Asymptotic Analysis of the Spatially Homogeneous Boltzmann Equation: Grazing Collisions Limit

Abstract

In the present work, we consider the asymptotic problem of the spatially homogeneous Boltzmann equation when almost all collisions are grazing, that is, the deviation angle $\theta$ of the collision is limited near zero (i.e., $\theta \le \epsilon$). We show that by taking the proper scaling to the cross-section which was used in [37], that is, assuming

$$\begin{aligned} B^\epsilon ( v-v_{*},\sigma )=2(1-s)|v-v_*|^{\gamma }\epsilon ^{-3}\sin ^{-1}\theta \left( \frac{\theta }{\epsilon }\right) ^{-1-2s}\mathrm {1}_{\theta \le \epsilon }, \end{aligned}$$

where $\theta = \langle \theta ={\frac{\upsilon -\upsilon _*}{|\upsilon -\upsilon _*|}}.\sigma \rangle ,$ the solution $f^\epsilon$ of the Boltzmann equation with initial data $f_0$ can be globally or locally expanded in some weighted Sobolev space as

$$\begin{aligned} f^\epsilon = f+ O(\epsilon ), \end{aligned}$$

where the function $f$ is the solution of Landau equation, which is associated with the grazing collisions limit of Boltzmann equation, with the same initial data $f_0$. This gives the rigorous justification of the Landau approximation in the spatially homogeneous case. In particular, if taking $\gamma =-3$ and $s=1-\epsilon$ in the cross-section $B^\epsilon$, we show that the above asymptotic formula still holds and in this case $f$ is the solution of Landau equation with the Coulomb potential. Going further, we revisit the well-posedness problem of the Boltzmann equation in the limiting process. We show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution $f^\epsilon$. Thanks to the weak convergence results on the grazing collisions limit in [37], in other words, we establish a unified framework to establish the well-posedness results for both Boltzmann and Landau equations.

Appendix

In this section, we shall give the error estimates between the Boltzmann collision operator $Q^\epsilon (g,h)$ and the Landau operator $Q_L(g,h)$. We have

Lemma 7.1

Suppose that $g$ and $h$ are Schwartz functions. If $\psi \in L^2$, one has

$$\begin{aligned} \bigg |\frac{1}{\epsilon }\langle Q^\epsilon (g,h)-Q_L(g,h), \langle \cdot \rangle ^l\psi \rangle _v\bigg |\lesssim \Vert g\Vert _{H^3_{\gamma +10+l}}\Vert h\Vert _{H^5_{\gamma +10+l}}\Vert \psi \Vert _{L^2}, \end{aligned}$$

or

$$\begin{aligned} \bigg |\frac{1}{\epsilon }\langle Q^\epsilon (g,h)-Q_L(g,h), \langle \cdot \rangle ^l\psi \rangle _v\bigg |\lesssim \Vert g\Vert _{H^5_{\gamma +10+l}}\Vert h\Vert _{H^3_{\gamma +10+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

If $\psi$ verifies

$$\begin{aligned} |\psi (v)|\le \langle v\rangle ^l, \end{aligned}$$

(81)

one has

$$\begin{aligned} \bigg |\frac{1}{\epsilon }\langle Q^\epsilon (g,h)-Q_L(g,h), \psi \rangle _v\bigg |\lesssim \Vert g\Vert _{H^3_{\gamma +12+l}}\Vert h\Vert _{H^5_{\gamma +10+l}}. \end{aligned}$$

Remark 7.1

As a direct consequence, we have

$$\begin{aligned} \bigg |\frac{1}{\epsilon }\langle Q^\epsilon (f,f)-Q_L(f,f), (sgn F^\epsilon _R)\langle \cdot \rangle ^l\rangle _v\bigg |\lesssim \Vert f\Vert _{H^3_{\gamma +12+l}}\Vert f\Vert _{H^5_{\gamma +10+l}}, \end{aligned}$$

and for $|\beta |\ge 3$,

$$\begin{aligned} \bigg |\frac{1}{\epsilon }\langle \partial _v^\beta \big [Q^\epsilon (f,f)-Q_L(f,f)\big ], \langle \cdot \rangle ^l\psi \rangle _v\bigg |\lesssim \Vert f\Vert ^2_{H^{|\beta |+3}_{\gamma +10+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

Proof

We shall borrow the idea of [17] to give the estimates. Introduce an orthogonal basis of ${\mathbb {R}}^3$:

$$\begin{aligned} \big (\frac{v-v_*}{|v-v_*|}, h^1_{v,v_*}, h^2_{v,v_*}\big ). \end{aligned}$$

Then one has

$$\begin{aligned} \sigma = \frac{v-v_*}{|v-v_*|}\cos \theta + (\cos \varphi h^1_{v,v_*}+\sin \varphi h^2_{v,v_*})\sin \theta . \end{aligned}$$

By change of variable: $\theta \rightarrow \epsilon \chi$, we have

$$\begin{aligned} v'=v+\frac{1}{2}A^\epsilon ,\quad v_{*}^{\prime }=v_*-\frac{1}{2}A^\epsilon , \end{aligned}$$

with

$$\begin{aligned} A^\epsilon =-(v-v_*)(1-\cos (\epsilon \chi ))+|v-v_*|(\cos \varphi h^1_{v,v_*}+\sin \varphi h^2_{v,v_*})\sin (\epsilon \chi ). \end{aligned}$$

The Boltzmann collision operator $Q^\epsilon$ can be rewritten as

$$\begin{aligned} Q^\epsilon (g,h)&= \frac{1}{\epsilon ^2}\int \limits _{{\mathbb {R}}^3}\int \limits _0^\frac{\pi }{\epsilon }\int \limits _0^{2\pi } \big [g(v_*-\frac{1}{2}A^\epsilon )h(v+\frac{1}{2}A^\epsilon )-g(v_*)h(v)\big ]\Psi (\chi )|v-v_*|^\gamma d\varphi d\chi dv_*. \end{aligned}$$

Here we set

$$\begin{aligned} \Psi (\chi )\mathop {=}\limits ^{\mathrm{def}}K\chi ^{-1-2s}\zeta (\chi ) \quad \text{ if }\quad \gamma >-3, \end{aligned}$$

and

$$\begin{aligned} \Psi (\chi )\mathop {=}\limits ^{\mathrm{def}}\epsilon K\chi ^{-3+2\epsilon }\zeta (\chi ) \quad \text{ if }\quad \gamma =-3. \end{aligned}$$

Then it gives

$$\begin{aligned} Q^\epsilon (g,h)\!=\!\frac{1}{\epsilon ^2}\int \limits _{{\mathbb {R}}^3}\int \limits _0^\frac{\pi }{2}\int \limits _0^{2\pi } \left[ \!g\left( \!v_*-\frac{1}{2}A^\epsilon \!\right) h\left( \!v\!+\!\frac{1}{2}A^\epsilon \!\right) \!-\!g(v_*)h(v)\!\right] \Psi (\chi )|v\!-\!v_*|^\gamma d\varphi d\chi dv_*. \end{aligned}$$

Note that

$$\begin{aligned} g(v_*-\frac{1}{2}A^\epsilon )=g(v_*)-\frac{1}{2}A^\epsilon \nabla _{v_*} g(v_*)+\frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2g(v_*)+r^1(v_*, v) \end{aligned}$$

and

$$\begin{aligned} h(v+\frac{1}{2}A^\epsilon )=h(v)+\frac{1}{2}A^\epsilon \nabla _{v} h(v)+\frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2h(v)+r^2(v_*, v), \end{aligned}$$

where

$$\begin{aligned} |r^1(v_*, v)|&\lesssim |A^\epsilon |^3\int \limits _0^1 |D^3 g|(v_*+\kappa (v_{*}^{\prime }-v_*))d\kappa ,\\ |r^2(v_*, v)|&\lesssim |A^\epsilon |^3\int \limits _0^1 |D^3 h|(v +\iota (v'-v))d\iota . \end{aligned}$$

Then we arrive at

$$\begin{aligned} Q^\epsilon (g,h)&= \frac{1}{\epsilon ^2}\int \limits _{{\mathbb {R}}^3}\int \limits _0^\frac{\pi }{2}\int \limits _0^{2\pi } \big [\frac{1}{2}A^\epsilon (\nabla _{v}-\nabla _{v_*})(g(v_*)h(v))+\frac{1}{8}A^\epsilon \otimes A^\epsilon :\\&\qquad (\nabla -\nabla _{v_*})^2(g(v_*)h(v))+R^1(v,v_*)\big ]\Psi (\chi )|v-v_*|^\gamma d\varphi d\chi dv_*, \end{aligned}$$

where

$$\begin{aligned} R^1(v,v_*)&= r^1(v_*, v)\big (h(v)+\frac{1}{2}A^\epsilon \nabla h(v)+\frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2h(v)+r^2(v_*, v) \big )\\&+\frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2g(v_*)\left( \frac{1}{2}A^\epsilon \nabla h(v)+\frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2h(v)+r^2(v_*, v) \right) \\&-\frac{1}{2}A^\epsilon \nabla _{v_*} g(v_*)\left( \frac{1}{8}A^\epsilon \otimes A^\epsilon :D^2h(v)+r^2(v_*, v) \right) +g(v_*)r^2(v_*, v). \end{aligned}$$

Suppose that

$$\begin{aligned}&T^\epsilon (v-v_*) \mathop {=}\limits ^{\mathrm{def}}\int \limits _{\chi ,\varphi } \frac{1}{2}A^\epsilon \Psi (\chi )|v-v_*|^\gamma d\varphi d\chi ,\\&\text{ and } \quad U^\epsilon (v-v_*) \mathop {=}\limits ^{\mathrm{def}}\int \limits _{\chi ,\varphi } \frac{1}{8}A^\epsilon \otimes A^\epsilon \Psi (\chi )|v-v_*|^\gamma d\varphi d\chi , \end{aligned}$$

then we have

$$\begin{aligned}&Q^\epsilon (g,h)=\frac{1}{\epsilon ^2}\int \limits _{{\mathbb {R}}^3}\big [T^\epsilon (v-v_*)(\nabla _{v}-\nabla _{v_*})(g(v_*)h(v))+U^\epsilon (v-v_*): \nonumber \\&\quad (\nabla -\nabla _{v_*})^2(g(v_*)h(v))dv_*+\frac{1}{\epsilon ^2}\int \limits _{{\mathbb {R}}^3}\int \limits _0^\frac{\pi }{2}\int \limits _0^{2\pi } R^1(v,v_*) \Psi (\chi )|v-v_*|^\gamma d\varphi d\chi dv_*.\qquad \end{aligned}$$

(82)

Observe that

$$\begin{aligned} T^\epsilon (v-v_*)=-\epsilon ^24\Lambda |v-v_*|^\gamma (v-v_*)+R^2(v,v_*), \end{aligned}$$

with

$$\begin{aligned} R^2(v,v_*)=-\pi \bigg [\int \limits _0^\frac{\pi }{2} (1-\cos (\epsilon \chi )-\frac{1}{2}(\epsilon \chi )^2)\Psi (\chi )d\chi \bigg ]|v-v_*|^\gamma (v-v_*), \end{aligned}$$

if $\gamma >-3$ and

$$\begin{aligned} R^2(v,v_*)&= -\pi \bigg [\int \limits _0^\frac{\pi }{2} (1-\cos (\epsilon \chi )-\frac{1}{2}(\epsilon \chi )^2)\Psi (\chi )d\chi \bigg ]|v-v_*|^\gamma (v-v_*)\\&+\,\frac{\pi }{2}\epsilon ^2|v-v_*|^\gamma (v-v_*)\big (\int \limits _0^\frac{\pi }{2} \Psi (\chi )\chi ^2d\chi -\frac{1}{2}K\big ), \end{aligned}$$

if $\gamma =-3$.

We also notice that

$$\begin{aligned} U^\epsilon (v-v_*)=\epsilon ^2a(v-v_*)+R^3(v,v_*), \end{aligned}$$

(83)

where

$$\begin{aligned} a(v-v_*)= \Lambda |v-v_*|^\gamma \big [|v-v_*|^2Id-(v-v_*)\otimes (v-v_*)\big ] \end{aligned}$$

and

$$\begin{aligned} R^3(v,v_*)&= |v-v_*|^\gamma (v-v_*)\otimes (v-v_*)\int \limits _{0}^{\frac{\pi }{2}}[1-\cos (\epsilon \chi )]^2\Psi (\chi )d\chi \\&+\frac{\pi }{8}|v-v_*|^\gamma \big [|v-v_*|^2Id-(v-v_*)\otimes (v-v_*)\big ]\\&\times \int \limits _{0}^{\frac{\pi }{2}}[ \sin ^2(\epsilon \chi )-(\epsilon \chi )^2]\Psi (\chi )d\chi , \end{aligned}$$

if $\gamma >-3$ and

$$\begin{aligned} R^3(v,v_*)&= |v-v_*|^\gamma (v-v_*)\otimes (v-v_*)\int \limits _{0}^{\frac{\pi }{2}}[1-\cos (\epsilon \chi )]^2\Psi (\chi )d\chi \\&+\,\frac{\pi }{8}|v-v_*|^\gamma \big [|v-v_*|^2Id-(v-v_*)\otimes (v-v_*)\big ]\\&\times \,\bigg [\int \limits _{0}^{\frac{\pi }{2}}\big (\sin ^2(\epsilon \chi )-(\epsilon \chi )^2\big )\Psi (\chi )d\chi +\epsilon ^2\big (\int \limits _0^\frac{\pi }{2} \Psi (\chi )\chi ^2d\chi -\frac{1}{2}K\big )\bigg ], \end{aligned}$$

if $\gamma =-3$.

Thanks to (82) and the fact

$$\begin{aligned} (\nabla -\nabla _{v_*})\cdot U^\epsilon (v,v_*)&= -4\epsilon ^2\Lambda |v-v_*|^\gamma (v-v_*)+(\nabla -\nabla _{v_*})R^3(v,v_*)\\&= T^\epsilon (v,v_*)-R^2(v,v_*)+(\nabla -\nabla _{v_*})R^3(v,v_*), \end{aligned}$$

we have

$$\begin{aligned} Q^\epsilon (g,h)&= \frac{1}{\epsilon ^2}\int \limits _{v_*} (\nabla -\nabla _{v_*})\cdot \big [U^\epsilon (v,v_*)(\nabla -\nabla _{v_*})(g_*h)\big ]dv_*\\&+\,\frac{1}{\epsilon ^2}\int \limits _{v_*}(\nabla -\nabla _{v_*})(g_*h) \cdot \big (-(\nabla -\nabla _{v_*})R^3(v,v_*)+R^2(v,v_*)\big )dv_*\\&+\,\frac{1}{\epsilon ^2}\int \limits _{v_*,\chi ,\varphi } R^1(v,v_*)\Psi (\chi )|v-v_*|^\gamma d\chi d\varphi dv_*. \end{aligned}$$

Noticing (83), we arrive at

$$\begin{aligned} \frac{1}{\epsilon }\big (Q^\epsilon (g,h)-Q_L(g,h)\big )&= \frac{1}{\epsilon ^3}\int \limits _{v_*} R^3(v,v_*):\big [(\nabla ^2-\nabla \nabla _{v_*}) (g_*h)\big ]dv_*\\&+\,\frac{1}{\epsilon ^3}\int \limits _{v_*}(\nabla -\nabla _{v_*})(g_*h) \cdot \big (\nabla _{v_*}R^3(v,v_*)+R^2(v,v_*)\big )dv_*\\&+\,\frac{1}{\epsilon ^3}\int \limits _{v_*,\chi ,\varphi } R^1(v,v_*)\Psi (\chi )|v-v_*|^\gamma d\chi d\varphi dv_*. \end{aligned}$$

We denote the righthand side by $E_1,E_2$ and $E_3$. Note that

$$\begin{aligned} |R^3(v,v_*)|\lesssim \epsilon ^3 |v-v_*|^{\gamma +2} \end{aligned}$$

and

$$\begin{aligned} |\nabla _{v_*} R^3(v,v_*)|+|R^2(v,v_*)|\lesssim \epsilon ^3 |v-v_*|^{\gamma +1}, \end{aligned}$$

one has that

$$\begin{aligned} |\langle E_1, \langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^1_{(\gamma +2)^++2}}\Vert h\Vert _{H^2_{(\gamma +2)^++l}}\Vert \psi \Vert _{L^2} \end{aligned}$$

and

$$\begin{aligned} |\langle E_2, \langle \cdot \rangle ^l \psi \rangle |\lesssim (\Vert g\Vert _{H^1_{(\gamma +1)^++2}}+\Vert g\Vert _{H^2})\Vert h\Vert _{H^2_{(\gamma +1)^++l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

When $\psi$ verifies the condition (81), we have

$$\begin{aligned} |\langle E_1, \psi \rangle |\lesssim \Vert g\Vert _{H^1_{(\gamma +2)^++2}}\Vert h\Vert _{H^2_{(\gamma +2)^++l+2}} \end{aligned}$$

and

$$\begin{aligned} |\langle E_2, \psi \rangle |\lesssim \Vert g\Vert _{H^1_{(\gamma +1)^++2}}\Vert h\Vert _{H^2_{(\gamma +1)^++l+2}}. \end{aligned}$$

Next we shall give the estimate to $E_3$. Due to the definition of $R^1(v,v_*)$ and the fact $|A^\epsilon |\lesssim \epsilon \chi |v-v_*|$, we obtain that

$$\begin{aligned} R^1(v,v_*)&\lesssim \sum _{|\beta _1|,|\beta _2|\le 2} (\epsilon \chi )^3|v-v_*|^3\langle v-v_*\rangle ^3 \bigg [|D^{\beta _1}_{v_*} g||D^{\beta _2}_{v} h|\\&+|D^{\beta _1}_{v_*} g|\int \limits _0^1|D^3h|(v+(v'-v)\iota )d\iota + |D^{\beta _2}_{v} h|\int \limits _0^1|D^3g|(v_*+(v_{*}^{\prime }-v_*)\kappa )d\kappa \\&+ \int \limits _0^1\int \limits _0^1|D^3g|(v_*+(v_{*}^{\prime }-v_*)\kappa )|D^3h|(v+(v'-v)\iota )d\kappa d\iota \bigg ]\mathop {=}\limits ^{\mathrm{def}}\sum _iR^1_i. \end{aligned}$$

From which, we deduce that $E_3$ can be bounded by the sum of $E_3^i$ with $i=1,2,3,4$ defined by

$$\begin{aligned} E_3^i\mathop {=}\limits ^{\mathrm{def}}\frac{1}{\epsilon ^3}\int \limits _{v_*,\chi ,\varphi } R^1_i(v,v_*)\Psi (\chi )|v-v_*|^\gamma d\chi d\varphi dv_*. \end{aligned}$$

It is easy to check that

$$\begin{aligned} |\langle E^1_3, \langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^2_{\gamma +8}}\Vert h\Vert _{H^2_{\gamma +6+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

And when $\psi$ verifies the condition (81), we have

$$\begin{aligned} |\langle E^1_3, \psi \rangle |\lesssim \Vert g\Vert _{H^2_{\gamma +8}}\Vert h\Vert _{H^2_{\gamma +8+l}}. \end{aligned}$$

Noting the fact

$$\begin{aligned} \langle v\rangle ^l\lesssim \langle v_*\rangle ^l+\langle v+(v'-v)\iota \rangle ^l, \end{aligned}$$

for $\iota \in [0,1]$ and by the change of variable: $v\rightarrow u= v+(v'-v)\iota$, one has

$$\begin{aligned} |\langle E^2_3,\langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^2_{\gamma +8+l}}\Vert h\Vert _{H^3_{\gamma +6+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

When $\psi$ verifies the condition (81), we have

$$\begin{aligned} |\langle E^2_3, \psi \rangle |\lesssim \Vert g\Vert _{H^2_{\gamma +8+l}}\Vert h\Vert _{H^3_{\gamma +8+l}}. \end{aligned}$$

The similar result can be obtained for $|\langle E^3_3,\langle \cdot \rangle ^l\psi \rangle |$, that is,

$$\begin{aligned} |\langle E^3_3,\langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^3_{\gamma +8+l}}\Vert h\Vert _{H^2_{\gamma +6+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

When $\psi$ verifies the condition (81), we have

$$\begin{aligned} |\langle E^3_3, \psi \rangle |\lesssim \Vert g\Vert _{H^3_{\gamma +8}}\Vert h\Vert _{H^2_{\gamma +8+l}}. \end{aligned}$$

Set $\iota (v)=v+(v'-v)\iota$ and $\kappa (v_*)=v_*+(v_{*}^{\prime }-v_*)\kappa$. Then one has

$$\begin{aligned}&|\langle E_3^4, \langle \cdot \rangle ^l\psi \rangle |\\&\quad \lesssim \int \limits _{v,v_*,\chi ,\varphi ,\kappa ,\iota } \langle v-v_*\rangle ^{6+\gamma }|D^3g|(\kappa (v_*))|D^3h|(\iota (v))\langle v\rangle ^l|\psi (v)|\Psi (\chi )\chi ^3 d\chi d\varphi d\kappa d\iota dvdv_*\\&\quad \lesssim \Vert h\Vert _{H^5_{\gamma +8+l}}\int \limits _{v,u} \langle v-u\rangle ^{-2}|D^3g|(u)\langle u\rangle ^{\gamma +8+l}|\psi (v)| dvdu\\&\quad \lesssim \Vert g\Vert _{H^3_{\gamma +10+l}}\Vert h\Vert _{H^5_{\gamma +8+l}}\Vert \psi \Vert _{L^2}, \end{aligned}$$

where we use the change of variable: $v_*\rightarrow u=\kappa (v_*)$ and the facts

$$\begin{aligned} \langle \kappa (v_*)\rangle ^{-1}\langle \iota (v)\rangle ^{-1} \lesssim \langle v-v_*\rangle ^{-1} \end{aligned}$$

and

$$\begin{aligned} \langle v\rangle \lesssim \langle \kappa (v_*)\rangle +\langle \iota (v)\rangle . \end{aligned}$$

We also note

$$\begin{aligned}&|\langle E_3^4, \langle \cdot \rangle ^l\psi \rangle |\\&\quad \lesssim \Vert g\Vert _{H^5_{\gamma +10+l}}\int \limits _{v,v_*,\chi ,\varphi ,\kappa ,\iota } \langle v-v_*\rangle ^{-4}(|D^3h|(\iota (v))\langle \iota (v)\rangle ^{\gamma +10+l})|\psi (v)|\Psi (\chi )\chi ^3d\chi d\varphi d\kappa d\iota dvdv_*\\&\quad \lesssim \Vert g\Vert _{H^5_{\gamma +10+l}}\Vert h\Vert _{H^3_{\gamma +10+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

The similar argument can be applied to the case that $\psi$ verifies the condition (81). One has

$$\begin{aligned} |\langle E_3^4, \psi \rangle |&\lesssim \int \limits _{v,v_*,\chi ,\varphi ,\kappa ,\iota } \langle v-v_*\rangle ^{6+\gamma }|D^3g|(\kappa (v_*))|D^3h|(\iota (v))\langle \cdot \rangle ^l\Psi (\chi )\chi ^3d\chi d\varphi d\kappa d\iota dvdv_*\\&\lesssim \Vert h\Vert _{H^5_{\gamma +10+l}}\int \limits _{v,u} \langle v-u\rangle ^{-4}|D^3g|(u)\langle u\rangle ^{\gamma +10+l} dvdu\\&\lesssim \Vert g\Vert _{H^3_{\gamma +12+l}}\Vert h\Vert _{H^5_{\gamma +10+l}}. \end{aligned}$$

Collecting the above estimates, we may arrive at

$$\begin{aligned} |\langle E_3, \langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^3_{\gamma +10+l}}\Vert h\Vert _{H^5_{\gamma +10+l}}\Vert \psi \Vert _{L^2} \end{aligned}$$

or

$$\begin{aligned} |\langle E_3, \langle \cdot \rangle ^l\psi \rangle |\lesssim \Vert g\Vert _{H^5_{\gamma +10+l}}\Vert h\Vert _{H^3_{\gamma +10+l}}\Vert \psi \Vert _{L^2}. \end{aligned}$$

When $\psi$ verifies the condition (81), one has

$$\begin{aligned} |\langle E_3, \psi \rangle |\lesssim \Vert g\Vert _{H^3_{\gamma +12+l}}\Vert h\Vert _{H^5_{\gamma +10+l}}. \end{aligned}$$

Patch together the estimates for $E_1, E_2$ and $E_3$, then we finally obtain the desired results. $\square$