Abstract
This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the \(L^\infty \)-setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in \(L^2 \cap L^\infty \) with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted \(L^2\cap L^\infty \)-space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable \(L^p\)-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in Alonso et al. (Rev Mat Iberoam, 2020), we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both \(L^2\) and \(L^\infty \)-spaces.
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1 Introduction
1.1 Setup and Objective
We consider in this paper the nonlinear Boltzmann equation
Solutions to this equation \(F=F(t,x,v)\ge 0\) are the mass density distribution of particles at a time-space point \((t,x)\in (0,\infty )\times \mathbb {T}^{3}\) with velocity \(v\in {{\mathbb {R}}}^{3}\). The equation is supplemented with the initial condition
The nonlinear operator Q(F, F) stands for the collision operator. It is defined by the integral formula
where the abbreviated notations are
and \((v, v_*)\) and \((v', v'_*)\) are the two pairs of velocity before and after the collision or vice versa. In the elastic collision case that we consider in this work, these velocities satisfy the conservation of momentum and energy during the collision process:
By introducing the parameter \(\sigma \) in \({\mathbb {S}}^2\), the scattering direction, one can write \((v', v'_*)\) in terms of \((v, v_*)\) as
In this paper we treat the hard potential case with the collision kernel taking the form
Following a convention, we assume without loss of generality that \(b(\cos \theta )\) is supported on \(\cos \theta \ge 0\). This is valid due to the structure of the collision operator. We consider the so-called non-cutoff kernels satisfying
Away from the region of grazing interactions \(\theta =0\), the scattering kernel b is assumed to be integrable in \(\mathbb {S}^{2}\).
The regime close to equilibrium is considered in this work as we seek solutions of the form
In such a situation, the unknown f satisfies the nonlinear Boltzmann equation
where \(\mathcal {L}\) stands for the linear operator
Physical solutions satisfy the laws of mass, momentum, and energy conservation, which translate to
for all \(t\ge 0\).
The goal of this work is to show the existence of solutions to the Boltzmann equation for any initial data \(f_0\) satisfying \((1 + |v|^2)^{k_0}f_0 \in L^{2}_{x,v}\cap L^{\infty }_{x,v}\) in the perturbative framework. We note that since the well-established work of constructing near-equilibrium solutions [3,4,5,6, 26] in the case of \(\mu ^{-1/2}f_0 \in L^{2}_{v}H^{2}_{x}\), a lot of efforts have been made to lower the regularity requirement on the initial data in seeking global solutions (cf. [20, 22, 40] and the references therein). To the best of our knowledge, our work is the first to obtain a global solution in the \(L^\infty \)-setting for the non-cutoff Boltzmann, thus adding a missing link to the studies of the global existence of solutions to the nonlinear Boltzmann equations.
1.2 Significance and Main Result
The problem of constructing solutions to the Boltzmann equation with initial data having minimal spatial regularity has been highly appealing to the community in both cutoff and without cutoff contexts of the Boltzmann equation. It is desirable to create mathematical tools that can deal with singularity creation/propagation since they may connect to the physical phenomena of shock formation and/or attenuation on the macroscopic scale.
For the cutoff Boltzmann equations, the development of the well-posedness theory for solutions near equilibrium in the \(L^\infty \)-framework can be traced back to Grad [25] for local existence and later by Ukai [44] for global existence under the Grad’s angular cutoff assumption. Ukai’s theory relies on the spectral analysis [23] of the linearized Boltzmann operator and a bootstrap argument. In addition to the \(L^\infty \)-framework, the well-posedness theory for the cutoff Boltzmann has also been well developed in other settings. For example, in the \(L^1\)-setting the classical theory on the renormalized solution was established by DiPerna-Lions in their seminal work [19] by making essential use of the famous H-theorem and the velocity averaging lemma. The \(L^2\)-framework based on energy methods has also been extensively explored ( [28, 38, 39]). Furthermore, an \(L^2-L^{\infty }\) interplay method has been introduced in [29, 45] and applied to various contexts (see [30] and the references therein) to obtain solutions with low regularity and close to equilibrium. Earlier theory on solutions near equilibrium has been focused on perturbations with Gaussian tails. More recently, a big step forward is made in [27] where the authors introduced a new framework of using spectral analysis to relax the velocity decay constraint from Gaussian to polynomial. In the cutoff context a key point for the control of the collision operator is to work in Banach spaces with an “algebraic structure” in the spatial variable.
For Boltzmann equations without angular cutoff, the well-posedness for large data in \(L^1\)-framework was obtained by Alexandre and Villani in [1], and the \(L^2\)-theory for perturbative solutions around an equilibrium was first established in [4,5,6,7, 26]. In [3, 8], the authors considered the space \(L^{2}_{v}H^{\beta }_{x}\) with \(\beta >3/2\) for local existence. The particular range of \(\beta \) seems almost optimal since the main idea is to use the Sobolev embedding \(H^{\beta }_{x}\subseteq L^{\infty }_{x}\) to handle the quadratic nonlinearity of the collision operator. This mimics the idea implemented for the cutoff case through the “algebraic” control \(\Vert f\,g\Vert _{H^{\beta }}\le \Vert f\Vert _{H^{\beta }}\Vert g \Vert _{H^{\beta }}\).
Contrary to the extensive studies in the \(L^2\)-setting, the \(L^\infty \)-theory for the wellposedness of the Boltzmann equation without angular cutoff has remained open. Recently in [20, 22], the authors were able to construct global-in-time solutions in a space based on the Wiener algebra in x with the norm
The key point in [22] is again the “algebraic” control \(\Vert f\,g\Vert _{\mathcal {W}} \le \Vert f\Vert _{\mathcal {W}} \Vert g\Vert _{\mathcal {W}}\). The Wiener algebra setting is more general than the Sobolev spaces \(H^\beta _x\) used in earlier works and is a considerable step toward \(L^\infty _x\)-spaces, but it is still more restrictive. On the other hand, its benefit of being a smaller space than \(L^\infty \) is that coercivity estimates obtained in such setting are strong enough to prove the uniqueness of solutions. In contrast, the method used in our paper is still insufficient to provide the desired uniqueness. We further note that the aforementioned works are in the context of velocity with Gaussian tails, in which spectral and coercivity properties of the collision operator appear naturally. For the recent development on the perturbation with a polynomial decay, one can refer to [12, 27, 31] and the references therein.
The goal of our paper is to give a global existence proof for the non-cutoff Boltzmann equation in the \(L^\infty \)-setting. Instead of following the path of exploring algebraic structures, we apply a different framework based on a De Giorgi argument [17]. The approach of our paper is inspired by the first part of [15] where the quasi-geostrophic equation was studied. In particular, we do not need the full machinery of the De Giorgi–Nash–Moser method but rely mainly on the level-set functions and energy estimates. Such an approach has been applied to the homogeneous Boltzmann equation in [9] and a linear radiative transfer equation in the forward-peak regime [11]. It is also applied to the inhomogeneous Landau equation [36]. Compared with Boltzmann, the Landau operator has a more localized structure which is closer to classical nonlinear parabolic operators. For example, the typical maximum principle argument holds for the Landau equation at least locally in time while it is unclear how this can be directly applied to the non-cutoff Boltzmann equation. As a consequence, application of the level-set method to the inhomogeneous Boltzmann equation is not straightforward.
We comment that the full machinery of the De Giorgi–Nash–Moser method has been used in a series of remarkable developments for the Landau and non-cutoff Boltzmann equations [24, 34,35,36, 41]. More specifically, solutions with bounded macroscopic densities are shown to have instantaneous \(C^\infty \)-regularization. Recently a proof of existence of \(L^\infty \)-solutions near equilibrium is given in [42] by combining such regularization with the long-time asymptotics shown in [18]. The proof in [42] is different in nature to the one presented here, since the former relies on the \(C^\infty \)-regularization and the lower bound of solutions while our solution stays in the weak sense.
With some details of the parameters left out, the main theorem can be summarized as:
Theorem 1.1
Suppose the cross section of the Boltzmann equation satisfies (1.3) with \(\gamma \in (0, 1]\) and \(s \in (0, 1)\) and the initial data \(F_0 \ge 0\) satisfies (1.4). Then for \(k_0, k\) large enough with \(k > k_0\), there exists \(\delta ^\natural _{*} > 0\) such that if
then there exists a unique non-negative solution \(F \in L^\infty (0, \infty ; L^2_x L^2_{k} (\mathbb {T}^3 \times {{\mathbb {R}}}^3))\) to (1.1). Moreover, for some \(\delta _0\) and \(\lambda '>0\), the solution F satisfies
Furthermore, for some \(C_*, {\widetilde{\lambda }} > 0\), the weighted \(L^\infty \)-norm of the perturbation decays exponentially in time:
1.3 Notations
We employ several notations for function spaces in this paper. First, \(L^2_{x,v}\) or \(L^2(\mathbb {T}^3 \times {{\mathbb {R}}}^3)\) denotes the usual \(L^2\)-space over \(\mathbb {T}^3 \times {{\mathbb {R}}}^3\) and \(L^2_x H^s_v(\mathbb {T}^3 \times {{\mathbb {R}}}^3)\) denotes the space where \((I - \Delta _v)^{s/2} f \in L^2_{x,v}\). Any weight in the subindex denotes a weight in v only. For example, \(L^2_x L^2_k\) denotes the space where \(\left\langle v\right\rangle ^k f \in L^2_{x,v}\) where \(\left\langle v\right\rangle \) is the Japanese bracket defined by \(\left\langle v\right\rangle ^2 = 1 + |v|^2\).
There are many parameters in this paper. Among them, we reserve the key ones for designated meanings that will not change throughout this paper:
-
\(\gamma \): power in the hard potential.
-
s: strength of the singularity in the collision kernel.
-
\(s'\): regularity in x (and t) derived from the averaging lemma.
-
\(\gamma _0\): dissipation coefficient in Lemma 2.6.
-
\(c_0\): dissipation coefficient in Proposition 2.5.
-
\(k_0\): moment in the \(L^\infty \)-bound (in t, x, v) of the solution.
-
\(\delta _0\): smallness of the \(L^\infty \)-norm (in t, x) of the solution for the energy estimates to close.
-
\(\epsilon \): strength of the regularizing operator \(\epsilon L_\alpha \) with \(L_\alpha \) defined in (3.2).
-
\({\mathcal {E}}_k\): k-th energy level.
-
\(\ell _0\): minimal order of moments needed for the inhomogeneous embedding in (3.60).
-
\(\lambda _0\): spectral gap of the linearized Boltzmann operator.
Other parameters such as \(p, q, p', k, \beta , \beta ', s'', \ell , \theta , \eta \) may change from statement to statement. Constants denoted by \(C, C_\ell , C_k\) may change from line to line.
Since we assume that the collision kernel \(b(\cos \theta )\) is supported on \(\cos \theta \ge 0\), the integration limits for b can be either \({\mathbb {S}}^2\) or \({\mathbb {S}}^2_+\) and we use them interchangeably.
1.4 Methodology and Organization
A brief outline of the strategy implemented in this paper is as follows: the underlying condition for the validity of the a priori estimates, in addition to sufficient regularity, is the smallness condition
where \(w_0>0\) is a threshold of polynomial decay and \(\delta _0>0\) is a sufficiently small quantity. With this condition, \(L^{2}_{x,v}\) and \(L^{2}_{x}H^{s}_{v}\) energy estimates with general weights in velocity can be proved. The bound in \(L^2_x H^s_v\) demonstrates the natural regularization in the velocity variable reminiscent of a fractional Laplace’s equation. Using a time-localized averaging lemma, one can “complete” the velocity energy estimate of the equation to include the regularization in the spatial variable using the norm \(H^{s'}_{x}L^{2}_{v}\) for some \(s' \in (0,s)\). This confirms the hypoelliptic properties of the equation as expected. The hypoellipticity paves the way to apply the De Giorgi argument through embeddings of Sobolev spaces into various \(L^p\) spaces. In particular, we construct the crucial energy functional
where \(0< s'' < s\) will be suitably chosen, K is any positive number, p is a parameter depending on s and \(f^{(\ell )}_{K, +}\) is the level-set function defined by
With \(K = K_n \rightarrow K_0\) for some \(K_0\) depending on the initial data, the main step in the De Giorgi argument is to show that \({\mathcal {E}}_p(K_n, T_1, T_2)\) satisfies an inhomogeneous (in degree) iterative relation (see (3.81) and (5.27)). This key iterative relation leads to the limit \({\mathcal {E}}_p(K_n, T_1, T_2) \rightarrow 0\) as \(n \rightarrow \infty \), thus proving the weighted \(L^\infty \)-bound. To enforce the smallness condition (1.6) when constructing the approximate solutions, we introduce a cutoff function \(\chi \) and consider the modified collision operator
so that the smallness condition is satisfied naturally for the approximate solutions. Such cutoff function automatically disappears after one applies the De Giorgi method and shows a posteriori that the smallness condition holds intrinsically when the initial data is small enough.
The strategy described above is applied first to the linearized equation and then to the nonlinear equation to obtain local solutions with \(L^\infty \)-bounds to the original Boltzmann equation. A major part of this paper is dedicated to the linear analysis. Although obtaining a solution to the linearized equation is fairly straightforward, significant effort has been carried out to show the \(L^\infty \)-bounds of the solution. A delicate issue is to handle the moments required in various estimates. Interestingly, the moment requirement imposed on solutions for linear and nonlinear estimates drastically differs, with the nonlinear case much easier to handle. The key factor at play is the quadratic structure of the collision operator. This structure reduces the moment needed on solutions for the \(L^{2}_{x,v}\) estimates which is essential for closing the argument. Finally, combining the local existence with the spectral gap we obtain a global solution to the original Boltzmann equation.
This paper is laid out as follows. After including a technical toolbox in Sect. 2, we establish the well-posedness for the linearized Boltzmann equation in Sects. 3 and 4: Sect. 3 consists of a priori \(L^2_{x,v}\) and \(L^2_x H^s_v\)-estimates and \(L^1\)-estimates for the collision term for the linear equation. These estimates are used in Sect. 4 to show the existence of solutions to the linear equation. In Sect. 5 we establish the nonlinear counterparts of the estimates of those in Sect. 3 and apply them to establish the local existence of the nonlinear Boltzmann equation. In Sect. 6 we combine the results in Sect. 5 and the spectral gap property of the linearized Boltzmann operator to establish the global existence of the nonlinear Boltzmann equation. The existence result proved in Sect. 6 is only for the weakly singular kernels. We extend the result to the strong singularity in Sect. 7.
To help the reader better understand the structure of the proof, we show a flow chart of the main steps in Fig. 1. Starting from the smallness assumption, the \(L^{2}\)-theory is performed. The velocity regularization appears in a standard way whereas spatial regularization is obtained through velocity averaging. The \(L^{2}\)-theory comprises both f and K-levels \(f^{(\ell )}_{K, +}\). A higher k-moment is needed (to be precise, \(k = k_0+\ell _0+2\) for some \(\ell _0\) depending on s) in the \(L^{2}\)-estimates to prove algebraic \(k_0\) moments in the \(L^{\infty }\)-estimates.
Flow chart of the strategy. Moments are related as \(k_0>w_0>0\) and so does regularity as \(s>s'>0\). The constant \(C(\mathcal {E}_0)\) is independent of the smallness parameter \(\delta _0\)
2 Technical Toolbox
2.1 Function Spaces
In this paper we use two classical function spaces, namely, Bessel potential and Sobolev-Slobodeckij spaces. Most of the work is based on the former, yet, the proof of some estimates is simpler if performed in the latter.
Definition
For \(p\in [1,\infty )\) and \(\beta \in \mathbb {R}\), the Bessel Potential space is
where \({\mathcal {F}}\) is the Fourier transform. The norm that equips \(H^{\beta ,p}(\mathbb {R}^{d})\) is naturally
Definition
For \(p\in [1,\infty )\) and \(\beta \in (0,1)\), the Sobolev-Slobodeckij space is
A natural norm that equips \(W^{\beta ,p}(\mathbb {R}^{d})\) is given by
The Bessel potential spaces and Sobolev-Slobodeckij spaces agree for \(p=2\). More generally, the following relation holds:
(i) For all \(p\in (1,2]\), \(\beta \in (0,1)\) it holds that \(W^{\beta ,p}(\mathbb {R}^{d}) \hookrightarrow H^{\beta ,p}(\mathbb {R}^{d})\).
(ii) For all \(p\in [2,\infty )\), \(\beta \in (0,1)\) it holds that \(H^{\beta ,p}(\mathbb {R}^{d}) \hookrightarrow W^{\beta ,p}(\mathbb {R}^{d})\).
The proof of this fact can be found in [43], Theorem 5 in Chapter V.
2.2 Useful Facts About Polynomial Weights
In this section, we list some useful estimates that are needed for later estimation. Recall that
First we present two lemmas related to commutator estimates of fractional derivatives. Since their proofs are technical and not directly associated with the Boltzmann operator, we leave them to Appendix 8.
Lemma 2.1
(cf., [32]) Let \(1\le p \le \infty \) and suppose \(\ell ,\, \theta \in {{\mathbb {R}}}\). Then there exists a generic constant C independent of f such that
that is, these two norms are equivalent. Here \(\left\langle D_v\right\rangle \) is the Fourier multiplier with the symbol \(\left\langle \xi \right\rangle \).
We will also need a homogeneous version related to fractional derivatives.
Lemma 2.2
Suppose \(\alpha \in (0, 1)\) and \(f \in H^\alpha _v({{\mathbb {R}}}^3)\) Then \(\left\langle v\right\rangle ^{-2} f \in H^\alpha _v({{\mathbb {R}}}^3)\) with the bound
Next we recall the now-classical trilinear estimate.
Proposition 2.3
([4, 40]) Denote \(a^+ = \max \{a, 0\}\). Then the bilinear operator Q satisfies
for any \(\sigma \in [\min \{s-1, -s\}, s]\), \(m \in {{\mathbb {R}}}\), \(\gamma \ge 0\) and \(0< s < 1\). Here, \(f,\, g,\, h\) are any functions so that the corresponding norms are well-defined. The constant C is independent of \(f,\, g,\, h\).
Lemma 2.4
([2]) Suppose f and b are functions that make sense of the integrals below. Then
(a) (Regular change of variables)
(b) (Singular change of variables)
Proposition 2.5
([7]) Suppose for some constants \(D_0, E_0 > 0\), the function F satisfies
Then there exist two constants \(c_0\) and C such that
Throughout this paper, we use \(\, \textrm{d} {\overline{\mu }}\) to denote the measure
For the convenience of the later analysis, we record a simple decomposition and bound related to the nonlinear Boltzmann operator:
Lemma 2.6
([12]) (a) Let G, h be functions that make sense of the integrals below and \({\mathbb {S}}^2_+\) be the upper half sphere with \(\cos \theta \ge 0\). Then for any \(s \in (0, 1)\) and \(\ell \ge 0\),
(b) Suppose in addition \(G \ge 0 \) and \(G = \mu + g\). Let \(\gamma _0, \gamma _1\) be the positive constants satisfying
and
Then we have
Proof
Part (a) follows from a direct addition-subtraction applied to the definition of Q. Part (b) follows from (3.15) and (3.16) in [12]. \(\square \)
Cancellation plays a vital role in dealing with the strong singularity. Let us recall a useful representation for \(|v'|\):
and, as a result,
where
By its definition, \(\omega \) satisfies that \(\omega \perp (v - v_*)\), thus, \(v \cdot \omega = v_*\cdot \omega \). Consequently, one has the freedom to choose \(v \cdot \omega \) or \(v_*\cdot \omega \) in the estimates. We also introduce the notation \({\widetilde{\omega }}\) for later use:
Lemma 2.7
(see [12]) Suppose \(\ell > 6\) and \((v, v_*), (v', v'_*)\) are the velocity pairs before and after the collision or vice versa. Let \(\omega \) be the vector defined in (2.8). Then,
where there exists a constant \(C_\ell \) only depending on \(\ell \) such that
We are ready to establish an all-important commutator estimate.
Proposition 2.8
(Weighted commutator) Suppose
(a) For general G, F, H making sense of the terms in the inequality, we have
where \({\widetilde{\omega }}\) is the unit vector defined in (2.9).
(b) Suppose \(G = \mu + g\) is non-negative and there exist \(\ell , K_0\) such that
Then for any \(s \in (0, 1)\) and the same \(\ell \) as in (2.13), we have
Proof
The proofs for both parts follow from a revision of the proof of Proposition 3.1 in [12], based on taking advantage of angular cancellations and using cutoff techniques such as in [10]. Applying Lemma 2.7, we decompose the integral as
The main difference between (2.12) and (2.14) is that in (2.12) the extra \(\gamma \)-weight falls on g while in (2.14) it is on H.
(a) Deriving the bound for \(\Gamma _1\) requires careful use of symmetry in the case of the strong singularity. The idea is similar to the proof of Proposition 3.1 in [12]. In particular, we decompose \(\Gamma _1\) as
By symmetry the first term \(\Gamma _{1,1}\) vanishes. This follows from the regular change of variables \(v \rightarrow v'\) and using \(v' - v_*\) as the new north pole. In this way we have that
where \({\widetilde{\omega }} = (\cos \phi , \sin \phi , 0)\) and the integration in \(\phi \) vanishes. The second term \(\Gamma _{1,2}\) is readily bounded by
where we have used the Young’s inequality and the regular change of variables \(v \rightarrow v'\). We will leave \(\Gamma _{1, 3}\) as is since in the later analysis, Proposition 2.9 will be applied in each specific case. Putting the components together gives the bound of \(\Gamma _1\) as
Next we show the estimate for \(\Gamma _2\). Start with the direct bound using Cauchy–Schwarz and a regular change of variables stated in Lemma 2.4:
A second way to estimate \(\Gamma _2\) is
where again we have applied the regular change of variables. Overall, we have
Estimate for \(\Gamma _3\) is similar to \(\Gamma _2\). By the bound on \({\mathfrak {R}}_1\) in Lemma 2.7, we have
Estimates for \(\Gamma _4\) and \(\Gamma _5\) are more straightforward. Using the upper bound of \({\mathfrak {R}}_2\) and a regular change of variables, we have
Similarly, we can bound \(\Gamma _5\) by
The desired estimate in (2.12) is obtained by adding all the bounds for \(\Gamma _1, \cdots , \Gamma _5\) in (2.16)-(2.20).
(b) The proof is similar to part (a) with a revision based on the extra condition (2.13) on g. In particular, we use the decomposition in (2.15):
where \(\Gamma _n\)’s are exactly the same as in (2.15). The estimates of \(\Gamma _1, \Gamma _4, \Gamma _5\) remain the same as in part (a), which give
To prove (2.14), we combine (2.13) with the positivity of G and the singular change of variables to estimate \(\Gamma _2\) and get
Similarly, using the positivity of G and (2.13), we have the bound of \(\Gamma _3\) as
Combining (2.21), (2.22) and (2.23) gives (2.14). \(\square \)
We summarize explicit bounds for the first term on the right-hand side of (2.12) and (2.14) in the following lemma:
Proposition 2.9
Let \({\widetilde{\omega }}\) be the unit vector defined in (2.9). Suppose G, F, H are functions that make sense of the integral below.
(a) If \(s \in [1/2, 1)\), then for any pair of \((s_1, \gamma _1)\) satisfying
we have
(b) If \(s \in (0, 1/2)\), then
(c) If \(F \in W^{1, \infty }({{\mathbb {R}}}^3_v)\), then for any \(s \in (0, 1)\) we have
Proof
Part (a) is an immediate application of (3.13) in [12] (with a reshuffle of the function names). Part (b) is a direct bound using the fact that \(b(\cos \theta ) \sin \tfrac{\theta }{2}\) is integral if \(s \in (0, 1/2)\). Hence,
Depending on the property of F, we can obtain two types of bounds here:
and
A combination of them gives (2.26).
Part (c) follows directly from the Mean Value Theorem, which gives the following bound:
Since on \({\mathbb {S}}^2_+\) it holds that
there exists a generic constant C such that
Hence, the (partial) integrand satisfies
when restricted on \({\mathbb {S}}^2_+\). Inserting such bound into the left-hand side of (2.27), we get
\(\square \)
We also record a proposition using the symmetry cancellation:
Proposition 2.10
[16, Lemma 2.1] Suppose \(H \in W^{2, \infty }({{\mathbb {R}}}^3)\). Then for any \(s \in (0, 1)\), it holds that
2.3 Interpolation Results
In this section we collect several results about interpolation in fractional Sobolev spaces that will be used in the sequel.
Lemma 2.11
Let \(\eta ,\eta '\in (0,1)\). Then for \(r = r(\eta , \eta ', d) > 2\) and \(\alpha = \alpha (\eta , \eta ', d)\in (0,1)\) defined in (2.33), it follows that
The constant C only depends on \(\eta , \eta ', d\).
Proof
By the Sobolev embedding in \({{\mathbb {R}}}^d\) and \(\mathbb {T}^d\) there exists c depending only \(\eta , \eta ', d\) such that
where
Set
One can readily check that
Then, using the Höder inequality we have
where the Minkowski’s integral inequality is used in the second last step. Inequality (2.28) then follows by the definition and property of \(\alpha \) in (2.33) and (2.34). \(\square \)
Observe that the estimates hold in the proof of Lemma 2.11, or equivalently, the existence of \(\alpha , \alpha _1, \alpha _2, r\) in the correct range is guaranteed as long as \(p, q > 2\). Based on such observation, we have a second interpolation similar as Lemma 2.11:
Lemma 2.12
Let \(\eta , \eta ' \in (0,1)\) and \(m \ge 1\). Then, for some \({\widetilde{r}} = r(\eta , \eta ', m, d) > 2\) and \({\widetilde{\alpha }} = {\widetilde{\alpha }} (\eta , \eta ', m, d) \in (0, 1)\), we have
The constant C only depends on \(\eta , \eta ',m, d\).
Proof
Using Sobolev imbedding we have that
where \(c=c(\eta ,\eta ',m,d)\) and
which are in a similar form as (2.29) and (2.30). By the comment before the statement of Lemma 2.12, the desired inequality holds with
\(\square \)
Next we show a “Leibniz” rule for fractional derivatives:
Lemma 2.13
Let \(p \in (1,2)\), \(0\le \beta '<\beta \in (0,1)\),
Then for any \(\varphi \) making sense of the terms of the inequality below, it follows that
where the constant C only depends on \(d, \beta ' , \beta , p\) and \({\dot{H}}^\beta (\mathbb {R}^{d})\) is the homogeneous Bessel potential space.
Proof
By the continuous embedding of the Bessel potential space in the fractional Sobolev-Slobodeckij space for \(p\in (1,2]\), it follows that
A simple computation shows that
where C depends on \(d,\beta \) and p. To estimate \(I_1\), decompose its integrand as the product
Then a direct application of the Hölder inequality with measure \(|x-y|^{-d} \, \textrm{d} x \, \textrm{d} y\) and pair \(\big ( \frac{2}{p},\frac{2}{2-p} \big )\) gives
which combined with the estimate for \(I_2\) proves the result. \(\square \)
2.4 Strong Averaging Lemma
The following result is a time-localised version of [14, Theorem 1.3] that is needed for the Cauchy problem.
Proposition 2.14
Fix \(0 \le T_{1} < T_{2}\), \(p\in (1,\infty )\), \(\beta \ge 0\), and assume that \(f \in C\big ([T_{1},T_{2}];L^{p}_{x,v}\big )\) with \(\Delta ^{\beta /2}_{v}f \in L^{p}_{t,x,v}\) satisfies
Then, for any \(r\in [0,\frac{1}{p}]\), \(m \in \mathbb {N}\), \(\beta _{-}\in [0,\beta )\), if we define
and
then it follows that
where the constant C only depends on \(d,\beta ,r,m,p\).
Proof
Multiplying the transport equation by \(1_{(T_{1},T_{2})}(t)\) we arrive at
Write the sources as
for \(0 \le r < {\widetilde{r}} \in (0,1]\) and \(m\in \mathbb {N}\). By [14, Theorem 1.3] and the additive contribution of the sources one has that
for \(s^\flat =\min \big \{\frac{(1-r)\beta }{m+1+\beta }, \frac{(1-{\widetilde{r}})\beta }{m+ 1+\beta }\big \}=\frac{(1-{\widetilde{r}})\beta }{m+ 1+\beta }\). It remains to estimate the term involving A on the right. First by [14, Lemma 2.3] we have
By the definition of A, we can explicitly compute
where \({\mathcal {B}}_{{\widetilde{r}} - r}\) is the Bessel kernel of order \({\widetilde{r}} - r\) in \({{\mathbb {R}}}\). The asymptotic behaviours of the Bessel kernel near 0 and \(\infty \) give
As a consequence, it follows that
under the condition \( {\widetilde{r}}> 1 - \frac{1}{p} + r >0\), which together with \(s^\flat =\frac{(1-{\widetilde{r}})\beta }{m+ 1+\beta }\) implies the choice (2.36). Analogous estimate follows at the point \(t=T_2\). \(\square \)
The next proposition is an estimate for \(Q(F, \mu )\):
Proposition 2.15
For any \(F \in L^1_{\gamma +2s}({{\mathbb {R}}}^3_v)\), the quantity \(Q(F, \mu )\) is in \(L^\infty ({{\mathbb {R}}}^3_v)\) with the bound
Proof
The proof follows from a similar line of argument as the proof of Proposition 2.1 in [4]. First we decompose \(|v - v_*|^\gamma \) as
where \(\Phi _{{\overline{c}}}\) is smooth and \(\Phi _{{\overline{c}}} = 0\) near \(v = v_*\) while \(\Phi _c = |v - v_*|^\gamma \) near \(v = v_*\). The main property of \(\Phi _c\) is
where \({\widehat{\Phi }}_c\) is the Fourier transform of \(\Phi _c\). Denote \(Q_c, Q_{{\overline{c}}}\) as the corresponding collision operators such that
Then by the trilinear estimate (2.1) in [4], we have
Hence we are left to bound \(Q_c (F, \mu )\). Take an arbitrary \(h \in L^1({{\mathbb {R}}}^3)\). In the Fourier space, we have
where
Note that \(\xi ^+\) is perpendicular to \(\xi ^-\). By Taylor’s theorem, we have
Similar as in [4], we decompose \(\xi ^-\) as
Inserting (2.42) into (2.41), we get
By symmetry \(\left\langle Q_c^{(1)}(F, \mu ), h\right\rangle \) vanishes. By the property that
we have
To bound \(\left\langle Q_c^{(3)}(F, \mu ), h\right\rangle \), we first use the property of \(\xi ^-\) to get
Make the change of variables
Since \(b(\cos \theta )\) is supported on \(\cos \theta \ge 0\), we have \(\theta \in [0, \pi /2]\) and
Similarly, by \(\sin \frac{\theta }{2} \le \sqrt{2}/2\) and the fact that \(\xi ^+ \perp \xi ^-\), we have
which gives
Applying the change of variables \((\xi , \xi _*) \rightarrow (w, z)\) in \(Q_c^{(3)}\), we have
Combining the estimates for \(Q_c^{(1)}, Q_c^{(2)}, Q_c^{(3)}\) and \(Q_{{\overline{c}}}\) gives the bound in (2.38). \(\square \)
Remark 2.16
It is clear from the proof of Proposition 2.15 that we can replace \(\mu \) by \(\mu \left\langle v\right\rangle ^\ell \) for any \(\ell \) and obtain that
Finally we state some elementary interpolations and the specific form of the Gronwall’s inequality used frequently in later sections.
Lemma 2.17
For any \(\alpha > 0\) and \(k \in {{\mathbb {R}}}\), we have
Lemma 2.18
Let \(C_1, C_2\) be two positive constants. Suppose \(u(t) \ge 0\) satisfies
Then
Note that the coefficient in the second term \(C_1^2\) is independent of \(C_2\).
Proof
First by the Cauchy–Schwarz inequality, we have
Then by the usual Gronwall’s inequality,
\(\square \)
3 Linear Local Theory: A Priori Estimates
We start with the theory for the linear equation
where \(T > 0\) is fixed and G is a fixed nonnegative function and we write
The operator \(L_{\alpha }\) is a regularising linear operator defined by
where \(\alpha > 0\) will be specified and fixed in the sequel.
The goal of this section is to establish a priori estimates in various \(L^2\)-based spaces. Hence we suppose F(t, x, v) is a sufficiently smooth nonnegative solution to (3.1) and let f(t, x, v) be its perturbation around the global Maxwellian, i.e.,
Then for any \(\epsilon \in (0,1]\), the pair (F, f) satisfies the equation
3.1 Local in Time \(L^{2}\)-Estimates
First we derive a uniform-in-\(\epsilon \) \(L^2\)-estimate for Eq. (3.3).
Proposition 3.1
(Bilinear uniform-in-\(\epsilon \) estimate) Suppose \(G = \mu + g \ge 0\) satisfies that
Suppose \(s \in (0, 1)\) and \(\ell > 8 + \gamma \). Let \(F = \mu + f\) be a solution to equation (3.3). Then
where \(\delta _2\) is a small enough constant satisfying (3.15), \(\gamma _0, c_0\) are the positive constants in Lemma 2.6 and Proposition 2.5 respectively, and \(b_0\) is a constant that only depends on \(s, \gamma \). All the coefficients \(c_0, \gamma _0, \delta _2, b_0, C_\ell \) are independent of \(\epsilon \). Furthermore, for any \(0 \le T_1< T_2 < T\) and \(0< s' < \frac{s}{2(s+3)}\), we have the regularisation
where the coefficient C is independent of \(\epsilon \).
Proof
Multiply (3.3) by \(\left\langle v\right\rangle ^{2\ell } f\) and integrate in x, v. The regularising term is bounded as
Decompose the integration of the collision term as
where by the trilinear estimate in Proposition 2.3, we have
To bound \(T_0\), we use (2.5) in Lemma 2.6 and (2.12) in Proposition 2.8 and get
Here we treat the mild and strong singularities separately. If \(s \in (0, 1/2)\), then we apply part (b) of Proposition 2.9 and bound the first term on the right-hand side of (3.11) as
If \(s \in [1/2, 1)\), then by part (a) of Proposition 2.9 and interpolation, we have
where \(\delta _1\) can be chosen arbitrarily small and \(b_0 = b_0(s, \gamma , s_1, \gamma _1)\). In fact, one can show that
Since the particular form of \(b_0\) is not needed we omit its derivation. Moreover, since the choices of \(s_1, \gamma _1\) only depend on \(s, \gamma \), we can view \(b_0\) as a constant that only depends on \(s, \gamma \). We can now combine both cases and get that for any \(s \in (0, 1)\),
Substituting this bound into (3.11) gives
where \(\delta _1\) can be taken arbitrarily small. Combining (3.7), (3.9) and (3.12) gives the energy estimate as
where all the constants are independent of \(\epsilon \).
To complete the basic energy estimate, we include the \(H^s\)-regularisation. To this end, we only need to perform the second kind of estimate for \(T_0\), as done in the proof of Proposition 3.2 in [12]. By Proposition 2.5, equation (2.3) in Lemma 2.6 and the same estimates in (3.10)-(3.12) for \(T_0\), we have
Choose \(\delta _1, \delta _2\) small enough such that
Multiplying (3.14) by \(\delta _2\) and adding it to (3.13) gives (3.5).
Finally we apply the averaging lemma in Proposition 2.14 to obtain the regularisation in x. In light of Eq. (3.3), if we invoke Proposition 2.14 with
then for any \(0\le T_{1} \le T_{2}<T\),
By the trilinear estimate in Proposition 2.3, it follows that
In this way, we are led to the desired inequality showing the spatial regularisation of f. \(\square \)
Applying the Gronwall’s inequality to Proposition 3.1, we obtain the following bound:
Corollary 3.2
Suppose \(G = \mu + g \ge 0\) satisfies that
Let \(F = \mu + f\) be a solution to Eq. (3.3) with \(s \in (0, 1)\). Assume that the following conditions hold:
Let
where \(b_0\) is the same exponent as in Proposition 3.1. Then it holds that
and
Furthermore, if in addition,
then for any \(0< s' < \frac{s}{2(s+3)}\) it holds that
where the exponent 10 is chosen such that \(10 > 8 + \gamma \).
Proof
Applying the additional bounds in (3.16) to (3.5) gives
Estimate (3.17) follows directly from applying the Gronwall’s inequality to (3.20). When integrating in time (3.20) one concludes that
This bound together with estimate (3.6), with \(T_{1}=0\) and \(T_{2}=t\), gives (3.19) for sufficiently large \(C>0\) under the condition \(\ell \ge 3+2\alpha \). \(\square \)
Our main \(L^\infty \)-bound will be based on various \(L^2\)-estimates of the level-set functions defined as follows: for any \(\ell \ge 0\) and \(K\ge 0\) define the levels \(f^{(\ell )}_{K}:=f\,\left\langle v\right\rangle ^{\ell } - K\) and
3.2 \(L^2\)-Estimates for Level Sets
The focus of this subsection is to prove the following natural a priori estimate for the level sets. It is a building block for the energy functional presented later in the argument.
Proposition 3.3
Suppose \(G = \mu + g \ge 0\), \(F = \mu + f\) and \(s \in (0, 1)\). Suppose in addition G satisfies that
Then for any \(\ell > 8 + \gamma \),
(a) the (bilinear) collision term satisfies,
where \(\delta _4\) satisfies the bound in (3.29).
(b) The regularising term satisfies
Proof
(a) As in the proof of Proposition 3.1, we make two estimates of the Q-term: one with \(H^s\)-norm in v and one without. To derive the one without the \(H^s\)-norm, make the decomposition
By the definition of Q and the positivity of G, the first term \(T_1\) satisfies
Continuing from here, the part of the integrand involving \(f^{(\ell )}_{K, +}\) satisfies
By the regular change of variables, (2.12) in Proposition 2.8 and Proposition 2.9, we bound \(T_1\) as
where \(b_0\) is the same exponent as in Proposition 3.1 and \(\delta _3 > 0\) can be arbitrarily small. In the estimate above we have combined the mild and strong singularities. Next we estimate \(T_2\). Writing out Q, we get
Applying the regular change of variables to the first term, then (2.12) in Proposition 2.8 and part (c) in Proposition 2.9 with \(F=1\) to the second term and a direct estimate to the third term, we get
Applying similar estimates and Remark 2.16 to \(T_3\), we have
Combining all the estimates, we obtain the first bound for the right-hand side as
Next we derive the second bound with the \(H^s\)-norm. To this end, we use Proposition 2.5, part (a) in Lemma 2.6, inequality (3.24) and similar bounds in the proof of (3.25) to re-estimate \(T_1\) as
Multiply (3.28) by a small enough \(\delta _4\), choose \(\delta _3\) small enough and add it to (3.27). This gives the desired bound in part (a). The specific requirements for \(\delta _3, \delta _4\) are
(b) To estimate the contribution of the \(\epsilon \)-regularising term to the energy estimate of the level set, denote
Decomposing F gives
Then \(T^1_R\) is directly bounded as
Carrying out the computation of differentiation, we get
where we have applied the positivity of \(f^{(\ell )}_{K, +}\) and the bound \(C_{\ell ,\alpha }\left\langle v\right\rangle ^{2\alpha -2} - \left\langle v\right\rangle ^{2\alpha } \le C'_{\ell , \alpha } {\varvec{1}}_{|v| \le V_0}\) for some constant \(V_0\) large enough.
To estimate \(T^3_{R}\), we break it into two parts:
Then \(T^{3,1}_{R} = -\left\Vert \left\langle v\right\rangle ^{\alpha } f^{(\ell )}_{K, +} \, \right\Vert _{L^2_{x,v}}^2\). Integrating by parts, we have
Hence,
Altogether we have
which concludes the proof of part (b). \(\square \)
To show that \(|f| \left\langle v\right\rangle ^k \le K\) in the later part, we will need to bound not only the level set function \(f^{(\ell )}_{K, +}\) but also the one for \((-f)^{(\ell )}_{K, +}\) since the former only gives \(f \left\langle v\right\rangle ^k \le K\). Given the linearity of the Boltzmann operator Q(G, F) in F, it is not surprising that estimate for \((-f)^{(\ell )}_{K, +}\) follows a similar line as that for \(f^{(\ell )}_{K, +}\). The equation for \(h = -f\) is
Proposition 3.4
Let \(h = -f\). Suppose
Then for any \(s \in (0, 1)\) and \(\ell > 8 + \gamma \),
(a) The bilinear collision term satisfies
where \(\delta _4\) is the same constant in (3.29).
(b) For the regularising term it holds that
Proof
(a) Make a similar decomposition as in (3.23):
Since \(J_1, J_2\) have the same forms as \(T_1, T_2\) in (3.23), by taking \(\delta _4\) with a bound in (3.29), we get
Decomposing \(J_3\) similarly as \(T_3\) in (3.26) and applying Proposition 2.15, inequality (2.12) in Proposition 2.8, we have
where note that inequality (2.12) in Proposition 2.8 does not require positivity of the functions in the integrand. Estimate in part (a) is a combination of the bounds for \(J_1, J_2, J_3\).
(b) Decompose the integral in part (c) in a similar way as in (3.30):
It is then clear that estimates for the three terms above are similar to those for \(T^1_R\), \(T^2_R\) and \(T^3_R\) in Proposition 3.3, since they rely on the absolute values of the terms. Hence we obtain a similar bound. \(\square \)
3.3 A Level Sets Estimate for the \(L^{1}\)-Norm of the Collisional Operator
In this part we estimate an \(L^1\)-norm related to Q(G, F), which provides the basis for a later application of the averaging lemma. By subtracting K from \(f\,\left\langle v\right\rangle ^{\ell }\) and multiplying Eq. (3.1) by \(f^{(\ell )}_{K, +}\), we obtain that
When applying the averaging lemma to the level sets in the next section, it will be important to estimate
One key observation is that the dominant part of the integrand above is its positive part.
Lemma 3.5
Let (F, f) be a pair satisfying the linearized Boltzmann equation (3.1). Then, for any \(j,\ell \ge 0\), \(\kappa \ge 0\), \(K\ge 0\) and \(0 \le T_1< T_2 < T\), it follows that
where \([\cdot ]^+\) denotes the positive part of the term.
Proof
First we fix \(v \in {{\mathbb {R}}}^3\) and integrate (3.33) in (t, x) to obtain that
for any \(v \in {{\mathbb {R}}}^3\).
An application of the Bessel potential in velocity to (3.34) then leads us to
Hence, if we denote
and \({\mathcal {G}}_-\) and \({\mathcal {G}}_+\) as its negative and positive parts respectively, then for any \(v \in {{\mathbb {R}}}^3\),
We thereby conclude that
which proves the lemma. \(\square \)
The counterpart for \(h = -f\) states
Lemma 3.6
Let h be a solution to the equation
Then, for any \(0 \le T_1< T_2 < T\), \(j,\ell \ge 0\), \(\kappa \ge 0\), \(K\ge 0\), it follows that
where \([\cdot ]^+\) denotes the positive part of the term.
The remainder of this subsection focuses on proving the following theorem:
Proposition 3.7
Suppose \(G= \mu + g \ge 0\) and \(F = \mu + f\) satisfy equation (3.1). Then, for any
it holds that
where \(C, C_\ell \) are independent of \(\epsilon \) and \(T_1, T_2\). Identical estimate holds for \({\widetilde{Q}}(G, -\mu + h)\) with \(f^{(\ell )}_{K, +}\) replaced by \(h^{(\ell )}_{K, +}\).
Proof
First note that for any \(\kappa \ge 0\),
which explains the first term in the right side of (3.36). Thus, using Lemma 3.5 we have that for \(j\ge 0\), \(\ell \ge 0\), \(\kappa \ge 0\),
where \(A_K\) is the set given by
Using that \(\widetilde{Q}(G,F) = Q(G,F) + \epsilon \,L_{\alpha }(F)\), we have
Denote in the following
Then, for \(\kappa > 2\) it holds for all derivatives up to second order that
with C independent of K. In fact, noting that the \(\kappa ^{th}\) Bessel kernel \({\mathcal {B}}_\kappa (w)\) in dimension d satisfies
and
we have
Since the inequality \(|\nabla _i {\mathcal {B}}_\kappa (w) | \le C'({\mathcal {B}}_\kappa (w) + {\mathcal {B}}_{\kappa -1} (w))\) holds (see [13, (4.5)]), we have the estimate of the first-order derivative. The estimate of second order is also obvious because similar inequality holds (see (4.4), (4.1) and (3.7) of [13]). In this way, we are led to estimate
We will estimate the main term \({\mathcal {Q}}\) and the regularising linear term \(T_R^+\) separately. The proofs align with those for Proposition 3.3. We start with \({\mathcal {Q}}\) and write it as
Then by the definition of Q and the positivity of G, the first term \(T_1^+\) satisfies
By Cauchy–Schwarz, the part of the integrand involving \(f^{(\ell )}_{K, +}\) satisfies
Write the upper bound of \(T_1^+\) correspondingly as
Similar as in the estimates of \(T_1\) in (3.25), the bounds for \(T_{1, 1}^+\) and \(T_{1, 2}^+\) follow from the regular change of variables, bound (2.12) in Proposition 2.8 together with Proposition 2.9:
Inserting the bound of \(W_K\) in (3.39), we get
Note that in the estimate above we have combined the cases of the mild and strong singularities.
The bound of \(T_{1,3}^+\) is derived by using Proposition 2.10, which gives
Combining the estimates for \(T_{1,1}^+, T_{1,2}^+, T_{1,3}^+\), we have
The estimates for \(T_2^+\) and \(T_3^+\) is similar to those for \(T_2\) and \(T_3\) in Proposition 3.3. In fact comparing the forms of \(T_2^+, T_3^+\) with \(T_2, T_3\) defined in (3.23), one can see that the only difference is that \(f^{(\ell )}_{K, +}\) in \(T_2, T_3\) is now replaced by \(f^{(\ell )}_{K, +}W_{K}\). Since no particular structure of \(f^{(\ell )}_{K, +}\) is used in the bounds of \(T_2\) and \(T_3\), we simply replace \(f^{(\ell )}_{K, +}\) in those bounds by \(f^{(\ell )}_{K, +}W_{K}\) and obtain
The bound of \({\mathcal {Q}}\) is the combination of the bounds of \(T_1^+, T_2^+, T_3^+\), which writes
Next we estimate \(T_R^+\) defined in (3.40). The estimate follows the same line as the one for \(T_R\) in part (b) of Proposition 3.3. We start with a similar decomposition as in (3.30):
It is then clear that the first two terms are bounded similarly as \(T_R^1\) and \(T_R^2\) which results in
The third term \(T^+_{R,3}\) needs more careful estimates due to the presence of \(W_K\). Via integration by parts once, we have
where the remainder Rem has five pieces
It is clear that \(Rem_1, Rem_2, Rem_3\) are directly bounded as
By integrating by parts, \(Rem_4\) satisfies
The last term \(Rem_5\) needs more careful treatment. Integrating by parts, we have
The main observation here is for any \(\kappa \ge 2\),
where the pointwise positivity is a consequence of the positivity of the Bessel potential. Hence for pointwise t, x, v we have
Applying such relation in the second term of (3.51), we obtain that
which is a leading order-term in moments. However, it is dominated by the dissipation because it has a smaller coefficient \(\frac{1}{2}\). Using integration by parts, the first term in (3.51) satisfies
These terms can be controlled similarly to the \(Rem_{1}\) and \(Rem_{2}\) as they are lower-order in moments. Thus,
The conclusion is that
Now combining the estimates for \(Rem_1, \ldots , Rem_5\) with the dissipation terms in (3.48), we have
Together with the bounds for \(T^+_{R, 1}\) and \(T^+_{R, 2}\), we obtain that
which, when combined with \({\mathcal {Q}}\), can be absorbed into the upper bound for \({\mathcal {Q}}\) in (3.46).
For \(h=-f\), all the previous estimates follow identically except \(T^+_3\), for which we apply a similar estimate for \(J_3\) in the proof of Proposition 3.4 instead of \(T_3\) in Proposition 3.3. Then the same bound follows. \(\square \)
3.4 Time–Space–Velocity Energy Functional
In this subsection we complete the \(L^{2}\)-energy estimate for the level-set function by adding the regularisation in the spatial variable. To such end we introduce the energy functional for \(s''\in (0,s) \subseteq (0, 1)\), \(\ell \ge 0\), \(p>1\),
The constant \(C_0\) does not play any essential role and the parameters \(s''>0\), \(p>1\) will be suitably chosen as the discussion progresses. We start with imposing one condition on p: let r(1) and r(p) be the exponents given in Lemma 2.12 such that
We require that p satisfies the condition
Such p exists, since by the continuity of \(r(\cdot )\),
Hence a sufficient condition for (3.58) to hold is by letting p be close enough to 1. Since such closeness is needed for later parts, we simply enforce it here: let \(p^\sharp \in (1, 2)\) be fixed and close enough to 1 such that
and in what follows we restrict to
The reason for imposing (3.58) or (3.56) will be clear in the proof of the following key interpolation lemma:
Lemma 3.8
(Energy functional interpolation) Let the parameters \(T_1, T_2, s, s'', \ell , n\) be given such that
Let \(\ell _0\) be large enough with the specification in (3.72) (\(\ell _0\) depends on n but is independent of \(\ell \)). Suppose
Let \(p > 1\) be fixed and satisfying (3.58) and let \({\mathcal {E}}_p(K, T_1, T_2)\) be the energy functional defined in (3.54). Then there exists a constant \(q_*\) which is independent of p and satisfies \(1< q_*< \frac{r(1)}{2}\) such that the following holds: for any \(1 < q \le q_*\), we can find a pair of parameters \((r_*, \xi _*)\) with the properties
such that for any \(0 \le M < K\) and \(0\le T_{1}\le T_{2}\le T\),
where C only depends on \((C_1, s, s'', q, p)\). The parameters \(q_*, r_*, \xi _*\) are defined in (3.65), (3.73) and (3.67) and they only depend on \((s, s'')\). In particular, all of these parameters are independent of K, M, \(T_1, T_2\) and f.
Proof
Recall the definitions of r(1), r(p) in (3.55). For any \((\theta ,\xi , q)\) satisfying the relation
which is depicted in Fig. 2, we define \(\beta \in (0, 1)\) by
Note that for a given pair of \((\theta , p)\), the parameters \(\beta \) and \(\xi \) are in one-to-one correspondence.
Choice of parameters
We observe that for any p satisfying (3.58), by the definition of \(\beta \), it holds that
and by (3.56),
The limits above are uniform in p as long as p satisfies (3.58). By continuity, there exist \(q_*, \theta _*\) such that if
then for \((\beta , \xi )\) satisfying (3.62), we have
and
As an example, we can choose
Such a choice guarantees that
It is then clear that the choices of \(q_*, \theta _*\) only depend on \(p^\sharp , s, s''\). By (3.63), if \(\xi _*\) is sufficiently close to \(2q_*\), then \(\beta \xi /2 < 1\). As a result, for any \(\zeta \in (0, 1)\), there exists \(\xi _*(\zeta )\) paired with \(\beta _*(\zeta )\) such that
The notations \(\xi _*(\zeta ), \beta _*(\zeta )\) are simply emphasizing the dependence of \(\xi _*, \beta _*\) on \(\zeta \) instead of indicating they are functions of \(\zeta \).
With the preparations above, we now fix \((q, \theta )\) satisfying (3.65) and let \(\zeta = {\widetilde{\alpha }}(s, s'', p, 3)\), where \({\widetilde{\alpha }}(s, s'', p, 3)\) is the parameter in Lemma 2.12. Next we fix a pair of parameters \(\xi _*, \beta _*\) satisfying \((\beta _*, \xi _*) = (\beta _*(\zeta ), \xi _*(\zeta ))\), such that (3.62) holds and
With these parameters chosen we carry out various interpolations. First,
where \(a_0 = \frac{1}{1-\beta _*} \left( \frac{n}{\xi _*} + 2 \beta _*\right) \). Application of Lemma 2.12 with \(({\widetilde{r}}, \eta , \eta ', m) = (r(p), s, s'', p)\) and Lemma 2.2 to (3.69) gives
where \(C = C(s, s'', p)\). By (3.68) and the Hölder’s inequality, the integral term in (3.70) is controlled by
Interpolating the \(L^{\theta }_{x,v}\)-norm with
it follows that
by taking
Overall, we have
with
We can make \(\beta '\) arbitrarily close to 1 by taking \(\theta _*\) in (3.65) close enough to 2. This way we have
hence the desired bound in (3.60). \(\square \)
Remark 3.9
The parameters \((\theta _*, q_*, r_*, \xi _*, \beta _*, \beta ')\) in Lemma 3.8 can be made explicit. Here we give an example of these parameters such that Lemma 3.8 holds. First we fix \(p^\sharp \) which satisfies (3.57) and let
Note that for each \(\theta \in (1,2)\), equations (3.62) and (3.68) provide a system that uniquely determines \((\xi , \beta )\) in terms of \(\theta \). We recall the system below
where \(\zeta ^\sharp = {\widetilde{\alpha }}(s, s'', p^\sharp , 3)\). For a fixed \(\theta \), we can solve and obtain
The condition for \(\theta \) comes from the combination of (3.71) and (3.73). At this moment we only need \(q \in (1, r(1)/2)\). Hence we require
Solving \((\theta , \beta ')\)-system above, we obtain the condition on \(\theta \) as
The existence issue is equivalent to whether there exists \(\theta \in (1,2)\) such that (3.75) and (3.77) hold simultaneously. In order to check this, we note that by (3.75), for any \(\theta \in (1, 2)\), it holds that
In particular, it holds that
Hence the right-hand side of (3.77) satisfies
while the middle term clearly satisfies \(\lim _{\theta \rightarrow 2} (2 - \theta ) = 0\). This shows there is a range of \(\theta \) values that satisfy all the desired properties. For a particular example we first introduce two parameters
where \(c_*\) is defined in (3.78). Then use the parameter \(c_{**}\) defined in (3.79) and let
By (3.75) we can solve and obtain
Now we check that (3.77) holds: by the definition of \(\alpha ^\sharp \), we have
Hence, repeating the previous estimate, we have
that is, inequality (3.77) holds. With such \((\theta _*, \xi )\), we obtain \(\beta , \beta ', r_*\) via formulas (3.74), (3.76), and (3.73).
Remark 3.10
We also make a comment regarding \(\ell _0\) in Lemma 3.8. Note that by Remark 3.9, \(\beta '\) and \(\xi _*, \beta _*\) are functions of \(\theta \). Hence \(\ell _0\) depends on \(n, \theta \), which in term depends on \(n, s, s''\).
With Lemma 3.8 at hand, we are ready to prove the precise estimate regarding the energy functional (3.54) in the context of the Boltzmann equation.
Proposition 3.11
(Energy functional interpolation inequality) Let \(T>0\) be fixed and let \(\ell _0>0\) be sufficiently large such that it satisfies (3.93). Assume that the given function G satisfies (3.4) and
Fix \(\ell \) such that
and assume that f is a solution of (3.3) which satisfies
Then, there exist \(s'' > 0\) and \(p >1\) such that for any
if we let \({\mathcal {E}}_p(M, T_1, T_2)\) be the energy functional in (3.54) with the parameters \(p, s''\), then it follows that
where the parameters \(\beta _i>1\) and \(a_i>0\) are defined in (3.96) and C is independent of \(K, M, f, T_1, T_2\). Furthermore, the estimate holds for \(h=-f\), solution to Eq. (3.31), with \(f^{(\ell )}_{K, +}\) replaced by \((-f)^{(\ell )}_{K,+}\).
Proof
We start with the bound of the term that involves the x-derivative on the left-hand side of (3.81). This constitutes the main part of the proof. To this end, fix \(\sigma \in (0, 1/2)\). From equation (3.3) one has that
that is, we have defined \({\mathcal {G}}^{(\ell )}_K\) as
where \(\kappa \) can be any number larger than 2. In what follows we take
Choose the parameters in Proposition 2.14 as
where \(1< p < 2\) is chosen to be close enough to 1 such that (3.58) holds and
where \(q_*\) is defined in (3.66) and the third condition guarantees that
With the choices of these parameters and (3.83), we now apply Proposition 2.14 and obtain that
In what follows, we bound the terms on the right-hand side of (3.86) in order with the bound for \(f^{(\ell )}_{K, +}(T_2)\) left to the end. Let \(n = 0\) in Lemma 3.8. Then the third term on the right-hand side of (3.86) is bounded as
For the fourth term one invokes Lemma 2.13 with \(p'=p/(2-p)\) to get
Since (3.84) holds, we can apply Lemma 3.8 in the p and \(p'\) norms with \(n=0\) to obtain that
where the parameters satisfy
So far for Lemma 3.8 to apply, we need
where \(\beta ', \beta _*\) are defined in the proof of Lemma 3.8.
Next we bound the last term on the right-hand side of (3.86). Using Lemma 2.1, the embedding in (3.85), the assumptions for G in (3.80) and Proposition 3.7 with \(j=4\), we get
Letting \(n=12\) and \(n=5\) respectively in Lemma 3.8, we can bound the last two terms in (3.91) as
where for such estimates to hold, we require that
where again \(\beta ', \beta _*\) are defined in the proof of Lemma 3.8. Then we are led to
Since \(\frac{K}{K-M}\ge 1\), we have that
Therefore, we conclude that
with constants \(C, {\widetilde{C}}_0\) independent of \(\epsilon \in [0,1]\).
Finally we bound the second term on the right-hand side of (3.86). By the positivity of the Bessel potential and Fubini’s theorem,
Integrating Eq. (3.82) first in x and then in t, v gives
where the last term satisfies the same bound as in (3.91). Hence the term involving \(f^{(\ell )}_{K, +}(T_2)\) does not add new terms to the bound. Overall, we obtain from (3.86), (3.87), (3.88), (3.94) that
where the constants \(C, C_\ell , C_0, {\widetilde{C}}_0\) are independent of \(f, K, M, T_1, T_2\) and as a summary,
Note that \(\beta _i > 1\) and \(a_i > 0\) for all \(i =1, \cdots , 4\).
For the first two terms on the left-hand side of (3.81) we invoke Proposition 3.3 to get
where the last step follows from similar bounds as in (3.92) and the subsequent procedure that led to (3.94). Together with (3.95) and by choosing \(C_0=C_0(s,\ell )>0\) sufficiently large such that
we obtain the desired estimate in (3.81).
Since \((-f)^{(\ell )}_{K, +}\) satisfies the same bound as \(f^{(\ell )}_{K, +}\) in Proposition 3.7, the same estimate for \((-f)^{(\ell )}_{K, +}\) as in (3.81) holds with Proposition 3.4 replacing Proposition 3.3. \(\square \)
Before showing the \(L^\infty \)-bound of f, we need a closed \(L^2\)-bound of the zeroth level energy \({\mathcal {E}}_0\) given by
where \(f_+\) denotes the positive part of f and
Proposition 3.12
Let \(T>0\) be fixed and \(\epsilon \in [0, 1]\), \(s \in (0, 1)\). Assume that the given function G satisfies (3.4) and
Fix \(\ell \) such that
and assume that f is a solution of (3.3) which satisfies \(\mu + f \ge 0\). Then for any \(0< s' < \frac{s}{2(s+3)}\), there exist \(s'' \in (0, \,\, s' \frac{\gamma }{2 \ell + \gamma })\) and \(p^\flat :=p^\flat (\ell ,\gamma ,s,s') > 1\) such that for any \(1< p < p^\flat \), we have
The same estimate holds for \((-f)^{\ell }_+\) and its associated \({\mathcal {E}}_0\).
Proof
The estimate for \({\mathcal {E}}_0\) follows from the basic energy estimates and the averaging lemma in earlier sections. By Corollary 3.2, the first two terms in \({\mathcal {E}}_0\) satisfy
since by (3.98) \(0\le \Sigma (g)\le 1+C\). Let us concentrate on the term with the spatial fractional differentiation. Invoking Lemma 2.13, it follows that for \(p\in (1,2)\), \(0<s''<\beta \in (0,s')\),
The controls of the \(L^{2p}\)- and \(L^{2p'}\)-norms of \(f^{(\ell )}_+\) are similar and both through suitable interpolations. First,
For any \(\beta > 0\), let \((\eta , \eta ') = (s, \beta )\) in Lemma 2.11. Then by choosing \(\xi (p) = r(s, \beta , 3)\) in that Lemma we have
Consequently, one is led to
If \(\beta \) is in the range
then we have the following interpolation
This can be seen by using the Plancherel and Young’s inequalities:
Take \(q = \frac{2 \ell + \gamma }{2 \ell }\). Then the restriction on \(\beta \) is that
Since \(\xi \) is an increasing function in \(\beta \), we obtain the corresponding range for \(\xi \) and for p by (3.102) as
where \(r(\cdot , \cdot , \cdot )\) is defined in Lemma 2.11. It is clear by its definition that \(p^\flat \) depends on \(\ell , \gamma , s, s'\). Using such parameters and combining the previous estimates, we obtain that
Integrating this estimate in \(t\in (0,T)\) and invoking Corollary 3.2, with \(\ell -\)moments, one is led to
Note that by making p close enough to 1, we have \(p' \in (1, p^\flat )\) where \(p'\) is defined in (3.101). Therefore (3.107) also holds with p replaced by \(p'\). Furthermore, integrating (3.105) in \(t\in (0,T)\) and invoking Corollary 3.2 once more, it holds that
Using the estimates (3.107)-(3.108) in the estimate (3.101), we conclude that
which combined with (3.100) gives (3.99).
The same estimate holds for \((-f)^{\ell }_+\) and its associated \({\mathcal {E}}_0\) since Corollary 3.2 applies to the absolute value of f, which dominates both the negative and positive parts of f. \(\square \)
We are now ready to build the main \(L^\infty \)-estimate for the linear equation (3.3).
Theorem 3.13
(Linear case) Suppose \(G = \mu + g \ge 0\) satisfies that
Let \(F = \mu + f\ge 0\) be a solution to Eq. (3.3) with \(s \in (0, 1)\). Assume the following holds:
Assume that the initial data satisfies
Additionally, assume that the solution satisfies
where \(\ell _0\) satisfies the bound in Proposition 3.11 (more precisely, (3.93)). Then it follows that
where
Proof
Choose \((p, s'')\) close enough to (1, 0) so that
where \(p^\sharp \) and \(p^\flat \) are defined in (3.58) and (3.106) respectively. Such \(p, s''\) guarantee that Lemma 3.8, Proposition 3.11 and Proposition 3.12 hold. We implement a classical iteration scheme to prove an estimation of the \(L^{\infty }\)-norm for solutions. To this end, fix \(K_0>0\) which will be specified later and introduce the increasing levels \(M_k\) as
Take \(T_{2}\in (0, T)\) with \(T>0\) fixed in the analysis. In order to simplify the notation, denote
Choose \(M=M_{k-1}<M_{k}=K\) and \(T_{1}=0\) in Proposition 3.11. Then
and
Taking supremum in \(T_{2}\in [0,T]\) one arrives at
Terms related to the initial data vanish by setting
Then we are led to
Let
and
Then one can check via a direct computation that \({\mathcal {E}}^*_k\) satisfies
By a comparison principle (since \({\mathcal {E}}_{0} = {\mathcal {E}}^{*}_{0}\)) one obtains that
since \(\beta _{i}>1\) (so that \(Q_0>1\)). In particular, we can infer that
which implies that
Thanks to the estimates on \({\mathcal {E}}_0\) given by Proposition 3.12, it follows that
Thus, given (3.114) and (3.115), one is led to
A similar bound is also valid for \(-f\) since Lemma 3.8, Propositions 3.11 and 3.12 all have their counterparts for \(-f\). \(\square \)
Remark 3.14
In fact, since the negative part \(f_-\) satisfies \(f_{-}\le \mu \), it has a Gaussian tail and
4 Linear Local Well-Posedness
In this section we establish the local well-posedness of a modified linearized Boltzmann equation. The ambient space for contraction is
where conditions on k will naturally appear along the argument and the weight is only in v. We will find a solution in the subset \({\mathcal {H}}_k\) defined by
The precise form of the equation under consideration in this section is
where \(g \in {\mathcal {H}}_k\) and we recall the definition of the operator \(L_\alpha \) defined in (3.2):
where \(\alpha \), to be specified later, is chosen to close the energy estimate. The cutoff function \(\chi \) satisfies
Note that since \(g \in {\mathcal {H}}_k\), we have \(\mu + g \chi (\left\langle v\right\rangle ^{k_0} g) \ge 0\).
The main well-posedness statement for the linear equation (4.3) is
Theorem 4.1
Suppose \(s \in (0, 1)\) and \(\epsilon \in [0, 1]\). Let \(g \in {\mathcal {H}}_k\) and let \(\chi \) be the cutoff function defined in (4.5).
(a) Let \(T > 0\) be arbitrary but fixed. Suppose the initial data \(f_0 \in {\mathcal {H}}_k\) and assume that
where \((k-\alpha )^+\) is the positive part of \(k-\alpha \). Suppose \(\delta _0\) is small enough such that (4.18) is satisfied. Then Eq. (4.3) has a unique solution \(f \in {\mathcal {H}}_k\).
(b) In addition to the assumptions in part (a) we further assume that \(\delta _0\) satisfies (4.26) and
where \(\ell _0\) is the weight chosen in Theorem 3.13 (more precisely, (3.93)). Then there exist \(\epsilon _*\) and \(\delta _{*}\) small enough such that for any \(T \in (0, 1)\), if the initial data satisfies
then for any \(0 \le \epsilon \le \epsilon _*\), the solution obtained in part (a) satisfies
The choice of \(\epsilon _*, \delta _{*}\) only depends on \(\gamma , s, k_0\).
Proof
(a) We will use a similar strategy of applying the Hahn-Banach theorem as in [12] to obtain a solution in \({\mathcal {H}}_k\). Denote \({\mathcal {T}}\) as the operator
where the adjoint is taken with respect to the inner product of \( L^2_x L^2_k(\mathbb {T}^3 \times {{\mathbb {R}}}^3)\) for each time t.
The main step is to show the coercivity of \({\mathcal {T}}\) on test functions. Let \(\mathcal {S}\) be the test function space given by
and for \(h \in \mathcal {S}\) denote
Then
The bound of each term is as follows. First,
For ease of notation, denote
It is clear that \(T_0^*\) has a similar structure as \(T_0\) in (3.8). Hence we first get a similar bound as in (3.10):
However, unlike (3.11), we cannot apply Proposition 2.8 directly since having a bound depending on an \(L^1_k\)-norm of g is unwarranted. Instead we revise the proof of Proposition 2.8 to obtain a proper bound. To this end, we make a similar decomposition as in (2.15) by using Lemma 2.7:
Estimates for \(\Gamma _1^*, \Gamma _4^*\) and \(\Gamma _5^*\) are the same as in the proof of Proposition 2.8, which combined with part (b) of Proposition 2.9 gives
by taking \(k_0\) large enough such that \(k_0 > 7 + \gamma \). The bounds for \(\Gamma _2^*\) and \(\Gamma _3^*\) are trickier, since as mentioned before we want to avoid introducing the \(L^1_k\)-norm of g. We do so by using the extra weight \(\left\langle v\right\rangle ^\alpha \) from the regularizing term in (4.3) to compensate for the loss of weights. The term \(\Gamma _2^*\) is given by
where \(3^+\) denotes any number close to and larger than 3. In the above estimate we have applied the bound
Since by (4.6),
it holds that
The same bound applied to \(\Gamma _3^*\), which combined with (4.14) and (4.16) gives
Combining estimates of all the three terms in \(\left\langle {\mathcal {T}}h, \, h \,\right\rangle _k\) we obtain that
by taking
Note that the restriction of \(\delta _0\) is independent of \(\epsilon \). By Gronwall’s inequaliy, we have for any \(t \in [0, T]\),
Note the following bounds:
and
These together with (4.19) give
which further implies that
As a consequence,
Moreover, we have
This also implies that
Define
Then \(\mathcal {W}\) is a subspace of
Note that if we let
Then
where the adjoint is taken in the weighted space \(L^2(0, T; L^2_x L^2_k(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\). Thus for any test function \(h \in {\mathcal {S}}\), we have shown that
Denote
Define the linear mapping on \(\mathcal {W}\)
Then by (4.20),
and
By the trilinear estimate in Proposition 2.3 and (4.20), the forcing term involving \(Q(g \chi , \mu )\) satisfies
provided
This shows G is a well-defined bounded linear functional on \(\mathcal {W}\), which then can be extended to \(\mathcal {Y}_1\) and \(\mathcal {Y}_2\). Therefore, there exists \(f \in {\mathcal {X}}^{(1)} \cap {\mathcal {X}}^{(2)}\) such that
with the norm of f satisfying
To show that \(f \in {\mathcal {H}}_k\), we need to prove that \(\mu + f \ge 0\). This can be done similarly as in [12]. Let \(F = \mu + f\) and \(G = \mu + g \chi \). Then \(G \ge 0\) and F satisfies
Let \(\eta : {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}^+ \cup \{0\}\) be the convex and decreasing \(W^{2, \infty }\)-function given by
Multiply (4.23) by \(\left\langle v\right\rangle ^{2k} \eta '(F) = \left\langle v\right\rangle ^{2k} F_-\). This gives
The term involving Q(G, F) is estimated in the same way as in Sect. 7.1 of [12]. We only need to check the regularizing terms. After integration they satisfy
and
which only adds to the lower-order term in the energy estimate. Hence the similar estimate as in [12] holds and gives \(F_- = 0\), that is, F is non-negative. Combined with (4.22) we have that \(f \in {\mathcal {H}}_k\). The uniqueness of f is guaranteed by the basic energy estimate in Proposition 3.1.
(b) Although (4.22) gives a regularization in v which can induce an \(L^\infty \)-bound of the solution by Theorem 3.13, the bound is undesirable since it depends on \(\epsilon \). Now we show the derivation of a uniform-in-\(\epsilon \) bound for a smaller weight by using the De Giorgi method in Theorem 3.13.
The main step is to show that by letting \(\ell = k_0 - \ell _0 - 7 - \gamma \) in Theorem 3.13, the solution from part (a) satisfies
where C is independent of \(\epsilon \). The main reason that \(\epsilon \) enters the energy estimate in part (a) is because, in the estimates of \(\Gamma _2^*\) and \(\Gamma _3^*\), we have to make use of the artificial regularization \(\epsilon L_\alpha \) to help us control the weighted \(L^\infty \)-norm of \(g \chi \). To avoid this difficulty, we lower the weight by introducing
By taking \(\ell = k_1\) in Proposition 3.1, we obtain the energy estimate
By the embedding of weighted \(L^1\)-norms into \(L^\infty _{k_0}\), we get
by letting
Therefore for \(T \le 1\), there exists \(C_1\) such that
The constant \(C_1\) only depends on \(k_0, \gamma , s\). By interpolation we obtain the bound
Given (4.7), or equivalently,
we now apply Theorem 3.13 to obtain that
where \(K^{lin}_0\) is defined in (3.110). From the definition of \(K^{lin}_0\), it is clear that there exist \(\epsilon _*, T, \delta _{*}\) such that if they are small enough, then
Specifically, we require that \(T < 1\) and
It is then clear that the bounds of \(T, \delta _{*}\) are all independent of \(\epsilon \). \(\square \)
5 Nonlinear Local Theory
In this section we establish the local existence of solutions to the nonlinear Boltzmann equation
The proof is divided into three steps: first, we show the local existence of the regularized modified nonlinear Boltzmann equation which has the same form as (4.3) with g replaced by f. Next, we use the De Giorgi method to show the \(L^\infty _{k_0}\)-bound of the solution, thus automatically removing the cutoff function. Finally, we use strong compactness to pass to the limit in \(\epsilon \) to recover the solution to the original Boltzmann equation. This whole process will be carried out into three subsections. Note that in this section we need to restrict ourselves to the weak singularity case with \(s \in (0, 1/2)\). This is due to insufficient regularization provided by the operator \(\epsilon L_\alpha \) in the contraction argument (see the last step in (5.5)).
5.1 Local Existence to the Modified Boltzmann Equation (MBE)
The modified equation has the form
where \(L_\alpha \) and \(\chi \) are the same as in the linear equation (4.3). The local existence of solutions to (5.1) will be shown by applying the fixed-point argument in \({\mathcal {X}}_k\) to the linear equation (4.3) with a suitable k.
Theorem 5.1
Suppose \(s \in (0, 1/2)\) and
where \(\ell _0\) is the same weight in Theorem 4.1 (precise statement in (3.93)). Suppose \(\epsilon , \delta _0, f_0\) satisfy the assumptions in both part (a) and part (b) in Theorem 4.1. For each such \(\epsilon \), if T is small enough (which may depend on \(\epsilon \)) then (5.1) has a solution \(f \in L^2_k((0, T) \times \mathbb {T}^3 \times {{\mathbb {R}}}^3)\). Moreover, f satisfies the bound
Proof
Let \(k > \max \{8 + \gamma , \, \alpha \}\) and \({\mathcal {H}}_k \subseteq {\mathcal {X}}_k\) be the set defined in (4.2). For a given \(g \in {\mathcal {H}}_k\), define the map
where \(f \in {\mathcal {H}}_k\) is the solution to (4.3). Theorem 4.1 guarantees that \(\Gamma \) is well-defined provided \(\delta _0, \epsilon , T\) are small enough. Moreover, if we choose \(k > \alpha \), then the assumptions in Theorem 4.1 require that
Our goal is to show that \(\Gamma \) is a contraction mapping on the space \({\mathcal {X}}_k = L^\infty (0, T; L^2_x L^2_k(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\) for T small enough. Let \(g, h \in {\mathcal {H}}_k\) and \(f_g, f_h\) be the corresponding solutions such that
The difference of the two equations reads
with the zero initial data for \(f_g - f_h\). Given sufficient regularity obtained in Theorem 4.1, we can now apply direct energy estimates. Multiply (5.3) by \((f_g - f_h) \left\langle v\right\rangle ^{2k}\) and integrate in x, v. Then by similar estimates as for (4.17), we have
By the trilinear estimate in Proposition 2.3, we have, for \(k > \alpha \),
where the last step is precisely the (only) reason that we have to restrict to the weak singularity in this section. The interpolations in the estimates above require that
A sufficient condition is
By Proposition 3.4 in [12], the last term in (5.4) is bounded as
where we have written \((f_g - f_h) \left\langle v\right\rangle ^{2k} = \left( (f_g - f_h) \left\langle v\right\rangle ^\gamma \right) \left\langle v\right\rangle ^{2k-\gamma }\) when applying Proposition 3.4 from [12]. Combining the inequalities above, we obtain
which, by choosing T small enough which may depend on \(\epsilon \), gives
Therefore, \(\Gamma \) is a contraction mapping and we obtain a unique solution to the modified equation (5.1). The uniform bound in (5.2) is a direct consequence of Theorem 4.1. \(\square \)
5.2 \(L^\infty _{k_0}\)-Bound of Solutions to MBE
In this part we show that the solution obtained in Theorem 5.1 is in fact a solution to the regularized Boltzmann equation
The main step is to prove that such a solution satisfies
This way the cutoff function automatically vanishes and we recover a solution to (5.7).
Note that f already satisfies a uniform-in-\(\epsilon \) bound in (5.2). Our goal is to enhance the weight to \(\left\langle v\right\rangle ^{k_0}\). For a large part, the proof of (5.8) parallels that of Theorem 3.13 for the linear case. The central difference, which will manifest itself repeatedly in the proofs below, is that the moment requirement on f for the quadratic problem (5.1) is substantially lessened in comparison to that of the linear equation (4.3). This is due to the quadratic structure of the collision operator which permits us to strategically allocate moments to the appropriate entry of the collision operator. Similar as in Sect. 3, the \(L^\infty _{k_0}\)-estimate is built upon various \(L^2\)-estimates of the solution f and its level-set functions. Hence we will need to lay the ground by proving several propositions before showing the \(L^\infty _{k_0}\)-estimate.
5.2.1 Local in time \(L^{2}\)-Estimates
As the first step we show a uniform-in-\(\epsilon \) weighted \(L^2\)-bound of f, the solution to (5.1). The following proposition is the analog to Proposition 3.1.
Proposition 5.2
(Nonlinear uniform-in-\(\epsilon \) estimate) Let f be a solution to equation (5.1) with singularity \(s \in (0, 1)\). Suppose
Then for any \(\ell \ge \frac{37 + 5\gamma }{2}\), the solution f satisfies, for \(\delta _{5}>0\) sufficiently small,
In particular, if the following additional conditions hold:
then for any \(\epsilon < 1\) and \(t \in [0,T)\), we have
where the constants \(c_0, \delta _5, C_\ell \) are all independent of \(\epsilon \). Furthermore, we have the regularisation in (t, x) as
for any \(s'< \frac{s}{2(s+3)}\).
Proof
This is a direct consequence of [12, Propositions 3.2, 3.4, and Step 1 in Theorem 6.1]. Note that the cutoff function \(\chi \) does not change the proofs in Propositions 3.2 and 3.4 in [12], since the coercivity is guaranteed by (5.9) and the upper bounds follow from
Bounds for the regularising term \(\epsilon L_\alpha \) and the (t, x)-smoothing in (5.13) are both handled in the same way as in the proofs of Proposition 3.1 and Corollary 3.2. \(\square \)
The uniform \(L^2\)-bound in Proposition 5.2 is the first place that one observes the weight difference in the \(\sup _x\)-norm compared with the linear case: the weight \(\left\langle v\right\rangle ^\ell \) does not appear in the \(\sup _x\)-norm in (5.10) as opposed to (3.5) in Proposition 3.1.
5.2.2 A priori \(L^2\)-estimates for level sets
Let us proceed to show the nonlinear counterpart for the a priori estimates for the level sets. We recall that it is a building block for the energy functional interpolation.
Proposition 5.3
Set \(F = \mu + f \ge 0\) and \(s \in (0, 1)\). Suppose \(k_0, \delta _0\) in the definition of \(\chi \) in (4.5) satisfy that \(k_0 > 8 + \gamma \) and \(\delta _0\) small enough such that (5.17) holds and
Then for any \(8 + \gamma < \ell \le k_0\),
where \(\epsilon _3\) is a constant with the bound in (5.22).
Proof
The proof follows from a similar argument to that of Proposition 3.3 for the linear case. We focus on removing the high moment dependence, such as in the norm \(L^\infty _x L^1_{\ell +\gamma }\), in estimate (3.21). First we make a similar decomposition to that of (3.23):
where we have abbreviated \(f \chi (\left\langle v\right\rangle ^{k_0} f)\) as \(f \chi \). Similar as for \(T_1\) in (3.23) (with G there now replaced by \(\mu + f \chi \)), by the regular change of variables together with (2.14) in Propositions 2.8 and 2.9, we bound \({\widetilde{T}}_1\) as
where \(s_1, \gamma _1\) are defined in (2.24) with \(s_1 < s\) and \(\gamma _1 < \gamma \). If we impose that
then
Next we estimate \({\widetilde{T}}_2\) by writing it as
By (2.14) in Proposition 2.8 and (2.27) in Proposition 2.9, we have
The third term \({\widetilde{T}}_{2,3}\) is directly bounded as
In order to bound \({\widetilde{T}}_{2,1}\), we use (2.44) and a regular change of variables to obtain that
Overall we have
Combining the estimates for \({\widetilde{T}}_1\) and \({\widetilde{T}}_2\), we obtain the first bound for the right-hand side as
Next we derive the second bound with the \(H^s\)-norm. To this end, we only need to re-estimate \({\widetilde{T}}_1\) as
Let \(\epsilon _3\) be a constant such that
The desired bound in (5.14) is obtained by multiplying (5.20) by a small enough \(\epsilon _3\), adding it to (5.20) and then interpolating \(L^2_x H^{s_1}_{\gamma _1/2}\) between \(L^2_x H^s_{\gamma /2}\) and \(L^2_{x,v}\). \(\square \)
Remark 5.4
Although in the proof of Proposition 5.3 it seems that the cutoff function plays an essential role in removing the \(\left\langle v\right\rangle ^\ell \)-dependence in the \(L^\infty _x\)-norm, the above estimates in fact hold (with some modifications) even when we treat the original Boltzmann operator Q(F, F). There are two ways to achieve this goal: first, if f is the solution to the modified equation obtained in Theorem 5.1 and \(\ell = k_0\) (which is the case when we apply Proposition 5.3 in the later analysis), we can use the \(L^\infty _{t,x,v}\)-bound of f with a lower weight \(k_0 - \ell _0 - \gamma - 6\). Then the majority of the weight can be transferred to the first component of Q(F, F). Thus it eliminates the need for a high moment in the \(L^\infty _x\) term. The second way is even more general, in the sense that we do not need any a prior \(L^\infty \)-bound on f. Instead we make use of the nonlinear structure and decompose the first entry in Q(F, F) into
and allocate all the \(\left\langle v\right\rangle ^\ell \) to such term and bound it using \(f^{(\ell )}_{\ell , +}\). The price to pay here is to have an extra K in the coefficient in the upper bound. It does not generate any essential problem since K is the upper bound of f which will eventually be small. However, it is more in line with the linear estimates to have homogeneity in K. Hence we opt to use the special structure of \(\chi \) in the proof of Proposition 5.3.
5.2.3 Level Estimate for \(-f\)
Similar as the linear case, we need to show that not only \(f \left\langle v\right\rangle ^\ell < \delta _0\) but also
Hence we establish the counterpart estimates for the level set of \(-f\).
Proposition 5.5
Let \(h = -f\). Suppose \(F = \mu - h \ge 0\). Suppose \(k_0, \delta _0\) in the definition of \(\chi \) in (4.5) satisfy that \(k_0 > 8 + \gamma \) and \(\delta _0\) small enough such that (5.17) holds and
Then for any \(s \in (0, 1)\) and \(8 + \gamma < \ell \le k_0\), the nonlinear estimate has the form
where \(\epsilon _3\) is the same constant in (5.22).
Proof
Decompose the term of interest in a similar way as in (5.15):
Since we have
the same estimates in (5.15) and (5.16) apply to obtain (5.23). \(\square \)
5.2.4 Level-Set Estimate for \(L^1\)-Norm of the Collisional Operator: Quadratic Version
Proposition 5.6
Let f be a solution to Eq. (5.1) and denote \(F = \mu + f\). Then, for any \(T > 0\) and
it holds that
where the coefficients \(C, C_\ell \) are independent of T and recall that
Furthermore, an identical estimate holds if \(f^{(\ell )}_{K, +}\) is replaced by \((-f)^{(\ell )}_{K,+}\).
Proof
The proof is a slight modification of that of Proposition 3.7. We only need to show the bound of
with the aim to remove the \(\ell \)-moment dependence in the \(\sup _x\)-norm in (3.36). The definition of \(W_K\) is in (3.38). Similar to the linear case, write
Decompose the upper bound of the first term \({\widetilde{T}}_1^+\) similarly as in (3.43) with G replaced by \(\mu +f \chi \):
The estimate for \({\widetilde{T}}_{1, 3}^+\) remains the same as for \(T_{1,3}^+\) in Proposition 3.7, which gives
Similar to the estimates of \({\widetilde{T}}_1\) in (5.16), the bounds for \({\widetilde{T}}_{1, 1}^+\) and \({\widetilde{T}}_{1, 2}^+\) follow from the regular change of variables together with (2.14) in Proposition 2.8 and Proposition 2.9, which has the form
Inserting the definition of \(W_K\), we get
Combining the estimates for \({\widetilde{T}}_{1,1}^+, {\widetilde{T}}_{1,2}^+, {\widetilde{T}}_{1,3}^+\), we have
The estimate for \({\widetilde{T}}_2^+\) is similar to those for \({\widetilde{T}}_2\) in (5.19) with \(f^{(\ell )}_{K, +}\) replaced by \(f^{(\ell )}_{K, +} W_K\). This gives
The bound of \({\mathcal {Q}}^{quad}\) is the combination of the bounds of \({\widetilde{T}}_1^+, {\widetilde{T}}_2^+\), which writes
The regularizing term \(L_\alpha \) is bounded in the same way as in (3.53), which will be absorbed into the estimate for \({\mathcal {Q}}^{quad}\). Combining the estimates for \({\mathcal {Q}}^{quad}\) and \(L_\alpha \) and integrating in t gives (5.24). \(\square \)
5.2.5 Time–Space–Velocity Energy Functional: Quadratic Version
Now we establish the key iterative inequality for the quadratic case which is the counterpart of Proposition 3.11.
Proposition 5.7
(Energy functional interpolation inequality) Let \(T > 0\) and \(\ell _0 > 0\) be the same weight as in Theorem 4.1 (precise statement in (3.93)). Let \(8 + \gamma < \ell \le k_0\). Suppose f is a solution to (5.1) which satisfies
where \(\delta _0\) satisfies the smallness condition in (5.17). Furthermore, suppose that
Then there exist \(p, s''\) such that for any \(0 \le T_1 < T_2 \le T\), \(\epsilon \in [0,1]\), \(\alpha \ge 0\) and \(0< M < K\),
for constants \(c_0:=c(\ell ,s,\gamma )\) and \(C:=C(\ell ,s,\gamma ,\alpha )\). In particular, C does not depend on \(T_1, T_2, T\). The parameters \(s'', p, \beta _i, a_i\) depend on \(\ell , \gamma , s\) in the same way as in Proposition 3.11.
Furthermore, the estimate holds for \((-f)\) with \(f^{(\ell )}_{K, +}\) replaced by \((-f)^{(\ell )}_{K,+}\).
Proof
By replacing Propositions 3.3 and 3.7 with Proposition 5.3 and Propositions 5.6, the proof is the same as that of 3.11. \(\square \)
5.2.6 Baseline Level \(\mathcal {E}_0\) and Level Set Iteration: Quadratic Case
Similar to Proposition 3.12, we now show the boundedness of the baseline case \({\mathcal {E}}_0\) which prepares the ground for the \({\mathcal {E}}_k\)-iteration.
Proposition 5.8
Suppose \(s \in (0,1)\), \(T > 0\) and \(\frac{37+5\gamma }{2} < \ell \le k_0\). Suppose f is a solution to Eqs. (5.1) and (5.9), (5.11) hold. Then the baseline energy functional \({\mathcal {E}}_0\) defined in (3.97) satisfies
Proof
The proof follows a similar line as for Proposition 3.12. We only need to replace the linear energy estimate in Corollary 3.2 with its nonlinear counterpart in Proposition 5.2. The proof of the x-regularizing term in Proposition 3.12 applies directly since it holds for general functions rather than merely solutions to any equation. \(\square \)
The \(L^\infty _{k_0}\)-bound now follows:
Proposition 5.9
Let \(T > 0\) and let \(f(t, \cdot , \cdot )\) be a solution to (5.1) with
Let \(\ell _0\) be the weight in Theorem 4.1 (precise statement in (3.93)). Suppose
Moreover, suppose
For any \(T < 1\), if \(\delta _0\) satisfying the assumptions for Theorem 4.1, Propositions 5.2 and 5.3 (more precisely, (4.18), (4.26), (5.11) and (5.17)) and \(\delta _*, \epsilon \) are chosen small enough (which depends on \(\delta _0\)), then we have
The smallness of \(\delta _*\) is independent of \(\epsilon \).
Proof
Take \(\ell = k_0\) in Propositions 5.7 and 5.8 and we only need to show that the assumptions in these two propositions hold. First, by the \(L^\infty \)-bound in (5.29), we have
and
since \(k_0 - \ell _0 - 7 - \gamma > 6 + \gamma + 2s\). We are left to show that
To this end, we apply (5.12) in Proposition 5.2 and get
Note that for (5.12) to hold, we only need the bound in (5.31). In particular, the weight in (5.31) is independent of \(k_0\), which again marks the essential difference between the linear equation and the nonlinear one. Combining Proposition 5.7 and Proposition 5.8 with the same argument in Theorem 3.13, we obtain that
where
Hence for any \(0< T < 1\), (5.30) holds by taking \(\delta _*\) and \(\epsilon \) small enough. \(\square \)
Remark 5.10
The result in Proposition 5.9 applies to the full range of singularity where \(s \in (0, 1)\). This provides the basis for extending the well-posedness result below from the mild to the strong singularity.
To pass in the limit in \(\epsilon \), we need to show that the time interval of existence is independent of \(\epsilon \). To this end, we need to find an explicit relation between the smallness of the initial data and the solution, that is, the relation between \(\delta _*\) and \(\delta _0\). This relation is derived from (5.33) by setting
Take \(T < 1\). Since \(\epsilon _*< 1\), \(\delta _0 < 1\), \(2/p > 1\) and \(2p'/p >1\), we get
Hence, we set
Denote the function
Then \({\mathfrak {H}}\) is invertible on [0, 1]. With this setup we have the following corollary of Proposition 5.9:
Corollary 5.11
Let \(0< T < 1\) and let \(f(t, \cdot , \cdot )\) be a solution to (5.1) with \(k = k_0 + \ell _0 + 2\), \(s \in (0, 1)\) and \(t \in [0, T]\). Let \(\ell _0\) be the weight in Theorem 4.1 (precise statement in (3.93)). Suppose
Suppose \(\delta _0\) satisfies the same bounds (4.18), (4.26), (5.11) and (5.17) as in Theorem 5.9. Let
where \({\mathfrak {H}}\) is defined in (5.34). Moreover, suppose
Then it holds that
We summarize the above results and state the local well-posedness of the regularized Eq. (5.7).
Theorem 5.12
Suppose \(s \in (0, 1/2)\) and
where \(\ell _0\) is the same weight in Theorem 4.1, which only depends on s (precise statement in (3.93)). Suppose
with \(\delta _*\) defined in (5.35) and \(\delta _0\) satisfying the same bounds as in Theorem 5.9. Then for any \(T < 1\), there exists \(\epsilon _*\) such that for any \(\epsilon \le \epsilon _*\), Eq. (5.7) has a solution f satisfying
Proof
Take \(k = k_0 + \ell _0 + 2\) in Theorem 5.1. Then the combination of Theorem 5.1 and Proposition 5.9 shows that there exists \(T_\epsilon \), which may depend on \(\epsilon \), such that (5.7) has a solution f which satisfies (5.38). We claim that such \(T_\epsilon \) can be extended to T independent of \(\epsilon \). Indeed, by Corollary 5.11 and Theorem 5.1 we first extend \(T_\epsilon \) to \({\widetilde{T}}_\epsilon \), where \({\widetilde{T}}_\epsilon \) is the largest interval such that
Such a bound, together with the basic \(L^2\)-estimate in Proposition 5.2, the \(L^2\)-level-set estimate in Proposition 5.3 and the \(L^\infty \)-estimate in Proposition 5.9 that are all independent of \(\epsilon \), gives
Hence \({\widetilde{T}}_\epsilon \) can be continued to the maximal interval [0, T) for any \(T < 1\). \(\square \)
We are ready to pass to the limit and obtain a local solution to the original Boltzmann equation (1.1).
Theorem 5.13
Suppose \(s \in (0, 1/2)\) and
where \(\ell _0\) is the same weight in Theorem 4.1 (precise statement in (3.93)). Suppose \(f_0\) satisfies
where \(\delta _{*}\) is defined in (5.35) with \(\delta _0\) satisfying the same bounds as in Theorem 5.9. Then for any \(T < 1\), the nonlinear Boltzmann equation (1.1) has a unique solution \(f \in L^\infty (0, T; L^2_{k_0+\ell _0+2}(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\). Moreover, f satisfies the bound
Proof
Denote \(f^\epsilon \) as the local solution to (5.7). By Proposition 5.2 and (5.38), we obtain the uniform-in-\(\epsilon \) bound of \(f^\epsilon \) in the following space:
We can extract a subsequence, still denoted as \(f^\epsilon \) such that
By the uniform polynomial decay and a diagonal argument, we have
Such strong convergence then implies the convergence of \(Q(f^\epsilon , f^\epsilon )\) to Q(f, f) as distributions. Indeed, if we take \(\phi \in C^\infty _c({{\mathbb {R}}}^3_v)\) as a test function, then
Therefore we obtain a solution f to the nonlinear Boltzmann equation (1.1), where f lives in the space
\(\square \)
6 Nonlinear Global Theory
In this section we extend the local-in-time result of the previous section to global, thus proving the main theorem for the weakly singular case. The key step is to use the spectral theory of the linearised Boltzmann operator for the hard potential case.
In the sequel \({\mathcal {L}}\) stands for the operator
The nonlinear Boltzmann equation is recast as
We recall the consequence of the spectral property of \({\mathcal {L}}\) shown in Theorem 5.8 in [12]:
Theorem 6.1
([12]) Let h be the solution to the linear equation
where \(h^{in}\) has zero mass, momentum and energy. Let \(\ell > \frac{5 \gamma + 37}{2}\) so that the spectral gap of \({\mathcal {L}}\) (Theorem 4.4 in [12]) holds. Then there exists \(T_0 > 0\) such that
and for any \(t \ge T_0\),
Here \(\lambda >0\) is the same decay rate as in the spectral gap estimate in Theorem 4.4 in [12].
Using Theorem 6.1 we show a lemma which is an intermediate step in establishing the global \(L^2\)-bound.
Lemma 6.2
Assume that \(h \in L^{2}\left( 0,T;L^{2}_x L^2_\ell \right) \) has zero total mass, momentum, and energy:
Then, for any \(s\in (0,1)\), \(\ell \ge \frac{5 \gamma + 37}{2}\) and \(t > 0\), it follows that
where \(\lambda >0\) is the spectral gap of \({\mathcal {L}}\) in \(L^{2}_x L^2_\ell \).
Proof
Assume first that \(t\le T_{0}\) with \(T_{0}\) defined in Theorem 6.1. Then
where for the latter inequality we used the time invariance of the semigroup and (6.2). For the case \(t>T_0\) split the integration as
The integral in \((0,T_0)\) falls into the previous case. For the interval \((T_0,t)\) one has
Note that for the first integral in the right side of (6.5) one has that \(w - \tau \ge T_0\). Invoking (6.3) one has
where \(\lambda >0\) is the spectral gap of \({\mathcal {L}}\). As a consequence,
where we have used the Cauchy–Schwarz inequality and changed the order of integration for the last two steps. Finally, for the latter integral in (6.5) one simply has that
Overall, we conclude for the case \(t>T_0\) that
Estimates (6.4) and (6.6) prove the theorem. \(\square \)
Proposition 6.3
Let \(F=\mathcal {\mu } + f\ge 0\) be a solution of the Boltzmann equation (6.1). Assume that
with \(\ell , \delta _0\) satisfying (5.11). In addition, suppose
Then it follows that
for a constant \(C:=C_{\ell }(\lambda ')\). The time relaxation rate \(\lambda ' \in (0,\lambda ]\) where \(\lambda >0\) is the spectral gap in \(L^{2}_x L^2_\ell \) of the linearised Boltzmann operator. Furthermore,
for a constant \(C:=C_{\ell }(\lambda ')\). All constants are independent of \(T>0\).
Proof
Set \(g(t) = e^{\lambda ' t}f(t) \) with \(\lambda '>0\) to be chosen. Then g satisfies
Since \(\ell , \delta _0\) satisfy (5.11), by multiplying estimate (5.10) (with \(\epsilon =0\)) by \(e^{2 \lambda ' t}\), we get
where \(c_2=\frac{c_0\delta _{8}}{8}\). Hence, integrating in time, one gets
Let us estimate the right side of (6.9). The equation for g can also be viewed as
Then, we can write
By [12, Theorem 4.4], the operator \({\mathcal {L}}\) has an spectral gap \(\lambda \) in \(L^{2}_x L^2_{\ell /2}\), provided
Then,
Furthermore, \(Q\big ( f(t) , g(t) \big )\) has total zero mass, momentum, and energy for all \(t\in (0,T)\). Then, Lemma 6.2 implies that for any \(\lambda '\in (0,\lambda )\) it holds that
By Proposition 2.3, it follows that
Consequently, from (6.10), (6.11), and (6.12) one is led to
Take \(\lambda '<\min \{\frac{\gamma _{0}}{8},\lambda \}\) and use estimate (6.13) in estimate (6.9) to conclude that
Choose \(\delta _{0}>0\) such that
Then (6.14) leads to
Plugging this estimate in (6.9) and (5.13) (with \(\epsilon = 0\)), one obtains (6.8) and concludes the proof. \(\square \)
We now have all the ingredients to show the main theorem for the weak singularity and it states
Theorem 6.4
(Global Existence) Let \(s \in (0, 1/2)\) and \(\gamma \in (0, 1]\). Suppose \(\delta _0\) is a constant small enough such that bounds in Theorem 5.9 and (6.15) are satisfied. Let \(\ell _0\) be the same weight in Theorem 4.1 and \(k_0\) be a constant satisfying
Let \(\delta _*^\natural \), defined in (6.17), be the constant measuring the smallness of the data. Suppose the initial data \(f_0\) has zero mass, momentum and energy and satisfies
Then the Boltzmann equation (1.1) has a unique solution \(f \in L^\infty (0, \infty ; L^2_x L^2_{k_0 + \ell _0 + 2}(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\). Moreover, f satisfies
Proof
The reason that Theorem 5.12 (or Theorem 5.13) can only treat a short-time existence is because that the bound in (5.33) (with \(\epsilon = 0\)) relies on T. It will exceed \(\delta _0\) if T is large, which will render the \(L^2\)-estimates invalid. Such dependence of T is through \(K_0^{quad}({\mathcal {E}}_0)\) since \({\mathcal {E}}_0\) grows with T (see (5.28)) when the spectral gap is not used. Equipped now with Proposition 6.3 we can replace Proposition 5.2 in the proof of (5.28) with (6.7) and (6.8) to get
As a result, there exists \(C_{k_0}\) independent of T such that
Similarly as in (5.34) and (5.35), define
Under the smallness assumption in (6.16), we obtain in the same way as in Theorem 5.12 that
This shows for any \(T > 0\), the solution can be extended beyond T, thus giving the global existence. With the weighted \(L^\infty \)-bound, uniqueness follows from a direct energy estimate similar to (5.10). \(\square \)
7 Strong Singularity
In this part we show the well-posedness for the nonlinear Boltzmann equation with a strong singularity. The only reason we have to restrict to the weak singularity in Sects. 5–6 is because that in the construction of solutions in Theorem 5.1, when using the fixed-point argument in (5.5), the regularizing term \(\epsilon L_\alpha \) needs to be used to control the \(H^{2s}\)-norm. All the a priori estimates are performed for the full range of \(s \in (0, 1)\).
To circumvent the difficulty mentioned above when constructing approximate solutions in the strong singularity case, our strategy is to smooth the collision kernel into a weakly singular one and repeat the process in Sects. 3–6 to find approximate solutions uniformly bounded in the smoothing parameter \(\eta \). Note that we can as well simply regularize the kernel into a cutoff one by removing all its singularities. But that will require introducing new estimates for cutoff kernels. Since all the tools are available for the weakly singular kernel in the previous sections, we take a weak-singularity smoothing. The weak singularity itself will not play an essential role.
Without extra means, we will not be able to obtain uniform bounds in \(\eta \). This is because the regularity gained by part of the collision term cannot compensate, uniformly in \(\eta \), the loss of derivatives in the rest of the terms. Consequently, many estimates will not close for the nonlinear Boltzmann operator with the smoothed collision term. To overcome this difficulty, we temporarily add a dissipation term \(\epsilon L_\alpha \) as in the previous sections and will remove it after obtaining a local well-posedness for the nonlinear Boltzmann equation (with \(\epsilon L_\alpha \)) with the strong singularity.
Recall the original Boltzmann equation with a strong singularity \(s \in [1/2, 1)\):
whose collision kernel satisfies
Fix \(s_*\in (0, 1/2)\) such that
For any \(\eta \in (0, 1)\), let \(Q_\eta \) be the approximate operator with the collision kernel
Although the lower bound will not be used in the subsequent proof, we note that the coefficient \(\alpha _0\) is independent of \(\eta \) since
The uniform upper bound in (7.3) is the key for uniform estimates in \(\eta \). Consider the regularized equation
where \(\epsilon \in (0, 1)\) and \(L_\alpha \) is the same operator as in (3.2). First we note that due to the uniform bounds in (7.3), the constant in the trilinear estimate is independent of \(\eta \). This is summarized as
Lemma 7.1
Let b be the original collision kernel with \(s \in [1/2, 1)\) and \(b_\eta \) be the one defined in (7.3). Then there exists C independent of \(\eta \) such that
for any \(\sigma \in [\min \{s-1, -s\}, s]\), \(m \in {{\mathbb {R}}}\), \(\gamma \ge 0\) and \(0< s < 1\).
As mentioned at the beginning of this section, our plan is to repeat the process of proving the well-posedness of (5.7), with the goal to obtain a local well-posedness result for (7.4) over a time interval uniform in \(\eta \). We show that the sequence of intermediate results from Proposition 3.1 to Corollary 5.11 can be modified (with indispensable help from \(\epsilon L_\alpha \)) in the way that their coefficients are all independent of \(\eta \). The main idea is that in all these estimates, we only rely on the upper bound of the collision kernel with no further structures required. We start with the modified equation with the cutoff function in (4.5) (with its solution still denoted as \(f_\eta \)):
Its linearized version is
with the initial \(f_\eta \big |_{t=0} = f_0(x, v)\). Choices of weights remain the same as in the previous sections. We will show the details for the basic energy estimates for the linearized equation to illustrate how to use (7.3) to derive uniform-in-\(\eta \) bounds. The rest of the steps are parallel to those in Sects. 3–6 and their details will be either sketched or omitted. The regularization \(\epsilon L_\alpha \) helps to simplify the estimates, since for each fixed \(\epsilon \), the gain of velocity regularity (and subsequently the hypoellipticity) now comes from \(\epsilon L_\alpha \) instead of Q.
Proposition 7.2
Suppose \(G = \mu + g \ge 0\) and \(\delta _0\) in the cutoff function is small enough such that \(G_\chi = \mu + g \chi \ge 0\) satisfies
Suppose \(s \in [1/2, 1)\). Let \(F_\eta = \mu + f_\eta \) be a solution to Eq. (7.7). Then for any
the solution \(f_\eta \) satisfies
where \(C_{\ell , \epsilon }\) is independent of \(\eta \) but does depend on \(\epsilon \) and \(C_\ell \) is independent of both \(\epsilon \) and \(\eta \). Furthermore, for any \(0 \le T_1< T_2 < T\) and any \(s' \le \frac{1}{8}\), we have the regularisation in t, x as
where the coefficient \(C_{\ell , \epsilon }\) is independent of \(\eta \) and C is independent of \(\epsilon \).
Proof
By (3.7), the regularizing term \(\epsilon L_\alpha \) gives
Since \(\epsilon L_\alpha \) will provide the dominating term in both the weight and the regularity, we can bound the collision term in a more direct way via the trilinear estimate in Lemma 7.1: for \(\ell < k_0 - 5 - \gamma \), it holds that
Combining the two estimates above and apply the Gronwall’s inequality gives (7.9).
Next, we apply the averaging lemma in Proposition 2.14 to obtain the regularisation in x. In light of equation (7.7), if we invoke Proposition 2.14 with
then for any \(0\le T_{1} \le T_{2}<T\),
By the trilinear estimate in Lemma 7.1, it follows that
Applying (7.9) we get
which is the desired inequality showing the spatial regularisation of \(f_\eta \). \(\square \)
The basic \(L^2\)-level-set estimate parallel to Proposition 3.3 is
Proposition 7.3
Suppose \(G = \mu + g \ge 0\) and
Then the level-set function satisfies
where the constants \(C_\epsilon , C\) are independent of \(\eta \).
Proof
Recall the decomposition in (3.23):
where by the positivity of \(\mu + g\chi \) and the same upper bound for \(T_1\) in (3.24), we have
By Lemma 7.1, the coefficient C in the inequality above is independent of \(\eta \). By interpolation,
The second term on the right-hand side of 7.13 satisfies the same bound as for \(T_2\) in (3.23) since only the upper bound of \(b_\eta \) is needed in the estimates. Moreover, the regularizing term \(\epsilon L_\alpha \) satisfies the same bound as in (3.22), which combined with (7.14) gives (7.3). \(\square \)
The counterpart of Proposition 3.7 states
Proposition 7.4
Let \(G= \mu + g \ge 0\) and \(F = \mu + f\) satisfying equation (7.7). Denote
Then, for any \(T > 0\) and
it follows that
where \(C, C_\ell \) are independent of \(\eta \). Identical estimate holds for \({\widetilde{Q}}^-_\eta (\mu +g\chi , -\mu + h)\) with \(f^{(\ell )}_{K, +}\) replaced by \(h^{(\ell )}_{K, +}\).
Proof
As in the proof of Proposition 3.7, we only need to control \({\mathcal {Q}}\) in (3.40) with b replaced by \(b_\eta \). By the same decomposition in (3.41) and a similar argument in Proposition 7.3, we have
The estimate for \(T_R^+\) in (3.40) remains the same. \(\square \)
Lemma 3.8 stays the same since it is independent of the collision kernel. Same with Proposition 3.11. Using the energy bound in Proposition 7.2 which is similar to Corollary 3.2, we obtain a similar bound for \({\mathcal {E}}_0\) (with \(s=1\) and \(s' < 1/8\)) as in Proposition 3.12:
Proposition 7.5
Let \(T>0\) be fixed. Suppose \(\epsilon \in [0, 1]\) and \(s \in [1/2, 1)\). Assume that the given function \(G = \mu + g \ge 0\). Suppose \(\ell \) satisfies
and assume that f is a solution of (7.7) which satisfies \(\mu + f \ge 0\). Then for any \(s' < 1/8\), there exist \(s'' > 0\) and \(p^{\flat }:=p^{\flat }(\ell ,\gamma ,s,s') > 1\) such that if \(s'' < s' \frac{\gamma }{2 \ell + \gamma }\) and \(1< p < p^{\flat }\), then we have
where \(C_\epsilon \) is independent of \(\eta \). The same estimate holds for \((-f)^{\ell }_+\) and its associated \({\mathcal {E}}_0\).
Since all the building blocks leading to Theorem 3.13 agree, we have a similar statement for the a priori \(L^\infty \)-bound:
Theorem 7.6
(Linear case) Suppose \(G = \mu + g \ge 0\). Let \(F = \mu + f_\eta \ge 0\) be a solution to Eq. (7.7) with \(s \in [1/2, 1)\). Assume that \(\ell \) satisfies
Assume that the initial data satisfies
Additionally, assume that the solution satisfies
where \(\ell _0\) satisfies the bound in Proposition 3.11 (or (3.93)). Then it follows that
where
Here \(C_\epsilon \) is independent of \(\eta \) and C is independent of both \(\epsilon \) and \(\eta \).
It is clear from Theorem 7.6 that for each \(\epsilon > 0\), if we let T be small enough (with smallness depending on \(\epsilon , \delta _0\) only) and \(\left\Vert \left\langle v\right\rangle ^{\ell } f_0 \, \right\Vert _{L^2_{x,v}\cap L^\infty _{x,v}}\) small enough (with smallness independent of both \(\epsilon \) and \(\eta \)), then
We can now combine the linear and nonlinear theory in Theorems 4.1 and 5.1 to obtain the local well-posedness of (7.6) as follows.
Theorem 7.7
Suppose \(s \in [1/2, 1)\) and let \(b_\eta \) be the regularized collision kernel. Suppose
where \(\ell _0\) is the same weight in Theorem 4.1 (precise statement in (3.93)). Suppose \(\epsilon , \delta _0, f_0\) satisfy the assumptions in both part (a) and part (b) in Theorem 4.1. Then for each such \(\epsilon \), if T is small enough (which only depends on \(\epsilon \)) then (7.6) has a solution \(f \in L^\infty _t ((0, T); L^2_x L^2_k(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\). Moreover, f satisfies the bound
Proof
The proof is the combination of the proofs of Theorems 4.1 and 5.1. When applying the fixed-point argument as in (5.5), we note that the coefficients obtained will depend on \(\eta \). This is the place that the regularization of b in (7.3) takes effect. Specifically, the counterpart of (5.5) is
Thus the first time interval of existence obtained depends on both \(\epsilon \) and \(\eta \). However, since the a priori estimates are independent of \(\eta \), such a solution can be extended to T independent of \(\eta \). \(\square \)
Once the existence of \(f_\eta \) is shown, we can pass to the limit and return to the original operator Q (with \(\chi \)).
Theorem 7.8
Suppose \(s \in [1/2, 1)\) and
where \(\ell _0\) is the same weight in Theorem 4.1 (precise statement in (3.93)). Suppose \(\epsilon , \delta _0, f_0\) satisfy the assumptions in both part (a) and part (b) in Theorem 4.1. Then for each such \(\epsilon \), if T is small enough (which only depend on \(\epsilon \)) then the equation
has a solution \(f \in L^2_{t, x} L^2_k((0, T) \times \mathbb {T}^3 \times {{\mathbb {R}}}^3)\). Moreover, f satisfies the bound
Proof
By Theorem 7.2 and Theorem 7.7, Eq. (7.6) has a solution \(f_\eta \) satisfying
Given the uniform polynomial decay and a diagonal argument, we can extract a subsequence, still denoted as \(f_\eta \), such that
Our goal is to show that
Using a test function \(\phi \), we consider the difference
By Proposition 2.10 and the upper bound of \(b_\eta \) in (7.3), \(E_1\) is bounded as
where C is independent of \(\eta \). The integrand in the inequality above satisfies
Therefore,
To estimate \(E_2\), note that by symmetry (or more precisely, anti-symmetry) and Taylor expansion, \(E_2\) satisfies
where by (7.3), the integrands of the last two terms satisfy the uniform bounds
and
Since the right-hand sides of the inequalities above are integrable, we can apply the Lebesgue Dominated Convergence Theorem and obtain that \(E_2 \rightarrow 0\) as \(\eta \rightarrow 0\). Hence (7.22) holds. \(\square \)
Recall that the only place that the restriction of a weak singularity enters is when we apply the fixed-point argument (see (5.5)) to obtain an approximate solution to Eq. (7.20). Once such restriction is bypassed via Theorem 7.8 , the rest of the results from Proposition 5.2 to Theorem 6.4 all hold, since they are all proved for \(s \in (0, 1)\). This leads us to the main theorem of this paper.
Theorem 7.9
(Global Existence) Let \(s \in (0, 1)\) and \(\gamma \in (0, 1]\). Suppose \(\delta _0\) is a constant small enough such that bounds in Theorem 5.9 and (6.15) are satisfied. Let \(\ell _0\) be the same weight in Theorem 4.1 and \(k_0\) be a constant satisfying
Let \(\delta _*^\natural \), defined in (6.17), be the constant measuring the smallness of the data. Suppose the initial data \(f_0\) has zero mass, momentum and energy and satisfies
Then the Boltzmann equation (1.1) has a unique solution \(f \in L^\infty (0, \infty ; L^2_x L^2_{k_0 + \ell _0 + 2}(\mathbb {T}^3 \times {{\mathbb {R}}}^3))\). Moreover, there exist \(\lambda ' > 0\) such that
Finally, based on the global result and the exponential decay of the \(L^2\)-norm in Theorem 7.9, we can show an exponential decay in the \(L^\infty \)-norm of the solution.
Theorem 7.10
Suppose \(k_0\) and the initial data \(f_0\) satisfy the same conditions in Theorem 7.9. Then there exists \(C_{k_0}, \eta _0 > 0\) such that for any \(t > 1\) the solution obtained in Theorem 7.9 satisfies
where \(\lambda '\) is the same decay rate in Theorem 7.9.
Proof
For any \(K, t_1 > 0\), let \({\mathcal {E}}_p\) be the energy functional similar as in (3.54):
Note that by its definition \({\mathcal {E}}_p\) is decreasing in \(t_1\) and K. The global bounds of f developed in Theorem 7.9 guarantee that \({\mathcal {E}}_p(K, t_1, \infty )\) is well-defined. Moreover, for any \(T \ge 0\) and \(\ell \le k_0 + \ell _0 + 2\),
where \(\lambda '\) is the decay rate in Theorem 7.9.
Our main goal is to remove the dependence on the weighted \(L^\infty \)-norm of \(f_0\) in (5.33) (with \(\epsilon = 0\)) so that the exponential decay in the weighted \(L^2\)-norm of f can be transferred to exponential decay in the \(L^\infty \)-norm. To this end, define the levels
Setting \( f_{k} := f^{(\ell )}_{M_{k},+}\) and proceeding as in the proof of Theorem 3.13, we arrive at
for \(k=1,2,\ldots \). The parameters \(a_i, \beta _i\) are the same as in Theorem 3.13. Fix \(T > 1\) and let \(T_k\) be the increasing time sequence
We further denote \({\mathcal {E}}_k\) as
Integrate (7.26) in \(t_{1}\in [T_{k-1},T_{k}]\) to obtain that
where we have applied the monotonicity of \({\mathcal {E}}_p(\cdot , \cdot , \cdot )\) in its first and second variables. By similar estimates as in (3.92) with the same definitions for \(r_*, \xi _*\), we have
and by (3.60), it holds that
Since we are interested in the long time behaviour we may take \(T\ge 1\) to derive that an analogous estimate to (3.112) with \(a_{i}, \beta _{i}\) defined in (3.96) holds:
The key difference between (7.27) and (3.112) is that \(K_0\) in (7.27) is independent of \(f_0\). Applying the De Giorgi iteration to (7.27) we conclude similarly as in (5.33) (with \(\epsilon = 0\)) that
where \({\mathcal {E}}_0:={\mathcal {E}}_{p}(0,T/2,\infty )\). Hence by the bound in (7.25), we have
In particular, the above inequality holds for \(\ell = k_0\), which is the desired bound in (7.24) for the positive part of f. Analogous computation can be performed for the negative part of f which finishes the bound in (7.24). \(\square \)
8 Proofs of Lemmas 2.1 and 2.2
In this appendix we show the proofs of Lemmas 2.1 and 2.2, starting with Lemma 2.1.
Proof of Lemma 2.1
For the proof it suffices to show, for \(u \in L^p({\mathbb {R}}^d)\),
To show (8.1), we use the expansion formula of pseudo-differential operators (Ex., [37, Theorem 3.1]),
where \(p_{(\beta )}^{(\alpha )}(v,\xi ) = \partial _\xi ^\alpha (-i\partial _v)^\beta p(v,\xi )\) for the symbol \(p(v,\xi )\). If \(N > d+1 + |\ell | + |\theta |\) then \(\widetilde{r}_N(v, \xi ) {\mathop {=}\limits ^{\Delta }}\left\langle v\right\rangle ^\ell r_N(v, \xi ) \left\langle \xi \right\rangle ^{-\theta }\) belongs to the symbol class \(S_{1,0}^{-d-1}\). In fact, it follows from [37, Theorem 3.1] that
Using the elementary identities
we have, for \(m, k \in {{\mathbb {N}}}\) sufficiently large,
Since \(\left\langle \xi \right\rangle \) and \(\left\langle \xi + \tau \eta \right\rangle \) are equivalent in \(I_1\), it follows that
and moreover the same bound for \(|I_2|\) holds if \(2m > N- \theta +d\). Using \( \left\langle v+y\right\rangle ^{-1} \left\langle y\right\rangle ^{-1} \le \left\langle v\right\rangle ^{-1}\), and taking k satisfying \(2k > N+ |\ell | +d\), we see that \({\widetilde{r}}_N(v, \xi )\) belongs to the desired symbol class. If we put \(K(v,z) = \int e^{i z \cdot \xi }{\widetilde{r}}_N(v, \xi )d\xi /(2\pi )^d\) then we have \({\widetilde{r}}_N(v,D_v) u (v) = \int K(v, v-y)u(y)dy\) and
which concludes that \({\widetilde{r}}_N(v, D_v)\) is \(L^p\) bounded operator for \(p \in [1,\infty ]\). Next we consider the the \(L^p\) boundedness of terms \(\left\langle v\right\rangle ^\ell (\left\langle v\right\rangle ^{-\ell } )_{(\alpha )}(\left\langle D_v\right\rangle ^\theta )^{(\alpha )} \left\langle D_v\right\rangle ^{-\theta }\) for \(0 \le |\alpha |< N-1\). Since the term for \(\alpha =0\) is identity, its \(L^p\) boundedness is trivial. Note that the multiplication \(\left\langle v\right\rangle ^\ell (\left\langle v\right\rangle ^{-\ell } )_{(\alpha )}\) is \(L^p\) bounded operator. If we put \(Q_\alpha (\xi ) = (\left\langle \xi \right\rangle ^\theta )^{(\alpha )} \left\langle \xi \right\rangle ^{-\theta }\) for \(\alpha \ne 0\), then the proof of (8.1) is completed by the fact that the Fourier multiplier \(Q_\alpha (D_v)\) is \(L^p\) bounded. Indeed, one can see that \(K_\alpha (z) {\mathop {=}\limits ^{\Delta }}\int e^{i z \cdot \xi } Q_\alpha (\xi ) d\xi /(2\pi )^d \in L^1\), more precisely, \(|K_{\alpha }(z)| \le C |z|^{-d+1}\) if \(|z| < 1\) and \(|K_\alpha (z)| \le C_m|z|^{-2m}\) if \(|z| \ge 1\) for any \(m \in {{\mathbb {N}}}\) satisfying \(2m > d\). To obtain these estimates, take a cutoff function \(\varphi (\xi ) \in C_0^\infty ({{\mathbb {R}}}^d)\) satisfying \(\varphi =1\) for \(|\xi | \le 1\) and \(\varphi =0\) for \(|\xi | \ge 2\), and decompose
for any \(A > 0\). Then we have
because \(\varphi (\xi /A) \in S^0_{1,0}\) and \(Q_\alpha (\xi ) \in S^{-1}_{1,0}\). Choosing \(A = |v|^{-1}\), we have the desired estimate for \(K_\alpha \) when \(|z| \le 1\), and another estimate is obvious by considering the same formula without the cutoff function \(\varphi \).
For the proof of (8.2) we use the expansion formula twice. First expansion is
where \(r_{1,N}(v, \xi )\) satisfies \(\left\langle v\right\rangle ^{\ell } \left\langle \xi \right\rangle ^{\ell } r_{1,N}(v,\xi ) \in S_{1,0}^{-d-1)}\) if N is chosen sufficiently large. This implies that the symbol of \(\left\langle D_v\right\rangle ^\theta \left\langle v\right\rangle ^{\ell } r_{1,N}(v,D_v)\) belongs to \(S_{1,0}^{-d-1}\), and hence one can show that \(\left\langle D_v\right\rangle ^\theta \left\langle v\right\rangle ^{\ell } r_{1,N}(v,D_v)\) is \(L^p\) bounded, by the same way as before. Since \(\left\langle D_v\right\rangle ^\theta \left\langle v\right\rangle ^\ell \left\langle v\right\rangle ^{-\ell } \left\langle D_v\right\rangle ^{-\theta } = Id\), it suffices to consider the \(L^p\) boundedness of \(\left\langle D_v\right\rangle ^\theta \left\langle v\right\rangle ^\ell (\left\langle v\right\rangle ^{-\ell } )_{(\alpha )}(\left\langle D_v\right\rangle ^{-\theta })^{(\alpha )}\) for \(\alpha \ne 0\). Use the expansion formula again
If \({\widetilde{N}}\) is large enough, then \(r_{2,{\widetilde{N}}}(v, D_v)(\left\langle D_v\right\rangle ^{-\theta })^{(\alpha )}\) is \(L^p\) bounded because its symbol belongs to \(S^{-d-1}_{1,0}\). On the other hand, since \(\Big (\left\langle v\right\rangle ^\ell (\left\langle v\right\rangle ^{-\ell } )_{(\alpha )}\Big )_{(\beta )}\) is bounded function and since \((\left\langle D_v\right\rangle ^\theta )^{(\beta )} (\left\langle D_v\right\rangle ^{-\theta })^{(\alpha )}\) is a Fourier multiplier with its symbol in \(S_{1,0}^{-1}\), we see their product is \(L^p\) bounded operator. Thus we obtain (8.2) . \(\square \)
Next we show the proof of Lemma 2.2.
Proof of Lemma 2.2
By one of the definitions of the fractional Laplacian, we have
where the first term on the right-hand side is readily bounded by \(C \left\Vert (-\Delta _v)^{\alpha /2} f \, \right\Vert _{L^2({{\mathbb {R}}}^3_v)}^2\). Hence we focus on the second term, which satisfies
For any \(v \in {{\mathbb {R}}}^3\), make the separation of the domain as
Then the \(v'\)-integration in (8.3) satisfies
where C is independent of v. Hence by letting \(p \in (2, 6)\) be the exponent in the Sobolev embedding
we can bound the term on the right-hand side of (8.3) as follows:
where \(q = (p/2)' = p/(p-2) > 3/2\) since \(p \in (2, 6)\). We therefore get
The lemma holds by a further integration in x. \(\square \)
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Acknowledgements
R. Alonso gratefully acknowledges the support from Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, grant Bolsa de Produtividade em Pesquisa (303325/2019-4). The research of Y. Morimoto was supported by JSPS Kakenhi Grant No.17K05318. The research of W. Sun was supported by the NSERC Discovery Grant R611626. T. Yang’s research was supported by a fellowship award from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. SRF2021-1S01).
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Alonso, R., Morimoto, Y., Sun, W. et al. De Giorgi Argument for Weighted \(L^2\cap L^\infty \) Solutions to the Non-cutoff Boltzmann Equation. J Stat Phys 190, 38 (2023). https://doi.org/10.1007/s10955-022-03053-8
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DOI: https://doi.org/10.1007/s10955-022-03053-8