On \(A_p\) weights and the Landau equation

Abstract

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrödinger operator whose coefficients are controlled by \(A_p\)-weights.

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Notes

  1. 1.

    so, \(w_2\) also being a \(\mathcal {A}_p\) weight would suffice, per Lemma 3.3

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Acknowledgements

MPG is supported by NSF DMS-1412748 and DMS-1514761. NG is partially supported by NSF-DMS 1700307. NG would like to thank the Fields Institute for Research in Mathematical Sciences, where part of the work in this manuscript was carried out in the Fall of 2014. MPG would like to thank NCTS Mathematics Division Taipei for their kind hospitality. The authors thank Luis Silvestre for many fruitful communications, as well as the anonymous referee for many useful remarks that helped improve this paper.

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Appendix A: Auxiliary computations

Appendix A: Auxiliary computations

A.1: Some properties of the weight \(|w|^{-m}\)

Lemma A.1

The following inequalities hold for any \(m < d \),

$$\begin{aligned} \fint _{B_r(v_0)} \frac{1}{|v|^{m}}\;dv \approx \left( \max \{ |v_0|,2r\}\right) ^{-m}. \end{aligned}$$

The implied constants being determined by m and d. The same result holds (with a different implied constant) if one uses cubes instead of balls.

Remark A.2

In light of the inclusions

$$\begin{aligned} B_r(v_0) \subset Q_r(v_0) \subset B_{r\sqrt{d}}(v_0), \end{aligned}$$

it is clear that the inequality above holds for cubes if and only if it holds balls.

Proof of Lemma A.1

In light of Remark A.2 it suffices to prove the lemma for balls \(B_r(v_0)\). In the integral under consideration, take the change of variables \(v=r(w_0+w)\), where \(w_0 := v_0/r\), then

$$\begin{aligned} \fint _{B_{r}(v_0)}|v|^{-m}\;dv = r^{-m}\fint _{B_1(0)}|w_0+w|^{-m}\;dw \end{aligned}$$

If \(|v_0|\le 2r\), that is if \(|w_0|\le 2\), then an elementary computation shows that

$$\begin{aligned} \fint _{B_1(0)}|w_0+w|^{-m}\;dw \approx 1. \end{aligned}$$

It follows that if \(|v_0|\le 2r\),

$$\begin{aligned} \fint _{B_{r}(v_0)}|v|^{-m}\;dv \approx r^{-m}. \end{aligned}$$

If \(|v_0|\ge 2r\), that is if \(|w_0|\ge 2\), then \(|w_0|/2\le |w_0+w|\le 2|w_0|\) for all \(w\in B_1(0)\), so

$$\begin{aligned} \fint _{B_1(0)}|w_0+w|^{-m}\;dw \approx |w_0|^{-m} = r^{m}|v_0|^{-m}. \end{aligned}$$

It follows that when \(|v_0|\ge 2r\),

$$\begin{aligned} \fint _{B_{r}(v_0)}|v|^m\;dv \approx |v_0|^{m}. \end{aligned}$$

\(\square \)

Remark A.3

If \(m\in [0,d)\), then \(|v_1|\le |v_0|+r\lesssim \max \{|v_0|,2r\}\) for all \(v_1\in B_r(v_0)\). Then, Lemma A.1 implies that

$$\begin{aligned} \fint _{B_r(v_0)}\frac{1}{|v|^m}\;dv \lesssim \frac{1}{|v_1|^m},\;\;\forall \;v_1\in B_r(v_0). \end{aligned}$$

The implied constant depending only on m and d. In particular, for every \(m\in [0,d)\) the function \(\frac{1}{|v|^{m}}\) is a \(\mathcal {A}_1\)-weight.

Lemma A.4

For every \(m \in \mathbb {R}\), \(v_0 \in \mathbb {R}^d\), and \(r\in (0,1)\) we have the estimate

$$\begin{aligned} \fint _{B_r(v_0)}(1+|v|)^{-m}\;dv \approx (1+\max \{|v_0|,2r\})^{-m}. \end{aligned}$$

The implied constants depending only on d and m. The same result holds (with a different implied constant) if one uses cubes instead of balls.

Proof

Again by Remark A.2, it is clear it suffices to prove the lemma for balls \(B_r(v_0)\). Using the same change of variables as in the previous lemma, it follows that

$$\begin{aligned} \fint _{B_r(v_0)}(1+|v|)^{-m}\;dv = \fint _{B_1(0)}(1+r|w_0+w|)^{-m}\;dw,\;\;w_0 = v_0/r. \end{aligned}$$

If \(|v_0|\ge 2r\), which is the same as \(|w_0|\ge 2\), we have as in the proof of the previous lemma that \(|w_0|/2\le |w_0+w|\le 2|w_0|\), thus

$$\begin{aligned} (1+r|w_0|)/2 \le 1+r|w_0+w| \le 2(1+r|w_0|),\;\;\forall \;w\in B_1(0). \end{aligned}$$

Since \(r|w_0|=|v_0|\), it follows that for \(|v_0|\ge 2r\),

$$\begin{aligned} \fint _{B_1(0)}(1+r|w_0+w|)^{-m}\;dw \approx (1+|v_0|)^{-m}. \end{aligned}$$

Next, given that \(r\in (0,1)\), we have \(1+2r \in (1,3)\). Therefore, for \(|v_0|\le 2r\) we have

$$\begin{aligned} (1+\max \{|v_0|,2r\})^{-m} = (1+2r)^{-m} \approx 1. \end{aligned}$$

At the same time, when \(|v_0|\le 2r\) we have \(0\le |v|\le 3r\) for all \(v\in B_r(v_0)\). Then, we can conclude that for \(|v_0|\le 2r\) and \(r\in (0,1)\) we have

$$\begin{aligned} \fint _{B_r(v_0)}(1+|v|)^{-m}\;dv \approx 1 \approx (1+2r)^{-m} = (1+\max \{|v_0|,2r) )^{-m}, \end{aligned}$$

with all the implied constants being determined by d and m, and the Lemma is proved. \(\square \)

In what follows, given \(e\in \mathbb {S}^{d-1}\), we shall write \([e] := \{ r e \mid r\in \mathbb {R} \}\).

Proposition A.5

Let \(\gamma \ge -d\), \(m\ge 0\) with \(m|2+\gamma |<d\), \(v_0 \in \mathbb {R}^d\), \(e\in \mathbb {S}^{d}\), and \(r>0\), then for any \(v_1 \in B_r(v_0)\) we have that

$$\begin{aligned} \fint _{B_r(v_0)}|v|^{(2+\gamma )m}(\Pi (v)e,e)^m\;dv\approx \left\{ \begin{array}{rl} |v_1|^{(2+\gamma )m}(\Pi (v_1)e,e)^m &{} \hbox { if } \hbox {dist}(v_1,[e]) \ge 2r,\\ \max \{ |v_1|,2r\}^{\gamma m}r^{2m} &{} \hbox { if } \hbox {dist}(v_1,[e])\le 2r. \end{array}\right. \end{aligned}$$

The implied constants being determined by dm, and \(\gamma \). The same result holds (with different implied constants) if one uses cubes instead of balls.

Proof

As before, we shall write the proof for the case of balls, noting the same arguments yield the respective result for cubes. Let us write \(p=-\,(2+\gamma )m\). The change of variables \(v=v_0+rw\) yields

$$\begin{aligned} \fint _{B_r(v_0)}|v|^{-p}(\Pi (v)e,e)^m\;dv&= \fint _{B_1(0)}|v_0+rw|^{-p}(\Pi (v_0+rw)e,e)^m\;dw. \end{aligned}$$

Then, writing \(v_0 = rw_0\), we have

$$\begin{aligned} |v_0+rw|^{-p}(\Pi (v_0+rw)e,e)^m = r^{-p} |w_0+w|^{-p}(\Pi (w_0+w)e,e)^m. \end{aligned}$$

We consider two cases, first, if \(|w_0|\le 2\), then

$$\begin{aligned} \fint _{B_1(0)}|w_0+w|^{-p}(\Pi (w_0+w)e,e)^m\;dw \approx \fint _{B_1(0)}(\Pi (w_0+w)e,e)^m\;dw \approx 1. \end{aligned}$$

While, if \(|w_0|\ge 2\), then

$$\begin{aligned} \frac{1}{2}|w_0| \le |w_0+w| \le \frac{3}{2}|w_0|,\;\;\forall \;w\in B_1(0). \end{aligned}$$

These inequalities together with \((\Pi (w_0+w)e,e)\ge 0\) lead to

$$\begin{aligned} \fint _{B_1(0)}|w_0+w|^{-p}(\Pi (w_0+w)e,e)^m\;dw \approx \max \{|v_0|,2r\}^{-p}\fint _{B_1(0)}(\Pi (w_0+w)e,e)^m\;dw. \end{aligned}$$

Next, we make use of the fact that

$$\begin{aligned} |v|^2(\Pi (v)e,e) = \hbox {dist}(v, [e] )^2. \end{aligned}$$

By the triangle inequality, if \(\hbox {dist}(w_0,[e]) \ge 2\), we have for every \(w\in B_1(0)\),

$$\begin{aligned} \frac{1}{2}\hbox {dist}(w_0, [e] )\le \hbox {dist}(w_0+w, [e] )\le \frac{3}{2}\hbox {dist}(w_0, [e] ). \end{aligned}$$

In this case we also have \(|w_0|\ge 2\), so

$$\begin{aligned} (\Pi (w_0+w)e,e) \approx (\Pi (w_0)e,e),\;\;\forall \;w\in B_1(0). \end{aligned}$$

Therefore, in the case \(\hbox {dis}(v_0,[e])\ge 2r\) we have

$$\begin{aligned} \fint _{B_r(v_0)}|v|^{-p}(\Pi (v)e,e)^m\;dv&\approx |v_0|^{-p}(\Pi (w_0)e,e)^m. \end{aligned}$$

It remains to consider the case \(|w_0|\ge 2\) and \(\hbox {dist}(w_0,[e])\le 2\). It is easy to see that

$$\begin{aligned} \fint _{B_1(0)}(\Pi (w_0+w)e,e)^m\;dw \approx \min \{1,|w_0|^{-1}\}^{2m} = |w_0|^{-2m}, \end{aligned}$$

from where it follows that,

$$\begin{aligned} \fint _{B_r(v_0)}|v|^{-p}(\Pi (v)e,e)^m\;dv&\approx |v_0|^{-p} (r/|v_0|)^{2m}. \end{aligned}$$

From the definition of p, \( |v_0|^{-p} \min \{1,r/|v_0|\}^{2m} = |v_0|^{(2+\gamma )m} (r^2|v_0|^{-2})^m\) thus

$$\begin{aligned} \fint _{B_r(v_0)}|v|^{-p}(\Pi (v)e,e)^m\;dv&\approx r^{2m}|v_0|^{\gamma m}. \end{aligned}$$

This proves the proposition for \(v_1 = v_0\). For \(v_1 \in B_r(v_0)\), we make use of the inclusions \(B_r(v_0)\subset B_{2r}(v_1)\) and \(B_{r}(v_1) \subset B_{2r}(v_0)\) to obtain the estimates in the general case. \(\square \)

The estimates on averages from Proposition A.5 yield that the function given by \(a_e(v)=(A_{f,\gamma }e,e)\) (\(e\in \mathbb {S}^{d-1}\) fixed) is almost in the \(\mathcal {A}_1\) class.

Proposition 6.6

Fix \(f\ge 0\), \(f\in L^1(\mathbb {R}^d)\), and \(v_0 \in \mathbb {R}^d, e\in \mathbb {S}^{d-1}\), and \(r>0\). Define

$$\begin{aligned} S(v,r,e) = \{ w \in \mathbb {R}^d \mid \; \hbox {dist}(w-v,[e]) \le 2r\},\;\;v \in \mathbb {R}^d. \end{aligned}$$

Then, for any \(v_1 \in B_r(v_0)\) we have the inequality

$$\begin{aligned} \fint _{B_r(v_0)}a_e(v)\;dv&\le Ca_e(v_1)+r^2\int _{S(v_1,r,e)}f(w)\max \{2r,|v_1-w|\}^\gamma \;dw, \end{aligned}$$

with \(C = C(d,\gamma )\). The same result holds if one uses cubes \(Q_r(v_0)\) in place of balls.

Proof

Throughout the proof let us write \(a_e(v)=(A_{f,\gamma }e,e)\). Recall that we always have,

$$\begin{aligned} \fint _{B_r(v_0)}a_e(v)\;dv&= C_{d,\gamma }\int _{\mathbb {R}^d}f(w)\fint _{B_r(v_0)}|v-w|^{2+\gamma }(\Pi (v-w)e,e)\;dvdw\\&= C_{d,\gamma }\int _{\mathbb {R}^d}f(w)\fint _{B_r(v_0)}K_e(v-w)\;dvdw, \end{aligned}$$

where for convenience we are writing

$$\begin{aligned} K_e(v) = |v|^{2+\gamma }(\Pi (v)e,e). \end{aligned}$$

Now, note that

$$\begin{aligned} \fint _{B_r(v_0)}K_e(v-w)\;dv = \fint _{B_r(v_0-w)}K_e(v)\;dv. \end{aligned}$$

Note that if \(v_1 \in B_r(v_0)\) then \(v_1-w\in B_r(v_0-w)\). Therefore, we may apply Proposition A.5 in \(B_r(v_0-w)\) with \(m=1\) and with respect to the point \(v_1-w\in B_r(v_0-w)\), which leads to a bound according to the location of w: If \(w\not \in S(v_1,r,e)\), then

$$\begin{aligned} \fint _{B_r(v_0)}K_e(v-w)\;dv \approx K_e(v_1-w), \end{aligned}$$

while for \(w \in S(v_1,r,e)\),

$$\begin{aligned} \fint _{B_r(v_0)}K_e(v-w)\;dv \approx r^{2}\max \{2r,|v_1-w|\}^{\gamma }. \end{aligned}$$

Combining these two estimates, it follows that

$$\begin{aligned} \fint _{B_r(v_0)}a_e(v)\;dv&\approx \int _{S(v_1,r,e)^c}f(w)K_e(v_1-w)\;dw\\&\quad +\,r^2\int _{S(v_1,r,e)}f(w)\max \{2r,|v_0-w|\}^{\gamma }\;dw. \end{aligned}$$

Since the first term is no larger than \(Ca_e(v_1)\), for \(C=C(d,\gamma )\), the proposition is proved. \(\square \)

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Gualdani, M., Guillen, N. On \(A_p\) weights and the Landau equation. Calc. Var. 58, 17 (2019). https://doi.org/10.1007/s00526-018-1451-6

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