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\(C^\infty \) Smoothing for Weak Solutions of the Inhomogeneous Landau Equation

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Abstract

We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth, and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.

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Notes

  1. Technically, Theorem 2.12 does not apply to f since it is not sufficiently regular; however, a standard mollification argument allows us to sidestep this potential issue. We omit the details.

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Correspondence to Stanley Snelson.

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Both authors were partially supported by National Science Foundation Grant DMS-1246999. CH was partially supported by NSF grant DMS-1907853. SS was partially supported by a Ralph E. Powe Award from ORAU.

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Communicated by C. Mouhot

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Both authors were partially supported by National Science Foundation grant DMS-1246999. CH was partially supported by NSF Grant DMS-1907853. SS was partially supported by a Ralph E. Powe Award from ORAU

Appendix A. Bounds on the Coefficients of the Landau Equation

Appendix A. Bounds on the Coefficients of the Landau Equation

In this appendix, we collect the available bounds on the coefficients \({{\overline{a}}}\), \({{\overline{b}}}\), and \({{\overline{c}}}\) in the Landau equation (1.3) with soft potentials (\(\gamma \in [-d,0)\)). The estimates in Propositions A.1 and A.2 were derived in [20] and [3]. Earlier, corresponding bounds in the case \(\gamma \geqq 0\) were shown in [6].

Proposition A.1

Let \(f:[0,T_0]\times {\mathbb {R}}^d\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}_+\) satisfy the bounds (1.8), (1.9), and (1.10), and let \({{\overline{a}}}\) be defined by (1.5). If \(\gamma \in [-d,0)\), then for unit vectors \(e\in {\mathbb {R}}^d\),

$$\begin{aligned} {{\overline{a}}}_{ij}(t,x,v) e_i e_j \geqq c{\left\{ \begin{array}{ll} (1+|v|)^{\gamma }, &{} e\in {\mathbb {S}}^{d-1}, \\ (1+|v|)^{\gamma +2},&{} e\cdot v = 0.\end{array}\right. } \end{aligned}$$
(A.1)

If \(\gamma \in [-2,0)\), then \({{\overline{a}}}\) satisfies the upper bound

$$\begin{aligned} {{\overline{a}}}_{ij}(t,x,v) e_i e_j \leqq C{\left\{ \begin{array}{ll} (1+|v|)^{\gamma +2}, &{}e\in {\mathbb {S}}^{d-1},\\ (1+|v|)^{\gamma }, &{} e\cdot v = |v|,\end{array}\right. } \end{aligned}$$
(A.2)

and if \(\gamma \in [-d,-2)\),

$$\begin{aligned} {{\overline{a}}}_{ij}(t,x,v)e_i e_j \leqq C\Vert f(t,x,\cdot )\Vert _{L^\infty ({\mathbb {R}}^d)}^{-(\gamma +2)/d}, \quad e\in {\mathbb {S}}^{d-1}. \end{aligned}$$
(A.3)

The constants c and C depend on d, \(\gamma \), \(m_0\), \(M_0\), \(E_0\), and \(H_0\).

Proposition A.2

Let f be as in Proposition A.1. The coefficients \({{\overline{b}}}\) and \({{\overline{c}}}\) defined by (1.6) and (1.7) respectively, satisfy the upper bounds

$$\begin{aligned}&\left| {{\overline{b}}}(t,x,v)\right| \nonumber \\&\quad \leqq C{\left\{ \begin{array}{ll} (1+ |v|)^{\gamma +1}, &{}-1 \leqq \gamma<0,\\ (1+|v|)^{\gamma +1}(1+\Vert f\Vert _{L^\infty (B_{1}(v))})^{-(\gamma +1)/d}, &{}\dfrac{-3d-2}{d+2}\leqq \gamma< -1,\\ (1+|v|)^{-2-2(\gamma +1)/d}\left( 1+\Vert f\Vert _{L^\infty (B_{1}(v))} \right) ^{-\gamma /d}, &{}-d \leqq \gamma < \dfrac{-3d-2}{d+2},\end{array}\right. }\nonumber \\ \end{aligned}$$
(A.4)

and

$$\begin{aligned} {{\overline{c}}}(t,x,v) \leqq C{\left\{ \begin{array}{ll} (1+|v|)^\gamma (1+\Vert f\Vert _{L^\infty (B_{1}(v))})^{-\gamma /d}, &{}\dfrac{-2d}{d+2}\leqq \gamma< 0,\\ (1+|v|)^{-2-2\gamma /d}\left( 1+\Vert f\Vert _{L^\infty (B_{1}(v))} \right) ^{-\gamma /d}, &{}-d< \gamma < \dfrac{-2d}{d+2},\end{array}\right. }\nonumber \\ \end{aligned}$$
(A.5)

where the constants depend on d, \(\gamma \), \(M_0\), and \(E_0\).

Finally, we show that when \(\gamma \in [-d,-2]\), the coefficients \({{\overline{a}}}\) and \({{\overline{c}}}\) still have the appropriate decay to prove Theorem 3.4, if sufficiently many moments of f are finite.

Lemma A.3

Let \(\gamma \in [-d,-2]\), and let \(f:[0,T_0]\times {\mathbb {R}}^d\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}\) be a bounded function satisfying (1.8) and (1.9). Assume in addition that

$$\begin{aligned} \int _{{\mathbb {R}}^d}|v|^p f(v)\, \mathrm {d}v \leqq P_0, \end{aligned}$$

where p is the smallest integer such that \(p>\dfrac{d|\gamma |}{2+\gamma +d}\). Then the upper bounds (A.2) hold, with constants depending on d, \(\gamma \), \(M_0\), \(E_0\), \(P_0\), and \(\Vert f\Vert _{L^\infty ([0,T_0]\times {\mathbb {R}}^{d}\times {\mathbb {R}}^d)}\).

If, in addition, \(\gamma > -d/2-1\), there is an \(\varepsilon >0\) depending on d and \(\gamma \) such that

$$\begin{aligned} {{\overline{c}}}(t,x,v) \leqq C(1+|v|)^{\gamma +2-\varepsilon }, \end{aligned}$$

with C depending on the same quantities.

Proof

For any \(e\in {\mathbb {S}}^{d-1}\), the formula (1.5) implies

$$\begin{aligned} {{\overline{a}}}_{ij}(t,x,v)e_i e_j&= a_{d,\gamma } \int _{{\mathbb {R}}^d} \left( 1 - \left( \frac{w\cdot e}{|w|}\right) ^2\right) |w|^{\gamma +2} f(v-w)\, \mathrm {d}w \\&\lesssim \int _{{\mathbb {R}}^d} |w|^{\gamma +2} f(v-w)\, \mathrm {d}w. \end{aligned}$$

Let \(r := \frac{1}{2} |v|^{(\gamma +2)/(\gamma +2+d)}\), \(R = |v|/2\), and define

$$\begin{aligned}&I_1 = \int _{B_{r}} |w|^{\gamma +2} f(v-w)\, \mathrm {d}w, \quad I_2 = \int _{B_{R}{\setminus } B_{r} } |w|^{\gamma +2} f(v-w)\, \mathrm {d}w, \\&\quad I_3 = \int _{{\mathbb {R}}^d {\setminus } B_{R}} |w|^{\gamma +2} f(v-w)\, \mathrm {d}w. \end{aligned}$$

We have

$$\begin{aligned} I_1\lesssim & {} \Vert f\Vert _{L^\infty } r^{d+\gamma +2} \lesssim |v|^{\gamma +2},\\ I_2\lesssim & {} r^{\gamma +2} |v|^{-p}\int _{B_{R}} |v-w|^p f(v-w) \, \mathrm {d}w \lesssim P_0 |v|^{-p+(\gamma +2)^2/(d+\gamma +2)}. \end{aligned}$$

Our choice of p implies \(-p< d(\gamma +2)/(d+\gamma +2)\), so that \(I_2 \lesssim |v|^{\gamma +2}\). Finally, for \(|w|\geqq |v|/2\), we have \(|w|^{\gamma +2} \lesssim |v|^{\gamma +2}\), and

$$\begin{aligned} I_3 \lesssim |v|^{\gamma +2} \int _{{\mathbb {R}}^d{\setminus } B_R}f(v-w)\, \mathrm {d}w \leqq M_0|v|^{\gamma +2}. \end{aligned}$$

If e is parallel to v, then proceeding as in [3, Lemma 2.1], we have

$$\begin{aligned}&\int _{{\mathbb {R}}^d} \left( 1 - \left( \frac{w\cdot e}{|w|}\right) ^2\right) |w|^{\gamma +2} f(v-w)\, \mathrm {d}w\\&\quad = \int _{{\mathbb {R}}^d} \left( 1 - \left( \frac{(v-z)\cdot e}{|v-z|}\right) ^2\right) |v-z|^{\gamma +2} f(z)\, \mathrm {d}z\\&\quad = \int _{{\mathbb {R}}^d} \left( |v-z|^2 - \left( |v|-z\cdot e\right) ^2\right) |v-z|^{\gamma } f(z)\, \mathrm {d}z\\&\quad = \int _{{\mathbb {R}}^d} \left( |z|^2 - (z\cdot e)^2\right) |v-z|^{\gamma } f(z)\, \mathrm {d}z\\&\quad = \int _{{\mathbb {R}}^d} |z|^2 \sin ^2\theta |v-z|^{\gamma } f(z)\, \mathrm {d}z, \end{aligned}$$

where \(\theta \) is the angle between v and z. We may assume \(|v|>2\). Let \(R = |v|/2\) and \(q = \dfrac{p(p-2)}{p+d}\). By our choice of p, we have \((\gamma +2)p/q > -d\). If \(z\in B_R(v)\), then \(|\sin \theta | \leqq |v-z|/|v|\), \(|z|\lesssim |v|\), and

$$\begin{aligned}&\int _{B_R(v)} |z|^2\sin ^2\theta |v-z|^\gamma f(z)\, \mathrm {d}z\\&\quad \leqq |v|^{-2}\int _{B_R(v)} |z|^2 |v-z|^{\gamma +2}f(z)\, \mathrm {d}z\\&\quad \leqq |v|^{-p+q}\Vert f\Vert _{L^\infty }^{q/p}\int _{B_R(v)} |z|^{p-q} f(z)^{(p-q)/p}|v-z|^{\gamma +2} \, \mathrm {d}z\\&\quad \lesssim |v|^{-p+q}\left( \int _{B_R(v)}|z|^pf(z)\, \mathrm {d}z\right) ^{(p-q)/p}\left( \int _{B_R(v)} |v-z|^{(\gamma +2)p/q}\, \mathrm {d}z\right) ^{q/p}\\&\quad \lesssim |v|^{-p+q} E_0^{(p-q)/p} \left( |v|^{(\gamma +2)p/q + d}\right) ^{q/p} \lesssim |v|^{\gamma }. \end{aligned}$$

If \(|v-z| \geqq R=|v|/2\), then \(|v-z|^\gamma \lesssim |v|^\gamma \), and we have

$$\begin{aligned} \int _{{\mathbb {R}}^d{\setminus } B_R(v)} |z|^2 \sin ^2\theta |v-z|^\gamma f(z)\, \mathrm {d}z&\lesssim |v|^{\gamma } \int _{{\mathbb {R}}^d{\setminus } B_R(v)} |z|^2 f(z)\, \mathrm {d}z \lesssim E_0 |v|^{\gamma }. \end{aligned}$$

For \({{\overline{c}}}\), our choice of p and the restriction that \(\gamma > -d/2 -1\) implies there is an \(\varepsilon >0\) such that \(-p + \dfrac{\gamma (\gamma +2-\varepsilon )}{d+\gamma } < \gamma + 2\). Define \(r = \frac{1}{2} |v|^{(\gamma +2-\varepsilon )/(d+\gamma )}\), \(R = |v|/2\), and \(I_1\), \(I_2\), \(I_3\) as above. The same method implies that \(I_1 + I_2 + I_3 \lesssim |v|^{\gamma +2-\varepsilon }\). \(\quad \square \)

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Henderson, C., Snelson, S. \(C^\infty \) Smoothing for Weak Solutions of the Inhomogeneous Landau Equation. Arch Rational Mech Anal 236, 113–143 (2020). https://doi.org/10.1007/s00205-019-01465-7

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