Abstract
In this note we study Boltzmann’s collision kernel for inverse power law interactions \(U_s(r)=1/r^{s1}\) for \(s>2\) in dimension \( d=3 \). We prove the limit of the noncutoff kernel to the hardsphere kernel and give precise asymptotic formulas of the singular layer near \(\theta \simeq 0\) in the limit \( s\rightarrow \infty \). Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.
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1 Introduction
The Boltzmann equation reads as
where \(f=f(t,x,v)\) is the velocity distribution of particles with position \(x\in \Omega \subset \mathbb {R}^3\) and velocity \(v\in \mathbb {R}^3\) at time \(t\in [0,\infty )\).
The equation has been considered as a fundamental model for the collisional gases that interact either under the hardsphere potential \(U(r)=\infty \) for \(r\le 2\epsilon \) and \(=0\) for \(r\ge 2\epsilon \), or under the longrange potential \(U_s(r)\simeq \frac{1}{r^{s1}}\) for \(s>2\). Here \(\epsilon \) is the radius of each hardsphere. The prototype of the model was suggested by Maxwell [1, 2] and Boltzmann [3].
In this note we consider the particular case of inverse power law interactions \( U_s(r)= 1/r^{s1} \) leading to noncutoff kernels [cf. formula (3)]
Here, \( b_s \) is the socalled angular part. We prove that the function \( B_s \) converges to the hardsphere kernel in the limit \( s\rightarrow \infty \). We give a precise study of the singularity as \( \theta \rightarrow 0 \) when \( s\rightarrow \infty \). Finally, we show that solutions to the homogeneous Boltzmann equation with collision kernel \( B_s \) converge to the solution to the equation for hardspheres. Such a limit result was suggested to exist in [4, Remark 1.0.1].
1.1 Boltzmann Collision Operator
The Boltzmann collision operator Q takes the form
where we used the standard notation \( f'=f(v'), \, f'_*= f(v'_*),\, f_*=f(v_*) \). Also \( (v',v'_*) \) are the postcollisional velocities and \( (v,v_*) \) the precollisional velocities. The function B is Boltzmann’s collision kernel and strongly depends on the microscopic interaction of two particles in the course of a collision. It only depends on the length of relative velocities \( vv_* \) and the socalled deviation angle \( \theta \in [0,\pi ] \) through \( n\cdot \sigma = \cos \theta \).
It is customary to distinguish two main classes of kernels, namely angular cutoff and noncutoff kernels. This refers to a possible singularity of the kernel when \( \theta \rightarrow 0 \). Such deviation angles correspond to grazing collisions, i.e. collisions such that \( v\approx v' \). They appear only for longrange or weak interactions.
1.2 Derivation of Boltzmann’s Collision Kernel for LongRange Interactions
Let us give here a derivation of the collision kernel for inverse power law interactions. We consider the collision of two particles (x, v) , \( (x_*,v_*) \) with equal mass \( m=1 \). Due to conservation of momentum and conservation of energy, both \( v_c=(v+v_*)/2 \) and \( vv_* \) are conserved. Here, \( v_c \) is the velocity of the center of mass \( x_c=(x+x_*)/2 \). It is convenient to use the coordinate system \( (\bar{x}, \bar{v})=(xx_*,vv_*) \), in which the center of mass is zero and at rest. In this coordinate system, the velocities after the collision have equal lengths but opposite directions due to the conservation of momentum and energy. Hence, they are given by \( \bar{v}\sigma /2 \) and \( \bar{v}\sigma /2 \), respectively, for \( \sigma \in S^2 \). In the original coordinate system, we thus get
In order to derive the distribution of \( \sigma \) in the scattering problem, we need to consider the interaction of both particles via the potential U. As is wellknown we can reduce it to a single particle problem in the center of mass coordinate system \( (\bar{x}, \bar{v}) \) with (reduced) mass \( \mu =1/2 \), see e.g. [5, Section 13]. The motion is planar and we can use polar coordinates. The Hamiltonian reads,
where \( \dot{r} \), \( \dot{\varphi } \) denote the velocity variables (i.e., derivatives with respect to the time variable t) corresponding to r, \( \varphi \), respectively. Both energy \( E= H(r,\varphi ,\dot{r}, \dot{\varphi }) \) and angular momentum \( L=\mu r^2\dot{\varphi } \) are conserved.
For the collision process we consider the particle \( (\bar{x},\bar{v})(t) \) passing the center of the potential with asymptotic velocity \( vv_* \) as \( t\rightarrow \infty \), \( r\rightarrow \infty \). The particle is scattered and moves away from the center with asymptotic velocity \( v'v'_*\) as \( t\rightarrow \infty \), \( r\rightarrow \infty \). The turning point (\( \dot{r}=0 \)) is given at distance \( r_m \), which is the largest root of
We can determine E and L by considering the asymptotic value \( t\rightarrow \infty \). This yields
where \( \psi \) is the angle between \( \bar{x} \) and \( \bar{v} \). Furthermore, \( \rho \) is the impact parameter, which is the distance of the closest approach if the particle is passing the center without the presence of an interaction, see Fig. . The formula for L can be obtained by a geometric argument.
The solution to the above problem is implicitly given by, see e.g. [5, Section 14],
In the limit \( t\rightarrow \infty \) the angle \( \varphi \) is zero. By a symmetry argument, one can see that the angle \( \varphi _0 \) of the line through the center and the point of closest approach is given by (see Fig. 1)
Now, we plug in the values for \( E, \, L \) and use the change of variables \( y=\rho /r_* \). Furthermore, we use \( U(r)=r^{(s1)} \) and define \( \beta = \rho (vv_*/2)^{2/(s1)} \) to get, cf. [6, page 69–71],
The deviation angle is given by \( \theta = \pi 2\varphi _0 \) for a given impact parameter \( \rho \).
The number of particles scattered with deviation angle close to \( \theta \) is proportional to \( vv_* \) and the corresponding crosssection, that is \( 2\pi \rho d\rho = 2\pi \rho (\theta )\rho '(\theta )\, d\theta \). Changing to the variable \( \beta \) and integrating via the solid angle yields the formula
Let us note that \( \beta '(\theta )>0 \). This completes the formal derivation of the Boltzmann collision operator for the longrange interactions.
1.3 Outline of the Article
We now provide a brief outline of the rest of the article. In Sect. 2, we give a proof of the limit of the noncutoff kernel to the hardsphere kernel as \(s\rightarrow \infty \). Then in Sect. 3, we study the asymptotics of the singular layer near \(\theta \simeq 0\) as \(s\rightarrow \infty \). Finally, in Sect. 4, we prove the convergence of the solution to the spatially homogeneous Boltzmann equation without angular cutoff to the solution to the hardsphere Boltzmann equation as \(s\rightarrow \infty \).
2 Limit of the Noncutoff Collision Kernel
In this section, we study the limit of the kernel (3) as \(s\rightarrow \infty \). Our first result contains the limit of the kernel as \(s\rightarrow \infty \) as well as some uniform estimates. These estimates together with the ones in Sect. 3 play a crucial role for the proof of the rigorous limit of a weak solution to the spatially homogeneous Boltzmann equation without angular cutoff to the one for the hardsphere interaction, see Sect. 4.
Theorem 1
Let us define the angular part of the collision kernel via

(i)
We have as \( s\rightarrow \infty \)
$$\begin{aligned} b_s(\cos \theta ) \rightarrow \dfrac{1}{4} \end{aligned}$$locally uniformly for \( \theta \in (0,\pi ] \).

(ii)
The following asymptotics holds
$$\begin{aligned} \lim _{\theta \rightarrow 0} \theta ^{1+2/(s1)} \, b_s(\cos \theta ) \sin \theta = C_s, \quad C_s := \dfrac{2^{4/(s1)}}{s1} \left( \dfrac{\sqrt{\pi } \Gamma \left( \frac{s}{2}\right) }{\Gamma \left( \frac{s1}{2}\right) }\right) ^{2/(s1)}. \end{aligned}$$ 
(iii)
Finally, we have the uniform bound
$$\begin{aligned} \sup _{s\ge 3}\, \sup _{\theta \in (0,\pi ]} \theta ^{1+2/(s1)} \, b_s(\cos \theta ) \sin \theta <\infty . \end{aligned}$$
Remark 2
Note that in (i) the limiting collision kernel corresponds to hardsphere interactions. Writing the kernel (3) in terms of the angle \( \varphi = (\pi \theta )/2 \) we get \( vv_*\cos \varphi \, \mathbb {I}_{\cos \varphi \ge 0} \) as \( s\rightarrow \infty \).
Furthermore, in (ii) we have \( C_s\rightarrow 0 \) as \( s\rightarrow \infty \). In fact,
where \( W_{s1} \) is the Wallis integral. It is known that \(\lim _{s\rightarrow \infty } \sqrt{s}W_{s1}=\sqrt{\pi /2} \).
Finally, compare (iii) with [5, Section 20].
2.1 Rearrangement of the Deviation Angle
It is convenient to rearrange (2)
Here, we dropped the index zero in \( \varphi _0, \, x_0 \), used the change of variables \( z=y/x_0 \) and the fact that \( x_0=x \) is the positive root of
We recall that the deviation angle \( \theta = \pi 2\varphi \). One can see that the mappings \( \beta \mapsto x, \, x\mapsto \varphi \) are strictly increasing and real analytic functions \( [0,\infty )\rightarrow [0,1)\rightarrow [0,\pi /2) \) for each \( s\ge 2 \). We will use the index s to indicate that we consider the variable as a function.
2.2 Proof of Theorem 1
Proof of Theorem 1 (i)
We first study the function \( \varphi _s(x) \). The integrand can be written
Here, we used that
This yields for any \( x\in \mathbb {C}\) with \( x\in [0,1\varepsilon ] \), \( \varepsilon >0 \) a uniform majorant, entailing locally uniform convergence,
As a consequence of the analyticity we have (\( x_s \) is the inverse of \( \varphi _s \))
locally uniformly for \( \varphi \in [0,\pi /2) \).
Next, we look at the functions [see (5)]
Hence, we have the locally uniform convergence for \( x\in [0,1) \) as \( s\rightarrow \infty \)
We conclude with the above analysis
locally uniformly for \( \theta \in (0,\pi ] \) as \( s\rightarrow \infty \). Notice that \( \varphi =(\pi \theta )/2 \) and the extra factor 1/2 results from \( d\varphi /d\theta =1/2 \).
Proof of Theorem 1 (ii)
We have the following equalities for \( \varphi \in [0,\pi /2) \) and some \( \psi \in (\varphi ,\pi /2) \)
Combining them yields
Let us note that
and as a consequence we have
Using a similar expression as in (6) we get the asserted asymptotics.
Proof of Theorem 1 (iii)
For the last estimate we use (7). Note that \( \varphi _s' \) is increasing for \( s\ge 3 \), so that
Note that
The last inequality follows from the fact that \( s\mapsto \varphi _s'(0) \) is a decreasing function and \( \varphi _s'(0)\rightarrow 1 \) as \( s\rightarrow \infty \). This implies \( x_s' \le 1 \). Using (7) for \( x\in [0,1) \) we obtain
Since \( \varphi _s' \) is increasing for \( s\ge 3 \) we have
We then obtain with the previous estimates
One can see that
for some constant \( c>0 \). All in all, the right hand side in (8) is uniformly bounded in \( s\ge 3 \) and \( \varphi \in [0,\pi /2] \). This implies the uniform bound. \(\square \)
This completes the proof of the limit of the noncutoff collision kernel to the hardsphere kernel. In the next section, we further study the behavior of \(b_s\) for \( \theta \rightarrow 0 \) when \(s\rightarrow \infty \).
3 Asymptotics of the Noncutoff Collision Kernel
We now study the asymptotics of the singular layer of \( b_s(\cos \theta ) \) near \( \theta \simeq 0 \) when \( s\rightarrow \infty \). To this end, we note that Theorem 1 (ii) in combination with Remark 2 yields
Thus, we need to look at the scaled function
with \( \theta = \psi /\sqrt{s} \). In the following, we use this scaling to compute the limit \( s\rightarrow \infty \). First, we derive a similar formula to (4). Note that
Let us define
where \( \xi _s \) is defined for \( \psi \in [0,\pi \sqrt{s}] \). The inverse function for \( \xi \in [0,2s] \) is given by
Notice that in the last equality we used the definition of \( \varphi _s \) in (4). Note that \( \psi _s \) is an analytic function on (0, 2s) . With this we can state the asymptotic behavior.
Theorem 3
The angular part \( b_s(\cos \theta ) \), \( s\ge 2 \), satisfies the following asymptotic limit
which holds locally uniformly for \( \psi \in (0,\infty ) \). Here, \( \Phi :(0,\infty ) \rightarrow \mathbb {R}\) is real analytic satisfying
Furthermore, we have
where \( \Phi _0: [0,\infty )\rightarrow \mathbb {R}\) is continuous.
Remark 4
Note that the singularity \( 1/\psi ^2 \) of \( \Phi \) for \( \psi \rightarrow 0 \) is consistent with the asymptotics in Theorem 1 (ii), since \( s C_s \rightarrow 1 \) as \( s\rightarrow \infty \). Furthermore, the result of the limit \( \psi \rightarrow \infty \) coincides with Theorem 1 (i).
Proof of Theorem 3
The proof consists of the following four steps.
Step 1 We first derive the limits
where
To this end we choose \( \xi \in [0,\infty ) \) and assume s large enough such that \( \xi \in [0,2s] \). Let us write
Since \( g_s\ge 1z^2 \) the second integral in (10) goes to zero as \( s\rightarrow \infty \). The first term in (10) can be rearranged to get
We now perform the change of variables \( z=1\zeta /s \) to get with
and the formula
Using that \( \zeta \le s \) and \( \xi \le 2\,s \) we can obtain
and
Hence, we have \(h_s(\zeta ,\xi ) \ge \zeta .\) In addition, we also have
Thus, the integrand in (15) can be estimated by
In conjunction with
we conclude the locally uniform convergence
where \( \psi _\infty \) is given in (13). Since the above estimates also hold in a neighborhood of \( \xi \in (0,\infty ) \) in the complex plane, the limit is real analytic. A calculation allows to derive the formula (14). Alternatively, one can compute the derivative of (10) and mimic the preceding computation.
Step 2 Since \( \psi _\infty '>0 \) we also have from the analyticity and the locally uniform convergence
locally uniformly for \( \psi \in (0,\infty ) \). Furthermore, by (9)
This yields with the definition of \( b_s(\cos (\psi /s)) \), cf. (6) and formulas (7),
Using a Taylor expansion we can replace \( \sin (\psi /\sqrt{s}) \) by \( \psi /\sqrt{s} \) without modifying the value of the limit. We use (9) and
which is a consequence of (9), to obtain
Step 3 We now use a Taylor approximation for (16). It is convenient to define
Here, \( J(\xi ) \) is the integral in (13). This yields
We then have
which defines \( \Phi _0 \). The following formulas hold
With this we derive
which yields the expression in (12).
The formulas (17) can be calculated without difficulty, since the integrals are welldefined. For instance,
Step 4 Finally, for the limit in (11) we have with (16)
We prove below that
which implies the assertion. For the preceding two limits we use the change of variables \( \zeta =\xi z \) to get
The integrand can be estimated by (we use here \( \xi \ge 1 \) say)
Hence, we can use the dominated convergence theorem to obtain the stated limit. A similar computation applies to \( \sqrt{\xi } J(\xi ) \). This concludes the proof. \(\square \)
This completes the proof of the asymptotics of the singularity for \(\theta \simeq 0\) as \(s\rightarrow \infty \). In the next section, we provide a proof of the limit of solutions to the spatially homogeneous Boltzmann equation without cutoff to solutions of the homogeneous Boltzmann equation for hardspheres using the estimates in Sects. 2 and 3.
4 Convergence of the Solution for the Homogeneous Boltzmann Equation
In this section, we consider the spatially homogeneous Boltzmann equation
with collision kernel \( B_s(vv_*, n\cdot \sigma ) \), \( s>2 \), given in (3). Let us first recall the following wellposedness result for cutoff kernels with hard potentials \( \gamma \in (0,1] \) (e.g. hardsphere corresponding to \( s=\infty \)), see [7, Theorem 1.1] and [8, Section 3.7, Theorem 3]. The first wellposedness results are due to Arkeryd [9, 10]. We use here the weighted spaces \( L^1_p \) with weight function \( (1+v^2)^{p/2} \).
Lemma 5
Let \( f_0\in L^1_2 \), then there is a unique solution \( f\in C([0,\infty ); L^1_2) \) to (18) which preserves energy, i.e. for all \( t\ge 0 \)
Remark 6
Let us mention that the condition of the energy conservation is essential for uniqueness [11, 12].
Next, we consider the noncutoff kernel \( B_s \). Since we are interested in the limit \( s\rightarrow \infty \), we can assume \( s>5 \) so that
where the constant \( c_0 \) is independent of \( s>5 \), see Theorem 1 (iii). In this case, we can use the weak formulation of (18) by testing with functions \( \psi \in C^1_b( [0,\infty )\times \mathbb {R}^3) \), see e.g. [8, Section 4.1]. The collision operator can be defined by means of the prepostcollisional change of variables
For the integral on the sphere we have, via a Taylor approximation,
for some constant \( C_0>0 \) independent of \( s>5 \). Let us also define the entropy of f
We also recall the existence of weak solutions to the homogeneous Boltzmann equation, which is the content of the following lemma, see e.g. [13, Section 4] and [8, Section 4.7, Theorem 9 (ii)]. With a slight abuse of notation we write \(f^s(t,v)\) and \(f^\infty (t,v)\) to describe the solutions to the Boltzmann equations with kernels \(B_s\) and \(B_\infty \), respectively.
Lemma 7
Let \( f_0\in L^1_{1+\gamma +\delta } \), for \( \delta >0 \) arbitrary, with finite entropy. Under the conditions (19) there is a weak solution \( f^s\in L^\infty ([0,\infty ); L^1_{1+\gamma +\delta }) \) to (18) which preserves energy. Furthermore, we have \( H(f^s(t))\le H(f_0) \) for all \( t\ge 0 \).
We finally have the following convergence result.
Theorem 8
Let \( f_0\in L^1_p \) with finite entropy and arbitrary \( p>2 \). Consider a sequence of weak solutions \( f^s \) to (18) as in Lemma 7 with collision kernel \( B_s \), \( s>5 \). Then, \( f^s(t)\rightharpoonup f^\infty (t) \) weakly in \( L^1 \) for all \( t\ge 0 \) as \( s\rightarrow \infty \), where \( f^\infty \) is the unique solution to (18) for hardsphere interactions.
Proof of Theorem 8
First of all, applying a version of the Povzner estimate (see e.g. [7, Lemma 2.2] which is also applicable for noncutoff kernels, cf. [8, Appendix]) we have
This estimate is independent of s as long as s is sufficiently large. Assume for example \( s>6 \). In fact, in the Povzner estimate we only need a uniform lower and upper bound on the angular part \( b_s(\cos \theta ) \). This is ensured by Theorem 1 items (i) and (iii). Also note that for, say, \( s>6 \) we have \( \gamma (s)\ge 1/5 \). Furthermore, from the weak formulation we also obtain
for all \( t_1,\, t_2\ge 0 \). Here, the constant C is independent of \( s>6 \) due to (19) and (20). By the uniform entropy bound
and the previous weak equicontinuity property we can apply the DunfordPettis theorem yielding
weakly in \( L^1 \) for all \( t\ge 0 \) for a subsequence \( s_n\rightarrow \infty \).
Using Theorem 1, items (i) and (iii), we can pass to the limit in the weak formulation. Hence, \( f^\infty \) is a weak solution to (18) for hardsphere interactions. Since there is no angular singularity, one can infer
By the uniform moment bound (20), the second moments also converge for all \( t\ge 0 \) as \( s_n\rightarrow \infty \). As a consequence \( f^\infty \) preserves energy and thus \( f^\infty \) is the unique solution in Lemma 5. This implies that the whole sequence converges \( f^s(t)\rightharpoonup f^\infty (t) \) as \( s\rightarrow \infty \). \(\square \)
5 Conclusion
We proved the convergence of the collision kernel for inverse power law interactions \( 1/r^{s1} \) to the hardsphere kernel as \( s\rightarrow \infty \). We furthermore studied the asymptotics of the angular singularity \( \theta \rightarrow 0 \). Finally, solutions to the homogeneous Boltzmann equation converge respectively.
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Acknowledgements
The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the collaborative research centre The mathematics of emerging effects (CRC 1060, ProjectID 211504053). J. W. Jang is supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIT) NRF2022R1G1A1009044 and by the Basic Science Research Institute Fund of Korea NRF2021R1A6A1A10042944. B. Kepka is funded by the Bonn International Graduate School of Mathematics at the Hausdorff Center for Mathematics (EXC 2047/1, ProjectID 390685813). J. J. L. Velázquez is also funded by DFG under Germany’s Excellence StrategyEXC2047/1390685813.
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Jang, J.W., Kepka, B., Nota, A. et al. Vanishing Angular Singularity Limit to the HardSphere Boltzmann Equation. J Stat Phys 190, 77 (2023). https://doi.org/10.1007/s10955023030894
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DOI: https://doi.org/10.1007/s10955023030894