Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates

Abstract

We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove the global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision “gain” term in Boltzmann’s equation, combined with a novel application of the Kaniel–Shinbrot iteration.

Appendices

Appendix A. An Endpoint Strichartz Estimate

We recall Theorem 10.1 from [15].

Theorem A.1

[15] Let $\sigma > 0$, H be a Hilbert space and $B_0, B_1$ be Banach spaces. Suppose that for each time t we have an operator $U(t) : H \rightarrow B_0^*$ such that

$$\begin{aligned}&\left\| U(t) \right\| _{H \rightarrow B_0^*} \lesssim 1 \end{aligned}$$

(A.1)

$$\begin{aligned}&\left\| U(t) \left( U(s) \right) ^* \right\| _{B_1 \rightarrow B_1^*} \lesssim |t-s|^{-\sigma }. \end{aligned}$$

(A.2)

Let $B_\theta$ denote the real interpolation space $\left( B_0, B_1 \right) _{\theta ,2}$. Then we have the estimates

$$\begin{aligned}&\left\| U(t) f \right\| _{L_t^q B_\theta ^*} \lesssim \left\| f \right\| _H \end{aligned}$$

(A.3)

$$\begin{aligned}&\left\| \int \left( U(s) \right) ^* F(s)\,\mathrm{d}s \right\| _H \lesssim \left\| F \right\| _{L_t^{q^\prime } B_\theta } \end{aligned}$$

(A.4)

$$\begin{aligned}&\left\| \int _{s < t} U(t) \left( U(s) \right) ^* F(s)\,\mathrm{d}s \right\| _{L_t^q B_\theta ^*} \lesssim \left\| F \right\| _{L_t^{\tilde{q}^\prime } B_{\tilde{\theta }}^*}, \end{aligned}$$

(A.5)

whenever $0 {\leqq }\theta {\leqq }1$, $2 {\leqq }q = \frac{2}{\sigma \theta }$, $(q,\theta ,\sigma ) \ne (2,1,1)$, and similarly for $( \tilde{q}, \tilde{\theta } )$. If the decay estimate is strengthened to

$$\begin{aligned} \left\| U(t) \left( U(s) \right) ^* \right\| _{B_1 \rightarrow B_1^*} \lesssim \left( 1 + |t-s| \right) ^{-\sigma }, \end{aligned}$$

(A.6)

then the requirement $q = \frac{2}{\sigma \theta }$ can be relaxed to $q {\geqq }\frac{2}{\sigma \theta }$, and similarly for $( \tilde{q}, \tilde{\theta } )$.

For our application, we will need to think of $\gamma _0 (x,x^\prime )$ as an arbitrary measurable complex-valued function of $x,x^\prime \in \mathbb {R}^2$. Let us take $H = L^2_{x,x^\prime } \left( \mathbb {R}^2 \times \mathbb {R}^2 \right)$, $B_0 = L^2_{x,x^\prime } \left( \mathbb {R}^2 \times \mathbb {R}^2 \right)$, and $B_1 = L^1_{x,x^\prime } \left( \mathbb {R}^2 \times \mathbb {R}^2 \right)$. We employ the notation $\Delta _{\pm } = \Delta _{x} - \Delta _{x^\prime }$. The energy estimate

$$\begin{aligned} \left\| e^{i t \Delta _{\pm }} \gamma _0 \right\| _{L^2_{x,x^\prime }} \lesssim \left\| \gamma _0 \right\| _{L^2_{x,x^\prime }} \end{aligned}$$

(A.7)

is immediate. The dispersive estimate

$$\begin{aligned} \left\| e^{i (t-s) \Delta _{\pm }} \gamma _0 \right\| _{L^\infty _{x,x^\prime }} \lesssim |t-s|^{-2} \left\| \gamma _0 (x,x^\prime ) \right\| _{L^1_{x,x^\prime }} \end{aligned}$$

(A.8)

follows from writing the fundamental solution of $\left( i \partial _t + \Delta _{\pm } \right) \gamma = 0$, that is

$$\begin{aligned} \frac{1}{t^2} e^{i \left( |x|^2 - |x^\prime |^2 \right) / t} \end{aligned}$$

for initial data $\delta (x) \delta (x^\prime )$, and applying Young’s inequality. The relevant parameters for Theorem A.1 are $q = 2$, $\theta = \frac{1}{2}$ and $\sigma = 2$. The real interpolation space $(B_0, B_1)_{\theta ,2}$ is the Lorentz space $L^{\frac{4}{3},2}_{x,x^\prime }$ ([5] Theorem 5.3.1), and its dual is $L^{4,2}_{x,x^\prime }$ ([10] Theorem 1.4.17 (vi)), so we obtain

$$\begin{aligned} \left\| e^{i t \Delta _{\pm }} \gamma _0 \right\| _{L^2_t L^{4,2}_{x,x^\prime } \left( \mathbb {R}^2 \times \mathbb {R}^2 \right) } \lesssim \left\| \gamma _0 \right\| _{L^2_{x,x^\prime } \left( \mathbb {R}^2 \times \mathbb {R}^2 \right) }, \end{aligned}$$

(A.9)

which is the desired inequality.

Appendix B. Fractional Leibniz Formulas

Theorem B.1

Let $s \in \left( 0,1 \right)$ and $n \in \left\{ 2, 3, 4, 5, \dots \right\}$. Then if $f(x),g(x) : \mathbb {R}^n \rightarrow \mathbb {R}$ are measurable functions such that $f \in H^s \left( \mathbb {R}^n \right)$ and $g \in L^\infty \left( \mathbb {R}^n \right)$, then $\left( - \Delta \right) ^{\frac{s}{2}} \left( fg \right)$ and $f \left( -\Delta \right) ^{\frac{s}{2}} g$ are canonically identified with well-defined tempered distributions, and their difference is in $L^2 \left( \mathbb {R}^n \right)$ and the following estimate holds:

$$\begin{aligned} \left\| \left( -\Delta \right) ^{\frac{s}{2}} \left( f g \right) - f \left( - \Delta \right) ^{\frac{s}{2}} g \right\| _{L^2 \left( \mathbb {R}^n \right) } {\leqq }C \left( n,s \right) \left\| \left( - \Delta \right) ^{\frac{s}{2}} f \right\| _{L^2 \left( \mathbb {R}^n \right) } \left\| g \right\| _{L^\infty \left( \mathbb {R}^n \right) }. \end{aligned}$$

(B.1)

Proof

The estimate follows formally from [17], Appendix A, Theorem A.12, in the one-dimensional case, for Schwartz functions f, g. (Also see [16] problem 5.1 and pp. 105–110 for the multidimensional case.) The objective here is to ensure that the result remains true in suitable inhomogeneous Sobolev spaces; the argument is broken into three parts.

(i) For f, g in the Schwartz class, the estimate (B.1) is true due to [17].

(ii) Keeping f fixed in the Schwartz class, we can pass to the distributional limit $g_n \rightharpoonup g \in L^\infty \left( \mathbb {R}^n \right)$ in (B.1), where each $g_n$ is Schwartz and uniformly bounded in $L^\infty$. Every term makes sense because g is a tempered distribution and f is Schwartz.

(iii) We need to pass to the limit $f_n \rightarrow f \in H^s \left( \mathbb {R}^n \right)$ in (B.1), where the $f_n$ are Schwartz and uniformly bounded in $H^s$, but $g \in L^\infty \left( \mathbb {R}^n \right)$ is now fixed. Now $f_n , f$ are uniformly bounded in $H^s \left( \mathbb {R}^n \right)$, hence uniformly bounded in $L^r \left( \mathbb {R}^n \right)$ where $\frac{1}{2} - \frac{s}{n} = \frac{1}{r}$, by the Sobolev embedding theorem. Hence $f_n g$ and fg are uniformly bounded in $L^r \left( \mathbb {R}^n \right)$, so they are well-defined tempered distributions, as is $\left( -\Delta \right) ^{\frac{s}{2}} \left( fg \right)$. For any Schwartz function $\psi$, the estimate

$$\begin{aligned} \int \psi \left( -\Delta \right) ^{\frac{s}{2}} \left( fg \right)&{\leqq }C \left\| \left( -\Delta \right) ^{\frac{s}{2}} \psi \right\| _{L^{r^\prime } \left( \mathbb {R}^n \right) } \left\| f \right\| _{L^r \left( \mathbb {R}^n \right) } \left\| g \right\| _{L^\infty \left( \mathbb {R}^n \right) } \nonumber \\&{\leqq }C \left\| \left( -\Delta \right) ^{\frac{s}{2}} \psi \right\| _{L^{r^\prime } \left( \mathbb {R}^n \right) } \left\| \left( -\Delta \right) ^{\frac{s}{2}} f \right\| _{L^2 \left( \mathbb {R}^n \right) } \left\| g \right\| _{L^\infty \left( \mathbb {R}^n \right) }, \end{aligned}$$

(B.2)

where $\frac{1}{2} - \frac{s}{n} = \frac{1}{r}$, follows from duality, Hölder’s inequality, and Sobolev’s inequality.

Finally we deal with the term $f \left( -\Delta \right) ^{\frac{s}{2}} g$. The idea is to re-write it as

$$\begin{aligned} f \left( -\Delta \right) ^{\frac{s}{2}} g = - \left\{ \left( -\Delta \right) ^{\frac{s}{2}} \left( f g \right) - f \left( - \Delta \right) ^{\frac{s}{2}} g \right\} + \left( -\Delta \right) ^{\frac{s}{2}} \left( fg \right) , \end{aligned}$$

(B.3)

so it is a difference of two things which apparently make sense. Using this difference formula and the commutator estimate of Kenig-Ponce-Vega from the theorem statement, we can prove the estimate

$$\begin{aligned}&\int \psi f \left( -\Delta \right) ^{\frac{s}{2}} g \nonumber \\&\quad {\leqq }C \left( \left\| \psi \right\| _{L^2 \left( \mathbb {R}^n \right) } + \left\| \left( -\Delta \right) ^{\frac{s}{2}} \psi \right\| _{L^{r^\prime } \left( \mathbb {R}^n \right) } \right) \left\| \left( -\Delta \right) ^{\frac{s}{2}} f \right\| _{L^2 \left( \mathbb {R}^n \right) } \left\| g \right\| _{L^\infty \left( \mathbb {R}^n \right) }, \end{aligned}$$

(B.4)

where $g \in L^\infty \left( \mathbb {R}^n \right)$ and $f , \psi$ are in the Schwartz class. We conclude (by density of Schwartz functions in $H^s \left( \mathbb {R}^n \right)$) that $f \left( -\Delta \right) ^{\frac{s}{2}} g$ is canonically identified with a well-defined tempered distribution whenever $f \in H^s \left( \mathbb {R}^n \right)$ and $g \in L^\infty \left( \mathbb {R}^n \right)$; moreover, we can take distributional limits in $f_n$ where needed (keeping $g \in L^\infty \left( \mathbb {R}^n \right)$ fixed) to derive the desired estimate in this class. $\square$

Appendix C. Some general estimates in $L^2 \bigcap L^1$

Assume throughout this appendix that $0 {\leqq }f_0 \in L^2_{x,v} \cap L^1_{x,v}$. As is typical for a kinetic equation, we will consider a suitable mollification (with the same, i.e. unmollified, initial data $f_0$), which takes the following form:

$$\begin{aligned} \left( \partial _t + v \cdot \nabla _x \right) f^n = \frac{Q(f^n,f^n)}{1+\frac{1}{n} \rho _{f^n}}. \end{aligned}$$

(C.1)

Here $n=1,2,3,\dots$ and $f^n (t=0) = f_0$. Note that we are not allowed to mollify the data in general, because that would change the profile of the data, and we are looking for local well-posedness in the critical space $L^2$ (with an auxiliary $L^1$ estimate). It is well-known that the mollified equation (C.1) is globally well-posed for initial data $f_0 \in L^1$; the proof is by a Picard iteration and time-stepping procedure. [9]

Since $Q = Q^+ - Q^-$ (both non-negative) and $\rho _{f^n} {\geqq }0$, we can conclude

$$\begin{aligned} \left( \partial _t + v \cdot \nabla _x \right) f^n {\leqq }Q^+ (f^n,f^n), \end{aligned}$$

(C.2)

which implies

$$\begin{aligned} f^n (t) {\leqq }T(t) f_0 + \int _0^t T(t-t^\prime ) Q^+ ( f^n,f^n), (t^\prime )\,\mathrm{d}t^\prime \end{aligned}$$

(C.3)

where $T(t) = e^{-t v \cdot \nabla _x}$. In particular, for $0 {\leqq }t {\leqq }T$,

$$\begin{aligned} f^n (t) {\leqq }T(t) f_0 + \int _0^T T(t-t^\prime ) Q^+ ( f^n,f^n). (t^\prime )\,\mathrm{d}t^\prime \end{aligned}$$

(C.4)

Apply $Q^+ (\cdot ,\cdot )$ to both sides of this inequality and apply monotonicity to obtain

$$\begin{aligned}&Q^+ (f^n,f^n) {\leqq }Q^+ \left( T(t) f_0, T(t) f_0 \right) \nonumber \\&\quad + \int _0^T Q^+ \left( T(t) f_0, T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ) \right) \,\mathrm{d}t^\prime \nonumber \\&\quad + \int _0^T Q^+ \left( T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ), T(t) f_0\right) \,\mathrm{d}t^\prime \nonumber \\&\quad + \int _0^T \int _0^T Q^+ \left( T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ), T(t-t^{\prime \prime }) Q^+ (f^n,f^n) (t^{\prime \prime }) \right) \,\mathrm{d}t^\prime \,\mathrm{d}t^{\prime \prime }. \end{aligned}$$

(C.5)

Now we take the $L^1_{t \in [0,T]} L^2_{x,v}$ norm of both sides (noting that this quantity might be infinite), and apply Minkowski’s inequality to get

$$\begin{aligned}&\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} {\leqq }\left\| Q^+ \left( T(t) f_0, T(t) f_0 \right) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} \nonumber \\&\quad + \int _0^T \left\| Q^+ \left( T(t) f_0, T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ) \right) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}}\,\mathrm{d}t^\prime \nonumber \\&\quad + \int _0^T \left\| Q^+ \left( T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ), T(t) f_0\right) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} \,\mathrm{d}t^\prime \nonumber \\&\quad + \int _0^T \int _0^T\,\mathrm{d}t^\prime \,\mathrm{d}t^{\prime \prime } \nonumber \\&\qquad \times \left\| Q^+ \left( T(t-t^\prime ) Q^+ (f^n,f^n) (t^\prime ), T(t-t^{\prime \prime }) Q^+ (f^n,f^n) (t^{\prime \prime }) \right) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}}. \end{aligned}$$

(C.6)

Apply Proposition 5.4 to the last term, and Proposition 5.6 to the first three terms, to obtain

$$\begin{aligned}&\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} {\leqq }\delta _{f_0} (T) \left\| f_0 \right\| _{L^2_{x,v}} \nonumber \\&\quad + 2 \int _0^T \delta _{f_0} (T) \left\| T(-t^\prime ) Q^+ (f^n,f^n) (t^\prime ) \right\| _{ L^2_{x,v}} \,\mathrm{d}t^\prime \nonumber \\&\quad + \int _0^T \int _0^T C \left\| T(-t^{\prime }) Q^+ (f^n,f^n) (t^{\prime }) \right\| _{ L^2_{x,v}} \left\| T(-t^{\prime \prime }) Q^+ (f^n,f^n) (t^{\prime \prime }) \right\| _{ L^2_{x,v}} \,\mathrm{d}t^\prime \,\mathrm{d}t^{\prime \prime }, \end{aligned}$$

(C.7)

where $\limsup _{T \rightarrow 0^+} \delta _{f_0} (T) = 0$ (note that $\delta _{f_0} (T)$ depends on the profile of the data for any fixed $T>0$).

Overall, we conclude that

$$\begin{aligned}&\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} {\leqq }\delta _{f_0} (T) \left\| f_0 \right\| _{L^2_{x,v}} \nonumber \\&\quad + 2 \delta _{f_0} (T) \left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}} + C \left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T]} L^2_{x,v}}^2, \end{aligned}$$

(C.8)

where $\limsup _{T \rightarrow 0^+} \delta _{f_0} (T) = 0$. In the case that $\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T_0]} L^2_{x,v}}$ is finite for some $T_0 > 0$, a standard continuity argument allows us to bound this quantity uniformly in n up to some other small time $T>0$ which depends on $f_0$. We can state this is an alternative: there are numbers $C(f_0),T(f_0)$ such that, for each n, exactly one of the following holds:

• (1) Case 1: $\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,\sigma ]} L^2_{x,v}} = \infty$ for every $\sigma >0$

• (2) Case 2: $\left\| Q^+ (f^n,f^n) \right\| _{L^1_{t \in [0,T (f_0)]} L^2_{x,v}} {\leqq }C (f_0)$

In particular $C(f_0),T(f_0)$ are independent of n; hence, as long as Case 2 holds for infinitely many n, we can hope for a compactness argument. Note that once $Q^+ (f^n,f^n)$ is placed uniformly in $L^1_{t \in [0,T (f_0)]} L^2_{x,v}$, the method of Section 9 implies that

$$\begin{aligned} T(-t) \frac{Q^- (f^n,f^n)(t)}{1+\frac{1}{n} \rho _{f^n (t)}} \end{aligned}$$

is uniformly bounded in $L^2_{x,v} L^1_{t \in [0,T (f_0)]}$; in particular, $\left( \partial _t + v \cdot \nabla _x \right) f^n$ is locally integrable in (t, x, v), boundedly with respect to n. Moreover, on $[0,T(f_0)]$, $f^n$ satisfies the full range of Strichartz estimates expected for $L^2$ solutions of the free transp1ort equation, uniformly in n.

Remark C.1

The classical $L^1$ velocity averaging lemma used in [9] requires that both $f^n$ and $\left( \partial _t + v \cdot \nabla _x \right) f^n$ are relatively weakly compact in $L^1 (K)$ for compact sets $K \subset [0,\infty ) \times \mathbb {R}^2_x \times \mathbb {R}^2_v$. However, a refinement cited as Lemma 4.1 in [3] states that, under the condition that $f^n$ is relatively weakly compact in $L^1 (K)$ for compact sets K, it suffices for $\left( \partial _t + v \cdot \nabla _x \right) f^n$ to be uniformly bounded in $L^1 (K)$ for compact K.