# On the convergence of the regularized entropy-based moment method for
kinetic equations

Graham W. Alldredge
Berlin, Germany
<EMAIL_ADDRESS>The first author was supported in part by by the German
Science Foundation (DFG), project ID AL 2030/1-1. Martin Frank
Department of Mathematics
Karlsruhe Institute of Technology
Karlsruhe, Germany
<EMAIL_ADDRESS>Jan Giesselmann
Department of Mathematics, Technical University of Darmstadt
Darmstadt, Germany
<EMAIL_ADDRESS>The third author is grateful for
financial support by the German Science Foundation (DFG) via grant TRR 154
(Mathematical modelling, simulation and optimization using the example of gas
networks) project C05.

###### Abstract

The entropy-based moment method is a well-known discretization for the
velocity variable in kinetic equations which has many desirable theoretical
properties but is difficult to implement with high-order numerical methods.
The regularized entropy-based moment method was recently introduced to remove
one of the main challenges in the implementation of the entropy-based moment
method, namely the requirement of the realizability of the numerical solution.
In this work we use the method of relative entropy to prove the convergence of
the regularized method to the original method as the regularization parameter
goes to zero and give convergence rates. Our main assumptions are the
boundedness of the velocity domain and that the original moment solution is
Lipschitz continuous in space and bounded away from the boundary of
realizability. We provide results from numerical simulations showing that the
convergence rates we prove are optimal.

##  1 Introduction

Kinetic equations model systems consisting of large numbers of particles that
interact with each other or with a background medium and arise in a wide
variety of applications including rarefied gas dynamics [6], neutron transport
[18], radiative transport [20], and semiconductors [19]. The numerical
solution of kinetic equations remains an area of active research. In this
work, we consider the entropy-based moment method [17], which is a
discretization of the velocity variable in the kinetic equation. It has many
desirable theoretical properties but is computationally expensive and
challenging to implement.

Recently in [2] a regularized version of the entropy-based moment equations
was proposed to simplify the implementation of numerical methods for the
entropy-based moment equations. These regularized entropy-based moment
equations require the selection of a regularization parameter, and the authors
in [2] proposed a rule for selecting the regularization parameter so that the
error introduced by the regularization was of the order of the error in the
spatiotemporal discretization. With this selection rule the authors produced
numerical results which showed that the regularized equations could be used to
compute accurate results of the original entropy-based moment equations.

In this work, we prove that exact solutions of the regularized entropy-based
moment equations converge to the solutions of the original equations. We
quantify the difference between these two solutions using the relative
entropy, and the convergence rate is quadratic in the regularization
parameter. Under some assumptions, this is equivalent to linear convergence in
the regularization parameter in the $L^{2}$ norm.

To obtain our results we require the following main assumptions:

  1. (i)

the set of velocities of the kinetic equation is bounded;

  2. (ii)

periodic boundary conditions;

  3. (iii)

Lipschitz continuity of the function defining the collision term in the
original entropy-based moment equations;

  4. (iv)

the solution of the original entropy-based moment equations is both Lipschitz
continuous in space and bounded away from the boundary of the set of
realizable moment vectors.

The relative-entropy techniques we use are very similar to what was done in
[3, 4, 11, 22]. In general, relative entropy estimates are a widely applicable
tool for comparing thermomechanical theories having the form of hyperbolic
balance laws that are endowed with a strictly convex entropy [10]. A general
limitation of this methodology is that it requires the solution to the
limiting system to be Lipschitz continuous - this property can (usually) only
be expected for short times since shocks may form. There is recent progress in
overcoming this limitation, at least in one space dimension, by using the
relative entropy with shifts methodology that was developed by Vasseur and co-
workers [16, 21]. However, this condition can most probably not be removed in
two or more space dimensions since it is connected to non-uniqueness of
entropy solutions to hyperbolic balance laws.

To present our results, we first introduce the entropy-based moment equations
and their regularized version in Section 2 and precisely state our
assumptions. Then in Section 3 introduce the technique of relative entropy and
give a general version of our main result. We prove the estimates upon which
our main result relies and give the subsets of the realizable set on which
they hold in Section 4. Next we present the results of numerical experiments
confirming the theoretically predicted rates of convergence in Section 5, and
finally we draw conclusions and discuss directions for future work in Section
6.

##  2 Entropy-based moment equations and regularization

###  2.1 The kinetic equation

Kinetic equations evolve the kinetic density function
${f\colon[0,\infty)\times X\times V\to[0,\infty)}$ according to

$\partial_{t}f(t,x,v)+v\cdot\nabla_{x}f(t,x,v)=\mathcal{C}(f(t,x,\cdot))(v)$
(1)

(when neglecting long-range interactions). The function $f$ depends on time
$t\in[0,\infty)$, position ${x\in X\subseteq\mathbb{R}^{d}}$, and a velocity
variable $v\in V\subseteq\mathbb{R}^{d}$. The operator $\mathcal{C}$
introduces the effects of particle collisions; at each $x$ and $t$, it is an
integral operator in $v$. In order to be well-posed, (1) must be accompanied
by appropriate initial and boundary conditions.

The results in this work depend strongly on the following assumption.

######  Assumption 1.

The set of velocities $V$ is bounded.

We will see the crucial consequences of this assumption in the next
subsection. For any $g\in L^{1}(V)$ we use the notation

$\displaystyle\left\langle g\right\rangle:=\int_{V}g(v)\,dv.$ (2)

We define

$\displaystyle|V|:=\left\langle 1\right\rangle=\int_{V}\,dv\,.$ (3)

Many of the constants below depend on $\max_{v\in V}\|v\|$, though for clarity
of exposition we do not give this dependence explicitly.

We consider kinetic equations where the collision operator satisfies an
entropy-dissipation property: Let $\eta:D\to\mathbb{R}$, where
$D\subseteq\mathbb{R}$, be a strictly convex function and
$\mathbb{F}(V):=\\{g\in L^{1}(V):\operatorname{Range}(g)\subseteq D\\}$. We
call $\eta$ the _kinetic entropy function_. Then the local entropy
$\mathcal{H}:\mathbb{F}(V)\to\mathbb{R}$ is given by

$\displaystyle\mathcal{H}(f):=\left\langle\eta(f)\right\rangle.$ (4)

This entropy is dissipated if the collision operator $\mathcal{C}$ satisfies

$\displaystyle\left\langle\eta^{\prime}(g)\mathcal{C}(g)\right\rangle\leq 0$
(5)

for all $g:V\to D$ such that the integral is defined. Furthermore, we assume
that $\eta$ is sufficiently smooth:

######  Assumption 2.

The kinetic entropy satisfies $\eta\in C^{3}(D)$, $\eta^{\prime\prime}>0$ on
$D$, and (5), i.e., $\eta$ is an entropy dissipated by the kinetic equation
(1).

###  2.2 The original entropy-based moment equations and realizability

The original entropy-based moment equations are a semidiscretization of the
kinetic equation (1) in the velocity variable $v$. For an overview, see [17].
The velocity dependence of $f$ at each point in time and space is replaced by
the vector of moments

$\displaystyle\mathbf{u}(t,x):=(u_{0}(t,x),u_{1}(t,x),\ldots,u_{N}(t,x))\in\mathbb{R}^{N+1},$
(6)

which contains the approximations of velocity integrals of $f$ multiplied by
the basis functions

$\displaystyle\mathbf{m}(v):=(m_{0}(v),m_{1}(v),\ldots,m_{N}(v)),$ (7)

that is, ${u_{i}(t,x)\simeq\langle m_{i}f(t,x,\cdot)\rangle}$ for all
${i\in\\{0,1,\ldots,N\\}}$. Usually the basis functions are polynomials. We
make the following assumptions on the basis functions:

######  Assumption 3.

  1. (i)

For every $i\in\\{0,1,\ldots,N\\}$ we have $m_{i}\in L^{\infty}(V)$. Without
loss of generality, we assume that $\|m_{i}\|_{L^{\infty}(V)}$ is bounded by
one:

$\displaystyle\|m_{i}\|_{L^{\infty}(V)}=\sup_{v\in V}|m_{i}(v)|\leq
1\qquad\text{for }i\in\\{0,1,\ldots,N\\}.$ (8)

  2. (ii)

The constant function is in the linear span of the basis functions. Without
loss of generality we assume $m_{0}(v)\equiv 1$.

For each moment vector, the entropy-based moment method reconstructs an ansatz
for the the kinetic density by solving the constrained optimization problem

$\operatorname*{minimize}_{g\in\mathbb{F}(V)}\>\mathcal{H}(g)\qquad\operatorname*{subject~{}to}\>\left\langle\mathbf{m}g\right\rangle=\mathbf{u}$
(9)

(recall the definition of $\mathcal{H}$ in (4)). Under Assumption 1, this
problem has a unique solution for every $\mathbf{u}\in\mathcal{R}$ [15], where

$\displaystyle\mathcal{R}:=\left\\{\mathbf{u}:\text{ there exists a
}g\in\mathbb{F}(V)\text{ such that
}\left\langle\mathbf{m}g\right\rangle=\mathbf{u}\right\\}$ (10)

is the set of all _realizable_ moment vectors, and consequently, the system of
moment equations is well-defined and hyperbolic for all realizable moment
vectors.111 When $V$ is not bounded, such as in entropy-based moment equations
for the Boltzmann equation for rarefied gas dynamics, where
$V=\mathbb{R}^{3}$, there are important examples of realizable moment vectors
for which the primal problem has no solution; see [14, 15, 13]. This is a
significant open problem for entropy-based moment equations, and neither the
regularization nor our work here can get around this issue.

The solution to (9) takes the form
$G_{\hat{\boldsymbol{\alpha}}(\mathbf{u})}$, where

$\displaystyle
G_{\boldsymbol{\alpha}}:=\eta^{\prime}_{*}(\boldsymbol{\alpha}\cdot\mathbf{m})$
(11)

and $\hat{\boldsymbol{\alpha}}:\mathcal{R}\to\mathbb{R}^{N+1}$ maps a moment
vector $\mathbf{u}$ to the solution of the dual problem

$\hat{\boldsymbol{\alpha}}(\mathbf{u})=\operatorname*{argmax}_{\boldsymbol{\alpha}\in\mathbb{R}^{N+1}}\left\\{\boldsymbol{\alpha}\cdot\mathbf{u}-\left\langle\eta_{\ast}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle\right\\};$
(12)

here $\eta_{\ast}$ is the Legendre dual222 See, e.g., [9, §3.3.2.] or [5,
§3.3], where what we call the Legendre dual is called the conjugate function.
of $\eta$ (and thus $\eta_{\ast}^{\prime}$ is the inverse function of
$\eta^{\prime}$). The components of $\boldsymbol{\alpha}$ are the Lagrange
multipliers for the primal problem (9).

Now we are ready to give the entropy-based moment equations:

$\displaystyle\partial_{t}\mathbf{u}+\nabla_{x}\cdot\mathbf{f}(\mathbf{u})$
$\displaystyle=\mathbf{r}(\mathbf{u}),$ (13a) where the flux function
$\mathbf{f}$ and relaxation term $\mathbf{r}$ are given by
$\displaystyle\mathbf{f}(\mathbf{u}):=\left\langle
v\mathbf{m}G_{\hat{\boldsymbol{\alpha}}(\mathbf{u})}\right\rangle\qquad\mbox{and}\qquad\mathbf{r}(\mathbf{u}):=\left\langle\mathbf{m}\mathcal{C}(G_{\hat{\boldsymbol{\alpha}}(\mathbf{u})})\right\rangle.$
(13b)

Classical solutions of the moment equations (13) satisfy the entropy
dissipation law

$\displaystyle\partial_{t}h(\mathbf{u})+\nabla_{x}\cdot
j(\mathbf{u})=h^{\prime}(\mathbf{u})\cdot\mathbf{r}(\mathbf{u})\leq 0$ (14)

for the entropy and entropy flux

$h(\mathbf{u}):=\left\langle\eta(G_{\hat{\boldsymbol{\alpha}}(\mathbf{u})})\right\rangle\quad\mbox{and}\quad
j(\mathbf{u}):=\left\langle
v\eta(G_{\hat{\boldsymbol{\alpha}}(\mathbf{u})})\right\rangle.$ (15)

Note that $h^{\prime}=\hat{\boldsymbol{\alpha}}$, and that $h$ is strictly
convex as a consequence of Assumption 2 [13]. For readers unfamiliar with the
derivations of the dual problem and the entropy dissipation law and related
properties of the entropy-based moment equations, we review these in Appendix
B.

We make the following assumption on the collision term:

######  Assumption 4.

The function $\mathbf{r}$ in the collision term of the original entropy-based
moment equations (13) is Lipschitz continuous with constant $C_{\mathbf{r}}$
and satisfies $\lim_{\mathbf{u}\rightarrow 0}\mathbf{r}(\mathbf{u})=0$.

This applies, for example, to linear collision operators like in the case of
isotropic scattering. This assumption cannot be expected to apply for the
Boltzmann collision operator, at least not globally, since it is quadratic.
The assumption $\lim_{\mathbf{u}\rightarrow 0}\mathbf{r}(\mathbf{u})=0$ is
natural since otherwise there would be collision effects in the moment
equations when no particles are present.

Notice that the flux $\mathbf{f}$ and the source $\mathbf{r}$ in (13) can only
be defined when the optimization problem (9) is feasible, i.e., when the
$\mathbf{u}\in\mathcal{R}$. When $D=[0,\infty)$, this set corresponds to the
set of vectors which contain the moments of a nonnegative density, which are
indeed the only moment vectors we want to consider since the kinetic density
$f$ should be nonnegative. This theoretical advantage is however impractical
for implementations because it is in general difficult to keep a high-order
numerical solution of (13) within the realizable set $\mathcal{R}$.

Finally, we note that a consequence of Assumption 1 is that
$\hat{\boldsymbol{\alpha}}$ is a smooth bijection from $\mathcal{R}$ to
$\mathbb{R}^{N+1}$. Its inverse is the function which gives the moment vector
of the entropy ansatz corresponding to a given multiplier vector:

$\displaystyle\hat{\mathbf{u}}(\boldsymbol{\alpha}):=\left\langle\mathbf{m}\eta_{\ast}^{\prime}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle.$
(16)

This function plays a role in the analysis later.

###  2.3 The regularized entropy-based moment equations

To work around the problem of realizability, the regularized entropy-based
moment equations were proposed in [2]. Let $\gamma\in(0,\infty)$ be the
regularization parameter. Then the regularized entropy-based moment equations
are given by

$\displaystyle\partial_{t}\mathbf{u}+\nabla_{x}\cdot\mathbf{f}_{\gamma}(\mathbf{u})$
$\displaystyle=\mathbf{r}_{\gamma}(\mathbf{u}),$ (17a) where
$\displaystyle\mathbf{f}_{\gamma}(\mathbf{u}):=\left\langle
v\mathbf{m}G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})}\right\rangle\qquad\mbox{and}\qquad\mathbf{r}_{\gamma}(\mathbf{u}):=\left\langle\mathbf{m}\mathcal{C}(G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})})\right\rangle.$
(17b)

The ansatz $G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})}$ has the same
form as above (i.e.,
${G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})}=\eta_{\ast}^{\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})}$)
but is the solution of the unconstrained optimization problem

$\displaystyle\operatorname*{minimize}_{g\in\mathbb{F}(V)}\>\left\langle\eta(g)\right\rangle+\frac{1}{2\gamma}\left\|\left\langle\mathbf{m}g\right\rangle-\mathbf{u}\right\|^{2},$
(18)

which is feasible for any moment vector $\mathbf{u}\in\mathbb{R}^{N+1}$ (again
under Assumption 1). The new multiplier vector
$\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})$ is the solution of the
corresponding dual problem

$\displaystyle\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}):=\operatorname*{argmax}_{\boldsymbol{\alpha}\in\mathbb{R}^{N+1}}\left\\{\boldsymbol{\alpha}\cdot\mathbf{u}-\left\langle\eta_{*}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle-\frac{\gamma}{2}\|\boldsymbol{\alpha}\|^{2}\right\\}$
(19)

and satisfies the first-order necessary conditions

$\displaystyle\mathbf{u}=\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))+\gamma\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}).$
(20)

Classical solutions of the regularized equations (17) satisfy

$\displaystyle\partial_{t}h_{\gamma}(\mathbf{u})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u})=h_{\gamma}^{\prime}(\mathbf{u})\cdot\mathbf{r}_{\gamma}(\mathbf{u})\leq
0.$

where

$h_{\gamma}(\mathbf{u}):=\left\langle\eta(G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})})\right\rangle+\frac{1}{2\gamma}\left\|\left\langle\mathbf{m}G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})}\right\rangle-\mathbf{u}\right\|^{2}\qquad\mbox{and}\qquad
j_{\gamma}(\mathbf{u}):=\left\langle
v\eta(G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})})\right\rangle.$ (21)

Analogously to the original case, we have
$h_{\gamma}^{\prime}=\hat{\boldsymbol{\alpha}}_{\gamma}$. In this work, we are
mostly concerned with entropy solutions of the regularized equations, i.e.
weak solutions of (17) satisfying the admissibility criterion

$\displaystyle\partial_{t}h_{\gamma}(\mathbf{u})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u})\leq
h_{\gamma}^{\prime}(\mathbf{u})\cdot\mathbf{r}_{\gamma}(\mathbf{u}).$ (22)

Note that any Lipschitz continuous solution of (17) is automatically an
entropy solution.

Like $h$, the entropy $h_{\gamma}$ of the regularized equations is convex, and
its Legendre dual $(h_{\gamma})_{*}$ will prove to be useful in the analysis.
The Legendre dual $(h_{\gamma})_{*}$ and its first and second derivatives are
given by [2]

$\displaystyle(h_{\gamma})_{*}(\boldsymbol{\alpha})$
$\displaystyle=\left\langle\eta_{\ast}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle+\frac{\gamma}{2}\|\boldsymbol{\alpha}\|^{2},$
(23a) $\displaystyle(h_{\gamma})^{\prime}_{*}(\boldsymbol{\alpha})$
$\displaystyle=\left\langle\mathbf{m}\eta_{\ast}^{\prime}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle+\gamma\boldsymbol{\alpha},\text{
and}$ (23b)
$\displaystyle(h_{\gamma})^{\prime\prime}_{*}(\boldsymbol{\alpha})$
$\displaystyle=\left\langle\mathbf{m}\mathbf{m}^{T}\eta_{\ast}^{\prime\prime}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle+\gamma
I,$ (23c)

where $I$ is the $(N+1)\times(N+1)$ identity matrix. Note that
$(h_{\gamma})^{\prime}_{*}\circ
h_{\gamma}^{\prime}=(h_{\gamma})^{\prime}_{*}\circ\hat{\boldsymbol{\alpha}}_{\gamma}=\operatorname{id}$,
so we also have
$h_{\gamma}^{\prime\prime}=((h_{\gamma})^{\prime\prime}_{*}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{-1}$,
where the inverse indicates the matrix inverse. One also immediately
recognizes from the form of $(h_{\gamma})^{\prime\prime}_{*}$ that
$(h_{\gamma})_{*}$ and thus $h_{\gamma}$ are strictly convex for any
$\gamma>0$.

##  3 Relative entropy for convergence

Let $X\subset\mathbb{R}^{d}$ be a $d$-cube and $T\in(0,\infty)$. We consider
the initial-value problem

$\displaystyle\partial_{t}\mathbf{u}+\nabla_{x}\cdot\mathbf{f}(\mathbf{u})$
$\displaystyle=\mathbf{r}(\mathbf{u})$ $\displaystyle(t,x)$
$\displaystyle\in(0,T]\times X,$ (24a) $\displaystyle\mathbf{u}(0,x)$
$\displaystyle=\mathbf{u}^{0}(x)$ $\displaystyle x$ $\displaystyle\in X,$
(24b)

where $\mathbf{u}^{0}$ are the given initial conditions, and we use periodic
boundary conditions in space. For any $\gamma\in(0,\infty)$, the regularized
moment equations are

$\displaystyle\partial_{t}\mathbf{u}_{\gamma}+\nabla_{x}\cdot\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})$
$\displaystyle=\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})$ $\displaystyle(t,x)$
$\displaystyle\in(0,T]\times X,$ (25a) $\displaystyle\mathbf{u}_{\gamma}(0,x)$
$\displaystyle=\mathbf{u}^{0}(x)$ $\displaystyle x$ $\displaystyle\in X,$
(25b)

and we also take periodic boundary conditions. Notice that the initial
conditions are the same.

Following Dafermos [8], we introduce the relative entropy and relative entropy
flux relative to $h_{\gamma}$:

######  Definition 1.

Given moments $\mathbf{u}_{\gamma},\,\mathbf{u}\in\mathbb{R}^{N+1}$ the
_relative entropy_ and _relative entropy flux_ are given by

$\displaystyle h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle:=h_{\gamma}(\mathbf{u}_{\gamma})-h_{\gamma}(\mathbf{u})-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\mathbf{u}_{\gamma}-\mathbf{u})\text{
and}$ (26) $\displaystyle j_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle:=j_{\gamma}(\mathbf{u}_{\gamma})-j_{\gamma}(\mathbf{u})-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{f}_{\gamma}(\mathbf{u})),$
(27)

respectively.

Note that using the relative entropy with respect to $h_{\gamma}$ allows us
(in principle) to use non-realizable moment vectors in both arguments of the
relative entropy. For our subsequent convergence result, however, there will
be a strong difference between the first and the second slot of the relative
entropy. In particular, in our convergence results, Corollaries 1 and 3, we
will require the function in the second slot of the relative entropy to have
values only in some compact subset of the set of realizable vectors.

######  Lemma 1.

Let $\mathbf{u}$ be a Lipschitz continuous solution of (24) and let
$\mathbf{u}_{\gamma}$ be an entropy solution of (25), i.e.
$\mathbf{u}_{\gamma}\in L^{\infty}([0,T)\times X,\mathbb{R}^{N+1})$ satisfies

$\displaystyle\partial_{t}\mathbf{u}_{\gamma}+\nabla_{x}\cdot\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})$
$\displaystyle=\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma}),$
$\displaystyle\partial_{t}h_{\gamma}(\mathbf{u}_{\gamma})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u}_{\gamma})$ $\displaystyle\leq
h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})$

in the sense of distributions. Then, for almost all $0\leq t\leq T$ the
following inequality holds:

$\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(t,x)|\mathbf{u}(t,x))\,dx\leq-\int_{0}^{t}\int_{X}(\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})(s,x)):\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}(s,x)|\mathbf{u}(s,x))\\\
-q_{\gamma}(\mathbf{u}_{\gamma}(s,x),\mathbf{u}(s,x))-J_{\gamma}(\mathbf{u}_{\gamma}(s,x),\mathbf{u}(s,x)):\nabla_{x}\mathbf{u}(s,x)dxds$
(28)

with

$\displaystyle\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u}):=\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{f}_{\gamma}(\mathbf{u})-\mathbf{f}_{\gamma}^{\prime}(\mathbf{u})(\mathbf{u}_{\gamma}-\mathbf{u})$
(29)

and

$\displaystyle
q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u}):=(h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})-h_{\gamma}^{\prime}(\mathbf{u}))\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{r}(\mathbf{u})\cdot(h_{\gamma}^{\prime\prime}(\mathbf{u})(\mathbf{u}_{\gamma}-\mathbf{u}))$
(30)

and

$J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u}):=(\mathbf{u}_{\gamma}-\mathbf{u})h_{\gamma}^{\prime\prime}(\mathbf{u})(\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}_{\gamma}^{\prime}(\mathbf{u}))$
(31)

###### Proof.

The proof follows the proof of [8, Thm 5.2.1]. We need, however, to account
for the difference of flux functions in (25) and (24). Therefore, we provide a
brief proof that is to be understood in the dual space of
$W^{1,\infty}_{0}([0,T)\times X,[0,\infty))$, i.e. nonnegative Lipschitz
continuous functions with compact support.

$\displaystyle\partial_{t}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle=\partial_{t}h_{\gamma}(\mathbf{u}_{\gamma})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u}_{\gamma})-h_{\gamma}^{\prime}(\mathbf{u})\cdot\partial_{t}\mathbf{u}-j_{\gamma}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u}-(h_{\gamma}^{\prime\prime}(\mathbf{u})\partial_{t}\mathbf{u})\cdot(\mathbf{u}_{\gamma}-\mathbf{u})$
$\displaystyle\qquad-(h_{\gamma}^{\prime\prime}(\mathbf{u})\cdot\nabla_{x}\mathbf{u})\cdot(\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{f}_{\gamma}(\mathbf{u}))-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\partial_{t}\mathbf{u}_{\gamma}+\nabla_{x}\cdot\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}))$
$\displaystyle\qquad+\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\partial_{t}\mathbf{u}+\mathbf{f}_{\gamma}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u})$
(32a) $\displaystyle\leq
h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})r_{\gamma}(\mathbf{u}_{\gamma})-h_{\gamma}^{\prime\prime}(\mathbf{u})(-\mathbf{f}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u}+\mathbf{r}(\mathbf{u}))\cdot(\mathbf{u}_{\gamma}-\mathbf{u})$
$\displaystyle\qquad-(h_{\gamma}^{\prime\prime}(\mathbf{u})\cdot\nabla_{x}\mathbf{u})\cdot(\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{f}_{\gamma}(\mathbf{u}))-h_{\gamma}^{\prime}(\mathbf{u})\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})$
(32b)
$\displaystyle=-(\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})):\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})+J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u}):\nabla_{x}\mathbf{u},$
(32c)

where in (32b) we have used

$\displaystyle\partial_{t}h_{\gamma}(\mathbf{u}_{\gamma})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u}_{\gamma})\leq
h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma}),$
(33a) $\displaystyle-
h_{\gamma}^{\prime}(\mathbf{u})\cdot\partial_{t}\mathbf{u}-j_{\gamma}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u}+\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\partial_{t}\mathbf{u}+\mathbf{f}_{\gamma}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u})=0,$
(33b)
$\displaystyle\partial_{t}\mathbf{u}=-\mathbf{f}^{\prime}(\mathbf{u}):\nabla_{x}\mathbf{u}+\mathbf{r}(\mathbf{u}),\text{
and}$ (33c)
$\displaystyle\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot(\partial_{t}\mathbf{u}_{\gamma}+\nabla_{x}\cdot\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}))=h_{\gamma}^{\prime}(\mathbf{u})\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma}).$
(33d)

We have also used the commutation property

$h_{\gamma}^{\prime\prime}\mathbf{f}_{\gamma}^{\prime}=(\mathbf{f}_{\gamma}^{\prime})^{T}h_{\gamma}^{\prime\prime}$

that follows by taking the derivative of the compatibility relation
$h_{\gamma}^{\prime}\mathbf{f}_{\gamma}^{\prime}=j_{\gamma}^{\prime}$.

Now we fix $0\leq t\leq T$ and for $\varepsilon>0$ we test

$\partial_{t}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+\nabla_{x}\cdot
j_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\leq-(\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})):\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})+J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u}):\nabla_{x}\mathbf{u},$

with $\psi_{\varepsilon}\in W_{0}^{1,\infty}([0,T)\times X,[0,\infty))$

$\psi_{\varepsilon}(s,x):=\left\\{\begin{array}[]{ccc}1&:&s<t\\\
1-\frac{s-t}{\varepsilon}&:&t<s<t+\varepsilon\\\
0&:&s>t+\varepsilon\end{array}\right.$

Sending $\varepsilon$ to zero, we obtain (28) for all $t$ that are Lebesgue
points of the map $[0,T)\rightarrow\mathbb{R}$,
${t\mapsto\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(t,x)|\mathbf{u}(t,x))\,dx}$.
Note that $\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(0,x)|\mathbf{u}(0,x))\,dx=0$
since $\mathbf{u}$, $\mathbf{u}_{\gamma}$ satisfy the same initial condition.
∎

At this point, with relative-entropy methods it is typical to look to bound
the integrand of the right-hand side using
$h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ so that Grönwall’s inequality can
be used.

It turns out that such bounds can be obtained if we make some assumption on
the original solution $\mathbf{u}$ which bounds it away from the boundary of
the realizable set $\mathcal{R}$. The exact form this assumption takes depends
on the entropy; we will make this precise in Section 4.

######  Theorem 1.

Let there be a subset ${\mathcal{S}}$ of the set of realizable vectors
$\mathcal{R}$ such that there exists $\gamma_{0}>0$ and constants
$C_{J},C_{\mathbf{f}},C_{q}>0$ and $D_{J},D_{\mathbf{f}},D_{q}>0$ so that

$\begin{aligned} \|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|&\leq
C_{J}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{J}\gamma^{2}\\\
\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|&\leq
C_{\mathbf{f}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{\mathbf{f}}\gamma^{2}\\\
q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})&\leq
C_{q}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{q}\gamma^{2}\end{aligned}\quad\forall\mathbf{u}_{\gamma}\in\mathbb{R}^{N+1},\mathbf{u}\in{\mathcal{S}},\gamma\in(0,\gamma_{0}).$
(34)

Let $\mathbf{u}$ be a Lipschitz continuous solution of (24) satisfying
$\mathbf{u}(t,x)\in{\mathcal{S}}$ for all $(t,x)\in[0,T]\times X$. Let
$\\{\mathbf{u}_{\gamma}\\}_{\gamma\in(0,\gamma_{0})}$ be a family of entropy
solutions of (25). Furthermore assume that
${\|\nabla_{x}\mathbf{u}\|_{L^{\infty}([0,T]\times X)}\leq C_{u}}$,
$\|\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|_{L^{\infty}([0,T]\times
X)}\leq C_{\hat{\boldsymbol{\alpha}}}$ uniformly in $\gamma$. Then for
$\gamma$ sufficiently small,

$\displaystyle\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(T,x)|\mathbf{u}(T,x))\,dx\leq\exp(CT)DT\gamma^{2},$
(35)

where $C:=C_{\hat{\boldsymbol{\alpha}}}C_{\mathbf{f}}+C_{u}C_{J}+C_{q}$ and
${D:=C_{\hat{\boldsymbol{\alpha}}}D_{\mathbf{f}}+C_{u}D_{J}+D_{q}}$.

###### Proof.

Inserting (34) into (28) implies

$\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(t,x)|\mathbf{u}(t,x))\,dx\\\
\leq\int_{0}^{t}\int_{X}(C_{\hat{\boldsymbol{\alpha}}}C_{\mathbf{F}}+C_{u}C_{J}+C_{q})h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+(C_{\hat{\boldsymbol{\alpha}}}D_{\mathbf{f}}+C_{u}D_{J}+D_{q})\gamma^{2}dxds.$
(36)

The assertion of the Theorem follows by applying Grönwall’s lemma. ∎

######  Remark 1.

It is important to note that we need to bound the norms of
$\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ and
$J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$ while it is sufficient to bound
$q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$ from above (no bound from below
is needed). This is due to the fact that, in (28),
$\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ and
$J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$ are multiplied by
$\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})$ and
$\nabla_{x}\mathbf{u}$, respectively, whose directions are unknown.

######  Remark 2.

We will work with sets $\mathcal{S}$ such that the derivative
$\hat{\boldsymbol{\alpha}}_{\gamma}^{\prime}(\mathbf{u})=((h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))^{-1}$
is uniformly bounded for $\mathbf{u}\in\mathcal{S}$ and
$\gamma\in(0,\gamma_{0})$. Thus
$\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\in
L^{\infty}([0,T]\times X)$ follows from $\nabla_{x}\mathbf{u}\in
L^{\infty}([0,T]\times X)$ and the chain rule, and
$\|\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|_{L^{\infty}([0,T]\times
X)}$ is bounded as $\gamma\to 0$.

##  4 Estimates

In this section we give sets $\mathcal{S}$ over which the flux and source
terms can be bounded as required in Theorem 1.

First, we give some basic properties of realizable moment vectors and the
functions $\hat{\mathbf{u}}$, $\mathbf{f}$, and their regularized counterparts
which will be used repeatedly when estimating the flux and source terms.

###  4.1 Basic properties

For basis functions satisfying Assumption 3, the components of any realizable
moment vector $\mathbf{u}\in\mathcal{R}$ satisfy

$\displaystyle|u_{i}|=|\langle m_{i}g\rangle|\leq\langle
g\rangle=u_{0},\qquad\text{for all }i\in\\{0,1,\ldots,N\\},$ (37)

thus there exists a constant $C_{0}\in(0,\infty)$ such that

$\displaystyle\|\mathbf{u}\|\leq C_{0}u_{0}\qquad\text{for all
}\mathbf{u}\in\mathcal{R}.$ (38)

Under Assumption 1 there is a $C_{1}\in(0,\infty)$ such that

$\displaystyle\|\mathbf{f}^{\prime}(\mathbf{u})\|\leq C_{1}$ (39)

for all $\mathbf{u}\in\mathcal{R}$ [1, Lemma 3.1], so $\mathbf{f}$ is globally
Lipschitz continuous on $\overline{\mathcal{R}}$. The same bound holds
globally for the regularized flux, i.e.,

$\displaystyle\|\mathbf{f}_{\gamma}^{\prime}(\mathbf{u})\|\leq C_{1}$ (40)

for all $\mathbf{u}\in\mathbb{R}^{N+1}$ (this also follows from the argument
used in [1, Lemma 3.1]). Since $\mathbf{f}(0)=0$ (in the limit sense, since
vector $0$ lies on the boundary of $\mathcal{R}$), we have

$\displaystyle\|\mathbf{f}(\mathbf{u})\|\leq C_{1}\|\mathbf{u}\|.$ (41)

The map $\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma}$ plays an
important role in our analysis. This map returns the realizable moment vector
that the regularization uses to compute the flux and source terms. It is
globally Lipschitz continuous.333 Indeed, the singular values of its Jacobian
are bounded by one: Let
$H:=\langle\mathbf{m}\mathbf{m}^{T}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\rangle$,
and recall that it is symmetric positive definite. Then
$(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})=H(H+\gamma
I)^{-1}$. For any eigenvalue-eigenvector pair $(\lambda,\mathbf{c})$ of $H$,
we have  $\displaystyle(H+\gamma I)^{-1}H^{2}(H+\gamma
I)^{-1}\mathbf{c}=\left(\frac{\lambda}{\lambda+\gamma}\right)^{2}\mathbf{c},$
(42) from which we concluded that the singular values have the form
$\lambda/(\lambda+\gamma)$ and thus are bounded by one.  If we let $C_{2}$ be
its Lipschitz constant, then in particular

$\displaystyle\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))\|\leq
C_{2}\|\mathbf{w}\|+\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0))\|$
(43)

In all cases we consider, $\lim_{\gamma\to
0}\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0))=0$ (see Appendix C),
so we define

$\displaystyle
C_{3}:=\sup_{\gamma\in(0,\gamma_{0})}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0))\|$
(44)

and generally use

$\displaystyle\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))\|\leq
C_{2}\|\mathbf{w}\|+C_{3}$ (45)

for appropriate values of $\gamma$. Furthermore, since
$\mathbf{f}_{\gamma}=\mathbf{f}\circ\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma}$,444To
see this, start from (17b) and apply the fact that $\hat{\boldsymbol{\alpha}}$
is the inverse function of $\hat{\mathbf{u}}$ to get
$\displaystyle\mathbf{f}_{\gamma}(\mathbf{u})=\left\langle
v\mathbf{m}G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})}\right\rangle=\left\langle
v\mathbf{m}G_{\hat{\boldsymbol{\alpha}}(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))}\right\rangle=\mathbf{f}(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})));$
(46) here we have used
$\hat{\boldsymbol{\alpha}}\circ\hat{\mathbf{u}}=\operatorname{id}$, and the
last equality comes from the definition of $\mathbf{f}(\mathbf{u})$ in (13b).
we can combine this bound with (41) to get

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{w})\|\leq
C_{1}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))\|\leq
C_{1}(C_{2}\|\mathbf{w}\|+C_{3}).$ (47)

We conclude this section by recalling a handy property of the regularized
problem. The partial derivative of $\hat{\boldsymbol{\alpha}}_{\gamma}$ with
respect to $\gamma$ was computed in [2, Thm 3]:

$\displaystyle\frac{\partial}{\partial\gamma}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})=-h_{\gamma}^{\prime\prime}(\mathbf{w})\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}).$
(48)

From the positive-definiteness of $h_{\gamma}^{\prime\prime}$ we can
immediately conclude that
$\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|^{2}$ is a decreasing
function of $\gamma$ for any $\mathbf{w}\in\mathbb{R}^{N+1}$. We can further
conclude that $\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|\to 0$ as
$\gamma\to\infty$: Since $\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|$
is a decreasing function of $\gamma$, it remains bounded for
$\gamma\in(\gamma_{0},\infty)$, where $\gamma_{0}\in(0,\infty)$. Thus the
moment vector
$\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))$ associated
with the ansatz $G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})}$ is also
bounded for $\gamma\in(\gamma_{0},\infty)$, since $\hat{\mathbf{u}}$ is a
continuous map. Let

$\displaystyle
u_{\gamma_{0},\mathbf{w}}:=\sup_{\gamma\in(\gamma_{0},\infty)}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))\|.$
(49)

Then by rearranging the first-order necessary conditions (20) we have

$\displaystyle\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|=\frac{\|\mathbf{w}-\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w}))\|}{\gamma}\leq\frac{\|\mathbf{w}\|+u_{\gamma_{0},\mathbf{w}}}{\gamma},$
(50)

from which we conclude $\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|\to
0$ as $\gamma\to\infty$.

We can also use (48) to show that the entropy $h_{\gamma}$ is a decreasing
function of $\gamma$. Using the formula
$h_{\gamma}(\mathbf{w})=h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})))+\frac{\gamma}{2}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|^{2}$
(obtained by inserting (15) and (20) into (21)), we have, after some basic
manipulations,

$\displaystyle\frac{\partial}{\partial\gamma}h_{\gamma}(\mathbf{w})=-\frac{1}{2}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|^{2}\leq
0.$ (51)

###  4.2 Two classes of kinetic entropy functions

To prove the estimates (34), we need to assume that the true solution
$\mathbf{u}$ is bounded away from the boundary of realizability. This can be
achieved in various ways, but the specific form of the assumption must depend
on properties of the kinetic entropy function $\eta$.

We consider two kinds of kinetic entropy functions. The first is defined with
the Maxwell–Boltzmann entropy,

$\displaystyle\eta(z)=z\log(z)-z,$ (52)

in mind. (The second term is purely for mathematical convenience.)

######  Definition 2.

Let $\eta:(0,\infty)\to\mathbb{R}$ satisfy Assumption 2. We call $\eta$
_superlinear_ if
${\operatorname{Range}(\eta^{\prime})=\operatorname{Dom}(\eta_{*})=\mathbb{R}}$.

Since $\eta^{\prime}$ is an increasing function, these entropies grow
superlinearly as $z\to\infty$. Furthermore, since
$0\in\operatorname{Range}(\eta^{\prime})$, the entropy $\eta$ has a global
minimum.

######  Remark 3.

Note that we use the term _superlinear_ even though the functions
$\eta(z)=z^{\alpha}$ for ${\alpha\in(1,\infty)}$ do not belong to our class of
superlinear entropy functions.

The second kind of entropy we consider includes the Bose–Einstein entropy,

$\displaystyle\eta(z)=z\log(z)-(1+z)\log(1+z),$ (53)

and the Burg entropy,

$\displaystyle\eta(z)=-\log(z).$ (54)

######  Definition 3.

Let $\eta:(0,\infty)\to\mathbb{R}$ satisfy Assumption 2. We call $\eta$
_sublinear_ if
${\operatorname{Range}(\eta^{\prime})=\operatorname{Dom}(\eta_{*})=(-\infty,0)}$
with $\lim_{z\to\infty}\eta^{\prime}(z)=0$ and $\lim_{z\to
0}\eta^{\prime}(z)=-\infty$.

These entropies are monotonically decreasing functions with no global minimum.
The decay as $z\to\infty$ is sublinear.

###  4.3 The superlinear case

For superlinear entropies we consider the family of sets

$\displaystyle\mathcal{R}^{M}:=\left\\{\hat{\mathbf{u}}(\boldsymbol{\alpha}):\|\boldsymbol{\alpha}\|\leq
M\right\\},$ (55)

for $M\in(0,\infty)$. For each $M$, we have
$\mathcal{R}^{M}\subset\subset\mathcal{R}$, and as $M\to\infty$, the set
$\mathcal{R}^{M}$ approaches the full realizable set $\mathcal{R}$ (since
under Assumption 1 we have $\mathcal{R}=\hat{\mathbf{u}}(\mathbb{R}^{N+1})$).
We will show that the assumptions of Theorem 1 hold for superlinear entropies
when $\mathcal{S}=\mathcal{R}^{M}$ for any $M$.

First we give some properties on $\mathcal{R}^{M}$. Since $\mathcal{R}^{M}$ is
a compact set, both

$\displaystyle
u_{M}:=\sup_{\mathbf{w}\in\mathcal{R}^{M}}\|\mathbf{w}\|\qquad\mbox{and}\qquad
h_{M}:=\sup_{\mathbf{w}\in\mathcal{R}^{M}}h(\mathbf{w})$ (56)

are finite. Since $h_{\gamma}$ is a decreasing function of $\gamma$ (recall
(51)), we have

$\displaystyle h_{M}=\sup_{\begin{subarray}{c}\mathbf{w}\in\mathcal{R}^{M}\\\
\gamma\in(0,\infty)\end{subarray}}h_{\gamma}(\mathbf{w}).$ (57)

Another important consequence of restricting $\mathbf{u}$ to $\mathcal{R}^{M}$
is that $h_{\gamma}^{\prime\prime}$ is bounded from above and below over
$\mathcal{R}^{M}$. First recall that
$h_{\gamma}^{\prime\prime}=((h_{\gamma})^{\prime\prime}_{*}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{-1}$,
so we work with $(h_{\gamma})^{\prime\prime}_{*}$. Let $\mathbf{c}$ be the
unit-length eigenvector associated with the largest eigenvalue of
$(h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))$
for some $\mathbf{u}\in\mathcal{R}^{M}$. Then we have

$\displaystyle\lambda_{\max}((h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))$
$\displaystyle=\mathbf{c}\cdot\left(\left\langle\mathbf{m}\mathbf{m}^{T}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma
I\right)\mathbf{c}$ (58a)
$\displaystyle=\left\langle(\mathbf{c}\cdot\mathbf{m})^{2}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma$
(58b)
$\displaystyle\leq|V|\left(\sup_{y\in[-M,M]}\eta_{\ast}^{\prime\prime}(y)\right)+\gamma_{0}$
(58c)

for $\gamma\leq\gamma_{0}$, where we have used that
$\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|$ is a decreasing function
of $\gamma$ (recall (48)). Note that, on $\mathcal{R}^{M}$, as
$\gamma\to\infty$, all eigenvalues of $h_{\gamma}^{\prime\prime}$ go to zero,
and indeed the function $h_{\gamma}$ becomes flat. Since the behavior of the
regularized equations for large $\gamma$ is not particularly interesting, we
rule out these problems by considering only $\gamma$ smaller than some
arbitrary $\gamma_{0}$.

On the other hand, if we now let $\mathbf{c}$ be the unit-length eigenvector
associated with the smallest eigenvalue of
$(h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))$
for $\mathbf{u}\in\mathcal{R}^{M}$ we have

$\displaystyle\lambda_{\min}((h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))=\left\langle(\mathbf{c}\cdot\mathbf{m})^{2}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma\geq\lambda_{\min}(\langle\mathbf{m}\mathbf{m}^{T}\rangle)\inf_{y\in[-M,M]}\eta_{\ast}^{\prime\prime}(y)$
(59)

(where $\lambda_{\min}(\langle\mathbf{m}\mathbf{m}^{T}\rangle)>0$ because
$\mathbf{m}$ span a basis). Note that Assumption 2 guarantees strict
positivity of $\eta_{\ast}^{\prime\prime}$ because
$\eta_{\ast}^{\prime\prime}(y)=1/\eta^{\prime\prime}(\eta_{\ast}^{\prime}(y))$.

Thus for $h_{\gamma}^{\prime\prime}$ we can conclude the existence of positive
constants $\lambda_{\min,h^{\prime\prime},M}$ and $C_{h^{\prime\prime},M}$
such that

$\displaystyle\begin{aligned} \mathbf{v}\cdot
h_{\gamma}^{\prime\prime}(\mathbf{u})\mathbf{v}&\geq\lambda_{\min,h^{\prime\prime},M}\|\mathbf{v}\|^{2}\\\
\|h_{\gamma}^{\prime\prime}(\mathbf{u})\|&\leq
C_{h^{\prime\prime},M}\end{aligned}\quad\text{for all
}\mathbf{v}\in\mathbb{R}^{N+1}\text{, }\mathbf{u}\in\mathcal{R}^{M}\text{, and
}\gamma\in(0,\gamma_{0})$ (60)

#### The flux terms

######  Lemma 2.

Let $\eta$ be a superlinear kinetic entropy function, $M\in(0,\infty)$, and
$\gamma_{0}\in(0,\infty)$. Then there exist positive constants
$C_{\mathbf{f}}$, $C_{J}$, and $D_{J}$ such that

$\begin{aligned} \|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|&\leq
C_{\mathbf{f}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\\\
\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|&\leq
C_{J}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{J}\gamma^{2}\end{aligned}\quad\forall\mathbf{u}_{\gamma}\in\mathbb{R}^{N+1},\mathbf{u}\in\mathcal{R}^{M},\gamma\in(0,\gamma_{0}).$
(61)

These constants depend on $M$ and $\gamma_{0}$ but are independent of
$\mathbf{u}_{\gamma}$, $\mathbf{u}$, and $\gamma$.

In the proof of Lemma 2 we often use the following elementary lemma.

######  Lemma 3.

Let $f:[0,\infty)\to\mathbb{R}$ be a twice continuously differentiable
function such that $f^{\prime\prime}(z)>0$ for all $z$, and let
$a\in(0,\infty)$, $b\in\mathbb{R}$, and $c\in\mathbb{R}$ be given. If
$K\in(0,\infty)$ and $C\in(0,\infty)$ satisfy

$\displaystyle f(K)+c>0,\quad f^{\prime}(K)>0,\quad\mbox{and}\quad
C\geq\max\left\\{\frac{aK+b}{f(K)+c},\frac{a}{f^{\prime}(K)}\right\\},$ (62)

then

$\displaystyle az+b\leq C(f(z)+c)$ (63)

holds for all $z\geq K$.

In the proof of Lemma 2 we also use the following lemma to get a nonzero lower
bound the relative entropy.

######  Lemma 4.

Let $M\in(0,\infty)$, $L\in(M,\infty)$, and $\gamma_{0}\in(0,\infty)$. Then

$\displaystyle
C_{h,M,L}:=\inf_{\begin{subarray}{c}\mathbf{v}\in\mathbb{R}^{N+1}\setminus\mathcal{R}^{L}\\\
\mathbf{u}\in\mathcal{R}^{M}\\\
\gamma\in(0,\gamma_{0})\end{subarray}}h_{\gamma}(\mathbf{v}|\mathbf{u})$ (64)

is strictly positive.

The proof of Lemma 4 can be found in Appendix D

###### Lemma 2.

We partition $\mathbb{R}^{N+1}$ into three subsets and consider
$\mathbf{u}_{\gamma}$ on each of these three subsets, which we illustrate in
Figure 1. On the first set, $\mathcal{R}^{L}$ for $L\in(M,\infty)$, we take
advantage of the fact that $\mathbf{f}_{\gamma}$, $J_{\gamma}$, and the
relative entropy all look like $\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}$
locally, up to an $\mathcal{O}(\gamma^{2})$ term. On the second set,
$B_{K}\setminus\mathcal{R}^{L}$, where $B_{K}$ is a norm ball in
$\mathbb{R}^{N+1}$, we use the fact that neither $\mathbf{f}_{\gamma}$ nor
$J_{\gamma}$ nor $q_{\gamma}$ blow up on compact sets. Finally, in the third
set we use that $\mathbf{f}_{\gamma}$ and $J_{\gamma}$ grow linearly in either
$\|\mathbf{u}_{\gamma}\|$ or
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ for
large $\|\mathbf{u}_{\gamma}\|$ while $h_{\gamma}(\mathbf{u}_{\gamma})$ grows
at least linearly in $\|\mathbf{u}_{\gamma}\|$ or
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$.

$B_{K}$$\mathcal{R}$$\mathcal{R}^{L}$$\mathcal{R}^{M}$$u_{0}$$u_{1}$ Figure 1:
The sets used in the proof of Lemma 2. For this figure, we consider the
$M_{1}$ case in slab geometry: $V=[-1,1]$, $\mathbf{m}(v)=(1,v)$. In this
case, the realizable set is given by
$\mathcal{R}=\\{(u_{0},u_{1}):|u_{1}|<u_{0}\\}$, and we used $M=1$, $L=1.3$,
and $K=8$.

  1. (i)

We begin with
$(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$ for
an $L\in(M,\infty)$. (The reason for choosing $L>M$ is given in case (ii)
below.)

When $\mathbf{f}_{\gamma}$ is sufficiently smooth there exists an $\mathbf{w}$
on the line connecting $\mathbf{u}_{\gamma}$ and $\mathbf{u}$ such that

$\displaystyle\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})=(\mathbf{f}_{\gamma}^{\prime\prime}(\mathbf{w})(\mathbf{u}_{\gamma}-\mathbf{u}))(\mathbf{u}_{\gamma}-\mathbf{u})\,.$
(65)

In our case, $\mathbf{f}_{\gamma}$ indeed possesses the requisite smoothness:
We can write $\mathbf{f}_{\gamma}$ as
${\mathbf{f}_{\gamma}=\mathbf{g}\circ\hat{\boldsymbol{\alpha}}_{\gamma}}$,
where $\mathbf{g}(\boldsymbol{\alpha})=\left\langle
v\mathbf{m}G_{\boldsymbol{\alpha}}\right\rangle$. The function $\mathbf{g}$ is
smooth and its derivatives are bounded over any bounded set. Note that
$\mathbf{w}\in\operatorname{Conv}(\mathcal{R}^{L})$, and since
$\mathcal{R}^{L}\subset\subset\mathcal{R}$ and $\mathcal{R}$ is convex, we
also know $\operatorname{Conv}(\mathcal{R}^{L})\subset\subset\mathcal{R}$.
Thus we define

$\displaystyle\widetilde{L}:=\sup_{\mathbf{w}\in\operatorname{Conv}(\mathcal{R}^{L})}\|\hat{\boldsymbol{\alpha}}(\mathbf{w})\|<\infty.$
(66)

Since $\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})$ is continuous with
respect to $\gamma$ for $\gamma\in[0,\infty)$ (where
$\hat{\boldsymbol{\alpha}}_{\gamma=0}=\hat{\boldsymbol{\alpha}}$ when
$\mathbf{w}\in\mathcal{R}$) [2, §3.1] and
$\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|$ is a decreasing function
of $\gamma$ (recall (48)) we have

$\displaystyle\widetilde{L}=\sup_{\begin{subarray}{c}\mathbf{w}\in\operatorname{Conv}(\mathcal{R}^{L})\\\
\gamma\in(0,\infty)\end{subarray}}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\|.$
(67)

Thus it is clear that we only consider $\mathbf{g}$ and its derivatives within
a bounded set. Furthermore, when bounded away from the boundary of
$\mathcal{R}$, $\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})$ is a smooth
function of $\mathbf{u}$, and this smoothness is uniform for
$\gamma\in(0,\gamma_{0})$. For example,
$\hat{\boldsymbol{\alpha}}_{\gamma}^{\prime}=h_{\gamma}^{\prime\prime}$
(recall (23c)), and from (60) we have
$\|h_{\gamma}^{\prime\prime}(\mathbf{w})\|\leq
C_{h^{\prime\prime},\widetilde{L}}$ for $\mathbf{w}\in\mathcal{R}^{L}$. The
second derivative $\hat{\boldsymbol{\alpha}}_{\gamma}^{\prime\prime}$ can be
similarly bounded, but we omit this more tedious computation.

So we define

$\displaystyle
C_{\mathbf{f}^{\prime\prime},L}:=\sup_{\begin{subarray}{c}\mathbf{w}\in\operatorname{Conv}(\mathcal{R}^{L})\\\
\gamma\in(0,\gamma_{0})\end{subarray}}\|\mathbf{f}_{\gamma}^{\prime\prime}(\mathbf{w})\|$
(68)

for some $\gamma_{0}\in(0,\infty)$ and conclude

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|\leq
C_{\mathbf{f}^{\prime\prime},L}\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}$ (69)

for $(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$.

We now turn to $J_{\gamma}$. The deciding factor in $J_{\gamma}$ is
$(\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}_{\gamma}^{\prime}(\mathbf{u}))$;
to estimate it we use
$\mathbf{f}_{\gamma}=\mathbf{f}\circ\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma}$,
the Lipschitz continuity of $\mathbf{f}^{\prime}$ on $\mathcal{R}^{M}$
(guaranteed by (68)), and the accuracy inequality

$\displaystyle\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))-\mathbf{u}\|\leq
M\gamma$ (70)

from [2, Thm. 2] as follows:

$\displaystyle\|\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}_{\gamma}^{\prime}(\mathbf{u})\|$
$\displaystyle=\|\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}^{\prime}(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})\|$
(71a)
$\displaystyle=\|\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}^{\prime}(\mathbf{u})(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})$
$\displaystyle\qquad+\mathbf{f}^{\prime}(\mathbf{u})(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})-\mathbf{f}^{\prime}(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})\|$
(71b)
$\displaystyle\leq\|\mathbf{f}^{\prime}(\mathbf{u})(I-(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u}))\|$
$\displaystyle\qquad+\|(\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}^{\prime}(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))))(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})\|$
(71c) $\displaystyle\leq
C_{1}\gamma\|h_{\gamma}^{\prime\prime}(\mathbf{u})\|+C_{\mathbf{f}^{\prime\prime},M}M\gamma\|(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})\|$
(71d)
$\displaystyle\leq(C_{1}C_{h^{\prime\prime},M}+C_{\mathbf{f}^{\prime\prime},M}MC_{2})\gamma$
(71e) $\displaystyle=:C_{4}\gamma,$ (71f)

where we define
$C_{4}:=C_{1}C_{h^{\prime\prime},M}+C_{\mathbf{f}^{\prime\prime},M}MC_{2}$. In
(71d) we have used

$\displaystyle
I-(\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma})^{\prime}(\mathbf{u})=\gamma
h_{\gamma}^{\prime\prime}(\mathbf{u}),$ (72)

which is a straightforward computation using
$\hat{\mathbf{u}}^{\prime}=h^{\prime\prime}_{*}$ and
${\hat{\boldsymbol{\alpha}}_{\gamma}^{\prime}(\mathbf{u})=(h^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))+\gamma
I)^{-1}}$ (see (23c)). In this step we are also able to use
$C_{\mathbf{f}^{\prime\prime},M}$ because
$\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|$ is a decreasing function
of $\gamma$ (recall (48)).

With (71) and the bound on $h_{\gamma}^{\prime\prime}$ from (60) we can bound
$J_{\gamma}$ by

$\displaystyle\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
$\displaystyle\leq\|\mathbf{u}_{\gamma}-\mathbf{u}\|\|h_{\gamma}^{\prime\prime}(\mathbf{u})\|\|\mathbf{f}^{\prime}(\mathbf{u})-\mathbf{f}_{\gamma}^{\prime}(\mathbf{u})\|$
(73a)
$\displaystyle\leq\|\mathbf{u}_{\gamma}-\mathbf{u}\|C_{h^{\prime\prime},M}C_{4}\gamma$
(73b)
$\displaystyle\leq\frac{C_{h^{\prime\prime},M}C_{4}}{2}\left(\gamma^{2}+\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}\right),$
(73c)

where for the last step we have applied Young’s inequality.

Finally, we must bound $h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ similarly
from below. This follows immediately from (60):

$\displaystyle
h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})=(\mathbf{u}_{\gamma}-\mathbf{u})\cdot
h_{\gamma}^{\prime\prime}(\mathbf{w})(\mathbf{u}_{\gamma}-\mathbf{u})\geq\lambda_{\min,h^{\prime\prime},\widetilde{L}}\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}$
(74)

for all
$(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$ and
$\mathbf{w}$ is the appropriate vector in
$\operatorname{Conv}(\mathcal{R}^{L})$ from the mean-value theorem.

Altogether we have

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|$
$\displaystyle\leq\frac{C_{\mathbf{f}^{\prime\prime},L}}{\lambda_{\min,h^{\prime\prime},\widetilde{L}}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
(75) $\displaystyle\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
$\displaystyle\leq\frac{C_{h^{\prime\prime},M}C_{4}}{2\lambda_{\min,h^{\prime\prime},\widetilde{L}}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+\frac{C_{h^{\prime\prime},M}C_{4}}{2}\gamma^{2}$
(76)

for $(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$.

  2. (ii)

We now consider $\mathbf{u}_{\gamma}\in B_{K}\setminus\mathcal{R}^{L}$ for the
ball $B_{K}:=\\{\mathbf{w}\in\mathbb{R}^{N+1}:\|\mathbf{w}\|\leq K\\}$. Since
$B_{K}$ is a compact set, we know that the constants

$\displaystyle C_{\mathbf{f},K}$
$\displaystyle:=\sup_{\begin{subarray}{c}\mathbf{u}_{\gamma}\in
B_{K}\setminus\mathcal{R}^{L}\\\ \mathbf{u}\in\mathcal{R}^{M}\\\
\gamma\in(0,\gamma_{0})\end{subarray}}\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|$
(77) $\displaystyle C_{J,K}$
$\displaystyle:=\sup_{\begin{subarray}{c}\mathbf{u}_{\gamma}\in
B_{K}\setminus\mathcal{R}^{L}\\\ \mathbf{u}\in\mathcal{R}^{M}\\\
\gamma\in(0,\gamma_{0})\end{subarray}}\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
(78)

are finite for any $\gamma_{0}\in(0,\infty)$. Indeed with (40), (47), (56),
and (60) we immediately have the crude upper bounds

$\displaystyle C_{\mathbf{f},K}\leq
C_{1}((C_{2}+1)(K+u_{M})+2C_{3})\quad\mbox{and}\quad C_{J,K}\leq
2(K+u_{M})C_{h^{\prime\prime},M}C_{1}.$ (79)

To bound $h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ from below we use
$C_{h,M,L}$ from Lemma 4. (Note that here $L>M$ is crucial: this lower bound
is not strictly positive for $L\leq M$.)

Altogether we have

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|\leq\frac{C_{\mathbf{f},K}}{C_{h,M,L}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\qquad\mbox{and}\qquad\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|\leq\frac{C_{J,K}}{C_{h,M,L}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u}).$
(80)

Up to now, $K$ is arbitrary; in the next and final case we give a lower bound
that $K$ must satisfy.

  3. (iii)

Finally, for $\mathbf{u}_{\gamma}\in\mathbb{R}^{N+1}\setminus B_{K}$, how we
proceed depends on which term in $h_{\gamma}$ dominates: when
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ is
large, we use the entropy term, otherwise we use the quadratic term. To
distinguish these two subcases, let $\delta\in(0,1)$. The following arguments
work for any value of $\delta\in(0,1)$; the particular choice of $\delta$
merely affects the value of the constants $C_{\mathbf{f}}$ and $C_{J}$.

    1. (a)

$\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|\leq\delta\|\mathbf{u}_{\gamma}\|$
Here, we use the quadratic term of the relative entropy to dominate the linear
term of the relative flux. We bound the relative flux by

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
$\displaystyle\leq
C_{1}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|+C_{1}\|\mathbf{u}_{\gamma}\|+C_{1}(C_{2}u_{M}+C_{3})+C_{1}u_{M}$
$\displaystyle\leq
C_{1}(1+\delta)\|\mathbf{u}_{\gamma}\|+C_{1}((C_{2}+1)u_{M}+C_{3}),$ (81)

$J_{\gamma}$ by

$\displaystyle\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|\leq
2C_{1}C_{h^{\prime\prime},M}(\|\mathbf{u}_{\gamma}\|+u_{M}),$ (82)

and the relative entropy by

$\displaystyle h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ $\displaystyle\geq
h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})))+\frac{1}{2\gamma}(\|\mathbf{u}_{\gamma}\|-\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|)^{2}-h_{M}-M(\|\mathbf{u}_{\gamma}\|+u_{M})$
$\displaystyle\geq|V|\eta_{\min}+\frac{(1-\delta)^{2}}{2\gamma_{0}}\|\mathbf{u}_{\gamma}\|^{2}-h_{M}-M(\|\mathbf{u}_{\gamma}\|+u_{M})$
(83)

where we use $\eta_{\min}:=\min_{z\geq 0}\eta(z)$, and in the first inequality
we have used that $\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|$ is a
decreasing function of $\gamma$ (recall (48)).

Then the application of Lemma 3 with $z=\|\mathbf{u}_{\gamma}\|$ and
${f(z)=(1-\delta)^{2}z^{2}/(2\gamma_{0})-Mz}$ gives the following conditions
on $K$, $C_{\mathbf{f}}$, and $C_{J}$:

$\displaystyle\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK>h_{M}+Mu_{M}-2\eta_{\min},\quad\frac{(1-\delta)^{2}}{\gamma_{0}}K>M,$
(84) $\displaystyle\begin{aligned}
C_{\mathbf{f}}&\geq\max\left\\{\frac{C_{1}(1+\delta)K+C_{1}((C_{2}+1)u_{M}+C_{3})}{\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK+|V|\eta_{\min}-h_{M}-Mu_{M}},\frac{C_{1}(1+\delta)}{\frac{(1-\delta)^{2}}{\gamma_{0}}K-M}\right\\},\text{
and}\\\
C_{J}&\geq\max\left\\{\frac{2C_{1}C_{h^{\prime\prime},M}(K+u_{M})}{\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK+|V|\eta_{\min}-h_{M}-Mu_{M}},\frac{2C_{1}C_{h^{\prime\prime},M}}{\frac{(1-\delta)^{2}}{\gamma_{0}}K-M}\right\\}\end{aligned}$
(85)

    2. (b)

$\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|>\delta\|\mathbf{u}_{\gamma}\|$
In this case, we know that
$\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ is
not arbitrarily small and therefore formulate our bounds in terms of
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$. First,
for the upper bounds we have, using (38),

$\displaystyle\|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
$\displaystyle\leq\left(C_{1}C_{0}+C_{1}\frac{C_{0}}{\delta}\right)\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+C_{1}(C_{2}u_{M}+C_{3})+C_{1}u_{M},\text{
and}$ (86) $\displaystyle\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|$
$\displaystyle\leq
2C_{1}C_{h^{\prime\prime},M}\left(\frac{C_{0}}{\delta}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+u_{M}\right).$
(87)

For the lower bound on the entropy, notice that by Jensen’s inequality

$\displaystyle\left\langle\eta(g)\right\rangle\geq|V|\eta\left(\frac{1}{|V|}\left\langle
g\right\rangle\right),$ (88)

from which it follows that

$\displaystyle h(\mathbf{u})\geq|V|\eta\left(\frac{1}{|V|}u_{0}\right),$ (89)

for any realizable $\mathbf{u}$. Thus for the relative entropy we have

$\displaystyle
h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\geq|V|\eta\left(\frac{1}{|V|}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\right)-h_{M}-M\left(\frac{C_{0}}{\delta}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+u_{M}\right).$
(90)

Now, with the lower bound on
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$,
namely,

$\displaystyle\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\geq\frac{1}{C_{0}}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|\geq\frac{\delta}{C_{0}}\|\mathbf{u}_{\gamma}\|\geq\frac{\delta
K}{C_{0}}$ (91)

we can apply Lemma 3 with
$z=\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ and
${f(z)=|V|\eta(z/|V|)-MC_{0}z/\delta}$. Here the conditions on $K$,
$C_{\mathbf{f}}$, and $C_{J}$ become

$\displaystyle|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK>h_{M}+Mu_{M},\quad\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)>\frac{MC_{0}}{\delta},$ (92) $\displaystyle\begin{aligned}
C_{\mathbf{f}}&\geq\max\left\\{\frac{C_{1}((\delta+1)K+(C_{2}+1)u_{M}+C_{3})}{|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK-
h_{M}-Mu_{M}},\frac{C_{1}C_{0}\left(1+\frac{1}{\delta}\right)}{\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)-\frac{MC_{0}}{\delta}}\right\\},\text{ and}\\\
C_{J}&\geq\max\left\\{\frac{2C_{1}C_{h^{\prime\prime},M}\left(K+u_{M}\right)}{|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK-
h_{M}-Mu_{M}},\frac{2C_{0}C_{1}C_{h^{\prime\prime},M}}{\delta\left(\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)-\frac{MC_{0}}{\delta}\right)}\right\\}\end{aligned}$ (93)

Altogether we get (61) for

$\displaystyle C_{\mathbf{f}}$
$\displaystyle=\max\left\\{\frac{C_{\mathbf{f}^{\prime\prime},L}}{\lambda_{\min,h^{\prime\prime},\widetilde{L}}},\frac{C_{\mathbf{f},K}}{C_{h,M,L}},\frac{C_{1}(1+\delta)K+C_{1}((C_{2}+1)u_{M}+C_{3})}{\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK+|V|\eta_{\min}-h_{M}-Mu_{M}},\right.$
$\displaystyle\qquad\qquad\left.\frac{C_{1}(1+\delta)}{\frac{(1-\delta)^{2}}{\gamma_{0}}K-M},\frac{C_{1}((\delta+1)K+(C_{2}+1)u_{M}+C_{3})}{|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK-
h_{M}-Mu_{M}},\frac{C_{1}C_{0}\left(1+\frac{1}{\delta}\right)}{\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)-\frac{MC_{0}}{\delta}}\right\\},$ (94) $\displaystyle
C_{J}$
$\displaystyle=\max\left\\{\frac{C_{h^{\prime\prime},M}C_{4}}{2\lambda_{\min,h^{\prime\prime},\widetilde{L}}},\frac{C_{J,K}}{C_{h,M,L}},\frac{2C_{1}C_{h^{\prime\prime},M}(K+u_{M})}{\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK+|V|\eta_{\min}-h_{M}-Mu_{M}},\right.$
$\displaystyle\qquad\qquad\left.\frac{2C_{1}C_{h^{\prime\prime},M}}{\frac{(1-\delta)^{2}}{\gamma_{0}}K-M},\frac{2C_{1}C_{h^{\prime\prime},M}\left(K+u_{M}\right)}{|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK-
h_{M}-Mu_{M}},\frac{2C_{0}C_{1}C_{h^{\prime\prime},M}}{\delta\left(\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)-\frac{MC_{0}}{\delta}\right)}\right\\},\text{ and}$ (95)
$\displaystyle D_{J}$ $\displaystyle=\frac{C_{h^{\prime\prime},M}C_{4}}{2},$
(96)

where $K$ satisfies (84) and (92). ∎

#### The source term

######  Lemma 5.

Let $\eta$ be a superlinear kinetic entropy function, $M\in(0,\infty)$, and
$\gamma_{0}\in(0,\infty)$. Then there exist positive constants $C_{q}$ and
$D_{q}$ such that

$q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\leq
C_{q}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{q}\gamma^{2}\quad\forall\mathbf{u}_{\gamma}\in\mathbb{R}^{N+1},\mathbf{u}\in\mathcal{R}^{M},\gamma\in(0,\gamma_{0}).$
(97)

###### Proof.

We use the same lower bounds just derived for the relative entropy
$h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ in Lemma 2. Thus we only need to
give upper bounds of $q_{\gamma}$ on the same decomposition of
$\mathbb{R}^{N+1}\times\mathcal{R}^{M}$ used in the proof of Lemma 2.

  1. (i)

Let $(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$.
To write down an upper bound of $q_{\gamma}$, we define the constants

$\displaystyle
C_{h^{\prime\prime\prime}}=\sup_{\begin{subarray}{c}\mathbf{v}\in\operatorname{Conv}(\mathcal{R}^{L})\\\
\gamma\in(0,\gamma_{0})\end{subarray}}\|h_{\gamma}^{\prime\prime\prime}(\mathbf{v})\|\qquad\mbox{and}\qquad
r_{M}=\sup_{\mathbf{v}\in\mathcal{R}^{M}}\|\mathbf{r}(\mathbf{v})\|,$ (98)

all finite by smoothness of $h_{\gamma}$ and $\mathbf{r}$ and compactness of
$\operatorname{Conv}(\mathcal{R}^{L})$ and $\mathcal{R}^{M}$, and we use

$\displaystyle\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))-\mathbf{u}\|\leq\|\mathbf{u}_{\gamma}-\mathbf{u}\|+M\gamma,$
(99)

which follows from $\mathbf{u}\in\mathcal{R}^{M}$ [2, Thm. 2]. Then we
rearrange $q_{\gamma}$ and straightforwardly get the estimate

$\displaystyle q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$
$\displaystyle=(h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})-h_{\gamma}^{\prime}(\mathbf{u}))\cdot(\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{r}(\mathbf{u}))$
$\displaystyle\qquad+(h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})-h_{\gamma}^{\prime}(\mathbf{u})-h_{\gamma}^{\prime\prime}(\mathbf{u})(\mathbf{u}_{\gamma}-\mathbf{u}))\cdot\mathbf{r}(\mathbf{u})\,$
(100a)
$\displaystyle\leq(C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}+r_{M}C_{h^{\prime\prime\prime}})\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}+C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}M\gamma\|\mathbf{u}_{\gamma}-\mathbf{u}\|$
(100b)
$\displaystyle\leq(C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}+r_{M}C_{h^{\prime\prime\prime}})\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}+\frac{C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}M}{2}\left(\gamma^{2}+\|\mathbf{u}_{\gamma}-\mathbf{u}\|^{2}\right),$
(100c)

which appropriately mirrors (69) and (73).

  2. (ii)

Now for the case $\mathbf{u}_{\gamma}\in B_{K}\setminus\mathcal{R}^{L}$ (where
again $B_{K}:=\\{\mathbf{w}\in\mathbb{R}^{n}:\|\mathbf{w}\|\leq K\\}$), we
first use
$h_{\gamma}^{\prime}(\mathbf{u}_{\gamma})\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})\leq
0$ to get

$\displaystyle q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\leq-
h_{\gamma}^{\prime}(\mathbf{u})\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{r}(\mathbf{u})\cdot(h_{\gamma}^{\prime\prime}(\mathbf{u})(\mathbf{u}_{\gamma}-\mathbf{u}))$
(101)

As with the flux, none of these terms blow up for
$(\mathbf{u}_{\gamma},\mathbf{u})\in B_{K}\times\mathcal{R}^{M}$ for any
finite $K$:

$\displaystyle C_{q,K,\gamma_{0}}$
$\displaystyle:=\sup_{\begin{subarray}{c}\mathbf{u}_{\gamma}\in
B_{K}\setminus\operatorname{Conv}(\mathcal{R}^{L})\\\
\mathbf{u}\in\mathcal{R}^{M}\\\
\gamma\in(0,\gamma_{0})\end{subarray}}-h_{\gamma}^{\prime}(\mathbf{u})\cdot\mathbf{r}_{\gamma}(\mathbf{u}_{\gamma})-\mathbf{r}(\mathbf{u})\cdot(h_{\gamma}^{\prime\prime}(\mathbf{u})(\mathbf{u}_{\gamma}-\mathbf{u}))$
(102a) $\displaystyle\leq
MC_{\mathbf{r}}(C_{2}K+C_{3})+r_{M}C_{h^{\prime\prime},M}(K+u_{M}).$ (102b)

  3. (iii)

For large $\mathbf{u}_{\gamma}$ we show, as with the flux, that $q_{\gamma}$
grows linearly with $\|\mathbf{u}_{\gamma}\|$ when
$\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ is
small and linearly with
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$
otherwise. Indeed, if for $\delta\in(0,1)$ we have
$\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|\leq\delta\|\mathbf{u}_{\gamma}\|$,
then

$\displaystyle q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\leq
MC_{\mathbf{r}}\delta\|\mathbf{u}_{\gamma}\|+r_{M}C_{h^{\prime\prime},M}(\|\mathbf{u}_{\gamma}\|+u_{M}),$
(103)

i.e., linear growth in $\|\mathbf{u}_{\gamma}\|$ as in (iiia) and (82). On the
other hand, when
${\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|>\delta\|\mathbf{u}_{\gamma}\|}$,
then

$\displaystyle q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\leq
MC_{\mathbf{r}}C_{0}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+r_{M}C_{h^{\prime\prime},M}\left(\frac{C_{0}}{\delta}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+u_{M}\right),$
(104)

which is linear growth in
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))$ as in
(86) and (87).

Now we simply need to change the numerators from the flux case above to get
(97) for

$\displaystyle C_{q}$
$\displaystyle\geq\max\left\\{\frac{(C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}+r_{M}C_{h^{\prime\prime\prime}})+\frac{C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}M}{2}}{\lambda_{\min,h^{\prime\prime},\widetilde{L}}},\frac{C_{q,K,\gamma_{0}}}{C_{h,M,L}},\right.$
$\displaystyle\qquad\qquad\left.\frac{MC_{\mathbf{r}}\delta
K+r_{M}C_{h^{\prime\prime},M}(K+u_{M})}{\frac{(1-\delta)^{2}}{\gamma_{0}}K^{2}-MK+|V|\eta_{\min}-h_{M}-C_{0}v_{0,M}},\frac{MC_{\mathbf{r}}\delta+r_{M}C_{h^{\prime\prime},M}}{\frac{(1-\delta)^{2}}{\gamma_{0}}K-M},\right.$
$\displaystyle\qquad\qquad\left.\frac{MC_{\mathbf{r}}\delta
K+r_{M}C_{h^{\prime\prime},M}(K+u_{M})}{|V|\eta\left(\frac{\delta
K}{|V|C_{0}}\right)-MK-
h_{M}-Mu_{M}},\frac{MC_{\mathbf{r}}C_{0}+r_{M}C_{h^{\prime\prime},M}\frac{C_{0}}{\delta}}{\eta^{\prime}\left(\frac{\delta
K}{|V|C_{0}}\right)-\frac{MC_{0}}{\delta}}\right\\}$ (105)

for some $K$ and $\delta$ satisfying the same conditions as for the flux term,
and

$\displaystyle
D_{q}:=\frac{C_{h^{\prime\prime},\widetilde{L}}C_{\mathbf{r}}M}{2}.$ (106)

∎

Lemmas 2 and 5 yield a more precise version of Theorem 1:

######  Corollary 1.

Let $\eta$ be a superlinear kinetic entropy function and $\mathbf{u}$ a
Lipschitz solution of the entropy-based moment equations (24) for which there
exists $M\in(0,\infty)$ such that $\mathbf{u}(t,x)\in\mathcal{R}^{M}$ for all
$(t,x)\in[0,T]\times X$. Let
$\\{\mathbf{u}_{\gamma}\\}_{\gamma\in(0,\gamma_{0})}$ be a family of entropy
solutions of (25). Furthermore assume that
$\|\nabla_{x}\mathbf{u}\|_{L^{\infty}([0,T]\times X)}\leq C_{u}$.

Then
${\|\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|_{L^{\infty}([0,T]\times
X)}\leq C_{h^{\prime\prime},M}C_{u}}$, and

$\displaystyle\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(T,x)|\mathbf{u}(T,x))\,dx\leq\exp(CT)DT\gamma^{2}$
(107)

for $C:=C_{h^{\prime\prime},M}C_{u}C_{\mathbf{f}}+C_{u}C_{J}+C_{q}$ and
$D:=C_{u}D_{J}+D_{q}$, where the constants $C_{\mathbf{f}}$, $C_{J}$, $C_{q}$,
$D_{J}$, and $D_{q}$ are given by Lemmas 2 and 5.

######  Corollary 2.

Let $\eta$, $\mathbf{u}$, and $\mathbf{u}_{\gamma}$ satisfy the conditions of
Corollary 1 and $L\in(M,\infty)$, and consider

$X_{L}:=\\{x\in X\,:\,\mathbf{u}_{\gamma}(T,x)\in\mathcal{R}^{L}\\}.$

Then the Lebesgue-measure of the complement of this set satisfies

$\displaystyle|X\setminus X_{L}|\leq\frac{\exp(CT)DT}{C_{h,M,L}}\gamma^{2},$
(108)

where $C_{h,M,L}$ is defined in (64). Furthermore on $X_{L}$ we have

$\displaystyle\|\mathbf{u}_{\gamma}(T,\cdot)-\mathbf{u}(T,\cdot)\|_{L^{2}(X_{L})}\leq\sqrt{\frac{\exp(CT)DT}{\lambda_{\min,h^{\prime\prime},\widetilde{L}}}}\gamma$
(109)

###### Proof.

Recall the definition of $C_{h,M,L}$ in (64). Then we have

$\displaystyle C_{h,M,L}|X\setminus X_{L}|\leq\int_{X\setminus
X_{L}}h_{\gamma}(\mathbf{u}_{\gamma}(T,x)|\mathbf{u}(T,x))\,dx\stackrel{{\scriptstyle\eqref{eq:cor-
super}}}{{\leq}}\exp(CT)DT\gamma^{2},$ (110)

from which we conclude (108). The inequality (109) follows immediately from
(74) (cf. (60)). ∎

###  4.4 The sublinear case

In contrast to the Maxwell–Boltzmann-like entropies, the entropies $\eta$ we
consider in this class are not bounded from below but
${\lim_{z\to\infty}\eta^{\prime}(z)=0}$. Consequently
${\operatorname{Dom}(\eta_{\ast})\subseteq(-\infty,0)}$, and the multipliers
must satisfy $\boldsymbol{\alpha}\cdot\mathbf{m}<0$ for all $v\in V$. For such
entropies we replace the assumption in (55) with

$\displaystyle\mathbf{u}\in\mathcal{R}^{M,m}:=\left\\{\hat{\mathbf{u}}(\boldsymbol{\alpha}):\|\boldsymbol{\alpha}\|\leq
M\text{ and }\boldsymbol{\alpha}\cdot\mathbf{m}(v)\leq-m\text{ for all }v\in
V\right\\},$ (111)

for some $M\in(0,\infty)$ and $m\in(0,\infty)$. Note that as $M\to\infty$ and
$m\to 0$, the set $\mathcal{R}^{M,m}$ approaches the full realizable set
$\mathcal{R}$. Related to the parameter $m$ is

$\displaystyle
p_{0}:=-\sup_{\begin{subarray}{c}\mathbf{w}\in\mathcal{R}^{M,m}\\\
\gamma\in(0,\gamma_{0})\\\ v\in
V\end{subarray}}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{w})\cdot\mathbf{m}(v)>0;$
(112)

for some $\gamma_{0}\in(0,\infty)$.

The additional condition parameterized by $m$ in $\mathcal{R}^{M,m}$ ensures
that the ansatz $G_{\boldsymbol{\alpha}}$ in the sublinear case is bounded
away from zero. In the superlinear case, the ansätze $G_{\boldsymbol{\alpha}}$
for $\boldsymbol{\alpha}\in\mathcal{R}^{M}$ are already bounded away from
zero, but this is not so for the sublinear case, where $\boldsymbol{\alpha}$
must also fulfill $\boldsymbol{\alpha}\cdot\mathbf{m}(v)<0$ for all $v\in V$.

Note that $u_{M,m}$ and $h_{M,m}$ can be defined as in (56):

$\displaystyle
u_{M,m}:=\sup_{\mathbf{w}\in\mathcal{R}^{M,m}}\|\mathbf{w}\|\qquad\mbox{and}\qquad
h_{M,m}:=\sup_{\mathbf{w}\in\mathcal{R}^{M,m}}h(\mathbf{w});$ (113)

both are finite. We can also derive similar bounds on
$h_{\gamma}^{\prime\prime}$. Let $\mathbf{c}$ be the unit-length eigenvector
associated with the largest eigenvalue of
$(h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))$
for some $\mathbf{u}\in\mathcal{R}^{M,m}$

$\displaystyle\lambda_{\max}((h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))$
$\displaystyle=\mathbf{c}\cdot\left(\left\langle\mathbf{m}\mathbf{m}^{T}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma
I\right)\mathbf{c}$ (114a)
$\displaystyle=\left\langle(\mathbf{c}\cdot\mathbf{m})^{2}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma$
(114b)
$\displaystyle\leq|V|\left(\sup_{y\in[-M,-p_{0}]}\eta_{\ast}^{\prime\prime}(y)\right)+\gamma_{0}$
(114c)

for $\gamma\leq\gamma_{0}$. Similarly, if we now let $\mathbf{c}$ be the unit-
length eigenvector associated with the smallest eigenvalue of
$(h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))$
for $\mathbf{u}\in\mathcal{R}^{M,m}$ we have

$\displaystyle\lambda_{\min}((h_{\gamma})^{\prime\prime}_{*}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))=\left\langle(\mathbf{c}\cdot\mathbf{m})^{2}\eta_{\ast}^{\prime\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m})\right\rangle+\gamma\geq|V|\inf_{y\in[-M,-p_{0}]}\eta_{\ast}^{\prime\prime}(y).$
(115)

Thus for the sublinear case there exist positive constants
$\lambda_{\min,h^{\prime\prime},M,m}$ and $C_{h^{\prime\prime},M,m}$ such that

$\displaystyle\begin{aligned} \mathbf{v}\cdot
h_{\gamma}^{\prime\prime}(\mathbf{u})\mathbf{v}&\geq\lambda_{\min,h^{\prime\prime},M,m}\|\mathbf{v}\|^{2}\\\
\|h_{\gamma}^{\prime\prime}(\mathbf{u})\|&\leq
C_{h^{\prime\prime},M,m}\end{aligned}\quad\text{for all
}\mathbf{v}\in\mathbb{R}^{N+1}\text{, }\mathbf{u}\in\mathcal{R}^{M}\text{, and
}\gamma\in(0,\gamma_{0});$ (116)

(cf. the superlinear case (60)).

######  Lemma 6.

Let $\eta$ be a sublinear kinetic entropy function, $M\in(0,\infty)$,
$m\in(0,\infty)$, and $\gamma_{0}\in(0,\infty)$. Then there exist positive
constants $C_{J}$, $D_{J}$, $C_{\mathbf{F}}$, and $D_{\mathbf{F}}$ such that

$\begin{aligned} \|\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\|&\leq
C_{\mathbf{f}}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})\\\
\|J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})\|&\leq
C_{J}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{J}\gamma^{2}\\\
q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})&\leq
C_{q}h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})+D_{q}\gamma^{2}\end{aligned}\quad\forall\mathbf{u}_{\gamma}\in\mathbb{R}^{N+1},\mathbf{u}\in\mathcal{R}^{M,m},\gamma\in(0,\gamma_{0}).$
(117)

###### Proof.

Because it is so similar to the superlinear case, we only sketch the proof for
the sublinear case.

For sublinear $\eta$, the estimates of
$\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$,
$J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$,
$q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$ and
$h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ on
${B_{K}\times\mathcal{R}^{M,m}}$ can be derived just as in the superlinear
case for ${B_{K}\times\mathcal{R}^{M}}$, as well as the estimates of
$\mathbf{f}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$,
$J_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$, and
$q_{\gamma}(\mathbf{u}_{\gamma},\mathbf{u})$ for large $\mathbf{u}_{\gamma}$
in (iiia), (82), (103), (86), (87), and (104). The lower bound of
$h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$, however, in (83) is no longer
possible when $\eta$ is not bounded from below, and Lemma 3 can no longer be
applied to (90) because the right-hand side does not grow as
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\to\infty$.

In the subcase
$\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|\leq\delta\|\mathbf{u}_{\gamma}\|$,
we first apply a combination of the bounds on individual terms from above to
get

$\displaystyle h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle\geq|V|\eta\left(\frac{1}{|V|}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\right)+\frac{(1-\delta)^{2}}{2\gamma_{0}}\|\mathbf{u}_{\gamma}\|^{2}-h_{M,m}-M\|\mathbf{u}_{\gamma}\|-Mu_{M,m}.$
(118a) Now we recognize that
$\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\leq\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|$
and then use that $\eta$ is a monotonically decreasing function to conclude
$\displaystyle h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle\geq|V|\eta\left(\frac{\delta}{|V|}\|\mathbf{u}_{\gamma}\|\right)+\frac{(1-\delta)^{2}}{2\gamma_{0}}\|\mathbf{u}_{\gamma}\|^{2}-h_{M,m}-M\|\mathbf{u}_{\gamma}\|-Mu_{M,m}.$
(118b)

Now thanks to the convexity of $\eta$, we can apply Lemma 3 with

$\displaystyle f(z)=|V|\eta\left(\frac{\delta
z}{|V|}\right)+\frac{(1-\delta)^{2}z^{2}}{2\gamma_{0}}-Mz.$ (119)

In the other subcase, where
${\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\|>\delta\|\mathbf{u}_{\gamma}\|}$,
we use the assumption (111) and the first-order necessary condition (20) to
get

$\displaystyle h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$
$\displaystyle=h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})))+\frac{\gamma}{2}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})\|^{2}-h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))-\frac{\gamma}{2}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|^{2}$
$\displaystyle\qquad-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\big{(}\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))+\gamma\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})-\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))-\gamma\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\big{)}$
(120a)
$\displaystyle=h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})))-h(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})))-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\left(\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))-\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}))\right)$
$\displaystyle\qquad+\frac{\gamma}{2}\left\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\right\|^{2}$
(120b)
$\displaystyle\geq|V|\eta\left(\frac{1}{|V|}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\right)-h_{M,m}-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))-Mu_{M,m}$
(120c)
$\displaystyle=|V|\eta\left(\frac{1}{|V|}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\right)-h_{M,m}-\left\langle\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\cdot\mathbf{m}\eta_{\ast}^{\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})\cdot\mathbf{m})\right\rangle-
Mu_{M,m}$ (120d)
$\displaystyle\geq|V|\eta\left(\frac{1}{|V|}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))\right)-h_{M,m}+p_{0}\hat{u_{0}}(\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma}))-Mu_{M,m}.$
(120e)

In the last step we have used $\eta_{\ast}^{\prime}\geq 0$. We are again ready
to apply Lemma 3 with ${f(z)=|V|\eta(z/|V|)+p_{0}z}$ to derive conditions on
$K$, $C_{\mathbf{f}}$, and $C_{J}$ to achieve the desired estimate (117).

∎

This immediately gives the following corollary.

######  Corollary 3.

Let $\eta$ be a sublinear kinetic entropy function and $\mathbf{u}$ a
Lipschitz continuous solution of the entropy-based moment equations (24) for
which there exist $M\in(0,\infty)$ and $m\in(0,\infty)$ so that
$\mathbf{u}(t,x)\in\mathcal{R}^{M,m}$ for all $(t,x)\in[0,T]\times X$. Let
$\\{\mathbf{u}_{\gamma}\\}_{\gamma\in(0,\gamma_{0})}$ be a family of entropy
solutions of (25) for $\gamma\in(0,\gamma_{0})$. Furthermore assume that
$\|\nabla_{x}\mathbf{u}\|_{L^{\infty}([0,T]\times X)}\leq C_{u}$.

Then
${\|\nabla_{x}\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|_{L^{\infty}([0,T]\times
X)}\leq C_{h^{\prime\prime},M,m}C_{u}}$, and

$\displaystyle\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(T,x)|\mathbf{u}(T,x))\,dx\leq\exp(CT)DT\gamma^{2}$
(121)

for $C:=C_{\hat{\boldsymbol{\alpha}}}C_{\mathbf{f}}+C_{u}C_{J}+C_{q}$ and
$D:=C_{u}D_{J}+D_{q}$, where the constants $C_{\mathbf{f}}$, $C_{J}$, $C_{q}$,
$D_{J}$, and $D_{q}$ are given by Lemma 6.

Under the assumptions of Corollary 3 a result analogous to that of Corollary 2
is easy to prove.

##  5 Numerical Results

We consider the toy problem from [12, 2]. There the authors considered the
moment equations for the linear kinetic equations in slab geometry (see e.g.,
[18]):

$\displaystyle\partial_{t}f+v\partial_{x}f=\sigma_{\mathrm{s}}\left(\frac{1}{2}\left\langle
f\right\rangle-f\right).$ (122)

where $V=[-1,1]$ and $\sigma_{\mathrm{s}}\in[0,\infty)$. For the spatial
domain we take $X=[0,1]$.

######  Remark 4.

More generally, the kinetic equation includes terms for absorption of
particles by a background medium as well as a source term, as in [12, 2].
While the resulting terms in the moment equations can be straightforwardly
incorporated into our analysis and do not affect our main results, we have
left them out in this work for clarity of exposition.

The corresponding entropy-based moment equations are

$\displaystyle\partial_{t}\mathbf{u}+\partial_{x}\mathbf{f}(\mathbf{u})=\sigma_{\mathrm{s}}R\mathbf{u},$
(123)

where $R=\operatorname{diag}\\{0,-1,\dots,-1\\}$. For the basis functions we
used the Legendre polynomials. We used the Maxwell–Boltzmann entropy (52). The
collision term $\mathbf{r}(\mathbf{u})=\sigma_{\mathrm{s}}R\mathbf{u}$ clearly
satisfies Assumption 4.

For numerical computations we used the fourth-order Runge–Kutta discontinuous
Galerkin (RKDG) method as in [2] with 160 spatial cells and no slope limiter.
With this spatial resolution the numerical solutions were accurate enough to
observe the convergence in $\gamma$.

The initial conditions we used are constructed as follows. Let
${\omega(x):=\frac{1}{2}M_{0}(1+\cos(2\pi x))}$ be a periodic function which
we use to define the multiplier vector

$\displaystyle\boldsymbol{\beta}(x)=\begin{pmatrix}\log\left(\frac{\omega(x)}{2\sinh(\omega(x))}\right)\\\
\omega(x)\\\ 0\\\ \vdots\\\ 0\end{pmatrix}$ (124)

Then the initial conditions are given by

$\displaystyle\mathbf{u}^{0}(x)=\left\langle\mathbf{m}\exp(\boldsymbol{\beta}(x)\cdot\mathbf{m})\right\rangle.$
(125)

Note that $\beta_{0}$ is chosen such that the zeroth order moment of the
initial condition is one, i.e., $u^{0}_{0}(x)\equiv 1$. The solution
$\mathbf{u}(t,x)$ of the original entropy-based moment equations with these
initial conditions satisfies the assumption
${\mathbf{u}(t,x)\in\mathcal{R}^{M}}$ of Corollary 1 for $M\approx M_{0}$.
Indeed the maximum value of $\|\hat{\boldsymbol{\alpha}}(\mathbf{u}(t,x))\|$
in space tends to decrease as time advances depending on the value of
$\sigma_{\mathrm{s}}$: the larger $\sigma_{\mathrm{s}}$ is, the faster the
norms of the multipliers decrease in time. For $\sigma_{\mathrm{s}}=0$, the
value of $\max_{x}\|\hat{\boldsymbol{\alpha}}(\mathbf{u}(t,x))\|$ is nearly
constant in time.

We ran the solutions until the final time $T=0.1$. We used various values of
$N$ up to 15 and found that the results did not depend qualitatively on the
value of $N$. We tried several values of $\sigma_{\mathrm{s}}$ from zero to
one and here did observe that the results depended on the value of
$\sigma_{\mathrm{s}}$: for very small values of $\sigma_{\mathrm{s}}$ the
solutions appear not to enter the regime of second-order convergence until
$\gamma$ is very small, on the order of $10^{-11}$. Such values of $\gamma$
are so small, that for these solutions the error due to the numerical
optimizer started to dominate errors due to the regularization.

We remind the reader that in order to evaluate the flux function
$\mathbf{f}_{\gamma}$ we compute the multiplier vector
$\hat{\boldsymbol{\alpha}}_{\gamma}$ by numerically solving the dual problem
(19). We used the numerical optimizer described in [2] but found that for
problems with $\sigma_{\mathrm{s}}=0$ the value of tolerance $\tau$ on the
norm of the dual gradient used in the stopping criterion, namely
$\tau=10^{-7}$, was not small enough to observe convergence in $\gamma$ for
the very small values of $\gamma$ where the equations enter the regime of
second-order convergence. The difficulty here is that, in our experience, one
cannot reliably bring the norm of the dual gradient below $10^{-7}$ when the
norm of the multipliers at the solution is about ten or bigger.

But for smaller values of $M_{0}$, we found that using a combination of the
smaller tolerance $\tau=10^{-8}$ as well as modifying the optimizer to make
efforts to further reduce the norm of the dual gradient when possible allowed
us to decrease the numerical errors from the optimizer enough so that we could
observe near second-order convergence. This modification is described in
pseudocode in Algorithm 1 and works as follows: The optimizer runs as usual
until the norm of the dual gradient is smaller than $\tau$. Then, the
optimizer continues to take up to $\ell_{\max}$ additional iterations to bring
the norm of the dual gradient under the smaller tolerance
${\tau_{d}\in(0,\tau)}$, which we call the _desired_ tolerance. If the
optimizer is unable to bring the norm of the dual gradient under $\tau_{d}$ in
$\ell_{\max}$ additional iterations, the optimizer still exits successfully
(as long as the current multiplier vector still satisfies the original
stopping criterion). In all of the results reported here, we used
${\tau_{d}=10^{-11}}$ and ${\ell_{\max}=10}$.

Algorithm 1 The optimizer with modified stopping criterion

$k\leftarrow 0$

$\ell\leftarrow 0$

${\tt acceptable\\_tolerance\\_achieved}=$ false

while $k<k_{\max}$ do

if
($\|\hat{\mathbf{u}}(\boldsymbol{\alpha}_{k})+\gamma\boldsymbol{\alpha}_{k}-\mathbf{u}\|<\tau_{d}$)
or
($\|\hat{\mathbf{u}}(\boldsymbol{\alpha}_{k})+\gamma\boldsymbol{\alpha}_{k}-\mathbf{u}\|<\tau$
and $\ell>\ell_{\max}$) then

return $\boldsymbol{\alpha}_{k}$

end if

if ${\tt acceptable\\_tolerance\\_achieved}=$ false and
$\|\hat{\mathbf{u}}(\boldsymbol{\alpha}_{k})+\gamma\boldsymbol{\alpha}_{k}-\mathbf{u}\|<\tau$
then

${\tt acceptable\\_tolerance\\_achieved}\leftarrow$ true

end if

Compute search direction $\mathbf{d}_{k}$

Perform backtracking line search to determine backtracking parameter $\xi_{k}$

$\boldsymbol{\alpha}_{k+1}\leftarrow\boldsymbol{\alpha}_{k}+\xi_{k}\mathbf{d}_{k}$

$k\leftarrow k+1$

if ${\tt acceptable\\_tolerance\\_achieved}=$ true then

$\ell\leftarrow\ell+1$

end if

end while

The results are given in Tables 1 to 3, which include the errors between the
solution of the original equations and the solutions of the regularized
equations measured in the relative entropy as well as in the $L^{2}$ and
$L^{\infty}$ norms. The error measured in the relative entropy is

$\displaystyle\mathcal{H}_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u}):=\int_{X}h_{\gamma}(\mathbf{u}_{\gamma}(T,x)|\mathbf{u}(T,x))\,dx,$
(126)

where we compute $h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ numerically
using the formula

$\displaystyle
h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})=\left\langle\eta(G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})}|G_{\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})})\right\rangle+\frac{\gamma}{2}\|\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u}_{\gamma})-\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})\|^{2},$
(127)

where

$\displaystyle\eta(G_{\boldsymbol{\alpha}}|G_{\boldsymbol{\beta}}):=$
$\displaystyle\;\eta(G_{\boldsymbol{\alpha}})-\eta(G_{\boldsymbol{\beta}})-\eta^{\prime}(G_{\boldsymbol{\beta}})(G_{\boldsymbol{\alpha}}-G_{\boldsymbol{\beta}})$
(128a) $\displaystyle=$
$\displaystyle\;\eta(G_{\boldsymbol{\alpha}})-\eta(G_{\boldsymbol{\beta}})-(\boldsymbol{\beta}\cdot\mathbf{m})(G_{\boldsymbol{\alpha}}-G_{\boldsymbol{\beta}}).$
(128b)

(For the second line we have used
$G_{\boldsymbol{\beta}}=\eta_{\ast}^{\prime}(\boldsymbol{\beta}\cdot\mathbf{m})$
and $\eta^{\prime}\circ\eta_{\ast}^{\prime}=\operatorname{id}$.) Formula (127)
for $h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ can be deduced by inserting
(21) into (26) and simplifying, and it ensures the positivity of
$h_{\gamma}(\mathbf{u}_{\gamma}|\mathbf{u})$ despite errors due to the
approximate computation of $\hat{\boldsymbol{\alpha}}_{\gamma}(\mathbf{u})$.
The spatial integrals are computed using an eight-point Gauss quadrature on
each of four subintervals in each spatial cell. The observed convergence order
$\nu$ between solutions computed with $\gamma_{1}$ and $\gamma_{2}$ is given
by

$\displaystyle\frac{\mathcal{H}_{\gamma_{1}}(\mathbf{u}_{\gamma_{1}}|\mathbf{u})}{\mathcal{H}_{\gamma_{2}}(\mathbf{u}_{\gamma_{2}}|\mathbf{u})}=\left(\frac{\gamma_{1}}{\gamma_{2}}\right)^{\nu}$
(129)

The $L^{2}$ norm is computed using the same spatial quadrature, and the
$L^{\infty}$ norm is approximated by taking the maximum over these spatial
quadrature points.

In Tables 1 and 2, second-order convergence is clear in the relative entropy
until the value of the relative entropy reaches about $10^{-17}$, which is
below machine precision. These tables include varying values of $M_{0}$ and
$N$. For $\sigma_{\mathrm{s}}\geq 10^{-5}$, we observed second-order
convergence for all values of $N$ up to 15 that we tried and for $M_{0}$ up to
200\. For values of $M_{0}$ larger than 200, it is too difficult to satisfy
the smaller optimization tolerance $\tau=10^{-8}$. In all cases, we observe
first-order convergence in the $L^{2}$ norm as well as the $L^{\infty}$ norm.

For $\sigma_{\mathrm{s}}=0$, we were only able to solve the equations for
smaller values of $M_{0}$ and observed second-order convergence for a smaller
range of values of $\gamma$. These results can be found in Table 3, where we
have included results from additional values of $\gamma$ between $10^{-9}$ and
$10^{-11}$ to highlight the regime of second-order convergence. In Table 3, we
see that the observed convergence orders increase monotonically to 1.99 and
stay there until the value of the relative entropy goes below machine
precision and the $L^{\infty}$ norm is smaller than the optimization tolerance
$\tau=10^{-8}$. Indeed, at the final time, the optimizer was not able to solve
many of the problems to the desired tolerance $\tau_{d}=10^{-11}$, and in many
of these problems, the tolerance $\tau$ is only barely fulfilled. Therefore
errors on the order of $10^{-8}$ in the $L^{2}$ and $L^{\infty}$ norms are not
surprising, and this error of course also affects the computation of the
relative entropy.

$\gamma$ | $\mathcal{H}_{\gamma}$ | $\nu$ | $L^{2}$ | $\nu$ | $L^{\infty}$ | $\nu$
---|---|---|---|---|---|---
$10^{-3}$ | 3.977e-05 | – | 2.010e-03 | – | 3.939e-03 | –
$10^{-4}$ | 5.319e-07 | 1.87 | 2.115e-04 | 0.98 | 3.970e-04 | 1.00
$10^{-5}$ | 5.439e-09 | 1.99 | 2.123e-05 | 1.00 | 3.948e-05 | 1.00
$10^{-6}$ | 5.454e-11 | 2.00 | 2.131e-06 | 1.00 | 3.971e-06 | 1.00
$10^{-7}$ | 5.504e-13 | 2.00 | 2.136e-07 | 1.00 | 3.667e-07 | 1.03
$10^{-8}$ | 5.557e-15 | 2.00 | 2.147e-08 | 1.00 | 4.422e-08 | 0.92
$10^{-9}$ | 5.886e-17 | 1.98 | 2.336e-09 | 0.96 | 5.671e-09 | 0.89
$10^{-10}$ | 2.082e-18 | 1.45 | 3.110e-10 | 0.88 | 1.002e-09 | 0.75
Table 1: Convergence test: $N=9$, $\sigma_{\mathrm{s}}=1$, $M_{0}=100$. $\gamma$ | $\mathcal{H}_{\gamma}$ | $\nu$ | $L^{2}$ | $\nu$ | $L^{\infty}$ | $\nu$
---|---|---|---|---|---|---
$10^{-3}$ | 1.196e-03 | – | 7.040e-03 | – | 2.040e-02 | –
$10^{-4}$ | 2.769e-04 | 0.64 | 2.490e-03 | 0.45 | 1.587e-02 | 0.11
$10^{-5}$ | 3.442e-05 | 0.91 | 7.394e-04 | 0.53 | 6.236e-03 | 0.41
$10^{-6}$ | 1.691e-06 | 1.31 | 1.336e-04 | 0.74 | 2.020e-03 | 0.49
$10^{-7}$ | 2.033e-08 | 1.92 | 1.475e-05 | 0.96 | 2.168e-04 | 0.97
$10^{-8}$ | 2.043e-10 | 2.00 | 1.481e-06 | 1.00 | 2.164e-05 | 1.00
$10^{-9}$ | 2.130e-12 | 1.98 | 1.504e-07 | 0.99 | 2.199e-06 | 0.99
$10^{-10}$ | 2.362e-14 | 1.96 | 1.569e-08 | 0.98 | 2.287e-07 | 0.98
$10^{-11}$ | 2.380e-16 | 2.00 | 1.639e-09 | 0.98 | 2.415e-08 | 0.98
Table 2: Convergence test: $N=5$, $\sigma_{\mathrm{s}}=0.01$, $M_{0}=150$. $\gamma$ | $\mathcal{H}_{\gamma}$ | $\nu$ | $L^{2}$ | $\nu$ | $L^{\infty}$ | $\nu$
---|---|---|---|---|---|---
$10^{-3}$ | 9.833e-06 | – | 1.112e-03 | – | 1.738e-03 | –
$10^{-4}$ | 8.398e-07 | 1.07 | 1.400e-04 | 0.90 | 2.146e-04 | 0.91
$10^{-5}$ | 6.116e-08 | 1.14 | 1.634e-05 | 0.93 | 2.483e-05 | 0.94
$10^{-6}$ | 3.136e-09 | 1.29 | 1.797e-06 | 0.96 | 2.636e-06 | 0.97
$10^{-7}$ | 1.092e-10 | 1.46 | 1.885e-07 | 0.98 | 2.674e-07 | 0.99
$10^{-8}$ | 2.255e-12 | 1.69 | 1.911e-08 | 0.99 | 2.679e-08 | 1.00
$10^{-9}$ | 2.639e-14 | 1.93 | 1.916e-09 | 1.00 | 2.679e-09 | 1.00
$10^{-9.25}$ | 8.420e-15 | 1.98 | 1.077e-09 | 1.00 | 1.509e-09 | 1.00
$10^{-9.5}$ | 2.676e-15 | 1.99 | 6.060e-10 | 1.00 | 8.477e-10 | 1.00
$10^{-9.75}$ | 8.499e-16 | 1.99 | 3.409e-10 | 1.00 | 4.776e-10 | 1.00
$10^{-10}$ | 2.704e-16 | 1.99 | 1.919e-10 | 1.00 | 3.017e-10 | 0.80
$10^{-10.25}$ | 8.711e-17 | 1.97 | 1.087e-10 | 0.99 | 1.886e-10 | 0.82
$10^{-10.5}$ | 2.950e-17 | 1.88 | 6.189e-11 | 0.98 | 1.391e-10 | 0.53
$10^{-10.75}$ | 9.893e-18 | 1.90 | 3.652e-11 | 0.92 | 1.182e-10 | 0.28
$10^{-11}$ | 4.185e-18 | 1.49 | 2.256e-11 | 0.84 | 9.941e-11 | 0.30
$10^{-12}$ | 1.801e-18 | 0.37 | 1.376e-11 | 0.21 | 6.393e-11 | 0.19
Table 3: Convergence test: $N=15$, $\sigma_{\mathrm{s}}=0$, $M_{0}=8$.

##  6 Conclusions

The regularized entropy-based moment method for kinetic equations keeps many
of the desirable properties of the original entropy-based moment method but
removes the requirement that the moment vector of the solution remains
realizable. This facilitates the design and implementation of high-order
numerical methods for the regularized moment equations. However, the
regularized equations require the selection of a regularization parameter, and
the error caused by regularization needs to be accounted for and balanced with
other error sources. Our contribution is to rigorously prove the convergence
as the regularization parameter goes to zero expected by formal arguments and
to provide convergence rates. Numerical experiments show that these rates are
indeed optimal.

Our results hold for wide classes of entropy functions including the
Maxwell–Boltzmann entropy and the Bose–Einstein entropy. Our analysis relies
on some key assumptions: The solution to the original moment equations needs
to be Lipschitz and bounded away from the boundary of the set of realizable
states.

Relaxing our assumptions would of course strengthen our results. One would
like to be able to work with kinetic equations with unbounded velocity
domains, but here the original moment equations have fundamental problems [14,
15, 13] which remain in the regularized equations. Nevertheless, the Euler
equations, which are a case of the entropy-based moment method, do not have
these problems and would be an interesting starting point for extending our
analysis. One would also like to allow the solution to have values arbitrarily
close to or even on the boundary of the realizable set, but not enough work
has been done to consider the behavior of the moment equations near or on the
boundary of the realizable set, such as in [7]. It is not even known whether
the realizable set is invariant under the time evolution of the original
entropy-based moment equations. Finally, requiring a Lipschitz continuous
solution to the limiting system is typical for relative entropy estimates, see
[8], and in multiple space dimensions this is connected with non-uniqueness of
entropy solutions for certain moment systems such as the Euler equations.

##  Appendix A Constants

Here we list some the constants which play the most significant roles
throughout the paper. Each constant is a strictly positive real number.

$C_{0}$

Used to control the norm of a realizable moment vector using its zeroth entry:
${\|\mathbf{u}\|\leq C_{0}u_{0}}$ for all $\mathbf{u}\in\mathcal{R}$, where
$u_{0}$ is the zeroth component of $\mathbf{u}$. Introduced in (38).

$C_{1}$

Global Lipschitz constant of $\mathbf{f}$. See (39).

$C_{2}$

Global Lipschitz constant of
$\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma}$; see (43).

$C_{3}$

$\sup_{\gamma\in(0,\gamma_{0})}\|\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0)\|$,
see (44). This is used to get the affine bound in (45).

$C_{4}$

Used when bounding $J_{\gamma}$ for
$(\mathbf{u}_{\gamma},\mathbf{u})\in\mathcal{R}^{L}\times\mathcal{R}^{M}$.
Equal to
$C_{1}C_{h^{\prime\prime},\max,M}+C_{\mathbf{f}^{\prime\prime},M}MC_{2}$; see
(71).

$u_{M}$

Upper bound on $\|\mathbf{u}\|$ in $\mathcal{R}^{M}$; see (57).

$h_{M}$

Upper bound on $h(\mathbf{u})$ in $\mathcal{R}^{M}$; see (57).

$\lambda_{\min,h^{\prime\prime},M}$

Lower bound on the smallest eigenvalue of $h_{\gamma}^{\prime\prime}$ over
$\mathcal{R}^{M}$ and $\gamma\in(0,\gamma_{0})$; see (60) and for the
corresponding constant in the sublinear case (116).

$C_{h^{\prime\prime},M}$

Upper bound on $\|h_{\gamma}^{\prime\prime}\|$ over $\mathcal{R}^{M}$ and
$\gamma\in(0,\gamma_{0})$; see (60) and for the corresponding constant in the
sublinear case (116).

$C_{\mathbf{f}^{\prime\prime},L}$

Bound on the $\mathbf{f}_{\gamma}^{\prime\prime}$ over
$\operatorname{Conv}(\mathcal{R}^{L})$ and $\gamma\in(0,\gamma_{0})$; see
(68).

$C_{h,M,L}$

Lower bound on $h_{\gamma}(\mathbf{v}|\mathbf{u})$ for
$(\mathbf{v},\mathbf{u})\in(\mathbb{R}^{N+1}\setminus\mathcal{R}^{L})\times\mathcal{R}^{M}$
see Lemma 4 and Appendix D.

$C_{\mathbf{r}}$

Lipschitz constant for $\mathbf{r}$; see Assumption 4.

##  Appendix B Entropy relationships

In this appendix we quickly review the computations from [17] showing the
relationships between $h$, $h_{*}$, $\hat{\mathbf{u}}$, and
$\hat{\boldsymbol{\alpha}}$, in particular the key result that
$h^{\prime}=\hat{\boldsymbol{\alpha}}$. Start with the definition of the
entropy $h$,

$\displaystyle
h(\mathbf{u}):=\min_{g\in\mathbb{F}(V)}\left\\{\left\langle\eta(g)\right\rangle:\left\langle\mathbf{m}g\right\rangle=\mathbf{u}\right\\},$
(130)

i.e., the minimal value of the primal problem (9) as a function of the moment
vector. The corresponding Lagrangian is given by

$\displaystyle
L(g,\boldsymbol{\alpha}):=\left\langle\eta(g)\right\rangle+\boldsymbol{\alpha}\cdot\left(\mathbf{u}-\left\langle\mathbf{m}g\right\rangle\right),$
(131)

and thus the dual problem is

$\displaystyle\max_{\boldsymbol{\alpha}\in\mathbb{R}^{N+1}}\min_{g\in\mathbb{F}(V)}L(g,\boldsymbol{\alpha})=\max_{\boldsymbol{\alpha}\in\mathbb{R}^{N+1}}\boldsymbol{\alpha}\cdot\mathbf{u}-\left\langle\eta_{\ast}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle,$
(132)

cf. (12), where to get this equality one takes the minimization inside the
integral and applies the definition of the Legendre dual of $\eta$. Because
the duality gap is zero [13, Thm. 16] we have

$\displaystyle
h(\mathbf{u})=\max_{\boldsymbol{\alpha}\in\mathbb{R}^{N+1}}\left\\{\boldsymbol{\alpha}\cdot\mathbf{u}-\left\langle\eta_{\ast}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle\right\\},$
(133)

so $h$ is the Legendre transformation of

$\displaystyle
h_{*}(\boldsymbol{\alpha}):=\left\langle\eta_{\ast}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle.$
(134)

Its derivative is readily computed:

$\displaystyle
h^{\prime}_{*}(\boldsymbol{\alpha})=\left\langle\mathbf{m}\eta_{\ast}^{\prime}(\boldsymbol{\alpha}\cdot\mathbf{m})\right\rangle=:\hat{\mathbf{u}}(\boldsymbol{\alpha}).$
(135)

Now, recall that $\hat{\boldsymbol{\alpha}}(\mathbf{u})$ in (12) is defined to
be the multiplier vector that solves the dual problem (132). Then the first-
order necessary conditions for the dual problem imply

$\displaystyle\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}(\mathbf{u}))=\mathbf{u}.$
(136)

The reverse, i.e.,
$\hat{\boldsymbol{\alpha}}(\hat{\mathbf{u}}(\boldsymbol{\alpha}))=\boldsymbol{\alpha}$,
is a consequence of the uniqueness of the solution to the dual problem (thanks
to convexity). Thus $\hat{\boldsymbol{\alpha}}$ is the inverse function of
$\hat{\mathbf{u}}$. Finally, since the derivative of Legendre duals are
inverses of each other, we have

$\displaystyle
h^{\prime}=(h^{\prime}_{*})^{-1}=\hat{\mathbf{u}}^{-1}=\hat{\boldsymbol{\alpha}}.$
(137)

##  Appendix C The regularized solution for zero vector as $\gamma\to 0$

In this section we quickly consider

$\displaystyle\lim_{\gamma\to
0}\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0)),$ (138)

where $0\in\mathbb{R}^{N+1}$. This comes up in Section 4.1 when deriving
global estimates on the function
$\hat{\mathbf{u}}\circ\hat{\boldsymbol{\alpha}}_{\gamma}$ using Lipschitz
continuity.

For convenience we assume that the basis functions are orthogonal to each
other, which since $m_{0}\equiv 1$ (recall Assumption 3) in particular implies
that $\langle m_{i}\rangle=\langle m_{0}m_{i}\rangle=0$ for all
$i\in\\{1,\ldots,N\\}$. As a consequence, most of the components of
$\hat{\boldsymbol{\alpha}}_{\gamma}(0)$ are easy to determine. Consider the
first-order necessary conditions:

$\displaystyle 0=\left\langle
m_{i}\eta_{\ast}^{\prime}(\hat{\boldsymbol{\alpha}}_{\gamma}(0)\cdot\mathbf{m})\right\rangle+\gamma\hat{\alpha}_{\gamma,i}(0),\qquad
i\in\\{0,1,\ldots,N\\}.$ (139)

If we set $\hat{\alpha}_{\gamma,i}(0)=0$ for $i\in\\{1,\ldots,N\\}$, then the
entropy ansatz is constant in $v$, and by orthogonality of $\\{m_{i}\\}$, we
see that the first-order necessary conditions are satisfied for
$i\in\\{1,\ldots,N\\}$. It remains to determine the zeroth component
$\hat{\alpha}_{\gamma,0}(0)$, for which we need to solve

$\displaystyle
0=|V|\eta_{\ast}^{\prime}(\hat{\alpha}_{\gamma,0}(0))+\gamma\hat{\alpha}_{\gamma,0}(0).$
(140)

From this equation, it is clear that $\hat{\alpha}_{\gamma,0}(0)<0$ and thus
that $\hat{\alpha}_{\gamma,0}(0)\to-\infty$ monotonically (recall (48)) as
$\gamma\to 0$. The limiting value must be unbounded because
$0\not\in\mathcal{R}$. Now we recall that for the $\eta$ considered in this
work we have
$\operatorname{Range}(\eta_{\ast}^{\prime})=\operatorname{Dom}(\eta^{\prime})=(0,\infty)$,
and furthermore that $\eta_{\ast}^{\prime}$ is a monotonically increasing
function because $\eta_{\ast}$ is convex. Therefore
$\eta_{\ast}^{\prime}(\hat{\alpha}_{\gamma,0}(0))\to 0$ as $\gamma\to 0$. It
follows that also ${\hat{\mathbf{u}}(\hat{\boldsymbol{\alpha}}_{\gamma}(0))\to
0}$.

##  Appendix D Proof of Lemma 4

Let $L\in(M,\infty)$. We want to show

$\displaystyle
C_{h,M,L}:=\inf_{\begin{subarray}{c}\mathbf{v}\in\mathbb{R}^{N+1}\setminus\mathcal{R}^{L}\\\
\mathbf{u}\in\mathcal{R}^{M}\\\
\gamma\in(0,\gamma_{0})\end{subarray}}h_{\gamma}(\mathbf{v}|\mathbf{u})>0.$
(141)

The basic idea is that, by strict convexity of
$h_{\gamma}(\mathbf{v}|\mathbf{u})$ in its first argument, it only achieves
its minimum value, zero, when $\mathbf{v}=\mathbf{u}$. But this is ruled out
on
${(\mathbf{v},\mathbf{u})\in(\mathbb{R}^{N+1}\setminus\mathcal{R}^{L})\times\mathcal{R}^{M}}$
because
${\overline{\mathbb{R}^{N+1}\setminus\mathcal{R}^{L}}\cap\mathcal{R}^{M}=\emptyset}$.

We can get a more explicit bound as follows. Let $Q\in(M,L)$. We claim that
for every $\mathbf{v}\in\mathbb{R}^{N+1}\setminus\mathcal{R}^{L}$ and
$\mathbf{u}\in\mathcal{R}^{M}$ there exists a $\lambda_{Q}\in(0,1)$ such that

$\displaystyle\mathbf{w}_{Q}:=(1-\lambda_{Q})\mathbf{u}+\lambda_{Q}\mathbf{v}\in\mathcal{R}^{L}\setminus\mathcal{R}^{Q}.$
(142)

For $\mathbf{v}\in\mathcal{R}\setminus\mathcal{R}^{L}$ this straightforward,
because the function

$\displaystyle
f(\lambda)=\|\hat{\boldsymbol{\alpha}}((1-\lambda)\mathbf{u}+\lambda\mathbf{v})\|$
(143)

is a continuous function with $f(0)\leq M$ and $f(1)\geq L$. For
$\mathbf{v}\in\mathbb{R}^{N+1}\setminus\mathcal{R}$, by convexity of
$\mathcal{R}$ there exists a unique $\lambda_{\mathcal{R}}\in(0,1)$ such that
$(1-\lambda_{\mathcal{R}})\mathbf{u}+\lambda_{\mathcal{R}}\mathbf{v}\in\partial\mathcal{R}$.
But then, since $\mathcal{R}^{L}\subset\subset\mathcal{R}$, there must also be
a $\lambda_{L}\in(0,\lambda_{\mathcal{R}})$ such that
$(1-\lambda_{L})\mathbf{u}+\lambda_{L}\mathbf{v}\in\mathcal{R}\setminus\mathcal{R}^{L}$,
and so the first argument can be applied again.

Now, for any
$(\mathbf{w},\mathbf{u})\in(\mathcal{R}^{L}\setminus\mathcal{R}^{Q})\times\mathcal{R}^{M}$
we have

$\displaystyle
h_{\gamma}(\mathbf{w}|\mathbf{u})\stackrel{{\scriptstyle\eqref{eq:rel-ent-bnd-
below-L}}}{{\geq}}\lambda_{\min,h^{\prime\prime},\widetilde{L}}\|\mathbf{w}-\mathbf{u}\|^{2}\geq\lambda_{\min,h^{\prime\prime},\widetilde{L}}\inf_{\begin{subarray}{c}\mathbf{w}\in\overline{\mathcal{R}^{L}\setminus\mathcal{R}^{Q}}\\\
\mathbf{u}\in\mathcal{R}^{M}\end{subarray}}\|\mathbf{w}-\mathbf{u}\|^{2}=:C_{L}>0$
(144)

where the strict positivity follows from
$\mathcal{R}^{M}\cap\overline{\mathcal{R}^{L}\setminus\mathcal{R}^{Q}}=\emptyset$.

Finally, let
$(\mathbf{v},\mathbf{u})\in(\mathbb{R}^{N+1}\setminus\mathcal{R}^{L})\times\mathcal{R}^{M}$
and $\mathbf{w}_{Q}$ be as in (142). By convexity of the relative entropy in
its first argument, we have

$\displaystyle(1-\lambda_{Q})h_{\gamma}(\mathbf{u}|\mathbf{u})+\lambda_{Q}h_{\gamma}(\mathbf{v}|\mathbf{u})\geq
h_{\gamma}(\mathbf{w}_{Q}|\mathbf{u})\geq C_{L}.$ (145)

But with $h_{\gamma}(\mathbf{u}|\mathbf{u})=0$ and $\lambda_{Q}\in(0,1)$, we
immediately have $h_{\gamma}(\mathbf{v}|\mathbf{u})\geq C_{L}$, and by taking
the infimum as in (141) we conclude $C_{h,M,L}\geq C_{L}$.

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