially homogeneous Boltzmann equation with angular cut-off, that is when $\int_{0}^{\pi/2}\sin\theta b(\cos\theta)d\theta<\infty$, had been investigated by many authors. For the hard potentials, Arkeryd [8] and Mischler-Wennberg [20] established the existence and uniqueness of the solutions in weighted $L^{1}$ space. Recently, Lu-Mouhot in [14] extended the results to the space of non-negative measure with finite non-increasing kinetic energy. For the well-posedness of the spatially homogeneous Boltzmann equation without angular cut-off, we refer to [10] and the references therein. As for the regularity theory of the equation, we refer to [18] for the analysis of the positive part of the collision operator and the propagation of smoothness in the case of angular cut-off and refer to [2], [4], [13] and [19] in the case of long-range interaction.

For any $0<\epsilon\leq\frac{\sqrt{2}}{2}$, let $b^{\epsilon}=b\mathbf{1}_{\sin\frac{\theta}{2}\geq\epsilon}$, and $Q^{\epsilon}$ be the operator associated to the angular cut-off kernel $B^{\epsilon}(v-v_{*},\sigma)=|v-v_{*}|^{\gamma}b^{\epsilon}(\cos\theta)$. That is,

[TABLE]

Then the angular cut-off Boltzmann equation

[TABLE]

is well-posed(see [12]). And moreover if $f$ and $f^{\epsilon}$ are solutions to the Boltzmann equation (1.1) and the cutoff Boltzmann equation (1.3) with the same initial datum $f_{0}$ respectively, then one has

[TABLE]

The cut-off Boltzmann operator $Q^{\epsilon}$ omits all grazing collisions and then results in an error of order $2-2s$. We emphasize that the cutoff Boltzmann equation (1.3) is not a good approximation to the Boltzmann equation (1.1) as the singularity parameter $s$ approaches to $1$.

The effect of grazing collisions has been studied extensively, and we refer to [5] and [9]. It is proved that the limit of concentrating grazing collisions leads to the Landau collision operator. Mathematically, if denote $b_{\epsilon}=b\mathbf{1}_{\sin\frac{\theta}{2}\leq\epsilon}$, and let $Q_{\epsilon}$ be the operator associated to $B_{\epsilon}(v-v_{*},\sigma)=|v-v_{*}|^{\gamma}b_{\epsilon}(\cos\theta)$, according to [9], we shall have

[TABLE]

where the Landau collision operator $Q_{L}$ is defined as

[TABLE]

Here the symmetrical matrix $a$ is given by

[TABLE]

where $\Lambda$ is a constant.

This motivates us to compensate the omission of grazing collisions by Landau operator. Specifically, we consider the operator

[TABLE]

and propose our approximate equation,

[TABLE]

If $\tilde{f}^{\epsilon}$ is the solution to equation (1.7), we will prove

[TABLE]

That is, by adding Landau operator to the cutoff Boltzmann equation, we increase the order of error from $2-2s$ to $3-2s$. The accuracy of approximation of the Boltzmann equation (1.1) by equation (1.7) remains even if the singularity parameter $s$ goes to $1$. Another motivation for studying equation (1.7) is the recent development of numerical methods. We believe that our approximate equation can be solved numerically. In this regard, see next subsection for a detailed discussion. We emphasize that the solutions of our approximate equation (1.7) also have the above mentioned properties, namely, conservation of mass, moment, energy and entropy dissipation.

In the current paper, we study the well-posedness of equation (1.7) and then give the error analysis of the approximate equation (1.7) and the original Boltzmann equation (1.1). There are two main difficulties in the current paper. One is to show the existence of a non-negative solution to equation (1.7). We proceed by constructing a sequence of convergent non-negative functions with its limit being the solution. Since we consider hard potentials ($\gamma>0$), there will be an increase of weight at each iteration. Observing the coefficient before the weight increased term is strictly less $1$, we prove that, on a whole level, the increased weight is limited. The other difficulty is related to the estimate of the error function $F^{\epsilon}_{R}$ as defined in (4.1). Again, weight increase problem happens here and another problem is no sign information of $F^{\epsilon}_{R}$. We circumvent the problem of lacking sign information by writing the equation of error function in a suitable way. The weight increase problem is dealt with by carefully separating the integration region such that either the increased weight is eliminated or the coefficient before the weight increased term is controlled as desired.

1.4. Existing numerical results and future work

Our approximate equation contains both the angular cut-off Boltzmann operator $Q^{\epsilon}$ and Landau operator $Q_{L}$. Numerical methods of the Boltzmann equation and Landau equation have been investigated extensively. The most famous one is Kac’s program. Kac started from the Markov process corresponding to collisions only, and try to prove the limit towards the spatially homogeneous Boltzmann equation. For Kac’s program approximating Boltzmann equation, we refer to the recent work [15] and the references therein. In [15], the authors proved the propagation of chaos quantitatively in an abstract framework by proving stability and convergence estimates between linear semigroups. They then applied their results to prove the propagation of chaos of Kac’s program in the cases of hard sphere model ($B(v-v_{*},\cos\theta)=|v-v_{*}|$) and true Maxwell molecules ($B(v-v_{*},\cos\theta)=b(\cos\theta)$).

As for particle system approximating the Landau equation, we refer to [8] and the references therein. The authors in [8] proved quantitatively the propagation of chaos for a $N$-particle continuous drift diffusion process under the cases of Maxwell molecules ($\gamma=0$) and hard potentials ($0<\gamma\leq 1$).

As one can see from above, the Boltzmann equation corresponds to the limit of jump processes, while the Lanau equation corresponds to the limit of continuous processes. If we are to numerically solve our approximate equation (1.7), we need some jump-diffusion processes. Actually, the method in [15] is general and robust to deal with mixture of jump and diffusion processes. As shown to be successful in [16], the authors considered the Boltzmann equation for diffusively excited granular media, used jump-diffusion processes to approximate it, and then proved the propagation of chaos. The jump part is the Boltzmann operator with an integrable kernel, while the diffusive part is a Laplace operator. We know that the Landau operator behaves like the Laplace operator, except with some compensation to conserve energy.

In the recent work [7], the authors replaced the small collisions by a small diffusion term to approximate the Kac equation without cutoff, and successfully built a stochastic particle system to approximate the solution of the Kac equation without cutoff. The Kac equation is a one-dimensional case of the Boltzmann equation.

Thanks to the above breakthroughs, our approximate equation (1.7) has great potential to be solved numerically. In our future work, we will build a particle system based on equation (1.7) and prove the propagation of chaos.

1.5. Notations and main results

Let us introduce the function spaces and notations which we shall use throughout the paper.

• •

For integer $N\geq 0$, we define the Sobolev space

[TABLE]

where the multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ with $|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}$ and $\partial_{v}^{\alpha}=\partial_{v_{1}}^{\alpha_{1}}\partial_{v_{2}}^{\alpha_{2}}\partial_{v_{3}}^{\alpha_{3}}$.

• •

For real number $m,l$, we define the weighted Sobolev space

[TABLE]

where $\langle v\rangle\buildrel\hbox{\footnotesize def}\over{=}(1+|v|^{2})^{\frac{1}{2}}$, and $a(D)$ is the pesudo-differential operator with symbol $a(\xi)$ defined by

[TABLE]

• •

We also introduce the standard notations

[TABLE]

• •

For the ease of notation, let us define a new norm $||\cdot||_{\epsilon,m,l}$ for any $\epsilon,l>0$ and $m\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ as:

[TABLE]

If $m=0$, we simply write $||\cdot||_{\epsilon,l}$ instead of $||\cdot||_{\epsilon,0,l}$. If $m=l=0$, we simply write $||\cdot||_{\epsilon}$ instead of $||\cdot||_{\epsilon,0}$. Then for any $\epsilon\leq 1,||\cdot||_{H^{m+s}_{l}}\leq||\cdot||_{\epsilon,m,l}\leq 2||\cdot||_{H^{m+1}_{l}}$.

• •

Let us define the symbol $W^{\epsilon}(\xi)$ by

[TABLE]

which comes from the coercivity estimate of the cut-off Boltzmann operator $Q^{\epsilon}$.

• •

For any $f,g\in L^{2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, we denote by $\langle f,g\rangle$ the inner product of $f$ and $g$.

• •

By $a\lesssim b$, we mean that there is a uniform constant $C,$ which may be different on different lines, such that $a\leq Cb$. We write $a\sim b$ if both $a\lesssim b$ and $b\lesssim a$.

We do not bother to distinguish a function and its value at a point. For example, we do not distinguish weight function $\langle\cdot\rangle^{l}$ and the value $\langle v\rangle^{l}$ it takes at a point $v$.

We recall Young’s inequality for use in future. For $a,b\geq 0$ and $p,q>1$, with $\frac{1}{p}+\frac{1}{q}=1$, there holds

[TABLE]

As a result, for any $\eta>0$, we have the basic inequality

[TABLE]

We also recall the Gronwall’s inequality. For any $a,b\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$, and a function $y$ defined on $\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}$ satisfying

[TABLE]

then

[TABLE]

There is also an integral type of Gronwall’s inequality. Let $y,\alpha,\beta$ be functions defined on $\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}$. If $\beta$ is nonnegative and for any $t>a\geq 0$, $y$ satisfies

[TABLE]

then

[TABLE]

If, in addition, the function $\alpha$ is non-decreasing, then

[TABLE]

Before stating our main results, let us give the definition of $\phi$ which is related to the weight function:

[TABLE]

where

[TABLE]

We begin with the first result concerns the propagation of the moments and smoothness for the solution to our approximate equation.

Theorem 1.1**.**

Let $\phi:\mathop{\mathbb{N}\kern 0.0pt}\nolimits\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}\rightarrow\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ be the function defined as in (1.14). Let $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ and $l\geq 0$. If $f_{0}\in L^{1}_{q}\cap H^{N}_{l}$ with $q\geq\phi(N,l)$, then (1.7) admits a non-negative and unique solution $f^{\epsilon}$ in $L^{\infty}([0,\infty];L^{1}_{q}\cap H^{N}_{l})$ and morevover there exists a constant $C$, depending only on $||f_{0}||_{L^{1}_{q}}$ and $||f_{0}||_{H^{N}_{l}}$, such that for any $t\geq 0$ and $\epsilon$ small enough,

[TABLE]

Remark 1.1**.**

The result of Theorem 1.1 is also true when $\epsilon=0$, which corresponds to the propagation of moments and smoothness of solution of the original Boltzmann equation (1.1).

The last two theorems describe the error between solutions of the Boltzmann equation and our approximate equation.

Theorem 1.2**.**

Let $l\geq 0$ such that $(\frac{4}{\pi})^{2l-2s}(l-s)\geq\frac{2^{4-2s}\pi K}{A_{2}}$ and $2l\geq\frac{s}{1-s}(\gamma+2)+\gamma$. Suppose $f_{0}\in L^{1}_{q}\cap H^{5}_{2l+\gamma+12}$ with $q\geq\phi(5,2l+\gamma+12)$. Let $f$ and $f^{\epsilon}$ be solutions to the Boltzmann equation (1.1) and the approximated equation (1.7) with the same initial datum $f_{0}$ respectively, then we have for any $t\geq 0$,

[TABLE]

where $C(f_{0},t)$ is a constant depending only on $||f_{0}||_{L^{1}_{q}},||f_{0}||_{H^{5}_{2l+\gamma+12}}$ and time $t$.

Let us introduce the definition of $\psi$:

[TABLE]

and $\varphi$:

[TABLE]

Then we have:

Theorem 1.3**.**

Let $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ and $l\geq 0$ such that $(\frac{4}{\pi})^{2l+5-2s}(2l+5-2s)\geq\frac{2^{5-2s}\pi K}{A_{2}}$ and $2l+5\geq\frac{s}{1-s}(\gamma+2)+\gamma$. Let $\psi,\varphi:\mathop{\mathbb{N}\kern 0.0pt}\nolimits\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}\rightarrow\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ be functions defined as in (1.19) and (1.20). Suppose $f_{0}\in L^{1}_{q}\cap H^{N+5}_{\psi(N,l)}$ with $q\geq\varphi(N,l)$. Let $f$ and $f^{\epsilon}$ be solutions to the Boltzmann equation (1.1) and the approximated equation (1.7) with the same initial datum $f_{0}$ respectively, then we have for any $t\geq 0$,

[TABLE]

where $C(f_{0},t)$ is a constant depending only on $||f_{0}||_{L^{1}_{q}},||f_{0}||_{H^{N+5}_{\psi(N,l)}}$ and time $t$.

1.6. Plan of the paper

In section 2, we state three estimates (upper bound, coercivity, commutator) of the operator $M^{\epsilon}$. Section 3 is devoted to the well-posedness theory of our approximate equation, namely, uniqueness and existence of non-negative solution. In the last section, we prove the high order convergence of solutions between the Boltzmann equation and our approximate equation.

2. Estimates of the collision operators

In this section, we state three estimates of the operator $M^{\epsilon}$, as defined in (1.6) which will used frequently in next sections. We begin with upper bound of the collision operator.

Theorem 2.1**.**

Suppose the collision kernel $B$ satisfies the Assumption (A-1)-(A-4), and $Q^{\epsilon}$ is the collision operator associated to the collision kernel $B^{\epsilon}$. Let $w_{1},w_{2}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ with $w_{1}+w_{2}\geq\gamma+2$, $a_{1},a_{2}\geq 0$ with $a_{1}+a_{2}=2s$ and $b_{1},b_{2}\geq 0$ with $b_{1}+b_{2}=2$. Then for smooth functions $g,h$ and $f$, the following estimate holds uniformly with respect to $\epsilon$:

[TABLE]

Proof.

For the cut-off Boltzmann operator $Q^{\epsilon}$, as in [11], for any $w_{1},w_{2}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ with $w_{1}+w_{2}\geq\gamma+2$, there holds

[TABLE]

Again from [11], we have

[TABLE]

Patching together the above two estimates, the estimate (2.1) follows accordingly. ∎

We now turn to coercivity estimate of the operator.

Theorem 2.2**.**

Suppose the collision kernel $B$ satisfies the Assumption (A-1)-(A-4), and $Q^{\epsilon}$ is the collision operator associated to the collision kernel $B^{\epsilon}$. Suppose function $g$ is nonnegative and satisfies

[TABLE]

then there exists constants $C_{1}(g)$ and $C_{2}(g)$ depending only on $||g||_{L^{1}_{1}}$ and $||g||_{LlogL}$ such that

[TABLE]

Proof.

For the cut-off Boltzmann operator $Q^{\epsilon}$, with a similar argument as in [1], one has

[TABLE]

For the Landau operator $Q_{L}$, by [6], there holds

[TABLE]

The coercivity estimate (2.5) follows by noting that

[TABLE]

∎

In the last, we move to commutator estimates. We first give the commutator estimate of the cut-off Boltzmann operator $Q^{\epsilon}$ as a lemma.

Lemma 2.1**.**

Suppose the collision kernel $B$ satisfies the Assumption (A-1)-(A-4), and $Q^{\epsilon}$ is the collision operator associated to the collision kernel $B^{\epsilon}$. Let $N_{2},N_{3}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ and $l\geq 0$ with $N_{2}+N_{3}\geq l+\gamma$, and let $N_{1}=|N_{2}|+|N_{3}|+\max\{|l-1|,|l-2|\}$. Then for smooth functions $g,h$ and $f$, the following estimate holds uniformly with respect to $\epsilon$:

[TABLE]

Proof.

One may refer to [4] for a proof. ∎

The next lemma is the commutator estimate of the Landau operator $Q_{L}$.

Lemma 2.2**.**

Let $N_{2},N_{3}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ and $l\geq 0$ with $N_{2}+N_{3}\geq l+\gamma$. Then for smooth functions $g,h$ and $f$, the following estimate holds true:

[TABLE]

where $C(l)=\max\{2l^{2}+12l,20l-2l^{2}\}$.

Proof.

We define as usual the following quantities in 3-dimension:

[TABLE]

Hence the Landau operator $Q_{L}$ can be rewritten as:

[TABLE]

Then we have

[TABLE]

It is easy to check

[TABLE]

Thus we have

[TABLE]

Considering the following facts

[TABLE]

and

[TABLE]

and

[TABLE]

we arrive at

[TABLE]

Thanks to

[TABLE]

we have

[TABLE]

provided $N_{2}+N_{3}\geq l+\gamma$. Similarly, if $N_{2}+N_{3}\geq l+\gamma$, there holds

[TABLE]

With the help of the fact

[TABLE]

we have

[TABLE]

provided $N_{2}+N_{3}\geq l-2+\gamma$. Patching together the above estimates, if $N_{2}+N_{3}\geq l+\gamma$, we have

[TABLE]

∎

In the end of this section, we state the commutator estimate of the operator $M^{\epsilon}$.

Theorem 2.3**.**

Suppose the collision kernel $B$ satisfies the Assumption (A-1)-(A-4), and $Q^{\epsilon}$ is the collision operator associated to the collision kernel $B^{\epsilon}$. Let $N_{2},N_{3}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$ and $l\geq 0$ with $N_{2}+N_{3}\geq l+\gamma$, and let $N_{1}=\max\{|N_{2}|+|N_{3}|+\max\{|l-1|,|l-2|\},\gamma+3\}$. Then for smooth functions $g,h$ and $f$, the following estimate holds uniformly with respect to $\epsilon$:

[TABLE]

Proof.

The commutator estimate (2.9) follows from lemma 2.1 and 2.2. ∎

3. Well-posedness for approximate equation (1.7): existence and uniqueness

In this section, we will show that (1.7) admits a non-negative, unique and smooth solution if the initial data is smooth. To do that, we separate the proof into three steps. In the first step, we prove that the linear equation to (1.7) admits a non-negative and smooth solution. Then in the next step, by using Picard iteration scheme, we get the well-posedness result. In the final step, we improve the well-posedness result by applying the symmetric property of the collision operators.

3.1. Well-posedness of linear equation to (1.7)

Throughout this subsection, $\epsilon>0$ is a fixed but small enough number. In the following, we construct a non-negative solution to the linear equation:

[TABLE]

Let us define two operators:

[TABLE]

[TABLE]

Then we have $Q^{\epsilon}=Q^{\epsilon+}-Q^{\epsilon-}$, so we call $Q^{\epsilon+}$ the gain operator and $Q^{\epsilon-}$ the loss operator.

We first give a proposition, which shall be used in both the current section and the next section.

Proposition 3.1**.**

Let $p\geq 2,n=\frac{v-v_{*}}{|v-v_{*}|},u=\frac{v+v_{*}}{|v+v_{*}|},j=\frac{u-(u\cdot n)n}{|u-(u\cdot n)n|},h=\sqrt{|v|^{2}|v_{*}|^{2}-(v\cdot v_{*})^{2}}$, and $E(\theta)=\langle v\rangle^{2}\cos^{2}\frac{\theta}{2}+\langle v_{*}\rangle^{2}\sin^{2}\frac{\theta}{2}$. Suppose $\omega$ is the vector such that $\sigma=\cos\theta n+\sin\theta\omega$, then there holds

[TABLE]

Proof.

It is easy to check $\langle v^{\prime}\rangle^{2}=E(\theta)+h(j\cdot\omega)\sin\theta$. By Taylor expansion, we have

[TABLE]

For the last term $\mathfrak{M}_{3}$, we have for any $\kappa\in[0,1]$:

[TABLE]

Together with $h^{2}\leq\langle v\rangle^{2}\langle v_{*}\rangle^{2}$, we arrive at

[TABLE]

For the term $\mathfrak{M}_{1}$, we have

[TABLE]

Combining $\mathfrak{M}_{1},\mathfrak{M}_{2},\mathfrak{M}_{3}$, we arrive at (3.1). ∎

We begin with an equation which shall be used to construct solution to the linear equation to (3.1).

Lemma 3.1**.**

Let $g,h\geq 0$ be smooth functions. Suppose $f^{\epsilon}$ is the solution to the following equation

[TABLE]

Then $f^{\epsilon}(t)\geq 0$ for any $t\geq 0$.

Proof.

Denote $f^{\epsilon}_{-}=\min\{0,f^{\epsilon}\}\leq 0$, then we have $f^{\epsilon}_{-}|_{t=0}=0$, and

[TABLE]

Since $g,h\geq 0$ and $f^{\epsilon}_{-}\leq 0$, it is clear that

[TABLE]

By the definition of $Q_{L}$, we have

[TABLE]

Since $a$ is a positive semi-definite matrix, we have $\mathfrak{I}_{1}\leq 0$. By assumption (A-2), there holds $\int_{\SS^{2}}b^{\epsilon}(\cos\theta)d\sigma\sim\frac{\epsilon^{-2s}}{s}$. Therefore, there exists $\epsilon_{*}>0$ such that, for any $0<\epsilon\leq\epsilon_{*}$,

[TABLE]

Finally, we arrive at

[TABLE]

Thus $||f^{\epsilon}_{-}(t)||_{L^{2}}=0$ for any $t\geq 0$, which implies $f^{\epsilon}(t)\geq 0$ for any $t\geq 0$. ∎

Now we are ready to construct a solution to the linear equation (1.7).

Lemma 3.2**.**

Let $l\geq 4,T>0$ be real numbers. Suppose the non-negative datum $f_{0}\in H^{5}_{l+3\gamma/2+10}\cap L^{1}_{l+5\gamma/2+16}$ with $||f_{0}||_{L^{1}}>0$. Suppose $g(t,v)$ is a non-negative function satisfying

[TABLE]

*then (3.1) admits a unique non-negative solution $f$ in $L^{\infty}([0,T];L^{1}_{l+5\gamma/2+16}\cap H^{5}_{l+\gamma+10})\cap L^{1}([0,T];L^{1}_{l+7\gamma/2+16})$. *

Proof.

Define a sequence of approximate solutions $\{f^{n}\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ by

[TABLE]

According to the previous lemma, we have $f^{n}\geq 0$.

Step 1: (Uniform Upper Bound)

Step 1.1: (Uniform Upper Bound in $L^{1}_{l}$)

In this step, we shall use the energy method to get the uniform upper bound of $L^{1}_{l}$ norm of $\{f^{n}\}_{n}$ with respect to $n$. Applying the basic inequality (1.10), for any $\eta>0$, there holds

[TABLE]

Also one has

[TABLE]

Thanks to the above two facts, we obtain

[TABLE]

where $A^{\epsilon}=\int_{\SS^{2}}b^{\epsilon}(\cos\theta)d\sigma$. It is easy to check

[TABLE]

That is, for any $\eta>0$, there holds

[TABLE]

Then we obtain

[TABLE]

For the Landau operator, referring to [6], there holds

[TABLE]

Patching together the above estimates, we arrive at

[TABLE]

Observing that

[TABLE]

thus we can take an $\eta>0$ small enough such that,

[TABLE]

With such a small $\eta$, let us denote $a=\frac{A^{\epsilon}}{(1+\eta)^{\gamma/2}}+\frac{1}{2}\epsilon^{2-2s}l\Lambda,~{}\delta=\frac{1}{2}\epsilon^{2-2s}l\Lambda,~{}K_{1}=(1+\frac{1}{\eta})^{\frac{\gamma}{2}}A^{\epsilon},~{}K_{2}=\sup_{0\leq s\leq T}C(l,\gamma,\eta)A^{\epsilon}||g(s)||_{L^{1}_{l}},~{}K_{3}=\sup_{0\leq s\leq T}\{\eta^{-\gamma/2}A^{\epsilon}||g(s)||_{L^{1}_{\gamma}}+\epsilon^{2-2s}(4l+2)l\Lambda||g(s)||_{L^{1}_{4}}\}$. Therefore, we arrive at a neater inequality on the interval $[0,T]$,

[TABLE]

By defining $y^{n}(t)=e^{-K_{3}t}||f^{n}(t)||_{L^{1}_{l}}$ and $x^{n}(t)=\int_{0}^{t}e^{-K_{3}s}||g(s)||_{L^{1}}||f^{n}(s)||_{L^{1}_{l+\gamma}}ds$ for any $0\leq t\leq T$ and $n\geq 0$, we derive that

[TABLE]

Now denote $S^{n}(t)=\sum_{i=0}^{n}(\frac{a}{a+\delta})^{i}y^{n-i}(t)$ for $n\geq 0$, by recursive derivation and noting that $y^{0}(t)\leq||f_{0}||_{L^{1}_{l}}$ and $x^{0}(t)\leq M\frac{1-e^{-K_{3}t}}{K_{3}}||f_{0}||_{L^{1}_{l+\gamma}}$, we obtain

[TABLE]

By further recursive derivation, we have

[TABLE]

Noting that

[TABLE]

and

[TABLE]

and recalling the definition of constants $K_{1},K_{2},K_{3}$, we obtain

[TABLE]

Step 1.2: (Uniform Upper Bound in $L^{2}_{l}$)

In this step, we show the uniform upper bound of $L^{2}_{l}$ norm of $\{f^{n}\}_{n}$ with respect to $n$. It is easy to check that

[TABLE]

By Cauchy-Schwartz inequality, there holds

[TABLE]

where we have used the estimate (3.6) and the usual change of variable $v\rightarrow v^{\prime}$. By direct calculation, we have

[TABLE]

By coercivity estimate (2.6) and commutator (2.8) estimate of the Landau operator, thus we have

[TABLE]

Now patching together the inequalities (3.1), (3.14) and (3.1), and using the basic inequality (1.10), we have

[TABLE]

where $C_{1},K_{1}$ are some positive constants depending on $m,M,\epsilon$. For any $\lambda,s>0$, one has

[TABLE]

With the help of the above inequality, we have

[TABLE]

for some new constant $K_{1}$. By the previous step, with the uniform upper bound of $||f^{n}||_{L^{1}_{l+\gamma/2}}$, we have

[TABLE]

where $K_{2}$ is some constant depending on $||f_{0}||_{L^{1}_{l+3\gamma/2}}$ and uniform upper bound of $||g||_{L^{1}_{l+3\gamma/2}}$. Now we use the same technique as in the previous step. Integrating both sides with respect to time, for any $t_{n}\in[0,t]$, we obtain

[TABLE]

Now denote $S^{n}(t_{n})=\sum_{i=0}^{n}(\frac{1}{2})^{i}||f^{n-i}(t_{n})||^{2}_{L^{2}_{l}}$ and $x^{n}(t_{n})=C_{1}\epsilon^{2-2s}\int_{0}^{t_{n}}||f^{n}(r)||^{2}_{L^{2}_{l+\gamma/2}}dr$ for $n\geq 0$, by recursive derivation and noting that $x^{0}(t_{n})\leq C_{1}\epsilon^{2-2s}t||f_{0}||^{2}_{L^{2}_{l+\gamma/2}}$, we obtain, for $n\geq 1$,

[TABLE]

By tracking the definitions of constants $K_{1},K_{2}$, we obtain

[TABLE]

Step 1.3: (Uniform Upper Bound in $H^{m}_{l}$ with $m\geq 1$)

Fix an $\alpha$ with $|\alpha|\leq m$, one has

[TABLE]

Then we have

[TABLE]

As the same as (3.1), we have

[TABLE]

As the same as (3.14), we have

[TABLE]

When $|\alpha_{2}|\leq|\alpha|-1\leq m-1$, by upper bound estimate (2.3) and commutator (2.8) estimate of the Landau operator, we have

[TABLE]

When $\alpha_{2}=\alpha$, as the same as (3.1), we have

[TABLE]

Now patching together the above estimates and taking sum over $|\alpha|\leq m$, we have

[TABLE]

where $C_{1},K_{1}$ are some positive constants depending on uniform upper bound of $||g||_{H^{m}_{2l+\gamma+2}}$ and uniform lower bound of $||g||_{L^{1}}$. Thanks to interpolation theory and the basic inequality (1.10), for any $\eta>0$, there exists some constant $C_{\eta}$ such that

[TABLE]

thus we have

[TABLE]

for some new constant $K_{1}$. By the previous step, with the uniform upper bound of $||f^{n}||_{L^{1}_{l+\gamma/2+2m+6}}$, we have

[TABLE]

where $K_{2}$ is some constant depending on $||f_{0}||_{L^{1}_{l+3\gamma/2+2m+6}}$ and uniform upper bound of $||g||_{L^{1}_{l+3\gamma/2+2m+6}}$. Noticing that inequality (3.19) has exactly the same structure as inequality (3.17), we have

[TABLE]

Step 2: (Cauchy Sequence)

In this step, we prove that $\{f^{n}(t)\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $L^{1}_{l}$ for any $t\geq 0$. Set $h^{n}=f^{n}-f^{n-1}$ for $n\geq 1$. Then for $n\geq 2$, we have

[TABLE]

Because we are uncertain about the sign of $h^{n}$, we have to introduce the sign function $sgn(h^{n})$. Similar as in (3.1), we obtain

[TABLE]

Similar as in (3.9)

[TABLE]

For the inner product $\langle Q_{L}(g,h^{n}),sgn(h^{n})\langle v\rangle^{l}\rangle$, we can approximate Landau operator by Boltzmann operators. Let $b_{\lambda}(\cos\theta)=\lambda^{2s-2}b(\cos\theta)\textbf{1}_{\theta\leq\lambda}$ for each $\lambda\leq\frac{\pi}{2}$, such that

[TABLE]

Let $Q_{\lambda}$ be the Boltzmann operator associated to the kernel $b_{\lambda}(\cos\theta)|v-v_{*}|^{\gamma}$, then by lemma 7.1 in [9], there holds

[TABLE]

By the uniform estimate (3.1) and our assumption on $g$ and $f_{0}$, we have

[TABLE]

Thanks to proposition 3.1, for $l\geq 4$, we derive that

[TABLE]

For $\lambda$ small enough, we have

[TABLE]

Thus we have

[TABLE]

and

[TABLE]

and finally

[TABLE]

With the help of the above three inequalities, we arrive at

[TABLE]

Let $\lambda$ tend to [math], by (3.24) and the uniform estimate (3.1), we have

[TABLE]

Choose $\eta$ small enough such that

[TABLE]

and denote $a=\frac{A^{\epsilon}}{(1+\eta)^{\gamma/2}}+\frac{1}{32}\epsilon^{2-2s}\Lambda,~{}\delta=\frac{1}{32}\epsilon^{2-2s}\Lambda$. Patch altogether (3.1), (3.1) and (3.1), we obtain

[TABLE]

For ease of notation, denote $K_{1}=(1+\frac{1}{\eta})^{\frac{\gamma}{2}}A^{\epsilon}$, $K_{2}=C(l,\gamma,\eta)A^{\epsilon}\sup_{0\leq s\leq t}||g(s)||_{L^{1}_{l}}$ and $K_{3}=(\eta^{-\gamma/2}A^{\epsilon}+\frac{\Lambda}{8}\epsilon^{2-2s}+C(l)\epsilon^{2-2s}\Lambda)\sup_{0\leq s\leq t}||g(s)||_{L^{1}_{l}}$. Then we have a much neater inequality on the interval $[0,t]$,

[TABLE]

Using the same technique as in the previous step, by defining $y^{n}(t_{n})=e^{-K_{3}t_{n}}||h^{n}(t_{n})||_{L^{1}_{l}}$ and $x^{n}(t_{n})=\int_{0}^{t_{n}}e^{-K_{3}s}||g(s)||_{L^{1}}||h^{n}(s)||_{L^{1}_{l+\gamma}}ds$, for $n\geq 1$ and $t_{n}\in[0,t]$. Then for $n\geq 2$, we derive that

[TABLE]

where we have used the initial condition $h^{n}(0)=0$. Now denote $S^{n}(s)=\sum_{i=0}^{n-1}(\frac{a}{a+\delta})^{i}y^{n-i}(s)$ for $n\geq 1$ and $s\in[0,t]$, by recursive derivation, we obtain

[TABLE]

By previous estimates (3.1), we have

[TABLE]

For ease of notation, for $n\geq 1$, let us define

[TABLE]

Thus, by further recursive derivation, for any $t_{n}\in[0,t]$, we obtain

[TABLE]

where we used the fact $S^{1}(t_{1})\leq C(t)\leq(a+\delta+1)C(t)=b^{1}(t)$. Note that $b^{n}(t)$ is a geometric sequence and $b^{n}(t)=b^{1}(t)(\frac{a}{a+\delta})^{n-1}$ for any $n\geq 1$, thus we have

[TABLE]

By recalling the definitions of $S^{n}$ and $x^{n}$, we arrive at

[TABLE]

Since the series $\sum_{n}(\frac{a}{a+\delta})^{n-1}$ is finite, we conclude that $\{f^{n}\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $L^{\infty}([0,t];L^{1}_{l})\cap L^{1}([0,t];L^{1}_{l+\gamma})$. Due to the arbitrariness of $t\in[0,T]$, there is a function $f\in L^{\infty}([0,T];L^{1}_{l})\cap L^{1}([0,T];L^{1}_{l+\gamma})$ such that

[TABLE]

It is obvious that $f$ is the solution to (3.1). Thus the non-positivity of $f$ is ensured by the non-positivity of $f^{n}$.

Step 3: (High Order Moments and Smoothness)

In this step, we prove the solution $f$ constructed in the previous step actually lies in $L^{\infty}([0,T];L^{1}_{l+5\gamma/2+16}\cap H^{5}_{l+\gamma+10})\cap L^{1}([0,T];L^{1}_{l+7\gamma/2+16})$. Let $q=l+5\gamma/2+16$. By lemma 3.1 and inequality (3.8), we first have

[TABLE]

Next, according to [6], one has

[TABLE]

Therefore we have

[TABLE]

By Gronwall’s inequality, it is not difficult to derive

[TABLE]

Recalling the uniform estimate (3.1) and the convergence of $\{f^{n}\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ in $L^{\infty}([0,T];L^{1}_{l})$, we also have $f\in L^{\infty}([0,T];H^{5}_{l+\gamma+10})$.

Step 4: (Uniqueness)

Suppose $f^{1},f^{2}\in L^{\infty}([0,T];L^{1}_{l+5\gamma/2+16}\cap H^{5}_{\gamma+10+l})\cap L^{1}([0,T];L^{1}_{l+7\gamma/2+16})$ are two non-negative solutions of equation (3.1), set $h=f^{1}-f^{2}$. Then $h$ is a solution to the following equation,

[TABLE]

Observe that the above equation is as the same as the equation (3.21) if $h^{n-1}=h^{n}$. With the same argument until inequality (3.1), we have

[TABLE]

where $C_{1}$ and $C_{2}$ are some positive constants depending on $M$ and $m$. Then we have

[TABLE]

which gives the uniqueness. ∎

3.2. First result on the well-posedness of approximate equation (1.7)

Based on the Picard iteration scheme, we derive that

Lemma 3.3**.**

Let $l\geq 4$ be a real number and $N$ be an nonnegative integer. Let $w_{H},w_{L},w$ be functions defined by

[TABLE]

[TABLE]

[TABLE]

Suppose the non-negative datum $f_{0}\in H^{(N+2)\vee 3}_{w_{H}(N,l)}\cap L^{1}_{w_{L}(N,l)}$ with $||f_{0}||_{L^{1}}>0$, then our approximate equation (1.7) admits a non-negative solution $f$ in $L^{\infty}([0,T^{*}];H^{N}_{l}\cap L^{1}_{w(N,l)})$ for some $T^{*}>0$. Moreover, if $N\geq 2$ and $l\geq 8+\gamma$, the solution is unique.

Proof.

Consider the sequence of functions $\{f^{n}\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ defined by

[TABLE]

We first mention that equation (3.32) conserves mass, that is, $||f^{n}(t)||_{L^{1}}=||f_{0}||_{L^{1}}$ for any $n\geq 0$ and $t\geq 0$. By previous lemma, $f^{n}\geq 0$ for any $n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$.

Step 1: (Uniform $L^{1}_{l}$ Upper Bound)

In this step we prove that $\{f^{n}\}_{n}$ has uniform upper bound in $L^{\infty}([0,T^{*}(l)];L^{1}_{l})$ with respect to $n$ for some $T^{*}(l)>0$ if $f_{0}\in L^{1}_{l+\gamma}$. Thanks to proposition 3.1, for any $l\geq 4$, we have

[TABLE]

In the following, denote $A^{\epsilon}_{2}=\int_{\SS^{2}}b^{\epsilon}\sin^{2}\frac{\theta}{2}d\sigma\leq\frac{A_{2}}{2}$, then we have

[TABLE]

and

[TABLE]

where we used $1-\cos^{l}\frac{\theta}{2}\geq 1-\cos^{4}\frac{\theta}{2}\geq\frac{3}{2}\sin^{2}\frac{\theta}{2}$ and $\sin^{l}\frac{\theta}{2}\leq\frac{1}{2}\sin^{2}\frac{\theta}{2}$. Together with $|v-v_{*}|^{\gamma}\geq\frac{3}{4}\langle v\rangle^{\gamma}-c_{1}\langle v_{*}\rangle^{\gamma}$, $|v-v_{*}|^{\gamma}\leq 2(\langle v\rangle^{\gamma}+\langle v_{*}\rangle^{\gamma})$, and $|v-v_{*}|^{\gamma}\leq\langle v\rangle^{\gamma}\langle v_{*}\rangle^{\gamma}$, we obtain

[TABLE]

Recalling (3.1), we have

[TABLE]

With (3.2) and (3.34) in hand, we have

[TABLE]

where we denote $C(\epsilon,l,\Lambda)=A^{\epsilon}_{2}+\frac{3}{2}c_{1}A^{\epsilon}_{2}+C(l)A^{\epsilon}_{2}+\epsilon^{2-2s}(4l+2)l\Lambda$. For simplicity, denote $m(l)=||f_{0}||_{L^{1}_{l}}$. For any $n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ and $l\geq 0$, define

[TABLE]

and

[TABLE]

We claim that for any $n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$,

[TABLE]

We will prove (3.36) by induction. First, it is obvious $C_{0,l}\leq 11m(l)$. Next, fix a $n\geq 1$, suppose $C_{n-1,l}\leq 11m(l)$, then on the interval $[0,T^{*}(l)]$, for any $1\leq k\leq n$, from (3.2), we have

[TABLE]

Thus for any $t\in[0,T^{*}(l)]$ and $1\leq k\leq n$, we derive that

[TABLE]

Multiplying the above inequality by $(\frac{8}{9})^{n-k}$ and taking sum over $1\leq k\leq n$, we obtain

[TABLE]

Observing that $\sum_{k=1}^{n}(\frac{8}{9})^{n-k}\leq 9$, we arrive at

[TABLE]

Thus we have

[TABLE]

by the definition of $T^{*}(l)$. That is, $C_{n}\leq 11m(l)$. Therefore the claim (3.36) is proved, which impiles

[TABLE]

Step 2: (Uniform $H^{N}_{l}$ Upper Bound)

In this step, we shall use the energy estimate to get the uniform upper bound of $L^{N}_{l}$ norm of $f^{n}$ with respect to $n$. Fix an $\alpha$ with $|\alpha|\leq N$, one has

[TABLE]

As before, we have

[TABLE]

By coercivity estimate (2.5) and commutator estimates (2.7), (2.8), we have

[TABLE]

By upper bound estimate (2.1) and commutator estimates (2.7), (2.8), for $|\alpha_{2}|\leq N-1$, we have

[TABLE]

When $N=0$, by (3.2), we have

[TABLE]

By (3.38), there holds

[TABLE]

so we have

[TABLE]

Thanks to the fact

[TABLE]

we have

[TABLE]

By Gronwall’s inequality, we obtain

[TABLE]

With the help of uniform $L^{2}_{l}$ norm and the above inequality, we can prove in a similar manner as in the second step in the proof of theorem 1.1,

[TABLE]

where $\phi(s,l)=\frac{(2l+4)(2+s)-2l}{s}$.

Now we turn to higher order regularity. Taking into account the fact $W^{\epsilon}(\xi)\leq\langle\xi\rangle$, for the fixed $\epsilon$, by (3.2) and (3.2), we have

[TABLE]

Thanks interpolation theory and Young’s inequality, one has

[TABLE]

[TABLE]

and finally

[TABLE]

we have used the Young’s inequality (1.10) with $p=\frac{2-s}{s},q=\frac{2-s}{s(1-s)}$ and $\eta=\frac{C_{1}(f_{0})}{32}\epsilon^{2-2s}$. Thus we arrive at for any $n\geq 1$,

[TABLE]

where $M$ is the uniform upper bound of $||f^{n}||_{H^{s}_{6}},||f^{n}||_{L^{2}_{l+5}}$ and $||f^{n}||_{L^{1}_{3l+6}}$ with respect to $n$ on the time interval $[0,T^{*}]$. Here $T^{*}=T^{*}(\max\{\phi(s,6),3l+6\})$. With the same technique as in dealing with (3.2), we obtain

[TABLE]

The above inequality is true for any $n\geq 1$ and $t\in[0,T^{*}]$, so we have the desired result

[TABLE]

Continuing the argument, there will be a function $q:\mathop{\mathbb{N}\kern 0.0pt}\nolimits\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+}\rightarrow\mathop{\mathbb{R}\kern 0.0pt}\nolimits$, such that

[TABLE]

Step 3: (Cauchy Sequence)

Now we are ready to prove $\{f^{n}\}_{n}$ is a Cauchy sequence in $L^{\infty}([0,T^{*}];L^{1}_{l})$. Set $h^{n}=f^{n+1}-f^{n}$ for $n\geq 0$. Then for $n\geq 1$, $h^{n}$ is the solution to the following equation

[TABLE]

As the same as (3.2), we have

[TABLE]

As the same as (3.1), we have

[TABLE]

Applying proposition 3.1 again, we obtain

[TABLE]

Recalling the Landau operator $Q_{L}$ can be rewritten as:

[TABLE]

we have

[TABLE]

Patch all together inequalities (3.2),(3.2),(3.2) and (3.2), we obtain

[TABLE]

where $K_{1}$ and $K_{2}$ are some constants depending at most on the uniform upper bound of $||f^{n}||_{H^{2}_{l+\gamma+4}}$, which is bounded by a constant depending on $||f_{0}||_{H^{3}_{l+3\gamma/2+4}},||f_{0}||_{L^{1}_{q(2,l+\gamma+4)+\gamma}}$. With a similar argument as in the previous lemma, for any $t\in[0,T^{*}(q(2,l+\gamma+4))]$, we can conclude

[TABLE]

where $M(t)=\frac{9}{8}A^{\epsilon}_{2}m(0)\int_{0}^{t}e^{-K_{1}s}||h^{0}(s)||_{L^{1}_{l+\gamma}}ds+22m(l)$. Thus $\sum_{n}||h^{n}(t)||_{L^{1}_{l}}$ is finite and $\{f^{n}(t)\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $L^{1}_{l}$. Due to the arbitrariness of $t\in[0,T^{*}(q(2,l+\gamma+4))]$, there is a function $f\in L^{\infty}([0,T^{*}];L^{1}_{l})$ such that

[TABLE]

In the following, we prove $\{f^{n}(t)\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $H^{N}_{l}$.

Fix an $\alpha$ with $|\alpha|\leq N$, one has

[TABLE]

Then we have

[TABLE]

As the same as (3.2), on the time interval $[0,T^{*}(2l+1)]$, we have

[TABLE]

and thus

[TABLE]

As the same as (3.2), for $|\alpha_{2}|\leq|\alpha|-1\leq N-1$ and any $\eta>0$, on the time interval $[0,T^{*}(q(N,2l+3))]$, we have

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

Similarly, on the time interval $[0,T^{*}(q(N+1,l+\gamma/2+2))]$, we have

[TABLE]

and so for any $\eta>0$,

[TABLE]

Taking a suitable $\eta$, we obtain

[TABLE]

Now taking sum over $|\alpha|\leq N$, we arrive at

[TABLE]

By interpolation theory, one has

[TABLE]

and

[TABLE]

where $w_{1}(N,l,s,\gamma)=l+\gamma/2+\frac{2(N+s+2)}{s}$ and $w_{2}(N,l,s,\gamma)=\frac{(N+s+2)(2l+3)-(N+2)(l+\gamma/2)}{s}$. It is easy to check $w_{1}\leq w_{2}$. Choosing suitable $\eta$ and $\lambda$, we have

[TABLE]

Thanks to (3.48), on the time interval $[0,T^{*}(q(2,w_{2}+\gamma+4))]$, there holds

[TABLE]

where $T^{*}=T^{*}(q(w_{2}+\gamma+4,2))$ and $M(T^{*})=\frac{9}{8}A^{\epsilon}_{2}m(0)\int_{0}^{T^{*}}e^{-K_{1}s}||h^{0}(s)||_{L^{1}_{w_{2}+\gamma}}ds+22m(w_{2})$. For ease of notation, let $K_{3}=C(\eta,||f_{0}||_{H^{N+1}_{2l+3+\gamma/2}},||f_{0}||_{L^{1}_{q(N,2l+3)+\gamma}})$ and

$K_{4}=C(\lambda,||f_{0}||_{H^{N+2}_{l+\gamma+2}},||f_{0}||_{L^{1}_{q(N+1,l+\gamma/2+2)+\gamma}})$. Then we have

[TABLE]

Integrating both sides with respect to time over $[0,t]$ for any $t\in[0,T^{*}]$, we have

[TABLE]

Thus $\sum_{n}||h^{n}(t)||_{H^{N}_{l}}$ is finite and $\{f^{n}(t)\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $H^{N}_{l}$. So there is a function $f\in L^{\infty}([0,T^{*}];H^{N}_{l})$ such that

[TABLE]

The condition on $f_{0}$ can be summarized by the definitions of $K_{3},K_{4}$ and the previous step as

[TABLE]

Under this condition, actually $\{f^{n}(t)\}_{n\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits}$ is a Cauchy sequence in $H^{N}_{l}\cap L^{1}_{w(N,l)}$.

It is obvious that $f$ is the solution to (1.7). Because $f^{n}$ is non-negative, the limit function $f$ is also non-negative.

Step 4: (Uniqueness)

Suppose $f,g\in L^{\infty}([0,T];H^{2}_{l+\gamma+4})$ are two non-negative solutions to (1.7). Set $F=f-g$ and $G=f+g$. Then $F$ is a solution to the following equation,

[TABLE]

Note that the above equation is as the same as equation (3.42) if $h^{n}=h^{n-1}$. Thus following the same argument until inequality (3.2), we have

[TABLE]

where $K$ is some constant depending on the uniform upper bound of $||G||_{H^{2}_{l+\gamma+4}}$. Note that the previous estimate holds true for $l\geq 4$. Therefore, our approximate equation (1.7) has at most one solution in the space $L^{\infty}([0,T];H^{N}_{l})$ if $N\geq 2$ and $l\geq 8+\gamma$.

∎

3.3. Improvement of the well-posedness result of approximate equation (1.7)

In this subsection, by using the symmetric property of the collision operators, we will prove the propagation of $L^{1}_{l}$ and $H^{N}_{l}$ norms of the solution $f$ to (1.7) and then extend the lifespan $T^{*}$ in Lemma 3.3 to be global. Thanks to Lemma 3.3, we may assume that solution $f^{\epsilon}$ to our approximate equation is non-negative and smooth. It means that in this subsection we only need to give the a priori estimates to the equation.

In order to prove the propagation of $L^{1}_{l}$ of the solution $f^{\epsilon}$, we first give two propositions. The first proposition is related to the Boltzmann operator, while the second deals with the Landau operator.

Proposition 3.2**.**

Let $p\geq 3$ and $k_{p}=[\frac{p+1}{2}]$. Suppose

[TABLE]

then one has

[TABLE]

Proof.

One may refer to Lemma 3.6 in [14] for the proof. ∎

Remark 3.1**.**

Lemma 3.6 in [14] only deals with the case $p\geq 3$, however, the conclusion is also valid in the case $2\leq p<3$ but with a different and smaller coefficient coming out instead of the constant $\frac{1}{4}$ before the highest order $2p+\gamma$.

Proposition 3.3**.**

Let $p>2$ and $f$ be a non-negative function, then

[TABLE]

Proof.

One may refer to [6] for the proof. ∎

Now we are ready to prove the propagation of moments and the smoothness.

Proof of Theorem 1.1: The proof will be divided into four steps.

Step 1: Propagation of the moments.

We consider the $2l$ moment. Assume $l\geq 3$, for the case $2\leq l<3$, the proof is similar thanks to remark 3.1. By the case By the definition of $M^{\epsilon}$, we have

[TABLE]

The term $\mathfrak{I}_{1}$ can be written as:

[TABLE]

Let $A^{\epsilon}_{2}=\int_{\SS^{2}}b^{\epsilon}(\cos\theta)\sin^{2}\theta d\sigma$, then by proposition 3.2, we have

[TABLE]

where we have used the assumption $\gamma\leq 2$. By interpolation, for any $2\leq p,q\leq 2l$ with $p+q=2l+2$, we have

[TABLE]

Using the fact $2\sum^{k_{l}}_{k=1}\binom{l}{k}\leq 2^{l}$, we can conclude:

[TABLE]

For the term $\mathfrak{I}_{2}$, we apply proposition 3.3 with $p=2l$ and obtain

[TABLE]

Let $0<\epsilon_{*}<\frac{\sqrt{2}}{2}$ be the point such that $A^{\epsilon_{*}}_{2}=\frac{A_{2}}{2}$, then for any $0<\epsilon\leq\epsilon_{*}$, we have

[TABLE]

For any $\eta>0$, there exists a constant $K_{1}(\eta,l)$ such that

[TABLE]

Thus we have $||f^{\epsilon}||_{L^{1}_{2l}}\leq\eta||f^{\epsilon}||_{L^{1}_{2l+\gamma}}+K_{1}(\eta,l)||f^{\epsilon}||_{L^{1}}$. With the preservation of mass and energy, by denoting $K_{2}(l)=2^{2l+1}A_{2}+2l(8l+2)\Lambda$ and taking $\eta(f_{0})=\frac{A||f_{0}||_{L^{1}}}{16K_{2}(l)||f_{0}||_{L^{1}_{2}}}$, we have

[TABLE]

Let $a=K_{2}(l)K_{1}(\eta(f_{0}),l)||f_{0}||_{L^{1}_{2}}||f_{0}||_{L^{1}}$ and $b=-\frac{A_{2}}{16}||f_{0}||_{L^{1}}$, by Gronwall’s inequality (1.11), we have the following:

[TABLE]

The constant $K(f_{0},l)$ depends only on $l$, $||f_{0}||_{L^{1}}$ and $||f_{0}||_{L^{1}_{2}}$.

Step 2: Propagation of $L^{2}_{l}$ norm.

By the definition of $M^{\epsilon}$, we have

[TABLE]

Applying coercivity estimates of (2.5) with $g=f^{\epsilon},f=f^{\epsilon}\langle v\rangle^{l}$, we have

[TABLE]

Applying commutator estimates (2.9) with $g=f^{\epsilon},h=f^{\epsilon},f=f^{\epsilon}\langle v\rangle^{l},N_{2}=l+\gamma/2,N_{3}=\gamma/2$ and $N_{1}=2l+5$, we have

[TABLE]

Thanks to the facts $||\cdot||^{2}_{H^{s}_{l+\gamma/2}}\leq||\cdot||^{2}_{\epsilon,l+\gamma/2}$ and $\epsilon^{2-2s}||\cdot||^{2}_{H^{1}_{l+\gamma/2}}\leq||\cdot||^{2}_{\epsilon,l+\gamma/2}$, we have

[TABLE]

Now patching together (3.53),and (3.54), we get

[TABLE]

where the existence of $C_{3}(f_{0},l)=C_{3}(||f_{0}||_{L^{1}_{2l+5}},||f_{0}||_{L\log L},l)$ is ensured by the previous step.

By applying (3.16) with $\lambda=\frac{C_{1}(f_{0})}{4C_{3}(f_{0},l)}$, we have

[TABLE]

Thanks to Gronwall’s inequality, there exists a constant $C(||f_{0}||_{L^{1}_{2l+5}},||f_{0}||_{L^{2}_{l}})$ such that for any $t\geq 0$,

[TABLE]

Inequality (1.17) is obtained in the case of $N=0$.

Step 3: Propagation of $H^{s}_{l}$ norm.

We first introduce some notations for the fractional derivative. We set

[TABLE]

with $(\tau_{h}f)(v)=f(v+h)$ and $0<s<1$. Then there holds

[TABLE]

Due to the definition of the fractional Sobolev space, we observe that:

[TABLE]

Moreover, we also have, for $|h|\leq\frac{1}{2}$ and $m\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits$,

[TABLE]

[TABLE]

and

[TABLE]

One may check the proof of (3.57), (3.58) and (3.3) in the appendix of [10].

Let $g^{\epsilon}=f^{\epsilon}\langle v\rangle^{l}$. It is easy to check that $\triangle_{s}g^{\epsilon}$ solves the following equation:

[TABLE]

By the upper bound estimate (2.1), noting $\gamma\leq 2$, we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

By the commutator estimates (2.7) and (2.8), we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

Similarly, we have

[TABLE]

which implies, for any $\eta_{2}>0$,

[TABLE]

Also by the upper bound estimate (2.1), we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

By the coercivity estimate (2.5), we have

[TABLE]

Thanks to

[TABLE]

patching together all the above estimates, taking $\eta_{1}=\frac{C_{1}(f_{0})}{6}$ in (3.60),(3.3),(3.3), we arrive at, for $|h|\leq\frac{1}{2}$,

[TABLE]

where we have used the fact $||\langle\cdot\rangle^{-2}||_{L^{2}}\leq\sqrt{2}\pi$. Integrating both sides from [math] to $t$ with respect to time, we obtain

[TABLE]

Integrating both sides on the Ball $B(0,\frac{1}{2})$ with respect to the variable $h$, noting that

$\int_{|h|\leq\frac{1}{2}}|h|^{-(1+2s)}dh$ is finite, thanks to the facts (3.56) and (3.3), taking a small enough $\eta_{2}$, we derive that

[TABLE]

Using the fact $||f^{\epsilon}||^{2}_{H^{s}_{2l+3}}||f^{\epsilon}||^{2}_{H^{s}_{l+1}}\leq||f^{\epsilon}||^{2}_{H^{s}_{l}}||f^{\epsilon}||^{2}_{H^{s}_{2l+4}}$, substituting into the uniform bound of $||f^{\epsilon}||_{L^{1}_{2l+5}}$ and $||f^{\epsilon}||_{L^{2}_{l}}$, we have

[TABLE]

Actually, inequality (3.3) holds true on any bounded interval. Therefore, for any $t_{1}<t_{2}$ with $t_{2}-t_{1}\leq 2$, we have

[TABLE]

By Gronwall’s inequality (1.12) and uniform upper bound (3.55) for integral of $||f^{\epsilon}||^{2}_{\epsilon,l}$ on any bounded interval, we arrive at

[TABLE]

Also from (3.55), we conclude that, in any unit interval $[t,t+1]$, there exists at least one point $t_{*}$ such that

[TABLE]

Combining (3.69) and (3.70), we have

[TABLE]

Together with (3.3), we finally arrive at

[TABLE]

By interpolation theory, there holds

[TABLE]

where $\phi(s,l)=\frac{(2l+4)(2+s)-2l}{s}\geq 4l+13$. Therefore we have

[TABLE]

Step 4: Propagation of $H^{N}_{l}$ norm when $N\geq 1$.

We prove the propagation by induction on $N$. Let $m\geq 1$ be an integer. Suppose inequality (1.17) holds true for all $N\leq m-1$, we now prove that it is also valid for $N=m$.

Set $g^{\epsilon}=\partial^{\alpha}_{v}f^{\epsilon}\langle v\rangle^{l}$ with $|\alpha|\leq m$, then $g^{\epsilon}$ solves

[TABLE]

By the coercivity estimate (2.5), we have

[TABLE]

By the commutator estimate (2.9), we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

For the remaining terms in the right hand of (3.3) with $|\alpha_{1}|\geq 1$, we split each of them into two terms:

[TABLE]

By the commutator estimate (2.9), for the case $|\alpha_{1}|=|\alpha|\leq m$, we have

[TABLE]

which implies, for any $\eta_{2}>0$,

[TABLE]

For the case $1\leq|\alpha_{1}|\leq|\alpha|-1\leq m-1$, we have

[TABLE]

which implies, for any $\eta_{2}>0$,

[TABLE]

By the upper bound estimate (2.1), for the case $|\alpha_{1}|=|\alpha|\leq m$, we have,

[TABLE]

which implies, for any $\eta_{3}>0$,

[TABLE]

While for the case $1\leq|\alpha_{1}|\leq|\alpha|-1\leq m-1$, we similarly have, for any $\eta_{3}>0$,

[TABLE]

Now choosing suitable $\eta_{1}$ in (3.3), $\eta_{2}$ in (3.77) and (3.3), and $\eta_{3}$ in (3.78) and (3.79), we have

[TABLE]

When $m=1$, inequality (3.3) reduces to

[TABLE]

Remembering that

[TABLE]

by interpolation theory and the basic inequality (1.10), for any $\eta>0$, we have

[TABLE]

where $x(l)=\frac{2l+7}{s}-\frac{1-s}{s}(l+\frac{\gamma}{2})$, and

[TABLE]

where $\tilde{x}(l)=(2l+7)(2-s)-(1-s)(l+\gamma/2)\leq x(l)$. Taking small enough $\eta$, we finally have

[TABLE]

Then by Gronwall’s inequality and the uniform upper bound (3.73) of $H^{s}$ norm, we arrive at

[TABLE]

Once again by interpolation theory, there holds

[TABLE]

where $y(l)=\frac{3x(l)-(s+2)l}{1-s}$. By setting $\phi(1,l)=\max\{\phi(s,x(l)),y(l)\}$, we have

[TABLE]

When $m\geq 2$, $||f^{\epsilon}||^{2}_{H^{1}_{2l+7}}$ has uniform bound by assumption. According to the interpolation inequality and the basic inequality (1.10), one has

[TABLE]

where $z(l)=2l+7+\frac{l+7}{s}$. With the fact $||f^{\epsilon}||_{\epsilon,l+3}\lesssim||f^{\epsilon}||_{H^{1}_{l+3}}$, we finally arrive at

[TABLE]

Then by Gronwall’s inequality and the assumed uniform bound of $H^{m-1}$ norm,

[TABLE]

By interpolation theory, there holds

[TABLE]

where $u(m,l)=(m+2)z(l)-(m+1)l$. Now by setting $\phi(m,l)=\max\{u(m,l),\phi(m-1,z(l))\}$, we arrive at

[TABLE]

The proof of theorem 1.1 is complete now.

Remark 3.2**.**

Since $L^{1}\subset H^{-m}$ if $m>3/2$, one can obtain lower weight requirement in the space $L^{1}$. We use $H^{-2}$ as one interpolation space just for a neat expression. For the same reason, we replace $\gamma$ with $2$.

4. Error estimates to the approximation

In this section, we prove the last two theorems stated in section 1. We first give a proof to theorem 1.2.

Proof of Theorem 1.2: For each $0<\epsilon\leq\sqrt{2}/2$, we define $F^{\epsilon}_{R}$ and $Q_{\epsilon}$ respectively as follows:

[TABLE]

[TABLE]

Take the difference between equations (1.1) and (1.7), and divide both sides by $\epsilon^{3-2s}$, we have

[TABLE]

where

[TABLE]

We now show that $L^{1}_{2l}$ norm of $F^{\epsilon}_{R}$ is bounded by the initial datum $f_{0}$ and time $t$. According to (4.3), we have

[TABLE]

Thanks to lemma (7.1) in the Appendix of [9], we have

[TABLE]

Now we deal with $\mathfrak{I}_{2}$, note that

[TABLE]

where

[TABLE]

and

[TABLE]

According to proposition 3.2, we have

[TABLE]

Now we turn to $\mathfrak{I}_{2,2}$. Recall that

[TABLE]

where $v(\kappa)=v_{*}+\kappa(v^{\prime}_{*}-v_{*})=v_{*}-\kappa(v^{\prime}-v)$. By symmetry,

[TABLE]

Observe that the matrix $\nabla^{2}\langle\cdot\rangle^{2l}$ is positive definite, we are only left with

[TABLE]

Split $\mathfrak{I}_{3}$ into two parts:

[TABLE]

where $\alpha=\frac{\gamma+2}{2-2s}$.

For $\mathfrak{I}_{3,1}$, we have

[TABLE]

By cancellation lemma,

[TABLE]

where $C(cancel)=2^{\frac{5+\gamma}{2}}A_{2}$. For the term $\mathfrak{I}_{3,1,2}$, apply Taylor expansion:

[TABLE]

where $v(\kappa)=v^{\prime}+\kappa(v-v^{\prime})$. For fixed $v_{*}$, it is easy to check

[TABLE]

Thus we are only left with

[TABLE]

Set $u=v^{\prime}+\kappa(v-v^{\prime})$, then we have

[TABLE]

and

[TABLE]

By the change of variable: $v\rightarrow u$, the Jacobian matrix is

[TABLE]

with its Jacobian

[TABLE]

Thanks to $|u-v_{*}|\leq|v-v_{*}|\leq\sqrt{2}|u-v_{*}|$, we obtain

[TABLE]

where we have used the fact $||\langle\cdot\rangle^{-2}||_{L^{2}}\leq\sqrt{2}\pi$.

Now we turn to $\mathfrak{I}_{3,2}$. Note that

[TABLE]

First look at the term $\mathfrak{I}_{3,2,1}$. Recall that $j=\frac{u-(u\cdot n)n}{|u-(u\cdot n)n|}$ in lemma 3.1, then we have $j\cdot n=0$, and thus

[TABLE]

Applying proposition 3.1 and the above equality, we obtain

[TABLE]

where $c_{l}=2^{l-3}(l(l-1)+4)$. Thanks to the following fact:

[TABLE]

and $|v-v_{*}|^{\gamma}\leq 2(\langle v\rangle^{\gamma}+\langle v_{*}\rangle^{\gamma})$, we have

[TABLE]

Due to $|v-v_{*}|^{\gamma}\leq\langle v\rangle^{2}\langle v_{*}\rangle^{2}$, we obtain

[TABLE]

For the term $\mathfrak{I}_{3,2,2}$, we have

[TABLE]

provided $2\alpha s+\gamma\leq 2l$.

Patch the above inequalities (4.6),(4.7),(4),(4.10), and (4)-(4), for those $l$ such that $\frac{2^{1-2s}\pi K}{l-s}(\frac{\pi}{4})^{2l-2s}\leq\frac{A_{2}}{8}$, we have the following desired result:

[TABLE]

where we have used the mass conservation property: $||f_{t}||_{L^{1}}=||f^{\epsilon}_{t}||_{L^{1}}=||f_{0}||_{L^{1}}$. The propagation of $L$ and $H$ norms of $f$ and $f^{\epsilon}$ allows us to conclude:

[TABLE]

Applying Gronwall’s inequality (1.11) with $a=C(||f_{0}||_{L^{1}_{\phi(5,2l+\gamma+12)}},||f_{0}||_{H^{5}_{2l+\gamma+12}})$

and $b=C(||f_{0}||_{L^{1}_{\max\{41,\phi(2,2l+2)\}}},||f_{0}||_{H^{2}_{2l+2}})$, we have

[TABLE]

We now prove theorem 1.3 in the rest of this section.

**Proof of Theorem 1.3:

**Step 1: (Case $N=0$)

Taking the difference between equations (1.1) and (1.7), and dividing both sides by $\epsilon^{3-2s}$, we have

[TABLE]

Then we have

[TABLE]

Thanks to lemma 7.1 in the Appendix of [9], we have

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

Splitting $\mathfrak{I}_{2}$ into two terms

[TABLE]

By coercivity estimate (2.5), we have

[TABLE]

By commutator estimate (2.9) with $N_{2}=l+\gamma/2,N_{3}=\gamma/2$, we have,

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

Splitting $\mathfrak{I}_{3}$ into two terms

[TABLE]

Applying upper bound estimate (2.1) with $w_{1}=\gamma/2+2,w_{2}=\gamma/2$, we have

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

By commutator estimate (2.9), we have

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

Now setting $\eta=\frac{C_{1}(f_{0})}{8}$ in (4.17),(4.19),(4.20),(4.21), and combining with (4.18), we have

[TABLE]

Now choosing $\lambda=\frac{C_{1}(f_{0})}{2}(C_{2}(f_{0})+\frac{1}{\eta}||f^{\epsilon}||^{2}_{L^{1}_{2l+5}})^{-1}$ in (3.16), we have

[TABLE]

According to theorem 1.2, we have

[TABLE]

The other terms of the right hand side of (4) are also bounded by some lower order or lower weight norm of initial datum $f_{0}$, thus we arrive at

[TABLE]

We remark that the dependence on $t$ is also at most exponential.

Step 2: (Case $N\geq 1$)

Suppose inequality (1.21) holds true for all $N\leq m-1$, we now prove that it is also valid for $N=m$.

Let $g^{\epsilon}_{\alpha,l}=\langle v\rangle^{l}\partial^{\alpha}_{v}F^{\epsilon}_{R}$ with $|\alpha|\leq m$, then $g^{\epsilon}_{\alpha,l}$ solves

[TABLE]

Therefore we have

[TABLE]

Again by lemma 7.1 in the Appendix of [9], we have

[TABLE]

which implies, for any $\eta>0$,

[TABLE]

Splitting $\mathfrak{I}_{1}(\alpha_{1},\alpha_{2})$ into two terms, we have

[TABLE]

By coercivity estimate (2.5), we have

[TABLE]

For $1\leq|\alpha_{1}|\leq|\alpha|\leq m$, by upper bound estimate (2.1) with $w_{1}=\gamma/2+2,w_{2}=\gamma/2$, we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

By commutator estimates (2.9) with $N_{2}=l+\gamma/2,N_{3}=\gamma/2$, we have

[TABLE]

which implies, for any $\eta_{2}>0$,

[TABLE]

Splitting $\mathfrak{I}_{2}(\alpha_{1},\alpha_{2})$ into two terms, we have

[TABLE]

Applying upper bound estimate (2.1) with $w_{1}=\gamma/2+2,w_{2}=\gamma/2$, we may have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

By commutator estimate (2.9) with $N_{2}=l+\gamma/2,N_{3}=\gamma/2$, we have

[TABLE]

which implies, for any $\eta_{1}>0$,

[TABLE]

Patching all together (4.25),(4.26),(4.27),(4.28),(4.29),(4.30), and taking $\eta_{1}$ small enough, we arrive at

[TABLE]

Let $a(m)=\sum_{r=0}^{m}\binom{r+2}{r}$. Summing over $|\alpha|\leq m$, by taking $\eta_{2}=\frac{C_{1}(f_{0})}{4}\frac{1}{a(m)}$, we have

[TABLE]

Thanks to (3.81), we may conclude

[TABLE]

By theorem 1.1 and remark 1.1, for any $t\geq 0$, we have

[TABLE]

By assumption, there holds

[TABLE]

On the other hand, by interpolation, we have

[TABLE]

where $\rho(m,l)=(m+7)\psi(m-1,z(l))-(m+6)(l+\gamma+10)$. By defining $\varphi(m,l)=\max\{\varphi(m-1,z(l)),\rho(m,l)\}$, we have

[TABLE]

Acknowledgments. The authors thank the first 2016 exchange program between Hong Kong and the mainland of China, supported by the Ministry of education of the People’s Republic of China. They also express their gratitude to the Department of Mathematical Sciences of Tsinghua University and the Department of Mathematics of Hong Kong Baptist University for the kindly hospitality.