# LOCAL CONDITIONAL REGULARITY FOR THE LANDAU EQUATION WITH COULOMB POTENTIAL

###### Abstract

This paper studies the regularity of Villani solutions of the space
homogeneous Landau equation with Coulomb interaction in dimension 3\.
Specifically, we prove that any such solution belonging to the Lebesgue space
$L_{t}^{\infty}L_{v}^{q}$ with $q>3$ in an open cylinder $(0,S)\times B$,
where $B$ is an open ball of $\mathbb{R}^{3}$, must have Hï¿œlder continuous
second order derivatives in the velocity variables, and first order derivative
in the time variable locally in any compact subset of that cylinder.

Immanuel Ben Porat111CMLS, Ecole polytechnique, 91128 Palaiseau cedex, France
<EMAIL_ADDRESS>
##  1 Introduction

The objective of this work is to derive conditional regularity of certain weak
solutions $f=f(t,v)\geq 0$ a.e. of the space homogeneous Landau equation with
Coulomb potential. At the formal level, this equation reads (with the Einstein
summation convention being used)

$\partial_{t}f-\overline{a_{ij}}^{f}\frac{\partial f}{\partial v_{i}\partial
v_{j}}=8\pi f^{2},$ (1)

where

$\overline{a_{ij}}^{f}(t,v)=a_{ij}\star f(t,\cdot),$

with

$a_{ij}(z)=\frac{1}{|z|}(\delta_{ij}-\frac{z_{i}z_{j}}{|z|^{2}}).$

A fundamental existence theorem due to Villani (theorem 3, (i) in [1])
provides the existence of a special class of weak solutions of the associated
Cauchy problem for (1). For an earlier approach to the existence of weak
solutions see [20]. These solutions are known as “H- solutions” or “Villani
solutions” (see definition (2.1)). In this work we will be concerned only with
this class of solutions.

We note the similarity between equation (1) and the semilinear heat equation

$u_{t}-\Delta u=u^{2},$ (2)

for which finite time blow up may occur (see e.g. theorem 1 in [9]). The term
$8\pi f^{2}$ on the r.h.s. of (1) clearly promotes finite time blow-up. On the
other hand, if $f$ increases at some point, the diffusion matrix
$\overline{a_{ij}}^{f}$ in (1) increases as well. This is an important
difference between the Landau equation (1) and the semilinear heat equation 2.
Since any increase in the diffusion matrix offsets the effect of the quadratic
source term $8\pi f^{2}$, whether $f$ blows up in finite time remains an major
open problem at the time of this writing.

One of the most major recent contributions in the study of equation (1) was
produced in [4], where it is proved that for all $T>0$ any H-solution $f$ of
(1) with finite mass, energy and entropy must satisfy the bound

$\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}\frac{|\nabla_{v}\sqrt{f}|^{2}}{(1+|v|^{2})^{\frac{3}{2}}}dvdt<\infty.$

Observe that the bound above offers a striking similarity with Leray’s theory
of weak solutions to the Navier-Stokes equation in space dimension $3$.
Indeed, $\sqrt{f}$ belongs to
$L^{\infty}((0,+\infty);L^{2}(\mathbf{\mathbb{R}}^{3}))$ while
$\nabla_{v}\sqrt{f}$ belongs to
$L^{2}((0,+\infty);L_{loc}^{2}(\mathbf{\mathbb{R}}^{3}))$. Apart from the
$(1+|v|^{2})^{-\frac{3}{2}}$ weight in the dissipation estimate, these bounds
on $\sqrt{f}$ are reminiscent of the bounds satisfied by Leray weak solutions
of the Navier-Stokes equation in space dimension $3$. This loose analogy
suggests the problem of proving conditional regularity results for H-solutions
of the Landau equation (1). For instance, one could seek conditional, local
regularity results analogous to those obtained in the work of Serrin [8] for
the Navier-Stokes equation. Namely, Serrin proves that weak solutions of the
Navier Stokes equations lying in an appropiate mixed Lebesgue space are smooth
with respect to the spatial variable and locally Hï¿œlder continuous with
respect to time. In view of the analogy observed above, it is reasonable to
expect that an analogous result holds for equation (1).

Silvestre’s result in [10] can be thought of as a global Serrin type theorem
for the Landau equation (1). This result is complemented in some sense by the
propagation estimates obtained in [15], which allow to get improved regularity
on the solution by imposing integrability and suitable smallness assumptions
on the initial data. Other conditional regularity results include e.g.
[16],[17], [18] and [19]. In [16] the authors prove regularity of radially
symmetric $L^{p}$ solutions with $p>\frac{3}{2}$ . In [19], local Hï¿œlder
continuity is proved for essentially bounded weak solutions of equation (1).
In [17] regularity is proved provided the solutions satisfy a certain variant
of Poincare inequality together with what is referred by the authors as the
“local doubling property”, which is a reminiscent of the weak Harnack
inequality for supersolutions to parabolic equations. Finally, in [18]
regularity is proved provided the solutions satisfy a variant of the entropy
inequality, which is a reminiscent of the Leray energy inequality, satisfied
by some suitable weak solutions Navier Stokes equations. Regularity is of
course related to uniqueness, and so we refer the reader to [21] for relevant
uniqueness results.

Our new contribution here is obtaining a local Serrin type theorem for
equation (1), that is, a localisation of Silvestre’s theorem. This is
formulated precisely in theorem 2.6, which is the main result of the paper.
Our approach here differs from the strategy introduced in [10], which is based
on apriori estimates for the Landau equation. It is not clear whether it is
possible to establish a local version of these apriori estimates, and we shall
not pursue this direction here. Instead, our approach relies crucially on
analyzing the regularity of the coefficients $\overline{a_{ij}}^{f}$, and can
be roughly summarized as follows:

1\. Obtaining regularity on the coefficients $\overline{a_{ij}}^{f}$(this is
the content of lemma 3.7).

2\. Once regularity on the coefficients is achieved, we may apply classical
existence and uniqueness results from the theory of linear parabolic PDE’s in
order to show that the solution $f$ enjoys 2 weak derivatives in space and one
weak derivative in time (this is the content of theorem 3.1).

3\. Once the solution is known to be (locally) in some appropriate space-time
Sobolev space, we may improve the regularity of the solution via a bootstrap
argument.

##  2 Preliminaries and Main Results

###  2.1 General Notations

We shall consider here space dimension $3$. Let us fix some notation and
recall some basic definitions. We shall denote by
$\omega\subset(0,\infty)\times\mathbb{R}^{3}$ some cylinder, that is, a set of
the form $J\times B$ where $J=(0,S)\subset(0,\infty)$ is some finite open
interval and $B\subset\mathbb{R}^{3}$ is some open ball. Of course, upon
shifting the spatial variable, it is sufficient to prove theorem 2.6 for $B$
centered at the origin. Let $1\leq p\leq\infty,1\leq
q\leq\infty,k\in\mathbb{R}$. We recall the following function spaces.

$L_{t}^{p}L_{v}^{q}(\omega)\coloneqq L^{p}L^{q}(\omega):$ The space of all
measurable maps $f:\omega\rightarrow\mathbb{R}$ with
$||t\mapsto||f(t,\cdot)||_{q}||_{p}<\infty$.

$L_{k}^{p}(\mathbb{R}^{3}):$ the space of all measurable maps
$f:\mathbb{R}^{3}\rightarrow\mathbb{R}$ with
$||v\mapsto(1+|v|^{2})^{\frac{k}{2}}f||_{p}<\infty$.

$L\log L(\mathbb{R}^{3})\coloneqq\\{f\in
L^{1}(\mathbb{R}^{3})|\underset{\mathbb{R}^{3}}{\int}|f(v)||\log|f(v)||dv<\infty\\}$.

$W_{q}^{2,1}(\omega):$ the Banach space consisting of all elements in
$L^{q}(\omega)$ with generalized derivatives of the form
$\partial_{t}^{r}\partial_{x}^{s}$ where $2r+s\leq 2$ and such that
$\partial_{t}^{r}\partial_{x}^{s}\in L^{q}(\omega)$. The norm on
$W_{q}^{2,1}(\omega)$ is defined by
$||u||_{q}^{(2)}=\stackrel{{\scriptstyle[}}{{j}}=0]{2}{\sum}\underset{2r+s=j}{\sum}||\partial_{t}^{r}\partial_{x}^{s}u||_{q}$.

$W_{2}^{1,0}(\omega):$ the Hilbert space consisting of all elements in
$L^{2}(\omega)$ with generalized derivatives of the form $\partial_{x}$ such
that $\partial_{x}\in L^{2}(\omega)$. The scalar product on
$W_{2}^{1,0}(\omega)$ is defined by
$(u,v)=\underset{\omega}{\int}uv+\partial_{x_{k}}u\partial_{x_{k}}vdxdt$.

Let $0<l<1$. We will use the following Hï¿œlder spaces.

$H_{l}^{\ast}(\overline{\omega})$: for each $P,Q\in\omega$ we introduce the
metric $|P-Q|=\max\\{|x^{P}-x^{Q}|,8|t^{P}-t^{Q}|^{\frac{1}{2}}\\}$. Write
$P\backsim Q$ to mean $P=Q+\eta e_{k}$ for some $\eta\in\mathbb{R},1\leq k\leq
3$ (as customary $e_{k}$ stands for the unit vector in direction $k$ in
$\mathbb{R}^{3}$). We consider the distances $d_{P}=\inf\\{|P-Q|\\}_{Q\in
T(P)}$ where $T(P)$ is the set of points on the boundary of $\omega$ for which
there exist a continuous arc connecting $P,Q$ along which the $t$ coordinate
is nondecreasing from $Q$ to $P.$ This gives rise to a distance $d_{PQ}$
defined by $d_{PQ}=\min\\{d_{P},d_{Q}\\}$. For $m\in\mathbb{N}$ we then
consider

$||d^{m}v||_{l}^{\ast}=\underset{P\in\omega}{\sup}d_{P}^{m}|v(P)|+\underset{P,Q\in\omega,P\thicksim
Q}{\sup}d_{PQ}^{m+l}\frac{|v(P)-v(Q)|}{|P-Q|^{l}}.$

$(H_{l}^{\ast}(\overline{\omega}),||d\cdot||_{l}^{\ast})$ is the Banach space
whose elements are all $v$ on $\omega$ admiting a (unique) $C^{0}$ extension
to $\overline{\omega}$ and such that $||dv||_{l}^{\ast}<\infty$ .

$H_{l+2}^{\ast}(\overline{\omega}):$ we consider the norm

$||v||_{l+2}^{\ast}=||v||_{l}+\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}||d\partial_{x_{i}}v||_{l}+\underset{1\leq
i,j\leq
3}{\sum}||d^{2}\partial_{x_{i}x_{j}}v||_{l}+||d^{2}\partial_{t}v||_{l}^{\ast},$

where

$||d^{m}v||_{l}=\underset{P\in\omega}{\sup}d_{P}^{m}|v(P)|+\underset{P,Q\in\omega}{\sup}d_{PQ}^{m+l}\frac{|v(P)-v(Q)|}{|P-Q|^{l}},m\in\mathbb{N}.$

$(H_{l+2}^{\ast}(\overline{\omega}),||\cdot||_{l+2}^{\ast})$ is the Banach
space whose elements are all $v$ on $\omega$ admiting a (unique) $C^{2}$
extension to $\overline{\omega}$ and such that $||v||_{l+2}^{\ast}<\infty$.

$H^{l,\frac{l}{2}}(\overline{\omega}):$ we consider the norm

$||v||^{l,\frac{l}{2}}=\underset{P\in\omega}{\sup}|v(P)|+\underset{(t,x),(t,y)\in\omega}{\sup}\frac{|v(t,x)-v(t,y)|}{|x-y|^{l}}+\underset{(t,x),(s,x)\in\omega}{\sup}\frac{|v(t,x)-v(s,x)|}{|t-s|^{\frac{l}{2}}}.$

$(H^{l,\frac{l}{2}}(\overline{\omega}),||\cdot||^{l,\frac{l}{2}})$ is the
Banach space whose elements are all $v$ on $\omega$ admiting a (unique)
$C^{0}$ extension to $\overline{\omega}$ and such that
$||v||^{l,\frac{l}{2}}<\infty$ .

$H^{1+l,\frac{1+l}{2}}(\overline{\omega}):$ we consider the norm

$||v||^{1+l,\frac{1+l}{2}}=\underset{P\in\omega}{\sup}|v(P)|+\underset{(t,x),(t,y)\in\omega}{\sup}\frac{|v(t,x)-v(t,y)|}{|x-y|^{l}}+\underset{(t,x),(s,x)\in\omega}{\sup}\frac{|v(t,x)-v(s,x)|}{|t-s|^{\frac{l}{2}}}+$

$\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\underset{P\in\omega}{\sup}|\partial_{x_{i}}v(P)|+\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\underset{(t,x),(t,y)\in\omega}{\sup}\frac{|\partial_{x_{i}}v(t,x)-\partial_{x_{i}}v(t,y)|}{|x-y|^{l}}+\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\underset{(t,x),(s,x)\in\omega}{\sup}\frac{|\partial_{x_{i}}v(t,x)-\partial_{x_{i}}v(s,x)|}{|t-s|^{\frac{l}{2}}}.$

We remark that the notations used for the above Hï¿œlder type spaces has
nothing to do with Sobolev spaces (which are frequently denoted by $H^{k}$).

###  2.2 The Landau Equation and H-Solutions

Following [1], let us briefly recall the Landau equation and the notion of
H-solutions (also know as Villani solutions). We refer the reader to [1] for a
elaborative and motivational disscusion.

Define $\Pi:\mathbb{R}^{3}\setminus\\{0\\}\rightarrow M_{3}(\mathbb{R})$ by
$\Pi(z)=I-(\frac{z}{|z|})^{\otimes 2}$, so that
$\Pi_{ij}(z)=\delta_{ij}-\frac{z_{i}z_{j}}{|z|^{2}}$. Define
$a_{ij}(z)=\frac{1}{|z|}\Pi_{ij}(z)$ and
$b_{i}(z)=\stackrel{{\scriptstyle[}}{{j}}=1]{3}{\sum}\partial_{j}a_{ij}(z)$.
For a function $f$ on $\mathbb{R}^{3}$ consider the convolutions
$\overline{a}_{ij}^{f}\coloneqq\underset{\mathbb{R}^{3}}{\int}a_{ij}(v-z)f(z)dz,\overline{b}_{i}^{f}\coloneqq\underset{\mathbb{R}^{3}}{\int}b_{i}(v-z)f(z)dz$.
As is customary, for a function $f$ on $\mathbb{R}^{3}$ we shall write
$M(f)=\underset{\mathbb{R}^{3}}{\int}f(v)dv,E(f)=\underset{\mathbb{R}^{3}}{\int}f(v)\frac{|v|^{2}}{2}dv,H(f)=\underset{\mathbb{R}^{3}}{\int}f(v)\log(f(v))dv$
whenever the quantity on the RHS is well defined. The quantities
$M(f),E(f),H(f)$ are called the mass, energy and entropy of $f$ respectively.

######  Definition 2.1.

Let $f_{0}(v)=f_{0}$ have finite mass, energy and entropy. A H-solution to
equation (1) with initial data $f_{0}$ on $[0,T]\times\mathbb{R}^{3}$ is an
element $f\in C([0,T],\mathcal{D}^{\prime}(\mathbb{R}^{3}))\cap
L^{1}((0,T),L_{-1}^{1}(\mathbb{R}^{3}))$ such that

1\. $f\geq 0$ and $\forall t\in[0,T]:f(t,\cdot)\in
L_{2}^{1}(\mathbb{R}^{3})\cap L\log L(\mathbb{R}^{3})$

2\. $f(0,\cdot)=f_{0}(\cdot)$

3\. $\forall t\in[0,T]$:
$\underset{\mathbb{R}^{3}}{\int}f(t,v)\log(f(t,v))dv\leq\underset{\mathbb{R}^{3}}{\int}f_{0}(v)\log(f_{0}(v))dv$
and
$\underset{\mathbb{R}^{3}}{\int}f(t,v)\psi(v)dv=\underset{\mathbb{R}^{3}}{\int}f_{0}(v)\psi(v)dv$
where $\psi=1,v_{i},|v|^{2}$

4\. $\forall\varphi\in C^{1}([0,T],C_{0}^{\infty}(\mathbb{R}^{3})),\forall
t\in[0,T]$

$\underset{\mathbb{R}^{3}}{\int}f(t,v)\varphi(t,v)dv-\underset{\mathbb{R}^{3}}{\int}f_{0}(v)\varphi(0,v)dv-\stackrel{{\scriptstyle[}}{{0}}]{t}{\int}\underset{\mathbb{R}^{3}}{\int}f(\tau,v)\partial_{t}\varphi(\tau,v)dvd\tau$

$=-\stackrel{{\scriptstyle[}}{{0}}]{t}{\int}\underset{\mathbb{R}^{3}}{\int}\underset{\mathbb{R}^{3}}{\int}\Pi(\nabla_{v}\sqrt{\frac{f(\tau,v)f(\tau,w)}{|v-w|}}-\nabla_{w}\sqrt{\frac{f(\tau,v)f(\tau,w)}{|v-w|}})\sqrt{\frac{f(\tau,v)f(\tau,w)}{|v-w|}})(\nabla_{v}\varphi(t,v)-\nabla_{w}\varphi(t,w))dvdwd\tau.$

The integral on the RHS in 4. is well defined, as explained in detail in [1].
Apriori, it is not clear what is the relation between the notion of a
H-solution and a weak solution in the classical sense. As we will recall in
the next section, it can be shown that in fact H-solutions are weak solutions
in the classical sense, but this is a consequence of a highly nontrivial
theorem of Desvillettes [4]. In what follows, we shall refer to H-solutions of
equation (1) simply as H-solutions.

###  2.3 Known and New Results

First and most foremost we recall that Villani proved the global existence of
H-solutions for all initial data with finite mass, energy and entropy. In
addition he proved that these solutions are weakly Hï¿œlder continuous in
time, as described in the following

######  Theorem 2.2.

([1], Theorem 3, (i)) Let $f_{0}:\mathbb{R}^{3}\rightarrow\mathbb{R}$ have
finite mass, energy and entropy. Then there exist a H-solution $f$ with
initial datum $f_{0}$. Moreover, if $f$ is a H-solution with initial datum
$f_{0}$, then, for all $\varphi\in W^{2,\infty}(\mathbb{R}^{3})$,
$t\mapsto\underset{\mathbb{R}^{3}}{\int}f(t,v)\varphi(v)dv$ is Hï¿œlder
continuous with exponent $\frac{1}{2}$.

The next most important result for our purposes is the following weighted
$L^{2}$ estimate on the distributional derivative of $\sqrt{f}$

######  Theorem 2.3.

([4], Theorem 1) Let $f$ be a H-solution on $[0,T]\times\mathbb{R}^{3}$. Then
$\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}\frac{|\nabla_{v}\sqrt{f(t,v)}|^{2}}{(1+|v|^{2})^{\frac{3}{2}}}dvdt<\infty$.

We remark that theorem 2.3 implies in particular that any $L^{2}(\omega)$
H-solution is in $W_{2}^{1,0}(\omega)$, as can be seen from the identity
$\partial_{v_{j}}f=2\sqrt{f}\partial_{v_{j}}\sqrt{f}$. A list of important
conclusions is derived from theorem 2.3 in [4]. The first one, is that
H-solutions are in fact “usual” weak solutions in the sense of integration
against test functions

######  Corollary 2.4.

([4], Corollary 1.1) Let $f$ be a H-solution with initial datum $f_{0}$. Then
$f\in L^{1}((0,T),L_{-3}^{3}(\mathbb{R}^{3}))$ and for all $\varphi\in
C_{0}^{2}([0,T)\times\mathbb{R}^{3})$ it holds that

$-\underset{\mathbb{R}^{3}}{\int}f_{0}(v)\varphi(0,v)dv-\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f(t,v)\partial_{t}\varphi(t,v)dvdt=$

$\frac{1}{2}\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\stackrel{{\scriptstyle[}}{{j}}=1]{3}{\sum}\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}\underset{\mathbb{R}^{3}}{\int}f(t,v)f(t,w)a_{ij}(v-w)(\partial_{ij}\varphi(t,v)+\partial_{ij}\varphi(t,w))dvdwdt$
$+\stackrel{{\scriptstyle[}}{{i}}=1]{3}{\sum}\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}\underset{\mathbb{R}^{3}}{\int}f(t,v)f(t,w)b_{i}(v-w)(\partial_{i}\varphi(t,v)-\partial_{i}\varphi(t,w))dvdwdt.$

######  Remark 2.5.

If $f$ satisfies the condition $||f(t,\cdot)||_{L^{q}(B)}\leq S_{0}$ for some
$S_{0}>0,q>3$ then for any $\varphi\in C_{0}^{\infty}((0,S)\times B)$ it holds
that

$-\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f\partial_{t}\varphi+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}\overline{a}_{ij}^{f}\partial_{i}f\partial_{j}\varphi=\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f\overline{b}_{i}^{f}\partial_{i}\varphi(t,v).$
(3)

As we will see in lemma (3.7), subject to the above condition
$\overline{a}_{ij}^{f},\overline{b}_{i}^{f}\in
L_{\mathrm{loc}}^{\infty}(\omega)$, which together with theorem (2.3),
justifies that the above integrals are well defined.

Corollary 2.4 and the above remark will enable us to apply classical
regularity, uniqueness and existence results from the theory of linear
parabolic PDEs, which have the above weak formulation. We finish this section
by stating the main result of this work

######  Theorem 2.6.

Let $f$ be a H-solution on $[0,T]\times\mathbb{R}^{3}$. Let
$\omega=(0,S)\times B$ be an open cylinder in $[0,T]\times\mathbb{R}^{3}$.
Suppose there exist $S_{0}>0,q>3$ such that for all $t\in(0,S)$ one has
$||f(t,\cdot)||_{L^{q}(B)}\leq S_{0}$ . Then for all $0<\alpha<1$ we have
$f\in H_{\alpha+2}^{\ast}(\overline{\Omega})$ for each $\Omega\Subset\omega$.

######  Remark 2.7.

We point out that theorem 2.6 implies that $f$ is in fact a classical solution
to equation (1) in $\omega$. Indeed, from the last assertion of theorem 3.8 we
know that $f$ is a strong solution to equation 1, while theorem 2.6 in
particular implies that the weak time derivative and the first and second
order weak spatial derivatives are continuous.

##  3 Conditional Regularity

###  3.1 From H-Solutions To $\mathbf{\mathbf{W}_{2}^{2,1}}$

This subsection is the first step towards the conditional local regularity of
H-solutions as stated in Theorem 2.6. We recall that we have adapted the
notation $\omega=J\times B=(0,S)\times B.$ We prove

######  Theorem 3.1.

Let $f$ be a H-solution. Suppose there exist $S_{0}>0$ and $q>3$ such that for
all $t\in J$ one has $||f(t,\cdot)||_{L^{q}(B)}\leq S_{0}$ . Then $f\in
W_{2}^{2,1}(\Omega)$ for all $\Omega\Subset\omega$. Moreover, $f$ is a strong
solution to equation (1) in $\omega$, that is

$\partial_{t}f-\overline{a_{ij}}^{f}\frac{\partial f}{\partial v_{i}\partial
v_{j}}=8\pi f^{2}$

for a.e. $(t,x)\in\omega$.

First, we recall that local coercivity for the coefficients
$\overline{a}_{ij}^{f}$ has been established in proposition 2.3 of [15]. A
straightforward conclusion of the latter is local ellipticity of the
coefficients, as summarized in the following

######  Proposition 3.2.

(Local Ellipticity) There are constants
$0<c=c(M_{0},E_{0},H_{0},K),C=C(M_{0},S_{0},q)$ with the following property.
Suppose $f\in L_{2}^{1}(\mathbb{R}^{3})\cap L\log L(\mathbb{R}^{3})$ satisfy
$f\geq 0$ a.e. and $M(f)=M_{0},E(f)\leq E_{0},H(f)\leq
H_{0},||f||_{L^{q}(B)}\leq S_{0}$ for some $\frac{3}{2}<q\leq\infty$. Let
$K\Subset\mathbb{R}^{3}$. Then: $\forall\xi\in\mathbb{R}^{n},v\in
K:c|\xi|^{2}\leq\overline{a_{ij}}^{f}(v)$$\xi_{i}\xi_{j}\leq C|\xi|^{2}$.

###### Proof.

The coercivity estimate $c|\xi|^{2}\leq\overline{a_{ij}}^{f}(v)\xi_{i}\xi_{j}$
was established in proposition 2.3 of [15]. In addition for all $\xi\in S^{1}$
we have

$|\overline{a}_{ij}^{f}(v)\xi_{i}\xi_{j}|\leq\underset{\mathbb{R}^{3}}{\int}\frac{2}{|z|}|f(v-z)|dz=\underset{B}{\int}\frac{2}{|z|}|f(v-z)|dz+\underset{\mathbb{R}^{3}-B}{\int}\frac{2}{|z|}|f(v-z)|dz\leq
A||f||_{L^{q}(B)}+2M_{0}$

$\leq AS_{0}+2M_{0}\coloneqq C(q,M_{0},S_{0}),$

where $A=A(q)$ is some constant. ∎

Since the mass and energy of H-solutions are constant in time and the entropy
is uniformly bounded in time we immediately get

######  Corollary 3.3.

There are constants $0<c=c(M_{0},E_{0},H_{0},K),C=C(M_{0},S_{0},q)$ with the
following property. Let $f$ be a H-solution with
$M(f_{0})=M_{0},E(f_{0})=E_{0},H(f_{0})=H_{0},||f(t,\cdot)||_{L^{q}(B)}\leq
S_{0}$ for some $\frac{3}{2}<q\leq\infty$. Let $K\Subset\mathbb{R}^{3}$. Then
for all
$\xi\in\mathbb{R}^{n}$:$c|\xi|^{2}\leq\overline{a_{ij}}^{f}(t,v)\xi_{i}\xi_{j}\leq
C|\xi|^{2}$ for all $(t,v)\in J\times K$.

The strategy of the proof of theorem 3.1 will be roughly as follows.We start
by obtaining regularity of the coefficients $\overline{a}_{ij}^{f}$ . Then we
localize the solution in order to obtain a linear parabolic PDE in divergence
form. This will allow us to apply classical existence and uniqueness results
from the theory of linear parabolic PDE’s. The local ellipticity of the
coefficients (Corollary 3.3) will be freely and frequently used in the sequel.
We start by recalling the following parabolic version of the celebrated De
Giorgi-Nash-Moser method

######  Theorem 3.4.

([11], Theorem 18) Let $V:B\rightarrow\mathbb{R}^{3}$ be a vector field such
that $|V|^{2}\in L^{p}L^{q}(\omega)$, where $1<p<\infty,1<q<\infty$ satisfy
$\frac{2}{p}+\frac{3}{q}<2$. Suppose $u\in L^{\infty}L^{2}(\omega)$ is a weak
subsolution to

$\partial_{t}u-\partial_{j}(\overline{a}_{ij}^{f}\partial_{i}u)+\nabla u\cdot
V\geq 0$ (4)

Then there is some $\alpha>0$ such that $u\in
H^{\alpha,\frac{\alpha}{2}}(\Omega)$ for all $\Omega\Subset\omega$.

We will also heavily rely on the following Calderon Zygmund type theorem

######  Theorem 3.5.

([14], Theorem 4.12) Suppose $\nu\in
C^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$ has the form
$\nu(y)=\mu(\frac{y}{|y|})$ where $\mu\in L^{q}(S^{2})$ for some $q>1$ is an
even function such that $\underset{S^{2}}{\int}\mu(z)d\sigma(z)=0$. Then for
each $1<p<\infty$ the operator $T:L^{p}(\mathbb{R}^{3})\rightarrow
L^{p}(\mathbb{R}^{3})$ defined by $Tf=(\frac{\nu(y)}{|y|^{3}}\ast f(y))(x)$ is
bounded.

We shall first verify that the conditions imposed in theorem 3.5 are indeed
verified for the second derivatives of $a_{ij}$. This verification is based on
elementary (yet somewhat tedious) calculations, and is the content of the
following

######  Lemma 3.6.

For each $1\leq k,l\leq 3$ it holds that
$\partial_{kl}a_{ij}(y)=\frac{\nu_{kl}(y)}{|y|^{3}}$ where $\nu_{kl}$ is as in
theorem 3.5.

Proof. We differentiate

$\partial_{k}a_{ij}(z)=\partial_{k}(\frac{\delta_{ij}}{|z|}+\frac{z_{i}z_{j}}{|z|^{3}})=-\frac{\delta_{ij}z_{k}}{|z|^{3}}+\frac{\delta_{ik}z_{j}+\delta_{jk}z_{i}}{|z|^{3}}-\frac{3z_{i}z_{j}z_{k}}{|z|^{5}}=\frac{\delta_{ik}z_{j}+\delta_{jk}z_{i}-\delta_{ij}z_{k}}{|z|^{3}}-\frac{3z_{i}z_{j}z_{k}}{|z|^{5}}$

.

$\partial_{l}(\frac{\delta_{ik}z_{j}+\delta_{jk}z_{i}-\delta_{ij}z_{k}}{|z|^{3}})=\frac{\delta_{ik}\delta_{lj}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}}{|z|^{3}}-\frac{3z_{l}(\delta_{ik}z_{j}+\delta_{jk}z_{i}-\delta_{ij}z_{k})}{|z|^{5}}$

$\partial_{l}(\frac{3z_{i}z_{j}z_{k}}{|z|^{5}})=\frac{3\delta_{il}z_{j}z_{k}+3\delta_{lj}z_{i}z_{k}+3\delta_{kl}z_{j}z_{i}}{|z|^{5}}-\frac{15z_{i}z_{j}z_{k}z_{l}}{|z|^{7}}.$

Hence

$\partial_{kl}a_{ij}(z)=\partial_{kl}(\frac{\delta_{ij}}{|z|}+\frac{z_{i}z_{j}}{|z|^{2}})=$

$\frac{\delta_{ik}\delta_{lj}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}}{|z|^{3}}-\frac{3z_{l}(\delta_{ik}z_{j}+\delta_{jk}z_{i}-\delta_{ij}z_{k})}{|z|^{5}}-\frac{3\delta_{il}z_{j}z_{k}+3\delta_{lj}z_{i}z_{k}+3\delta_{kl}z_{j}z_{i}}{|z|^{5}}+\frac{15z_{i}z_{j}z_{k}z_{l}}{|z|^{7}}=$

$=\frac{1}{|z|^{3}}(\delta_{ik}\delta_{lj}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}-\frac{3z_{l}(\delta_{ik}z_{j}+\delta_{jk}z_{i}-\delta_{ij}z_{k})+3\delta_{il}z_{j}z_{k}+3\delta_{lj}z_{i}z_{k}+3\delta_{kl}z_{j}z_{i}}{|z|^{2}}+\frac{15z_{i}z_{j}z_{k}z_{l}}{|z|^{4}}).$

Denote by $P_{i},1\leq i\leq 3$ the projection on the $i$-th coordinate. With
this notation we recognize the following identity

$\partial_{kl}(a_{ij})=\frac{1}{|z|^{3}}(\delta_{ik}\delta_{lj}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}-3\delta_{ik}P_{l}(\frac{z}{|z|})P_{j}(\frac{z}{|z|})-3\delta_{jk}P_{i}(\frac{z}{|z|})P_{l}(\frac{z}{|z|})+3\delta_{ij}P_{l}(\frac{z}{|z|})P_{k}(\frac{z}{|z|})$

$-3\delta_{il}P_{j}(\frac{z}{|z|})P_{k}(\frac{z}{|z|})-3\delta_{lj}P_{i}(\frac{z}{|z|})P_{k}(\frac{z}{|z|})-3\delta_{kl}P_{j}(\frac{z}{|z|})P_{i}(\frac{z}{|z|})+15P_{i}(\frac{z}{|z|})P_{j}(\frac{z}{|z|})P_{l}(\frac{z}{|z|})P_{k}(\frac{z}{|z|}))\coloneqq\frac{\mu_{kl}(\frac{z}{|z|})}{|z|^{3}}.$

It is apparent that $\mu_{kl}$ is even and $\mu_{kl}\in L^{q}(S^{2})$ for
$q>1$. In addition, the following calculations are an elementary exercise in
calculus (see e.g. [13])

$\underset{S^{2}}{\int}\delta_{ik}\delta_{lj}d\sigma(z)=4\pi\delta_{ik}\delta_{lj}.$

$\underset{S^{2}}{\int}P_{i}(z)P_{j}(z)d\sigma(z)=\frac{4\pi}{3}\delta_{ij}.$

$\underset{S^{2}}{\int}P_{i}(z)P_{j}(z)P_{l}(z)P_{k}(z)d\sigma(z)=-\frac{4\pi}{15}(\delta_{ik}\delta_{lj}+\delta_{jk}\delta_{il}-\delta_{ij}\delta_{kl}-\delta_{ik}\delta_{lj}-\delta_{jk}\delta_{il}+\delta_{ij}\delta_{lk}-\delta_{il}\delta_{jk}-\delta_{lj}\delta_{ik}-\delta_{kl}\delta_{ij}).$

With the aid of the above identities it is readily checked that
$\underset{S^{2}}{\int}\mu(z)d\sigma(z)=0$. $\square$

######  Lemma 3.7.

Let $f$ be a H-solution with $||f(t,\cdot)||_{L^{q}(B)}\leq S_{0}$ where
$3<q\leq\infty$. Then:

1\. There is some $\alpha>0$ such that $f\in
H^{\alpha,\frac{\alpha}{2}}(\Omega)$ for all $\Omega\Subset\omega$.

2\. (i) $a_{ij}\in C^{0}(\omega)$.

2\. (ii) For each $t\in J$ the function $a_{ij}(t,\cdot)$ is differentiable on
$B$ and $\partial_{k}a_{ij}\in C^{0}(\omega)$.

###### Proof.

We assume with out loss of generality $q<\infty$. We start by showing
$\overline{b_{i}}^{f}\in L_{\mathrm{loc}}^{\infty}(\omega)$.

Let $B^{\prime}\Subset B$ be a ball and $J^{\prime}\Subset J$. Pick
$B^{\prime\prime}\Subset B$ to be a ball such that $B^{\prime}\Subset
B^{\prime\prime}$ and any $2\epsilon-$neighborhood of $B^{\prime}$ is $\Subset
B^{\prime\prime}$, for some sufficiently small $\epsilon>0$. Pick $\chi\in
C_{0}^{\infty}(\mathbb{\mathbb{R}}^{3})$ with $\chi(z)\equiv 1$ on
$|z|\leq\epsilon$, $\chi\equiv 0$ on $|z|>2\epsilon$ . Then for all $(t,v)\in
J^{\prime}\times B^{\prime}$ we have

$|\overline{b_{i}}^{f}(t,v)|\lesssim\underset{\mathbb{R}^{3}}{\int}\frac{1}{|v-w|^{2}}f(t,w)dw=\underset{\mathbb{R}^{3}}{\int}\frac{\chi(v-w)}{|v-w|^{2}}f(t,w)dw+\underset{\mathbb{R}^{3}}{\int}\frac{1-\chi(v-w)}{|v-w|^{2}}f(t,w)dw=$

$\underset{|v-w|\leq
2\epsilon}{\int}\frac{\chi(v-w)}{|v-w|^{2}}f(t,w)dw+\underset{\mathbb{R}^{3}}{\int}\frac{1-\chi(v-w)}{|v-w|^{2}}f(t,w)dw\leq\underset{B^{\prime\prime}}{\int}\frac{\chi(v-w)}{|v-w|^{2}}f(t,w)dw+||\frac{1-\chi(w)}{|w|^{2}}||_{\infty}||f(t,\cdot)||_{L^{1}(\mathbb{R}^{3})}$

$\lesssim||f(t,\cdot)||_{L^{q}(B^{\prime\prime})}+||f(t,\cdot)||_{L^{1}(\mathbb{R}^{3})}.$

where the last inequality follows from Hï¿œlder nequality and the assumption
$q>3$. Taking the double supremum on both sides gives $\overline{b_{i}}^{f}\in
L_{\mathrm{loc}}^{\infty}(\omega)$.

1\. Now we show that $f$ is locally Hï¿œlder continuous in $\omega$. To this
aim we wish to show that $f$ (which by assumption is
$L^{\infty}L^{q}(\omega),q>3$) is a subsolution to an inequality of the form
(4). Let $\omega^{\prime}=(0,S^{\prime})\times B^{\prime}\Subset\omega$.
Corollary 2.4 and Remark 2.5 imply in particular the following equation for
all $0\leq\varphi\in C_{0}^{2}(\omega^{\prime})$

$-\underset{\omega^{\prime}}{\int}f\partial_{t}\varphi+\underset{\omega^{\prime}}{\int}(\overline{a}_{ij}^{f}\partial_{i}f)(\partial_{j}\varphi)=\underset{\omega^{\prime}}{\int}f\overline{b}_{i}^{f}\partial_{i}\varphi(t,v).$
(5)

That $f\in W_{2}^{1,0}(\omega^{\prime})$ is a particular byproduct of theorem
2.3. Furthermore, in step 2.i we will prove (independently) that
$\partial_{kl}\overline{a_{ij}}^{f}(t,\cdot)\in L_{\mathrm{loc}}^{q}(B)$ for
each fixed $t$, so that $\overline{b}_{i}^{f}(t,\cdot)\in
W_{2}^{1}(B^{\prime})$ for each fixed $t$.

Therefore we may integrate by parts the RHS of equation (5) and arrive at the
equation

$-\underset{\omega^{\prime}}{\int}f\partial_{t}\varphi+\underset{\omega^{\prime}}{\int}(\overline{a}_{ij}^{f}\partial_{i}f)(\partial_{j}\varphi)=-\underset{\omega^{\prime}}{\int}(\overline{b}_{i}^{f}\partial_{i}f+f\partial_{i}\overline{b}_{i}^{f})\varphi.$

Hence

$-\underset{\omega^{\prime}}{\int}f\partial_{t}\varphi+\underset{\omega^{\prime}}{\int}(\overline{a}_{ij}^{f}\partial_{i}f)(\partial_{j}\varphi)+\underset{\omega^{\prime}}{\int}\nabla
f\cdot
V\varphi=-\underset{\omega^{\prime}}{\int}f\varphi\partial_{i}\overline{b}_{i}^{f}.$
(6)

where $V=(\overline{b}_{1}^{f},\overline{b}_{2}^{f},\overline{b}_{3}^{f})$. By
an elementary calculation $-\partial_{i}(b_{i})$ is positively proportional to
the Dirac distribution, which in turn implies that
$-\partial_{i}(\overline{b}_{i}^{f})$ is positively proportional to $f$.
Therefore the RHS of (6) is nonnegative. Thus

$-\underset{\omega^{\prime}}{\int}f\partial_{t}\varphi+\underset{\omega^{\prime}}{\int}(\overline{a}_{ij}^{f}\partial_{i}f)(\partial_{j}\varphi)+\underset{\omega^{\prime}}{\int}\nabla
f\cdot V\varphi-\underset{B^{\prime}}{\int}f\varphi(0,v)dv\geq 0.$

Since $\overline{b_{i}}^{f}\in L_{\mathrm{loc}}^{\infty}(\omega)$ we know in
particular that $|V|^{2}\in L^{\infty}L^{q}(\omega^{\prime})$ for some
$q>\frac{3}{2}$ which obviously implies the integrability condition imposed
for $V$ in theorem 3.4. Therefore $f$ is locally Hï¿œlder continuous in
$\omega$.

2\. i. We show that for each fixed $t$, $\overline{a_{ij}}^{f}(t,\cdot)$ is
differentiable.

Let $B^{\prime}\Subset B,J^{\prime}\Subset J$. Denote by $\rho>0$ the radius
of $B^{\prime}$ and pick $\epsilon>0$ so small so that
$B^{\prime\prime}\coloneqq B{}_{\rho+2\epsilon}\Subset B$.

$\mathbf{1}_{B^{\prime}}(v)(\partial_{kl}\overline{a_{ij}}^{f})(t,v)=\mathbf{1}_{B^{\prime}}(v)\underset{\mathbb{R}^{3}}{\int}\partial_{kl}a_{ij}(v-w)f(t,w)dw=$

$=\mathbf{1}_{B^{\prime}}(v)\underset{\mathbb{R}^{3}}{\int}\partial_{kl}a_{ij}(v-w)\mathbf{1}_{B^{\prime\prime}}(w)f(t,w)dw+\mathbf{1}_{B^{\prime}}(v)\underset{\mathbb{R}^{3}}{\int}\partial_{kl}a_{ij}(v-w)\mathbf{1}_{\mathbb{R}^{3}-B^{\prime\prime}}(w)f(t,w)dw=$

$=\mathbf{1}_{B^{\prime}}(v)\underset{\mathbb{R}^{3}}{\int}\partial_{kl}a_{ij}(v-w)\mathbf{1}_{B^{\prime\prime}}(w)f(t,w)dw+\mathbf{1}_{B^{\prime}}(v)\underset{\mathbb{R}^{3}}{\int}\partial_{kl}a_{ij}(v-w)\mathbf{1}_{|v-w|\geq
2\epsilon}\mathbf{1}_{\mathbb{R}^{3}-B^{\prime\prime}}(w)f(t,w)dw\coloneqq
g_{t}(\cdot)+h_{t}(\cdot).$

Lemma 3.6, theorem 3.5 and the assumption on $f$ imply that $g_{t}\in
L^{q}(B^{\prime})$. Furthermore, by Young’s convolution inequality:

$||h_{t}(\cdot)||_{L^{q}(B^{\prime})}=||h_{t}(\cdot)||_{L^{q}(\mathbb{R}^{3})}=||(|\mathbf{1}_{|\cdot|\geq
2\epsilon}(\partial_{kl}a_{ij})|)\ast(\mathbf{1}_{\mathbb{R}^{3}-B^{\prime\prime}}f)||_{L^{q}(\mathbb{R}^{3})}\leq||\mathbf{1}_{|\cdot|\geq
2\epsilon}\partial_{kl}a_{ij}||_{L^{q}(\mathbb{R}^{3})}||\mathbf{1}_{\mathbb{R}^{3}-B^{\prime\prime}}f(t,\cdot)||_{L^{1}(\mathbb{R}^{3})}<\infty.$

We move to show that $\overline{a_{ij}}^{f}(t,\cdot)\in L^{q}(\omega)$. The
proof is similar to the argument presented at the begining. Denote by $R$ the
radius of the ball $B.$ We have

$\underset{\mathbb{R}^{3}}{\int}|a_{ij}(v-z)|f(t,z)dz\leq
2\underset{\mathbb{R}^{3}}{\int}\frac{1}{|v-z|}f(t,z)dz=2\underset{\mathbb{R}^{3}}{\int}\frac{1}{|z|}f(t,v-z)dz=$

$2\underset{|z|\leq
R}{\int}\frac{1}{|z|}f(t,v-z)dz+2\underset{|z|>R}{\int}\frac{1}{|z|}f(t,v-z)dz\leq
C(||f(t,\cdot)||_{L^{q}(B)}+||f(t,\cdot)||_{1}).$

for some constant $C=C(q)$ (The finiteness of $\underset{|z|\leq
R}{\int}\frac{1}{|z|^{\frac{q}{q-1}}}dz$ is guaranteed because of the
assumption $q>3$). So
$\underset{\omega}{\sup}\underset{\mathbb{R}^{3}}{\int}|a_{ij}(v-z)|f(t,z)dz\leq
C(q)(||f||_{L^{\infty}L^{q}(\omega)}+||f||_{L^{\infty}L^{1}(\mathbb{R}^{3})})$,
and in particular $\overline{a_{ij}}^{f}(t,\cdot)\in L^{q}(\omega)$. Thus we
have proved $\overline{a_{ij}}^{f}(t,\cdot)\in W^{2,q}(B^{\prime})$ for $q>3$
which by Sobolev embedding implies that $\overline{a_{ij}}^{f}(t,\cdot)$ is
$C^{1}(B^{\prime})$.

We prove continuity with respect to $t$. Pick $\chi\in
C_{0}^{\infty}(\mathbb{R}^{3})$ with $\chi\equiv 1$ on $B_{\epsilon}(0)$ and
$\mathrm{supp}(\chi)\Subset B_{2\epsilon}(0)$.

$\mathbf{1}_{B^{\prime}}(v)\overline{a_{ij}}^{f}(t,v)=\mathbf{1}_{B^{\prime}}\underset{\mathbb{R}^{3}}{\int}a_{ij}(v-w)f(t,w)dw=\mathbf{1}_{B^{\prime}}(\underset{\mathbb{R}^{3}}{\int}\chi(v-w)a_{ij}(v-w)f(t,w)dw+\underset{\mathbb{R}^{3}}{\int}(1-\chi(v-w))a_{ij}(v-w)f(t,w)dw)=$

$=\mathbf{1}_{B^{\prime}}(\underset{|v-w|\leq
2\epsilon}{\int}\chi(v-w)a_{ij}(v-w)f(t,w)dw+\underset{\mathbb{R}^{3}}{\int}(1-\chi(v-w))a_{ij}(v-w)f(t,w)dw)=$

$=\mathbf{1}_{B^{\prime}}\underset{|v-w|\leq
2\epsilon}{\int}\chi(v-w)a_{ij}(v-w)\mathbf{1}_{B^{\prime\prime}}(w)f(t,w)dw+\mathbf{1}_{B^{\prime}}\underset{|v-w|\leq
2\epsilon}{\int}\chi(v-w)a_{ij}(v-w)\mathbf{1}_{\mathbb{R}^{3}\setminus
B^{\prime\prime}}(w)f(t,w)dw$

$+\mathbf{1}_{B^{\prime}}\underset{\mathbb{R}^{3}}{\int}(1-\chi(v-w))a_{ij}(v-w)f(t,w)dw=$

$\mathbf{1}_{B^{\prime}}\underset{|v-w|\leq
2\epsilon}{\int}\chi(v-w)a_{ij}(v-w)\mathbf{1}_{B^{\prime\prime}}(w)f(t,w)dw+\mathbf{1}_{B^{\prime}}\underset{\mathbb{R}^{3}}{\int}(1-\chi(v-w))a_{ij}(v-w)f(t,w)dw\coloneqq\widetilde{g_{t}}(v)+\widetilde{h_{t}}(v).$

Keep $v\in B^{\prime}$ fixed. That $t\mapsto\widetilde{g_{t}}(v)$ is
continuous is an immediate consequence of 1 and the CS inequality. Furthermore
it is clear that $w\mapsto(1-\chi(v-w))a_{ij}(v-w)\in
W^{2,\infty}(\mathbb{R}^{3})$, which by theorem 2.2 implies that
$t\mapsto\widetilde{h_{t}}(v)$ is continuous. So
$\overline{a_{ij}}^{f}(\cdot,v)$ is continuous on $J$ as a sum of such
functions.

ii. We already know that $\partial_{k}a_{ij}(t,\cdot)$ is continuous by 2.i.
Continuity with respect to $t$ is achieved as in 2.i ( here we use Hï¿œlder’s
inequality instead of CS). ∎

Before giving the proof of theorem 3.1, we will need the following existence
and uniqueness results, which will also prove themselves useful in section 4.

######  Theorem 3.8.

(Theorem 6.1,III in [6]) Suppose $A_{ij}(t,v)$ satisfies the following
conditions:

1\. $A_{ij}$ are locally uniformly elliptic on $\omega$.

2\. For all $t\in J,$ $A_{ij}(t,\cdot)$ are differentiable with respect to $v$
and $\underset{J}{\mathrm{ess}\sup}|\partial_{v_{k}}A_{ij}|<\infty$ for all
$1\leq k\leq 3$.

Suppose $F\in L^{2}(\omega)$. Then the problem

$\left\\{\begin{array}[]{cc}\partial_{t}u-\partial_{j}(A_{ij}\partial_{i}u)=F&\omega\\\
u=0&[0,S]\times\partial B\\\ u=0&\\{0\\}\times B\end{array}\right.$ (7)

has a unique solution from $W_{2}^{2,1}(\omega)$. Moreover, this solution is a
strong solution.

######  Theorem 3.9.

(Theorem 3.3, III [6]) Suppose $A_{ij}(t,v)$ are locally uniformly elliptic.
Let $F\in L^{2}(\omega)$. Then problem (7) cannot have more than one weak
solution in $W_{2}^{1,0}(\omega)$.

Proof of theorem 3.1. Suppose $\Omega=(T_{1},T_{2})\times
B^{\prime}\Subset\omega$. Let
$\omega^{\prime}\coloneqq(T_{1}^{\prime},T_{2}^{\prime})\times
B^{\prime\prime}$ such that $\Omega\Subset\omega^{\prime}\Subset\omega$. Let
$\zeta\in C_{0}^{\infty}(\omega^{\prime})$ such that $\zeta\equiv 1$ on
$\Omega$ and put $u=\zeta f$. Fix some test function $\chi\in
C_{0}^{2}([0,T)\times\mathbb{R}^{3})$. We compute

$\underset{\omega^{\prime}}{\int}-u\partial_{t}\chi+(\overline{a}_{ij}^{f})\partial_{i}u\partial_{j}\chi=\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}-u\partial_{t}\chi+(\overline{a}_{ij}^{f})\partial_{i}u\partial_{j}\chi=\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}-\zeta
f\partial_{t}\chi+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})(\partial_{i}\zeta
f+\partial_{i}f\zeta)\partial_{j}\chi$

$=\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}-\zeta
f\partial_{t}\chi+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})\partial_{i}f(\partial_{j}(\zeta\chi)-\chi\partial_{j}\zeta)+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})f\partial_{j}\chi\partial_{i}\zeta=$

$=-\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f\partial_{t}(\zeta\chi)+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})\partial_{i}f\partial_{j}(\zeta\chi)+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f\chi\partial_{t}\zeta-\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})\chi\partial_{i}f\partial_{j}\zeta+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})f\partial_{j}\chi\partial_{i}\zeta.$

By equation (3) the last sum is

$=\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}f\overline{b}_{i}^{f}\partial_{i}(\zeta\chi)+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\partial_{t}\zeta)f\chi-\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})\chi\partial_{i}f\partial_{j}\zeta+\stackrel{{\scriptstyle[}}{{0}}]{T}{\int}\underset{\mathbb{R}^{3}}{\int}(\overline{a}_{ij}^{f})f\partial_{j}\chi\partial_{i}\zeta.$

Since $f\in W_{2}^{1,0}(\omega^{\prime})$ and
$\overline{a_{ij}}^{f}(t,\cdot),\overline{b}_{i}^{f}(t,\cdot)\in
W_{2}^{1}(B^{\prime})$ for each fixed $t$, the last expression may be
integrated by parts and recasted as

$=-\underset{\omega^{\prime}}{\int}(\zeta
f\partial_{i}(\overline{b}_{i}^{f})+\zeta\overline{b}_{i}^{f}\partial_{i}f)\chi+\underset{\omega}{\int}\chi(f\partial_{t}\zeta-f\partial_{j}(\overline{a}_{ij}^{f})\partial_{i}\zeta-\overline{a}_{ij}^{f}(f\partial_{i}\partial_{j}\zeta+\partial_{j}f\partial_{i}\zeta)-\overline{a}_{ij}^{f}\partial_{i}f_{i}\partial_{j}\zeta)=$

$=\underset{\omega^{\prime}}{\int}(f\partial_{t}\zeta-f\partial_{j}(\overline{a}_{ij}^{f})\partial_{i}\zeta-\overline{a}_{ij}^{f}(f\partial_{i}\partial_{j}\zeta+\partial_{j}f\partial_{i}\zeta)-\overline{a}_{ij}^{f}\partial_{i}f\partial_{j}\zeta-\zeta
f\partial_{i}(\overline{b}_{i}^{f})-\zeta\overline{b}_{i}^{f}\partial_{i}f)\chi\coloneqq\underset{\omega^{\prime}}{\int}F\chi.$
(8)

Thus, we find that $u$ is a $W_{2}^{1,0}(\omega^{\prime})$ solution to the
linear equation

$\left\\{\begin{array}[]{cc}\partial_{t}u-\partial_{j}(\overline{a}_{ij}^{f}\partial_{i}u)=F&\omega^{\prime}\\\
u=0&[T_{1}^{\prime},T_{2}^{\prime}]\times\partial B^{\prime\prime}\\\
u=0&\\{T_{1}^{\prime}\\}\times B^{\prime\prime}\end{array}\right..$ (9)

Keeping in mind Step 1 of lemma 3.7 (in particular $f\in
L_{loc}^{\infty}(\omega)$ and $f\in L^{\infty}L^{s^{\prime}}(\omega)$ where
$s^{\prime}>3$), we make the following key observations

i. $f\in L^{2}(\omega^{\prime})$

ii. By theorem 2.3 $(\sqrt{f})_{i}\in L^{2}(\omega^{\prime})$ and thus
$f_{i}=2\sqrt{f}(\sqrt{f})_{i}\in L^{2}(\omega^{\prime})$

iii. $\overline{a}_{ij}^{f},\overline{b}_{i}^{f}\in
L^{\infty}(\omega^{\prime})$

iv. $(\overline{b}_{i}^{f})_{i}f$ is proportional to $f^{2}$.

From i-iv we immediately see that $F\in L^{2}(\omega^{\prime})$. Moreover,
lemma 3.7 shows that the condition
$\underset{(T_{1}^{\prime},T_{2}^{\prime})}{\mathrm{ess}\sup}|(\overline{a}_{ij}^{f})_{k}|<\infty$
required in theorem 3.8 is verified, and so we know that equation (9) has a
unique $W_{2}^{2,1}(\omega^{\prime})$ solution, call it $\widetilde{u}$. In
particular $\widetilde{u}$ is a $W_{2}^{1,0}(\omega^{\prime})$ solution. Now,
viewing equation (9) as an equation for the space
$W_{2}^{1,0}(\omega^{\prime})$ and owing to the uniqueness provided by theorem
3.9, it follows that $u=\widetilde{u}\in W_{2}^{2,1}(\omega^{\prime})$, which
implies $f\in W_{2}^{2,1}(\Omega)$. $\square$

With the aid of the following estimate we can improve the Lebesgue exponent of
the first order spatial derivatives of $f$

######  Lemma 3.10.

(Lemma 3.3, II in [6]) Suppose $u\in W_{q}^{2,1}(\omega)$ and $1\leq q\leq
r<\infty,\frac{1}{q}-\frac{1}{r}\leq\frac{1}{5}$. Then

$||\partial_{v_{i}}u||_{r,\omega}\leq
C_{1}(||u||_{q,\omega}+||\partial_{v_{i}}u||_{q,\omega}+||\partial_{v_{i}v_{j}}u||_{q,\omega})+C_{2}||u||_{q,\omega}.$
(10)

######  Corollary 3.11.

Let $f$ be a H-solution. Suppose there exist $S_{0}>0,q>3$ such that for all
$t\in J$ one has $||f(t,\cdot)||_{L^{q}(B)}\leq S_{0}$ .Then
$\partial_{v_{i}}f\in L^{\frac{10}{3}}(\Omega)$ for all $\Omega\Subset\omega$.

###### Proof.

By Theorem 3.1 $f\in W_{2}^{2,1}(\Omega)$ and so taking $q=2$ in Lemma 3.10 we
find that $\partial_{v_{i}}f\in L^{r}(\Omega)$ as long as $1\leq
r\leq\frac{10}{3}$. ∎

In the next section we will iterate lemma 3.10 in order to show that $f$ lies
locally in $W_{r}^{2,1}(\Omega)$ for arbitrary $1\leq r<\infty$.

##  4 From $\mathbf{\mathbf{W}_{2}^{2,1}}$ to $\mathbf{H_{\alpha+2}^{\ast}}$

We now wish to push further the main result obtained in the previous section,
by proving that $f$ is locally $H_{\alpha+2}^{\ast}$, which in particular
implies that $f$ is a classical solution to equation (1). To finish the proof
of theorem 2.6 we will need

######  Theorem 4.1.

([6], IV, Theorem 9.1) Suppose that $A_{ij}$ are locally uniformly elliptic
and bounded continuous on $\omega$. Suppose $\mathcal{F}\in
L^{q}(\omega),1<q<\infty$. Then the problem

$\left\\{\begin{array}[]{cc}\partial_{t}U-A_{ij}\partial_{i}\partial_{j}U=\mathcal{F}&\omega\\\
U=0&[T_{1},T_{2}]\times\partial B\\\ U=0&\\{T_{1}\\}\times
B\end{array}\right.$ (11)

has a unique solution $U\in W_{q}^{2,1}(\omega)$.

######  Lemma 4.2.

([6], Page 343) Let $5<q<\infty,$ $0<\alpha<1-\frac{5}{q}$ and let the
conditions of theorem 4.1 hold. The solution $u\in W_{q}^{2,1}(\omega)$ of
equation (11) has $u\in H^{1+\alpha,\frac{1+\alpha}{2}}(\omega)$.

In addition we need to slightly refine the conclusion of 2.i of lemma 3.7 by
showing that $\overline{a_{ij}}^{f}$ are Hï¿œlder continuous in space
uniformly with respect to time.

######  Lemma 4.3.

Suppose $f\in H^{\alpha,\frac{\alpha}{2}}(\omega)$. Let $\Omega\Subset\omega$.
Then there is some $C>0$ such that for all $(t,v_{1}),(t,v_{2})\in\Omega$ it
holds that
$|\overline{a_{ij}}^{f}(t,v_{1})-\overline{a_{ij}}^{f}(t,v_{2})|\leq
C|v_{1}-v_{2}|^{\alpha}$. In particular $\overline{a_{ij}}^{f}\in
H_{\alpha}^{\ast}(\overline{\Omega})$.

###### Proof.

With the same notation of lemma 3.7 we have

$\mathbf{1}_{B^{\prime}}(v)\overline{a_{ij}}^{f}(t,v)=\underset{B^{\prime\prime}}{\int}\chi(v-w)a_{ij}(v-w)f(t,w)dw+\mathbf{1}_{B^{\prime}}\underset{\mathbb{R}^{3}}{\int}(1-\chi(v-w))a_{ij}(v-w)f(t,w)dw\coloneqq
g_{t}(\cdot)+h_{t}(\cdot).$

Let $B_{v}^{\prime\prime}$ be the ball centered at $v$ with the same radius as
$B^{\prime\prime}$.

$|g_{t}(v_{1})-g_{t}(v_{2})|\leq\underset{B_{v}^{\prime\prime}}{\int}|\chi(w)a_{ij}(w)||f(t,v_{1}-w)-f(t,v_{2}-w)|dw\leq
C_{1}|v_{1}-v_{2}|^{\alpha}.$

In addition by the mean value theorem

$|h_{t}(v_{1})-h_{t}(v_{2})|\leq||\nabla h_{t}(\xi)||\times|v_{1}-v_{2}|\leq
C_{2}|v_{1}-v_{2}|,$

where $\xi$ is an intermidiate point and the second inequality is because
$||\nabla h_{t}||_{\infty}$ is easily seen to be bounded indepnedently of $t$.
∎

At the last step of the proof we will apply the following parabolic version of
interior Schauder estimates.

######  Theorem 4.4.

([12], Theorem 1) Suppose:

1\. There are constants $\nu,\mu>0$ such that for all $\xi\in\mathbb{R}^{3}$
it holds that $\nu|\xi|^{2}\leq A_{ij}(t,v)\xi_{i}\xi_{j}\leq\mu|\xi|^{2}$ for
all $(t,v)\in\omega$.

2\. There is a constant $\kappa>0$ such that
$||A_{ij}||_{H_{\alpha}^{\ast}(\overline{\omega})}\leq\kappa$ and
$||d^{2}\mathcal{F}||_{\alpha}^{\ast}<\infty$.

If $u\in H_{\alpha+2}^{\ast}(\overline{\omega})$ is a solution to the equation

$\partial_{t}u-A_{ij}\partial_{i}\partial_{j}u=\mathcal{F},$

then $||u||_{H_{\alpha+2}^{\ast}(\overline{\omega})}\leq
c(||d^{2}\mathcal{F}||_{H_{\alpha}^{\ast}(\overline{\omega})}+||u||_{0})$
where $c=c(\Omega,\kappa,\nu,\alpha)$.

Using the above Schauder estimates we can obtain the following uniqueness and
existence result. The proof is based on a standard method of continuity
argument. We include it here only for the sake of completeness, since it does
not appear explicitly in the literature.

######  Corollary 4.5.

Let the conditions 1+2 of theorem 4.4 hold. Write $\omega=(T_{1},T_{2})\times
B$. The equation

$\left\\{\begin{array}[]{cc}\partial_{t}u-A_{ij}\partial_{i}\partial_{j}u=\mathcal{F}&(T_{1},T_{2})\times
B\\\ u=0&[T_{1},T_{2}]\times\partial B\\\ u=0&\\{T_{1}\\}\times
B\end{array}\right.$ (12)

has a unique $H_{\alpha+2}^{\ast}(\overline{\omega})$ solution.

###### Proof.

Denote $L=\partial_{t}u-\overline{a}_{ij}^{f}\partial_{i}\partial_{j}u$.
Consider the Banach spaces
$\mathfrak{B}_{1}=H_{\alpha+2}^{\ast}(\overline{\omega})\cap\\{u=0$ on
$[T_{1},T_{2}]\times\partial B\cup\\{T_{1}\\}\times B\\}$ and
$\mathfrak{B}_{2}=H_{\alpha}^{\ast}(\overline{\omega})$. For each $0\leq s\leq
1$ consider the operator $L_{s}:\mathfrak{B}_{1}\rightarrow\mathfrak{B}_{2}$
defined by $L_{s}u=sLu+(1-s)(\partial_{t}u-\Delta u)$. The solvability of
equation (12) for arbitrary $\mathcal{F}\in
H_{\alpha}^{\ast}(\overline{\omega})$ is equivalent to the fact that $L_{1}$
is onto. Let $u\in\mathfrak{B}_{1}$ and write $L_{s}u=\mathscr{\mathcal{F}}\in
H_{\alpha}^{\ast}(\overline{\omega})$ . By theorem 4.4

we have

$||u||_{\mathfrak{B}_{1}}=||u||_{H_{\alpha+2}^{\ast}(\overline{\omega})}\leq
c(||d^{2}\mathcal{F}||_{\alpha}^{\ast}+||u||_{0})\leq
c(||\mathcal{F}||_{H_{\alpha}^{\ast}(\overline{\omega})}+\underset{\omega}{\sup}|\mathcal{F}|)\leq
c||\mathcal{F}||_{H_{\alpha}^{\ast}(\overline{\omega})}=c||L_{s}u||_{H_{\alpha}^{\ast}(\overline{\omega})},$

where the second inequality is by the maximum principle and $c$ is independent
of $s$. Indeed, note that if $\nu$ stands for the lower ellipticity constant
of $L$, then for each $s$ the lower ellipticity constant of $L_{s}$ can be
taken to be $\nu_{s}=\min(1,\nu)$, which is independent of $s$. It is also
apparent that the $H_{\alpha}^{\ast}(\overline{\omega})$ norm of the
coefficients of the second order derivatives of $L_{s}$ is bounded
independently of $s$. Therefore the constant $c$ from theorem 4.4 can be taken
to be the same for all the $L_{s}$. Furthermore, it is classical that the heat
equation

$\left\\{\begin{array}[]{cc}\partial_{t}u-\Delta
u=\mathcal{F}&(T_{1},T_{2})\times B\\\ u=0&[T_{1},T_{2}]\times\partial B\\\
u=0&\\{T_{1}\\}\times B\end{array}\right.$

has a $H_{\alpha+2}^{\ast}(\overline{\omega})$ solution provided
$\mathcal{F}\in H_{\alpha}^{\ast}(\overline{\omega})$ (in fact less regularity
on $\mathcal{F}$ is required here). Otherwise put, $L_{0}$ is onto
$\mathfrak{B}_{2}$. By the method of continuity (see e.g. theorem 5.2 in [3])
it follows that $L_{1}$ is onto, as desired. Uniqueness of the solution is a
consequence of the maximum principle. ∎

Proof of theorem 2.6. 1. Given a cylinder $(T_{1},T_{2})\times
B^{\prime}\Subset\omega$ pick a cylinder $(T_{1},T_{2})\times
B^{\prime}\Subset(T_{1}^{\prime},T_{2}^{\prime})\times
B^{\prime\prime}\Subset\omega$. Set
$\omega^{\prime}=(T_{1}^{\prime},T_{2}^{\prime})\times B^{\prime\prime}$. We
localize the solution $f$ by defining $u=\zeta f$ for some $\zeta\in
C_{0}^{\infty}((T_{1}^{\prime},T_{2}^{\prime})\times B^{\prime\prime})$
satisfying $\zeta\equiv 1$ on $(T_{1},T_{2})\times B^{\prime}$. First we wish
to show that $u$ satisfy an equation of the form (11). As in theorem 3.1 we
get that $u$ satisfies the equation

$\left\\{\begin{array}[]{cc}\partial_{t}u-\overline{a}_{ij}^{f}\partial_{i}\partial_{j}u=\mathcal{F}&\omega^{\prime}\\\
u=0&[T_{1}^{\prime},T_{2}^{\prime}]\times\partial B^{\prime\prime}\\\
u=0&\\{T_{1}^{\prime}\\}\times B^{\prime\prime}\end{array}\right.$ (13)

where
$\mathcal{F}=\zeta_{t}f-\overline{a}_{ij}^{f}(\zeta_{ij}f+f_{j}\zeta_{i})-\overline{a}_{ij}^{f}f_{i}\zeta_{j}-\zeta(\overline{b}_{i}^{f})_{i}f$.
By corollary 3.11 we see that $\mathcal{F}\in
L^{\frac{10}{3}}(\omega^{\prime})$ (recall that $(\overline{b}_{i}^{f})_{i}$
is propotional to $f$) and so by theorem 4.1 $u\in
W_{\frac{10}{3}}^{2,1}(\omega^{\prime})$. We proceed by iterating lemma 3.10:
utilizing lemma 3.10 with $q=\frac{10}{3},r=10$ we find that $u\in
W_{10}^{2,1}(\omega^{\prime})$. Utilizing lemma 3.10 once again with
$q=10,10\leq r<\infty$ we get $u\in W_{r}^{2,1}(\omega^{\prime})$. Thus, we
have shown $u\in W_{r}^{2,1}(\omega^{\prime})$ for all $1\leq r<\infty$. Lemma
4.2 entails that $u\in H^{1+\alpha,\frac{1+\alpha}{2}}(\omega^{\prime})$ for
all $0<\alpha<1$. We have thus shown that $f\in
H^{1+\alpha,\frac{1+\alpha}{2}}(\Omega)$ for any $\Omega\Subset\omega$ and
$0<\alpha<1$, which in turn implies that $\mathcal{F}$ verifies the Hï¿œlder
condition 2 in theorem 4.4 for all $0<\alpha<1$. In addition lemma 4.3
guarantees that $\overline{a}_{ij}^{f}$ verifies the Hï¿œlder condition 2
imposed in Theorem 4.4 for all $0<\alpha<1$. Corollary 4.5 ensures that
equation (13) has a unique $H_{\alpha+2}^{\ast}(\overline{\omega^{\prime}})$
solution (for arbitrary fixed $0<\alpha<1$) , and a fortiori this solution
must identify with $f$. $\square$

Acknowledgement.

This work is part of the author’s PhD thesis. I would like to express my
deepest graditute towards my supervisor Franï¿œois Golse for exposing me to
this research direction, for many fruitful discussions and for carefully
reading previous versions of this manuscript and providing insightful comments
and improvements. I would also like to thank an anonymous referee for
providing comments which improved the quality of this work.

## References

  * [1] C. Villani, On a new class of weak solution to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 (1998). 273–307.
  * [2] L.Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials part I. Comm. in Partial Diff. Eq. 25 (2000). 179–259.
  * [3] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. Springer. 1998.
  * [4] L. Desvillettes, Entropy dissipation estimates for the Landau equation. J. Funct. Anal. 269 (2015), 1359–1403.
  * [5] J.C. Robinson, J.L. Rodrigo and W. Sadowski, The three dimensional Navier-Stokes equations. Cambridge studies in advanced mathematics. 2016.
  * [6] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uraltseva, Linear and quasi-linear equations of parabolic type. American Mathematical Society.1968\.
  * [7] L.C. Evans, Partial differential equations. American Mathematical Society. 2009.
  * [8] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal. 9 (1962), 187–195.
  * [9] F. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^{p}$. Indiana Univ. Math. J. 29 (1980), 79–102.
  * [10] L. Silvestre, Upper bounds for parabolic eqyations and the Landau equation. J. Differential Equations 262 (2017), 3034–3055.
  * [11] A. Vasseur, The De Giorgi method for elliptic and parabolic equations and some applications. Lectures on the analysis of nonlinear partial differential equations. Part 4, 195–222, Morningside Lect. Math., 4, Int. Press, Somerville, MA, 2016.
  * [12] B.F. Knerr, Parabolic interior Schauder estimates by the maximum principle. Arch. Rational Mech. Anal. 75 (1980). 51–58.
  * [13] G.B. Folland, How to integrate a polynomial over a sphere. The American Mathematical Monthly. 108 (2001). 446–448.
  * [14] J. Duoandikoetxea, Fourier analysis. American Mathematical Society. 29 (2000).
  * [15] R. Alexandre, J. Liao and C. Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinet. Relat. Models 8 (2015), no. 4, 617–6.
  * [16] M. Gualdani and N. Guillen. Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential. Anal. PDE 9 (2016), no. 8, 1772–1809.
  * [17] M. Gualdani and N. Guillen. On Ap weights and the Landau equation. Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 17, 55 pp.
  * [18] F. Golse, M. Gualdani, C. Imbert and A. Vasseur. Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, arXiv:1906.0284.
  * [19] F.Golse, C. Imbert, C. Mouhot and A.F. Vasseur. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253–295.
  * [20] A. A. Arsen’ev, N. V. Peskov. The existence of a generalized solution of Landau’s equation. (Russian) Z. Vycisl. Mat i Mat. Fiz. 17 (1977), no. 4, 1063–1068, 1096\.
  * [21] N.Fournier. Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential. Comm. Math. Phys. 299 (2010), no. 3, 765–782.


# Initial 56Ni Masses in Type Ia Supernovae

Zsófia Bora ELTE Eötvös Loránd University, Department of Astronomy, Pázmány
Péter sétány 1/A, Budapest, 1117 Hungary  József Vinkó Konkoly Observatory,
CSFK, Konkoly-Thege M. út 15-17, Budapest, 1121, Hungary  ELTE Eötvös Loránd
University, Institute of Physics, Pázmány Péter sétány 1/A, Budapest, 1117
Hungary  Department of Optics & Quantum Electronics, University of Szeged, Dóm
tér 9, Szeged, 6720, Hungary  Réka Könyves-Tóth Konkoly Observatory, CSFK,
Konkoly-Thege M. út 15-17, Budapest, 1121, Hungary

###### Abstract

We infer initial masses of the synthesized radioactive nickel-56 in a sample
of recent Type Ia supernovae applying a new formalism introduced recently by
Khatami & Kasen (2019). It is shown that the nickel masses we derive do not
differ significantly from previous estimates based on the traditional Arnett-
model. We derive the $\beta$ parameter for our sample SNe and show that these
are consistent with the fiducial value of $\sim 1.6$ given by Khatami & Kasen
(2019) from SN Ia hydrodynamical simulations.

Type Ia supernovae — explosive nucleosynthesis – nickel masses

††journal: PASP††software: Numpy (Harris et al., 2020), Matplotlib (Hunter,
2007), Astropy (Astropy Collaboration et al., 2013)

##  1 Introduction

Type Ia supernovae (SNe Ia) are among the most suitable objects for
extragalactic distance measurements (see e.g. Jha et al., 2019, for a recent
review). Although the well-known correlation between their peak absolute
brightness (e.g. $M_{B}$ in the $B$-band) and the light curve decline rate
(e.g. $\Delta m_{15}$) is still based on empirical calibrations, it was
realized decades ago that the peak brightness is connected to the amount of
radioactive nickel (56Ni) synthesized during the thermonuclear explosion of
the carbon-oxygen white dwarf (C/O WD; Arnett 1982; Goldstein & Kasen 2018).
Understanding the physics behind the peak brightness - decline rate - nickel
mass ($M_{\rm Ni}$) connection may have a crucial importance in improving the
distance measurement methods.

The nickel yield of SNe Ia may also be the smoking gun for the explosion
mechanism triggering the thermonuclear runaway in the C/O white dwarf (see
e.g. Seitenzahl & Townsley, 2017, and references therein). For example,
delayed-detonation models that assume the explosion of near-Chandrasekhar mass
($M_{Ch}$) WDs predict higher 57Ni/56Ni abundance ratio than double-detonation
or violent merger models in which the exploding WD has sub-$M_{Ch}$ mass.
Graur et al. (2016) found about twice of the solar value for the 57Ni/56Ni
mass ratio from the late-time light curve decline rate of SN 2012cg, which led
to the conclusion that the exploding star in SN 2012cg was a near-$M_{Ch}$ WD.
On the other hand, Scalzo et al. (2014b) pointed out that the observed
diversity of both the 56Ni- and the ejecta masses of SNe Ia ($0.3<M_{\rm
Ni}<0.8$ M⊙, and $0.8<M_{\rm ej}<1.5$ M⊙, respectively) suggests that
sub-$M_{Ch}$ explosion mechanisms, such as the double detonation or the
violent merger scenario are responsible for the bulk of the observed SNe Ia
(see e.g. Maoz et al., 2014, for a review of explosion models). Similar
conclusions have been presented by numerous other studies, e.g. in Childress
et al. (2015), Dhawan et al. (2018), Scalzo et al. (2019), Wygoda et al.
(2019) and Könyves-Tóth et al. (2020).

The estimation of the initial amount of $M_{\rm Ni}$ synthesized during the
explosion is traditionally based on Arnett’s rule (Arnett, 1982), which states
that at the moment of maximum luminosity the instantaneous energy input from
the radioactive 56Ni $\rightarrow$ 56Co $\rightarrow$ 56Fe decay chain is
equal to the peak bolometric luminosity. Since the energy generation rate from
radioactive Ni-decay depends linearly on the initial amount 56Ni, it provides
a relatively easy way of inferring $M_{\rm Ni}$ (Stritzinger et al., 2006,
e.g.). Modeling the unblended Fe and Co features in the nebular spectra of SNe
Ia is an alternative method of deriving the Ni-mass (e.g. Mazzali et al.,
1997; Childress et al., 2015). Stritzinger et al. (2006) showed that these two
methods, i.e. Arnett’s rule and the nebular spectral modeling, provide
consistent Ni-mass estimates.

Recently, Khatami & Kasen (2019) found that Arnett’s rule has limited
accuracy. The main reason for this is that in the classical Arnett-model the
energy density profile of the ejecta is assumed to be self-similar immediately
after explosion. In reality, however, if the heating source (56Ni) is central,
like in core-collapse SNe, then some time is needed to reach self-similarity.
For more evenly mixed sources (like SNe Ia), self-similarity is reached
earlier, so Arnett’s rule applies to these objects quite well. Khatami & Kasen
(2019) found that for a central heating source, Arnett’s rule underestimates
the peak luminosity, while for more evenly mixed heating it may give an
overestimate. They also presented a new relation between the peak luminosity
and peak time of SN light curves that does not assume self-similarity.

In this paper we compare the nickel mass estimates from the formulae of
Khatami & Kasen (2019) to ones derived from recent observations of SNe Ia
based on Arnett’s rule (Könyves-Tóth et al., 2020). In Section 2 we review the
methodology of getting nickel mass estimates from the observations, then the
results from the two methods are compared to each other in Section 3. In
Section 4 we calibrate the value of $\beta$ using two other methods for
determining the nickel mass of Type Ia SNe, which are independent from
Arnett’s rule. Finally, Section 5 summarizes our conclusions.

##  2 Methods

In the original, self-similar model of a SN Ia, assuming homologous expansion
of a constant density ejecta, Arnett’s rule takes the following form:

$L_{\rm peak}=\alpha L_{\rm heat}(t_{\rm peak}),$ (1)

where $L_{\rm heat}(t_{\rm peak})$ is the heating function at the moment of
the luminosity peak, and $\alpha\sim 1$ is a correction factor that accounts
for small deviations from the assumptions of the Arnett-model.

For SNe Ia $L_{heat}$ is assumed to be the usual exponential form of the decay
chain of 56Ni $\rightarrow$ 56Co $\rightarrow$ 56Fe. If $\varepsilon_{\rm
Ni}=7.9\times 10^{43}$ erg s-1 and $\varepsilon_{\rm Co}=1.45\times 10^{43}$
erg s-1 are the heating rates and $t_{\rm Ni}=8.8$ days and $t_{\rm Co}=111.3$
days are the e-folding timescales of 56Ni and 56Co (e.g. Branch & Wheeler,
2017), then the heating function takes the form of

$L_{\rm heat}(t)=\frac{M_{\rm Ni}}{\rm M_{\odot}}\cdot\left[(\varepsilon_{\rm
Ni}-\varepsilon_{\rm Co})e^{-t/t_{\rm Ni}}+\varepsilon_{\rm Co}e^{-t/t_{\rm
Co}}\right].$ (2)

Since the heating function depends linearly on $M_{\rm Ni}$, it can be
expressed simply as

$M_{\rm Ni}~{}=~{}\frac{L_{\rm peak}}{\alpha Q(t_{\rm peak})},$ (3)

where $Q(t)$ is the time-dependent part of the $L_{\rm heat}(t)$ function in
Eq.2.

The caveat of this simple approach is that the inferred $M_{\rm Ni}$ depends
critically on $t_{\rm peak}$ and $L_{\rm peak}$, which must be known
accurately to get a reasonable result. If the photometric sampling is sparse,
it may be difficult to get a precise $M_{\rm Ni}$. To overcome this
difficulty, one possibility is to fit the entire bolometric light curve with
the prediction of the Arnett-model (e.g. Valenti et al., 2008; Chatzopoulos et
al., 2012). In this case the theoretical luminosity can be expressed as

$L(t)=\frac{2}{t_{\rm
m}^{2}}(1-e^{-(t_{\gamma}/t)^{2}})\int_{0}^{t}t^{\prime}L_{\rm
heat}(t^{\prime})e^{(t^{\prime}-t)^{2}/t_{\rm m}^{2}}dt^{\prime},$ (4)

where $t_{\rm m}$ is the mean light curve timescale (close to, but not equal
with $t_{\rm peak}$) and $t_{\gamma}$ is the timescale for the gamma-ray
leaking (Arnett, 1982; Valenti et al., 2008; Chatzopoulos et al., 2012).
Inserting Equation 2 into Equation 4 and fitting it to the bolometric light
curve around the peak, one can get the best-fit estimate for $M_{\rm Ni}$.

Khatami & Kasen (2019) inferred a new relation between the peak luminosity of
a SN and the spatial and temporal distribution of its radioactive heating
source:

$L_{\rm peak}=\frac{2}{\beta^{2}t_{\rm peak}^{2}}\int_{0}^{\beta t_{\rm
peak}}t^{\prime}L_{\rm heat}(t^{\prime})dt^{\prime},$ (5)

where $\beta$ is a numerical parameter that is related to the spatial
distribution of heating, while $L_{\rm heat}(t)$, again, describes the type of
heating that powers the SN ejecta. It can have many forms depending on the SN
type (see Khatami & Kasen, 2019, for a list of different sources).

Inserting $L_{heat}(t)$ from Equation 2 into Equation 5, $L_{\rm peak}$ can be
expressed as111In the original publication of Khatami & Kasen (2019) this
formula was written with $1-\frac{\beta t_{\rm peak}}{t_{\rm Ni}}$ instead of
$1-\left(\frac{\beta t_{\rm peak}}{t_{\rm Ni}}+1\right)$ in the first part of
the sum within the square brackets. Later it has been corrected in a new
version uploaded to arxiv.org. We present and use the correct version here.

$\displaystyle L_{\rm peak}$ $\displaystyle=\frac{2M_{\rm Ni}\varepsilon_{\rm
Ni}t_{\rm Ni}^{2}}{\beta^{2}t_{\rm
peak}^{2}}\cdot\left[\left(1-\frac{\varepsilon_{\rm Co}}{\varepsilon_{\rm
Ni}}\cdot\frac{t_{\rm Co}}{t_{\rm Co}-t_{\rm Ni}}\right)\right.\cdot$ (6)
$\displaystyle\left(1-\left(\frac{\beta t_{\rm peak}}{t_{\rm
Ni}}+1\right)e^{-\frac{\beta t_{\rm peak}}{t_{\rm Ni}}}\right)+\frac{t_{\rm
Co}^{2}}{t_{\rm Ni}^{2}}\frac{\varepsilon_{\rm Co}}{\varepsilon_{\rm
Ni}}\cdot$ $\displaystyle\left.\frac{t_{\rm Co}}{t_{\rm Co}-t_{\rm
Ni}}\cdot\left(1-\left(\frac{\beta t_{\rm peak}}{t_{\rm
Co}}+1\right)e^{-\frac{\beta t_{\rm peak}}{t_{\rm Co}}}\right)\right]$

Equation 6 describes the dependence of the peak luminosity on $t_{\rm peak}$,
i.e. the rise time to the bolometric maximum light, if the SN is powered by
the radioactive decay of 56Ni and 56Co. Again, the peak luminosity is linearly
proportional to $M_{\rm Ni}$, but the connection between $L_{\rm peak}$ and
$t_{\rm peak}$ is not as simple as in Equation 1. Figure 1 shows the $L_{\rm
peak}$ vs. $t_{\rm peak}$ relation for $\beta=1.666\pm 0.188$ (solid and
dotted lines) assuming different values of $M_{\rm Ni}$, as well as the
observed data given in Table 1.

Figure 1: $L_{\rm peak}$ as a function of $t_{\rm peak}$ (Eq. 6) for $\beta$ =
1.666 $\pm$ 0.188 (as derived in Section 4) for different values of $M_{Ni}$.
$L_{peak}$ is marked with solid lines, the dotted lines are showing the
uncertainty caused by the standard deviation of $\beta$. The $L_{peak,obs}$
values from Table 1 are shown as well.

##  3 Comparison with observations

In this paper we use the peak time - luminosity relation expressed in Equation
6 to infer the initial 56Ni mass of 16 SNe Ia studied recently by Könyves-Tóth
et al. (2020). They estimated ejecta masses, initial nickel masses and other
parameters by fitting bolometric light curves of 16 SNe Ia with the Arnett-
model, as described in Section 2.

Object | $t_{\rm peak}$ | $L_{\rm peak,obs}$ | $t_{\gamma}$
---|---|---|---
| [days] | [$10^{43}$ erg/s] | [days]
SN 2011fe | 16.59 $\pm$0.06 | 1.22 $\pm$0.12 | 37.603 $\pm$0.670
Gaia16alq | 19.92 $\pm$0.42 | 1.71 $\pm$0.17 | 46.669 $\pm$0.717
SN 2016asf | 15.08 $\pm$2.58 | 1.59 $\pm$0.16 | 39.192 $\pm$1.329
SN 2016bln | 17.42 $\pm$0.34 | 1.83 $\pm$0.18 | 44.508 $\pm$1.125
SN 2016coj | 14.17 $\pm$0.26 | 1.15 $\pm$0.11 | 32.967 $\pm$0.863
SN 2016eoa | 14.07 $\pm$3.05 | 1.38 $\pm$0.14 | 39.038 $\pm$0.935
SN 2016ffh | 14.03 $\pm$1.16 | 1.86 $\pm$0.19 | 40.521 $\pm$0.926
SN 2016gcl | 17.79 $\pm$2.18 | 1.50 $\pm$0.15 | 43.623 $\pm$1.182
SN 2016ixb | 15.64 $\pm$2.09 | 1.14 $\pm$0.11 | 30.520 $\pm$1.345
SN 2017cts | 14.23 $\pm$1.10 | 1.58 $\pm$0.16 | 42.485 $\pm$0.959
SN 2017erp | 17.96 $\pm$0.09 | 1.94 $\pm$0.19 | 37.603 $\pm$1.084
SN 2017fgc | 16.21 $\pm$0.24 | 1.74 $\pm$0.17 | 45.398 $\pm$0.941
SN 2017fms | 14.04 $\pm$0.50 | 1.02 $\pm$0.10 | 34.612 $\pm$0.731
SN 2017hjy | 16.29 $\pm$0.49 | 1.65 $\pm$0.16 | 39.484 $\pm$0.840
SN 2017igf | 19.58 $\pm$0.93 | 0.91 $\pm$0.09 | 34.554 $\pm$1.193
SN 2018oh | 14.86 $\pm$0.86 | 1.57 $\pm$0.16 | 44.654 $\pm$0.928
Table 1: Parameters of the sample SNe Ia, adopted from Könyves-Tóth et al.
(2020).

Table 1 summarizes the parameters for the sample SNe collected from Könyves-
Tóth et al. (2020): here $t_{\rm peak}$ is the rise time from explosion to
maximum luminosity in days, $L_{\rm peak,obs}$ is the peak luminosity
determined from the observations.

Applying Equation 6 to these data, we derived new $M_{Ni}$ values using the
$t_{\rm peak}$ and $L_{\rm peak,obs}$ values from Table 1. We adopted
$\beta=1.6$ as recommended by Khatami & Kasen (2019) for SNe Ia based on
radiation hydrodynamical simulations.

The results are shown in Table 2 as $M_{Ni}^{KK}$. Uncertainties are derived
via propagating the errors listed in Table 1.

Object | $M_{\rm Ni}^{\rm Arnett}$ | $M_{\rm Ni}^{\rm tail}$ | $M_{\rm Ni}^{\rm t15}$ | $M_{\rm Ni}^{\rm avg}$ | $M_{\rm Ni}^{\rm KK}$
---|---|---|---|---|---
| $[M_{\rm\odot}]$ | $[M_{\rm\odot}]$ | $[M_{\rm\odot}]$ | $[M_{\rm\odot}]$ | $[M_{\rm\odot}]$
SN 2011fe | 0.567 $\pm$0.042 | 0.581 $\pm$0.049 | 0.59 $\pm$0.051 | 0.579 $\pm$0.047 | 0.496 $\pm$0.051
Gaia16alq | 0.744 $\pm$0.055 | 0.768 $\pm$0.098 | 0.781 $\pm$0.07 | 0.764 $\pm$0.074 | 0.796 $\pm$0.08
SN 2016asf | 0.597 $\pm$0.149 | … | 0.64 $\pm$0.055 | 0.618 $\pm$0.102 | 0.602 $\pm$0.064
SN 2016bln | 0.789 $\pm$0.097 | 0.802 $\pm$0.094 | 0.822 $\pm$0.071 | 0.804 $\pm$0.087 | 0.771 $\pm$0.088
SN 2016coj | 0.401 $\pm$0.053 | … | 0.398 $\pm$0.035 | 0.4 $\pm$0.044 | 0.416 $\pm$0.047
SN 2016eoa | 0.482 $\pm$0.103 | … | 0.507 $\pm$0.05 | 0.494 $\pm$0.076 | 0.497 $\pm$0.127
SN 2016ffh | 0.573 $\pm$0.078 | … | 0.527 $\pm$0.125 | 0.55 $\pm$0.102 | 0.668 $\pm$0.107
SN 2016gcl | 0.689 $\pm$0.164 | 0.692 $\pm$0.102 | 0.719 $\pm$0.098 | 0.7 $\pm$0.121 | 0.642 $\pm$0.123
SN 2016ixb | 0.483 $\pm$0.064 | … | 0.525 $\pm$0.098 | 0.504 $\pm$0.081 | 0.443 $\pm$0.088
SN 2017cts | 0.539 $\pm$0.063 | 0.558 $\pm$0.081 | 0.553 $\pm$0.099 | 0.55 $\pm$0.081 | 0.573 $\pm$0.089
SN 2017erp | 0.975 $\pm$0.083 | … | 1.074 $\pm$0.12 | 1.024 $\pm$0.102 | 0.836 $\pm$0.087
SN 2017fgc | 0.692 $\pm$0.047 | … | 0.701 $\pm$0.061 | 0.696 $\pm$0.054 | 0.695 $\pm$0.077
SN 2017fms | 0.36 $\pm$0.029 | … | 0.385 $\pm$0.034 | 0.372 $\pm$0.032 | 0.367 $\pm$0.046
SN 2017hjy | 0.688 $\pm$0.057 | 0.69 $\pm$0.065 | 0.7 $\pm$0.069 | 0.693 $\pm$0.064 | 0.661 $\pm$0.081
SN 2017igf | 0.42 $\pm$0.051 | 0.409 $\pm$0.043 | 0.447 $\pm$0.04 | 0.425 $\pm$0.045 | 0.418 $\pm$0.057
SN 2018oh | 0.598 $\pm$0.059 | 0.614 $\pm$0.081 | 0.566 $\pm$0.073 | 0.593 $\pm$0.071 | 0.588 $\pm$0.084
Table 2: Initial 56Ni masses from the different methods used in this paper.
$M_{\rm Ni}^{\rm Arnett}$ is based on Eq. 3, collected from Könyves-Tóth et
al. (2020). $M_{\rm Ni}^{\rm tail}$ and $M_{\rm Ni}^{\rm t15}$ refers to the
tail-mass and $t_{15}$-mass, respectively, as shown in Section 4. $M_{\rm
Ni}^{\rm avg}$ is the average of the previous three columns, while $M_{\rm
Ni}^{\rm KK}$ refers to the masses inferred from Equation 6.

Figure 2 shows the comparison between the masses derived from Arnett’s rule,
and the ones estimated using Equation 6.

Figure 2: Comparison of the new nickel masses derived from Equation 6 to the
original ones from Arnett’s rule in Table 1. The dotted line shows the 1:1
relation, while the solid line is the best-fit linear relation between the two
datasets.

These new nickel masses scatter somewhat around the ones derived by Könyves-
Tóth et al. (2020) from Arnett’s rule, but the differences are within the
uncertainties, so the two datasets are generally consistent. The SN that
deviates the most from the previous estimates is SN 2017erp, which may have an
overestimated $M_{\rm Ni}$ due to its peculiar reddening (see also Könyves-
Tóth et al., 2020). Its early red color was also reported by Li et al. (2021),
and even though it was classified as a normal-velocity (NV) supernova, they
also noted that it had the highest early-phase velocity within the group.
Thus, if the nickel mass of SN 2017erp was previously overestimated, the new
$M_{\rm Ni}$ based on equation 6 might be a better estimate, since it is lower
than the previous one from Arnett’s rule.

The best-fit linear relation between $M_{\rm Ni}^{\rm Arnett}$ and $M_{\rm
Ni}^{\rm KK}$, plotted with a solid blue line in Figure 2, has a slope of
$0.880\pm 0.068$, but if we omit SN 2017erp from the sample the slope becomes
$0.967\pm 0.075$, which is very close to the 1:1 relation (shown by the dotted
line).

In Figure 3 we show the linear relationship between $L_{\rm peak,obs}$ and the
new nickel masses ($M_{\rm Ni}^{\rm KK}$). It also demonstrates the diversity
of the peak luminosities and the corresponding nickel masses, similar to
Scalzo et al. (2014a, b, 2019) and Könyves-Tóth et al. (2020). A linear fit to
these data resulted in a relation of

$M_{\rm Ni}^{\rm KK}~{}=~{}0.425(\pm 0.038)\cdot L_{\rm
peak,obs}~{}-~{}0.041(\pm 0.053),$ (7)

which can be used to estimate the initial nickel masses directly from the
measured peak luminosities.

The validity of the new nickel masses estimated above is probed by using the
published data on SN 2011fe. SN 2011fe was one of the most thoroughly studied
SNe Ia in the last decade, because it was a nearby, very bright event
discovered only a few hours after explosion. There are numerous nickel mass
estimates for SN 2011fe in the literature based on different methods. For
example, Pereira et al. (2013) estimated $M_{\rm Ni}$ from spectrophotometric
observations, similar to Mazzali et al. (2015), who used optical and NIR
spectra to determine the initial nickel and iron masses of the ejecta of SN
2011fe. Scalzo et al. (2014a) determined $M_{\rm Ni}$ from the peak bolometric
luminosity, similar to our approach, while Childress et al. (2015) used the
flux of the [CoIII] $\lambda$5893 nebular emission feature. More recently,
Dhawan et al. (2016) related the phase of the secondary maximum of the near-
infrared (NIR) light curves to the bolometric peak luminosity, from which they
applied Arnett’s rule and delayed-detonation models to determine the initial
$M_{\rm Ni}$. As noted earlier, Könyves-Tóth et al. (2020) also gave an
estimate for $M_{\rm Ni}$ by fitting the Arnett-model to the whole bolometric
light curve. All of these values are collected in Table 3, and their mean is
$M_{\rm Ni}=0.50\pm 0.08M_{\rm\odot}$. This is within a $1\sigma$ agreement
with the new value of $M_{\rm Ni}^{KK}=0.496\pm 0.051M_{\rm\odot}$ given above
in Table 2.

$M_{\rm Ni}$ | Source
---|---
$[M_{\rm\odot}]$ |
0.53 $\pm$ 0.11 | Pereira et al. (2013)
0.42 $\pm$ 0.08 | Scalzo et al. (2014a)
0.47 $\pm$ 0.05 | Mazzali et al. (2015)
0.500 $\pm$ 0.026 | Childress et al. (2015)
0.52 $\pm$ 0.15 | Dhawan et al. (2016)
0.567 $\pm$ 0.054 | Könyves-Tóth et al. (2020)
0.50 $\pm$ 0.08 | mean
0.496 $\pm$ 0.05 | this work
Table 3: Estimates for the initial 56Ni mass of SN 2011fe from different
methods. Figure 3: The new 56Ni masses against the observed peak luminosities.

##  4 A closer look at the Beta parameter

The new formula by Khatami & Kasen (2019) (Equation 6) also introduces the
$\beta$ parameter, which is connected with the spatial distribution of
heating, recombination effects, and opacity. They found that the different
distributions of heating only changes the value of $\beta$, thus, Equation 6
still holds true. This means that the same formula, with different $\beta$
values and heating functions, can be used to describe the peak luminosity of a
wide variety of objects.

If we have independent measurements for $M_{\rm Ni}$ and $L_{\rm peak}$ then
we can apply Equation 6 to infer $\beta$ for each object. The original value
of $\beta$, given by Khatami & Kasen (2019) for Type Ia SNe as $\beta\sim
1.6$, is based on numerical simulations of SN Ia explosions.

In this section, we use two independent methods to determine the nickel
masses, and consequently, the values of $\beta$ for our sample SNe.

###  4.1 Tail luminosity method

It is well-known (Valenti et al., 2008; Scalzo et al., 2014a; Afsariardchi et
al., 2021) that the late-phase light curve (at $t>60$ days after explosion)
for a SN ejecta powered by the Ni-Co-Fe radioactive decay can be expressed as:

$L=L_{\gamma}(1-e^{-t_{\gamma}^{2}/t^{2}})+L_{\rm pos,KE},$ (8)

where $t_{\gamma}$ is, again, the timescale for gamma-ray leakage, while
$L_{\gamma}$ is the luminosity released in the form of gamma-rays:

$L_{\gamma}=\frac{M_{Ni}}{\rm
M_{\odot}}\left(C_{Ni}e^{-\frac{t}{t_{Ni}}}+0.968\cdot
C_{Co}e^{-\frac{t}{t_{Co}}}\right),$ (9)

$L_{\rm pos,KE}$ gives the luminosity due to the thermalization of the kinetic
energy of positrons released during the Co-decay:

$L_{\rm pos,KE}=\frac{M_{Ni}}{\rm M_{\odot}}\left[0.032\cdot
C_{Co}\left(e^{-\frac{t}{t_{Co}}}-e^{-\frac{t}{t_{Ni}}}\right)\right].$ (10)

Comparing these formulae with Equation 2, we set $t_{Ni}=8.8$ days,
$t_{Co}=111.3$ days, $C_{Ni}=\varepsilon_{\rm Ni}-\varepsilon_{\rm
Co}=6.45\cdot 10^{43}$ erg s-1 and $C_{Co}=\varepsilon_{\rm Co}=1.45\cdot
10^{43}$ erg s-1 (see also Branch & Wheeler, 2017). $t_{\gamma}$ was given in
Table 3 by Könyves-Tóth et al. (2020), so the only free parameter in fitting
the light curve tail is $M_{Ni}$. A caveat of this method is that it provides
appropriate nickel masses only for data beyond $t\sim 60$ days, thus, we could
obtain nickel masses only for those objects that had the late part of their
light curve covered by observations. We list the results of fitting Equation 8
to our data in Table 2 as $M_{\rm Ni}^{\rm tail}$.

###  4.2 The $t_{15}$ method

Sukhbold (2019) found that the bolometric luminosity is equal to $L_{\gamma}$
at $t_{15}=t_{peak}+15$ days, similar to Arnett’s rule. Using this, it is
possible to determine the nickel mass by measuring the bolometric luminosity
at $t_{15}$:

$L_{bol}(t_{15})\approx\frac{M_{Ni}}{\rm
M_{\odot}}\left(C_{Ni}e^{-\frac{t_{15}}{t_{Ni}}}+C_{Co}e^{-\frac{t_{15}}{t_{Co}}}\right)(1-e^{-t_{\gamma}^{2}/t_{15}^{2}})$
(11)

$L_{bol}(t_{15})$ for each SN was determined by interpolating the bolometric
light curves to $t_{15}$. The nickel masses found this way can be seen in
Table 2 as $M_{\rm Ni}^{\rm t15}$.

It is seen from Table 2 that the nickel masses inferred from both the tail
luminosity method and the $t_{15}$ method are very similar to those obtained
from Arnett’s rule. Their consistency is further illustrated in Figure 4,
where $M_{\rm Ni}^{\rm tail}$ and $M_{\rm Ni}^{\rm t15}$ are plotted against
$M_{\rm Ni}^{\rm Arnett}$. All data points are closer to the 1:1 relation
(shown as a dotted line) within their uncertainties. The slope of the best-fit
linear relationship, $1.047\pm 0.038$, is consistent with the identity
relation (see Fig. 4).

Figure 4: Values of $M_{Ni}$ from the tail luminosity and the $t_{15}$ methods
compared to the masses from Arnett’s rule.

Since all nickel masses found from different methods agree within their
uncertainties, we use their mean value (shown as $M_{\rm Ni}^{\rm avg}$ in
Table 2) for measuring the $\beta$ parameter. We find that $\beta$ varies
between 1.2 to 2.1 (Table 4), while the mean value is 1.666 $\pm$ 0.188, which
is reasonably close (within $1\sigma$) to 1.6 proposed by Khatami & Kasen
(2019). The inferred $\beta$ values are plotted in Figure 5 for each SN, while
their distribution is shown in Figure 6.

Object | $\beta$
---|---
SN 2011fe | 1.973 $\pm$ 0.209
Gaia16alq | 1.515 $\pm$ 0.166
SN 2016asf | 1.658 $\pm$ 0.359
SN 2016bln | 1.693 $\pm$ 0.214
SN 2016coj | 1.514 $\pm$ 0.206
SN 2016eoa | 1.587 $\pm$ 0.004
SN 2016ffh | 1.207 $\pm$ 0.247
SN 2016gcl | 1.798 $\pm$ 0.202
SN 2016ixb | 1.902 $\pm$ 0.160
SN 2017cts | 1.509 $\pm$ 0.198
SN 2017erp | 2.106 $\pm$ 0.279
SN 2017fgc | 1.604 $\pm$ 0.145
SN 2017fms | 1.632 $\pm$ 0.136
SN 2017hjy | 1.705 $\pm$ 0.161
SN 2017igf | 1.636 $\pm$ 0.156
SN 2018oh | 1.618 $\pm$ 0.172
Table 4: New individual $\beta$ values for the sample SNe Figure 5: The new
$\beta$ values inferred for the sample SNe. Figure 6: Distribution of $\beta$
values. The vertical thick and dotted lines show the mean value and the
standard deviation of the sample, respectively.

Comparing the results shown in Figures 2 and 3 with that of a similar analysis
published recently by Afsariardchi et al. (2021) for stripped-envelope
supernovae (SESNe, see their Figure 4), one can see the essential difference
between SESNe and Type Ia SNe: Arnett’rule significantly overestimates $M_{\rm
Ni}$ with respect to $M_{\rm Ni}^{\rm tail}$ and $M_{\rm Ni}^{\rm KK}$ for
SESNe, while all Ni masses agree very well with each other in Type Ia SNe.
Khatami & Kasen (2019) also noted that, unlike in other SNe, Arnett’s rule
seems to work quite well for Type Ia SNe due to their moderate (not too
central, not too shallow) 56Ni distribution. Our empirical results presented
here confirm this statement.

##  5 Conclusions

We use previously published data (Könyves-Tóth et al., 2020) to give estimates
for the initial masses of radioactive nickel in 16 Type Ia supernovae using a
new formula published by Khatami & Kasen (2019), which relies on the
relationship between the peak luminosity and peak time without assuming self-
similar energy distribution within the ejecta. We compare our results with
previous nickel mass estimates for SN 2011fe from the literature (see Table
3), and find very good agreement. Our new nickel masses are in a $1\sigma$
agreement with those derived by others.

Previous estimates for the initial nickel mass in SNe Ia were mostly carried
out by using Arnett’s rule. Our results (Figure 3) show that the new formula
by Khatami & Kasen (2019) gives consistent nickel masses with those estimated
from a radiation-diffusion Arnett-model, while also taking into account the
spatial distribution of heating that can be different in each case. Similar to
Scalzo et al. (2019), and Könyves-Tóth et al. (2020), we find that the 56Ni
masses show diversity, suggesting that the ejecta masses are also
inhomogeneous.

Finally, we give an approximate estimate for the $\beta$ parameter of each
studied SN (Figures 5 and 6), and find good agreement with the mean value
(1.6) given by Khatami & Kasen (2019) from SN Ia simulations.

This work is part of the project “Transient Astrophysical Objects” GINOP
2.3.2-15-2016-00033 of the National Research, Development and Innovation
Office (NKFIH), Hungary, funded by the European Union, and it was also
supported by the NKFIH/OTKA FK-134432 grant. We thank the anonymous referee
for the useful comments and suggestions that helped improving our paper.

## References

  * Afsariardchi et al. (2021) Afsariardchi, N., Drout, M. R., Khatami, D. K., et al. 2021, ApJ, 918, 89. doi:10.3847/1538-4357/ac0aeb
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33. doi:10.1051/0004-6361/201322068
  * Arnett (1982) Arnett, W. D. 1982, ApJ, 253, 785. doi:10.1086/159681
  * Baade & Zwicky (1934) Baade, W., & Zwicky, F. 1934, Contributions from the Mount Wilson Observatory, 3, 73
  * Chatzopoulos et al. (2012) Chatzopoulos, E., Wheeler, J. C., & Vinko, J. 2012, ApJ, 746, 121. doi:10.1088/0004-637X/746/2/121
  * Branch & Wheeler (2017) Branch, D. & Wheeler, J. C. 2017, Supernova Explosions: Astronomy and Astrophysics Library, ISBN 978-3-662-55052-6. Springer-Verlag GmbH Germany, 2017. doi:10.1007/978-3-662-55054-0
  * Childress et al. (2015) Childress, M. J., Hillier, D. J., Seitenzahl, I., et al. 2015, MNRAS, 454, 3816. doi:10.1093/mnras/stv2173
  * Dhawan et al. (2016) Dhawan, S., Leibundgut, B., Spyromilio, J., et al. 2016, A&A, 588, A84. doi:10.1051/0004-6361/201527201
  * Dhawan et al. (2018) Dhawan, S., Bulla, M., Goobar, A., et al. 2018, MNRAS, 480, 1445. doi:10.1093/mnras/sty1908
  * Flörs et al. (2020) Flörs, A., Spyromilio, J., Taubenberger, S., et al. 2020, MNRAS, 491, 2902. doi:10.1093/mnras/stz3013
  * Goldstein & Kasen (2018) Goldstein, D. A. & Kasen, D. 2018, ApJ, 852, L33. doi:10.3847/2041-8213/aaa409
  * Graur et al. (2016) Graur, O., Zurek, D., Shara, M. M., et al. 2016, ApJ, 819, 31. doi:10.3847/0004-637X/819/1/31
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357. doi:10.1038/s41586-020-2649-2
  * Hoyle & Fowler (1960) Hoyle, F. & Fowler, W. A. 1960, ApJ, 132, 565. doi:10.1086/146963
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90. doi:10.1109/MCSE.2007.55
  * Jha et al. (2019) Jha, S. W., Maguire, K., & Sullivan, M. 2019, Nature Astronomy, 3, 706. doi:10.1038/s41550-019-0858-0
  * Khatami & Kasen (2019) Khatami, D. K. & Kasen, D. N. 2019, ApJ, 878, 56. doi:10.3847/1538-4357/ab1f09
  * Könyves-Tóth et al. (2020) Könyves-Tóth, R., Vinkó, J., Ordasi, A., et al. 2020, ApJ, 892, 121. doi:10.3847/1538-4357/ab76bb
  * Li et al. (2021) Li, W., Wang, X., Bulla, M., et al. 2021, ApJ, 906, 99. doi:10.3847/1538-4357/abc9b5
  * Maoz et al. (2014) Maoz, D., Mannucci, F., & Nelemans, G. 2014, ARA&A, 52, 107. doi:10.1146/annurev-astro-082812-141031
  * Mazzali et al. (1997) Mazzali, P. A., Chugai, N., Turatto, M., et al. 1997, MNRAS, 284, 151. doi:10.1093/mnras/284.1.151
  * Mazzali et al. (2015) Mazzali, P. A., Sullivan, M., Filippenko, A. V., et al. 2015, MNRAS, 450, 2631. doi:10.1093/mnras/stv761
  * Pereira et al. (2013) Pereira, R., Thomas, R. C., Aldering, G., et al. 2013, A&A, 554, A27. doi:10.1051/0004-6361/201221008
  * Scalzo et al. (2014a) Scalzo, R., Aldering, G., Antilogus, P., et al. 2014, MNRAS, 440, 1498. doi:10.1093/mnras/stu350
  * Scalzo et al. (2014b) Scalzo, R. A., Ruiter, A. J., & Sim, S. A. 2014, MNRAS, 445, 2535. doi:10.1093/mnras/stu1808
  * Scalzo et al. (2019) Scalzo, R. A., Parent, E., Burns, C., et al. 2019, MNRAS, 483, 628. doi:10.1093/mnras/sty3178
  * Seitenzahl & Townsley (2017) Seitenzahl, I. R. & Townsley, D. M. 2017, Handbook of Supernovae, 1955. doi:10.1007/978-3-319-21846-5_87
  * Stritzinger et al. (2006) Stritzinger, M., Mazzali, P. A., Sollerman, J., et al. 2006, A&A, 460, 793. doi:10.1051/0004-6361:20065514
  * Sukhbold (2019) Sukhbold, T. 2019, ApJ, 874, 62. doi:10.3847/1538-4357/ab0701
  * Valenti et al. (2008) Valenti, S., Benetti, S., Cappellaro, E., et al. 2008, MNRAS, 383, 1485. doi:10.1111/j.1365-2966.2007.12647.x
  * Wygoda et al. (2019) Wygoda, N., Elbaz, Y., & Katz, B. 2019, MNRAS, 484, 3941. doi:10.1093/mnras/stz145