# Higher regularity for entropy solutions of conservation laws with
geometrically constrained discontinuous flux

Shyam Sundar Ghoshal<EMAIL_ADDRESS>Stéphane Junca
<EMAIL_ADDRESS>Akash Parmar<EMAIL_ADDRESS>Centre for
Applicable Mathematics, Tata Institute of Fundamental Research, Post Bag No
6503, Sharadanagar, Bangalore - 560065, India.  Université Côte d’Azur, LJAD,
Inria & CNRS, Parc Valrose, 06108 Nice, France.

###### Abstract

For the Burgers equation, the entropy solution becomes instantly
$\displaystyle BV$ with only $\displaystyle L^{\infty}$ initial data. For
conservation laws with genuinely nonlinear discontinuous flux, it is well
known that the $\displaystyle BV$ regularity of entropy solutions is lost.
Recently, this regularity has been proved to be fractional with $\displaystyle
s=1/2$. Moreover, for less nonlinear flux the solution has still a fractional
regularity $\displaystyle 0<s\leq 1/2$. The resulting general rule is the
regularity of entropy solutions for a discontinuous flux is less than for a
smooth flux. In this paper, an optimal geometric condition on the
discontinuous flux is used to recover the same regularity as for the smooth
flux with the same kind of non-linearity.

###### keywords:

Conservation laws, Interface , Discontinuous flux, Cauchy problem, Regularity
, $\displaystyle BV$ functions , Fractional $\displaystyle BV$ spaces.

###### MSC:

[2020] 35B65, 35L65 , 35F25 , 35L67 , 26A45 , 35B44.

###### Contents

  1. 1 Introduction
  2. 2 Main Results
  3. 3 Preliminaries on scalar conservation laws with an interface
  4. 4 Proof of main results
    1. 4.1 Smoothing effect for compatible fluxes with $\displaystyle L^{\infty}$ initial data
    2. 4.2 Propagation of the regularity and smoothing for compatible fluxes
  5. 5 Acknowledgement

##  1 Introduction

Scalar conservation laws with discontinuous flux arise in traffic flow
modeling of vehicular traffic on highways and use to predict congestion and
control them ([32, 33]). Another example is the continuous sedimentation of
solid particles in the liquid ([16, 17]). It occurs in the model of two-phase
flow in the porous media which is significant to many scientific and
industrial explorations such as petroleum engineering and many others [28].
With this brief discussion of the applications of scalar conservation laws, we
can continue to discuss some literature. For more details, one can see [10,
11, 36].

The well-posedness of the scalar conservation laws with discontinuous flux has
been studied extensively over the last few decades [4, 6, 8, 26, 34]. The
existence of the solution given by the several numerical schemes [3, 5, 9, 22,
25, 37]. However, the regularity of the entropy solution has not been studied.
In general, for conservation laws with discontinuous flux, the entropy
solutions are not regular, not in $\displaystyle BV$ even if the initial data
is in $\displaystyle BV$ [1]. The discontinuity of the flux usually forbids
the decay of the total variation. Even for constant initial data, the entropy
solution is not constant.

However, a fractional regularity occurs [23]. The fractional regularity
discovered has been proven to be optimal in many cases. The maximal regularity
belongs to some fractional $\displaystyle BV$ spaces $\displaystyle BV^{s}$.
For the general genuinely nonlinear case it is at most $\displaystyle
BV^{1/2}$.

Fractional $\displaystyle BV$ spaces have been introduced [7] to precisely
quantify the regularity of entropy solutions with shocks that do not belong to
$\displaystyle BV$ [12, 15]. Such regularity is optimal [20], for smooth
nonlinear flux as for the Burgers equation and less nonlinear flux. In the
recent work [23] the fractional $\displaystyle BV$ spaces are useful to
apprehend with precision the regularity of entropy solutions with
discontinuous convex flux.

In this paper, the unexpected regularity of entropy solutions has been given
for a special family of discontinuous flux. This family has been highlighted
in [18]. It provides solutions as regular as continuous flux. If the initial
data belongs to $\displaystyle BV$ the entropy solutions keep this
$\displaystyle BV$ regularity. Moreover, with the strongest non-linearity and
$\displaystyle L^{\infty}$ initial data, the solutions become $\displaystyle
BV$ as for the Burgers equation. In this article, the same family is
considered with more general non-linearity. In this case, the $\displaystyle
BV^{s}$ framework is needed. Propagation of $\displaystyle BV^{s}$ regularity
or smoothing effect in these spaces is similar to continuous flux although the
fractional total variation is not decreasing.

We proceed to describe the precise form of the discontinuous flux in detail.
The scalar conservation law analyzed in this paper focuses on a discontinuous
flux that consists of one interface. The specific form of the equation is as
follows:

$\displaystyle\displaystyle\left\\{\begin{array}[]{rlll}u_{t}+f(u)_{x}&=0,&\mbox{
if }&x>0,t>0,\\\ u_{t}+g(u)_{x}&=0,&\mbox{ if }&x<0,t>0,\\\
u(x,0)&=u_{0}(x),&\mbox{ if }&x\in\mathbb{R},\end{array}\right.$ (1.4)

where $\displaystyle u:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}$ is
unknown, $\displaystyle u_{0}(x)\in L^{\infty}(\mathbb{R})$ is the initial
data and $\displaystyle f$, $\displaystyle g$ are the fluxes. The family of
fluxes studied satisfies the following assumptions:

A1.

Fluxes $\displaystyle f$ and $\displaystyle g$ belongs to $\displaystyle
C^{1}(\mathbb{R},\mathbb{R})$ and are strictly convex.

A2.

Compatible fluxes: $\displaystyle\min f=\min g$, i.e., $\displaystyle
f(\theta_{f})=g(\theta_{g})$, where $\displaystyle\theta_{f}$,
$\displaystyle\theta_{g}$ are the critical points of $\displaystyle f,g$
respectively.

A3.

Non-degeneracy condition, there exist two numbers $\displaystyle p\geq 1,q\geq
1$ such that, for any compact set $\displaystyle K$, there exist positive
numbers $\displaystyle C_{1},C_{2}$ for all $\displaystyle u\not=v$, and
$\displaystyle u,v\in K$,

$\displaystyle\displaystyle\frac{|f^{\prime}(u)-f^{\prime}(v)|}{|u-v|^{p}}>C_{1}>0$
and
$\displaystyle\displaystyle\frac{|g^{\prime}(u)-g^{\prime}(v)|}{|u-v|^{q}}>C_{2}>0.$
(1.5)

$\displaystyle f(u)$$\displaystyle
g(u)$$\displaystyle\theta_{f}$$\displaystyle\theta_{g}$

Figure 1: An illustration of two compatible fluxes $\displaystyle f$ and
$\displaystyle g$.

The assumption (A2) is fundamental, else loss of regularity immediately occurs
near the interface. Examples with less regularity have been built in [23].
That means that geometric assumption (A2) is optimal to keep the same
regularity as for the case with smooth flux.

The nonlinear assumption (A3) implies strict convexity and is the necessary
condition to have a smoothing effect with at least a fractional derivative for
only $\displaystyle L^{\infty}$ data [12, 14, 20, 27]. Without this assumption
but still, with a nonlinear convex flux, the regularity has to be quantified
in bigger generalized $\displaystyle BV$ spaces [13, 24].

It is well-known about scalar conservation laws that it does not admit the
classical solution for all time $\displaystyle t$. Thus one needs the
following notion of a weak solution.

######  Definition 1.1 (Weak solution).

A function $\displaystyle u\in C(0,T;L^{1}_{loc}(\mathbb{R}))$ is said to be a
weak solution of the problem (1.4) if

$\int\limits_{0}^{\infty}\int\limits_{\mathbb{R}}u\frac{\partial\phi}{\partial
t}+F(x,u)\frac{\partial\phi}{\partial x}\mathrm{d}x\
\mathrm{d}t+\int\limits_{\mathbb{R}}u_{0}(x)\phi(x,0)\mathrm{d}x=0,$

for all $\displaystyle\phi\in D(\mathbb{R}\times\mathbb{R}^{+})$, where the
flux $\displaystyle F(x,u)$ is given as $\displaystyle
F(x,u)=H(x)f(u)+(1-H(x))g(u)$, and $\displaystyle H(x)$ is Heaviside function.

By using the definition of the weak solution one can derive the condition
called the Rankine-Hugoniot condition, at $\displaystyle x=0$, $\displaystyle
u$ satisfies the following for almost all $\displaystyle t$,

$f(u^{+}(t))=g(u^{-}(t)),$ (1.6)

where $\displaystyle u^{+}(t)=\lim\limits_{x\rightarrow 0+}u(x,t)$ and
$\displaystyle u^{-}(t)=\lim\limits_{x\rightarrow 0-}u(x,t)$. The existence of
traces like a $\displaystyle BV$ function is an important usual feature for
weak solutions of conservation laws [34]. The left and right traces
$\displaystyle u^{-}$ and $\displaystyle u^{+}$ are crucial for the
wellposedness as well as the regularity of the weak solution. The existence of
the interface traces has been established in [4] by using the Hamilton-Jacobi
type equation.

Due to the fact that the weak solutions are not unique one needs to take into
account the “entropy condition” to get the uniqueness of the weak solution.
For $\displaystyle f=g$, Kružkov [30] gave a generalized entropy condition.
However, due to the discontinuity of flux at the interface the Kružkov entropy
condition is insufficient for each side of the interface. Therefore, an
additional ”interface entropy condition” is required, which was introduced in
[4].

######  Definition 1.2 (Entropy solution [4]).

A weak solution $\displaystyle u$ of the problem (1.4) is said to be an
entropy solution if it satisfies Kruzkov entropy solutions on each side of the
interface $\displaystyle x=0$ and if the following “interface entropy
condition” is fulfilled for almost all $\displaystyle t>0$,

$|\\{t:f^{\prime}(u^{+}(t))>0>g^{\prime}(u^{-}(t))\\}|=0,$ (1.7)

Let us give a summary of this paper. The main results are given and commented
on in the next Section 2. Then, important tools are recalled in Section 3
before proving the main results. The proof of the main results is given in
Section 4 along with many useful lemmas.

##  2 Main Results

As we have mentioned the fluxes are convex and precisely satisfy the
assumptions (A1), (A2) and (A3). Let us denote the $\displaystyle g_{-}^{-1}$,
$\displaystyle f_{+}^{-1}$ denote the inverse of $\displaystyle g,f$ for
domain where $\displaystyle g^{\prime}(u)\leq 0$ and $\displaystyle
f^{\prime}(u)\geq 0$ respectively. Notice that the existence of a minimum for
$\displaystyle f$ and $\displaystyle g$ are always assumed in this paper as it
allows the critical behavior of the entropy solution. If we are assuming that
$\displaystyle f$ and $\displaystyle g$ have no minimum it implies that the
fluxes are strictly increasing or decreasing which is handled in [1]. In this
case, when the minimum of the two fluxes coincide, $\displaystyle
f(\theta_{f})=g(\theta_{g})$, it is similar to the case $\displaystyle f=g$
where there is no interface, except that the proof and the estimates are more
complicated and need the theory of conservation laws with discontinuous flux.
In particular, if $\displaystyle f$ and $\displaystyle g$ are uniformly convex
then a $\displaystyle BV$ smoothing effect occurs. The fractional
$\displaystyle BV$ spaces $\displaystyle BV^{s},0<s\leq 1$ are recalled in the
next section. For $\displaystyle s=1$, $\displaystyle BV^{1}$ is
$\displaystyle BV$.

######  Theorem 2.1 (Smoothing effect for compatible fluxes with
$\displaystyle L^{\infty}$ initial data).

Let $\displaystyle f$ and $\displaystyle g$ be convex fluxes with the same
minimum and satisfy the non-degeneracy condition (1.5), i.e., condition (A3)
along with assumptions (A1) and (A2). Let $\displaystyle u(\cdot,t)$ be the
entropy solution of (1.4) corresponding to an initial data $\displaystyle
u_{0}\in L^{\infty}(\mathbb{R})$. Then $\displaystyle u(t,\cdot)\in
BV^{s}_{loc}(\mathbb{R})$ for $\displaystyle t>0$ where $\displaystyle
s=\min\\{1/p,1/q\\}$.

######  Remark 2.1.

Note that the exponent $\displaystyle s=\min\\{1/p,1/q\\}$ in Theorem 2.1 is
optimal due to the counter-examples in [2, 12, 20]. In particular, for uniform
convex and compatible fluxes $\displaystyle f$ and $\displaystyle g$, there is
a smoothing effect in $\displaystyle BV$ as for the case $\displaystyle f=g$
[31, 35].

Once we know about the regularity of entropy solution in Theorem 2.1, it is
natural to see whether $\displaystyle TV^{s}(u(t,\cdot))$ satisfies an
explicit estimate in terms of time variable $\displaystyle t$. In the next
proposition, we provide a sufficient condition to obtain a $\displaystyle
1/t^{\gamma}$ estimate for an appropriate choice of $\displaystyle\gamma$.

######  Proposition 2.1 (Explicit $\displaystyle BV^{s}$ estimates).

Let $\displaystyle f$ and $\displaystyle g$ be two $\displaystyle C^{2}$
fluxes such that $\displaystyle f(\theta_{f})=g(\theta_{g})$ and satisfying
the non-degeneracy condition (1.5) with exponents $\displaystyle p,q$
respectively. Let $\displaystyle u(\cdot,t)$ be the entropy solution of (1.4)
corresponding to an initial data $\displaystyle u_{0}\in
L^{\infty}(\mathbb{R})$. We assume that there exist $\displaystyle\delta,r>0$
such that

$\operatorname*{ess-sup}\limits_{(-\delta,0)}u_{0}>\theta_{g}+r\mbox{ and
}\operatorname*{ess-sup}\limits_{(0,\delta)}u_{0}<\theta_{f}-r,$ (2.1)

then we have for all $\displaystyle M>0$,

$TV^{s}(u(\cdot,t),[-M,M])\leq\frac{C(f,g,M)}{\min\\{t^{\frac{1}{qs}},t^{\frac{1}{ps}}\\}}\left[\max\\{1,t/\delta,1/r\\}\right]^{\frac{pqs+1}{s^{2}}}+2(2||u_{0}||_{\infty})^{1/s}\mbox{
for }t>0,$ (2.2)

where $\displaystyle s=\min\\{1/p,1/q\\}$ and $\displaystyle C$ is depending
only on $\displaystyle f,g,M$.

Another nice property of $\displaystyle BV^{s}$ solutions is known for all
Lipschitz flux, $\displaystyle BV^{s}$ regularity in space of the initial data
is propagated for all time [7, 29]. It means the corresponding entropy
solution $\displaystyle u(x,t)$ has the $\displaystyle BV^{s}$ regularity in
space for all time $\displaystyle t>0$. For discontinuous flux, the situation
is more complicated due to the interface. To use the Lax-Oleinik formula with
an interface [4] we limit ourselves to convex $\displaystyle C^{1}$ fluxes but
the non-degeneracy condition (A3) is not required.

######  Theorem 2.2 (Propagation of the regularity for compatible fluxes).

Let $\displaystyle f$ and $\displaystyle g$ be the $\displaystyle C^{1}$
convex fluxes such that $\displaystyle\min f=\min g$, i.e. they satisfy the
assumptions (A1) and (A2). Let $\displaystyle u(\cdot,t)$ be the entropy
solution of (1.4) corresponding to an initial data $\displaystyle u_{0}\in
BV^{s}(\mathbb{R})$ for $\displaystyle s\in(0,1)$. Then $\displaystyle
u(\cdot,t)\in BV^{s}(\mathbb{R})$ for $\displaystyle t>0$.

For $\displaystyle s=1$, the propagation of the $\displaystyle BV$ regularity
already proved in [18] is recovered for less nonlinear fluxes.

Now, the two previous theorems can be mixed to get another result. As a
consequence of the Theorems 2.1 and 2.2, if $\displaystyle u_{0}$ belongs to
$\displaystyle BV^{s}$ then two cases occur. If $\displaystyle s$ is too small
then $\displaystyle u_{0}$ is regularized. If $\displaystyle s$ is too big
then the initial regularity is propagated. Precisely, the following corollary
is stated. To use Theorem 2.1, the flux has to satisfy the non-degeneracy
condition (A3) to get the $\displaystyle BV^{s}$ smoothing effect.

######  Corollary 2.1 (Optimal regularity for $\displaystyle BV^{s}$ initial
data and compatible fluxes).

Let $\displaystyle f$ and $\displaystyle g$ be the fluxes such that
$\displaystyle f(\theta_{f})=g(\theta_{g})$ and the fluxes satisfy the non-
degeneracy condition (1.5). Let $\displaystyle u(\cdot,t)$ be the entropy
solution of (1.4) corresponding to an initial data $\displaystyle u_{0}\in
BV^{s}(\mathbb{R})$ for $\displaystyle s\in(0,1)$. Then, $\displaystyle
u(\cdot,t)\in BV^{s_{1}}([-M,M])$ for $\displaystyle t>0,M>0$ where
$\displaystyle s_{1}=\min\\{\max\\{p^{-1},s\\},\max\\{q^{-1},s\\}\\}$.

##  3 Preliminaries on scalar conservation laws with an interface

The fundamental paper used here is [4] where Adimurthi and Gowda settle an
important foundation of the theory on scalar conservation laws with an
interface and two convex fluxes. In this paper, the author proposed the
natural entropy condition which means that no information comes only from the
interface but crosses or goes toward the interface. Such entropy condition is
in the spirit of Lax-entropy conditions for shock waves. To make this paper
self-contained, we recall some definitions and results. The following theorem
can be found in [4] Lemma 4.9 on page 51. It is a Lax-Oleinik or Lax-Hopf
formula for the initial value problem (1.4).

######  Theorem 3.1 ([4]).

Let $\displaystyle u_{0}\in L^{\infty}(\mathbb{R})$, then there exists the
entropy solution $\displaystyle u(\cdot,t)$ of (1.4) corresponding to an
initial data $\displaystyle u_{0}$. Furthermore, there exist Lipschitz curves
$\displaystyle R_{1}(t)\geq R_{2}(t)\geq 0$ and $\displaystyle L_{1}(t)\leq
L_{2}(t)\leq 0$, monotone functions $\displaystyle z_{\pm}(x,t)$ non-
decreasing in $\displaystyle x$ and non-increasing in $\displaystyle t$ and
$\displaystyle t_{\pm}(x,t)$ non-increasing in $\displaystyle x$ and non-
decreasing in $\displaystyle t$ such that the solution $\displaystyle u(x,t)$
can be given by the explicit formula for almost all $\displaystyle t>0$,

$\displaystyle\displaystyle
u(x,t)=\left\\{\begin{array}[]{lllll}(f^{\prime})^{-1}\left(\frac{x-z_{+}(x,t)}{t}\right)&\mbox{
if }&x\geq R_{1}(t),\\\
(f^{\prime})^{-1}\left(\frac{x}{t-t_{+}(x,t)}\right)&\mbox{ if }&0\leq
x<R_{1}(t),\\\ (g^{\prime})^{-1}\left(\frac{x-z_{-}(x,t)}{t}\right)&\mbox{ if
}&x\leq L_{1}(t),\\\
(g^{\prime})^{-1}\left(\frac{x}{t-t_{-}(x,t)}\right)&\mbox{ if
}&L_{1}(t)<x<0.\end{array}\right.$

Furthermore, if $\displaystyle f(\theta_{f})\geq g(\theta_{g})$ then
$\displaystyle R_{1}(t)=R_{2}(t)$ and if $\displaystyle f(\theta_{f})\leq
g(\theta_{g})$ then $\displaystyle L_{1}(t)=L_{2}(t)$. We also have only three
cases and following formula to compute the solution:

Case 1:

$\displaystyle L_{1}(t)=0$ and $\displaystyle R_{1}(t)=0$,

$\displaystyle\displaystyle
u(x,t)=\left\\{\begin{array}[]{lllll}u_{0}(z_{+}(x,t))&\mbox{ if }&x>0,\\\
u_{0}(z_{-}(x,t))&\mbox{ if }&x<0.\end{array}\right.$

Case 2:

$\displaystyle L_{1}(t)=0$ and $\displaystyle R_{1}(t)>0$, then

$\displaystyle\displaystyle
u(x,t)=\left\\{\begin{array}[]{lllll}f_{+}^{-1}g(u_{0}(z_{+}(x,t)))&\mbox{ if
}&0<x<R_{2}(t),\\\ f_{+}^{-1}g(\theta_{g})&\mbox{ if }&R_{2}(t)\leq x\leq
R_{1}(t),\\\ u_{0}(z_{-}(x,t))&\mbox{ if }&x<0.\end{array}\right.$

Case 3:

$\displaystyle L_{1}(t)<0$, $\displaystyle R_{1}(t)=0$, then

$\displaystyle\displaystyle
u(x,t)=\left\\{\begin{array}[]{lllll}g_{-}^{-1}f(u_{0}(z_{-}(x,t)))&\mbox{ if
}&L_{2}(t)<x<0,\\\ u_{0}(z_{-}(x,t))&\mbox{ if }&x\leq L_{1}(t),\\\
g_{-}^{-1}f(\theta_{f})&\mbox{ if }&L_{1}(t)<x<L_{2}(t).\end{array}\right.$

$\displaystyle t_{+}(x,t)$$\displaystyle
R_{1}(t)$$\displaystyle\bullet$$\displaystyle\bullet$$\displaystyle
R_{2}(t)$$\displaystyle\bullet$$\displaystyle R_{1}(t)$$\displaystyle
L_{1}(t)=L_{2}(t)=0$$\displaystyle z_{+}(x,t)$$\displaystyle
z_{-}(x,t)$$\displaystyle z_{+}(x,t)$

Figure 2: An illustration of solution for Case 2 and $\displaystyle L_{i}(t)$
and $\displaystyle R_{i}(t)$ curves

There is a maximum principle for such entropy solutions, but more complicate
than for $\displaystyle f=g$,

$\|u\|_{\infty}\leq\max\left(\|u_{0}\|_{\infty},\sup_{|v|\leq\|u_{0}\|_{\infty}}|f^{-1}_{+}(g(v))|,\sup_{|v|\leq\|u_{0}\|_{\infty}}|g^{-1}_{-}(f(v))|\right)=:S_{f,g,\left|\left|u_{0}\right|\right|_{\infty}}.$
(3.5)

In the above theorem, the curves $\displaystyle R_{i}$ play an important role
to understand the structure inherited by the entropy solution. They separate
the regions of three types of ‘characteristics’ as introduced in [4].

Now, the definition of $\displaystyle BV^{s}(I)$ is recalled for
$\displaystyle I$ an interval of $\displaystyle\mathbb{R}$. First, the
$\displaystyle s$-fractional total variation is, for $\displaystyle 0<s\leq 1$
and $\displaystyle p=1/s$,

$TV^{s}(u,I)=\sup_{\sigma}\sum_{i=1}^{n}|u(x_{i})-u(x_{i-1})|^{p},$ (3.6)

where the supremum is taken over all subdivisions $\displaystyle\sigma$ of the
interval $\displaystyle I$. Such a subdivision is a finite ordered subset of
$\displaystyle I$, $\displaystyle\sigma=\\{x_{i},\;i=0,\ldots n\\}$ for a
$\displaystyle n\in\mathbb{N}$ and $\displaystyle x_{0}<x_{1}<\ldots<x_{n}$.

Second, $\displaystyle BV^{s}(I)$ is the space of real functions on I with
bounded $\displaystyle s$-fractional total variation, called also
$\displaystyle p$-variation. The exponent $\displaystyle s$ means that
$\displaystyle BV^{s}$ is almost but different than the Sobolev spaces
$\displaystyle W^{s,p}(I)$ (when $\displaystyle I$ is bounded) [7]. Thus the
exponent $\displaystyle s$ corresponds to the fractional derivative
$\displaystyle s$.

Some applications of fractional $\displaystyle BV$ spaces for scalar
conservation laws with convex flux can be found in the non-exhaustive list
[14, 12, 13, 20]. $\displaystyle BV^{s}$ is compactly embedded in
$\displaystyle L^{1}_{loc}$ and capture the shock structure as $\displaystyle
BV$ but with less regularity.

Before we prove the main results in the next section, we need to state some
important elementary lemmas. In these lemmas, we prove the non-Lipschitz and
Lipschitz regularity of the singular map. The two first lemmas are in [23],
thus the proofs are omitted.

######  Lemma 3.1 ( Hölder continuity of the inverse [23]).

Let $\displaystyle g\in C^{1}(\mathbb{R})$ be satisfying the non-degeneracy
(1.5) with exponent $\displaystyle q$. Then $\displaystyle(g^{\prime})^{-1}$
is Hölder continuous with exponent $\displaystyle 1/q$.

######  Lemma 3.2 (Flatness of the antiderivative [23]).

Let $\displaystyle g$ be a $\displaystyle C^{2}$ function satisfying (1.5)
with exponent $\displaystyle q$ then $\displaystyle g_{+}$ satisfies (1.5)
with exponent $\displaystyle q+1$ on domain
$\displaystyle(\theta_{g},\infty)$.

The following lemma establishes the Lipschitz continuity of the “singular
maps” away from $\displaystyle\theta_{g}$ and $\displaystyle\theta_{f}$.
Before stating the lemma, let us define the singular maps. Let $\displaystyle
g_{-}^{-1}$ and $\displaystyle f_{+}^{-1}$ are the inverse of $\displaystyle
g$ and $\displaystyle f$ respectively, with appropriate domains given by:

$\displaystyle\displaystyle\left\\{\begin{array}[]{rlllll}g_{-}^{-1}:((g^{\prime})^{-1}(-\infty),(g^{\prime})^{-1}(0)]\rightarrow\mathbb{R}\\\
f_{+}^{-1}:[(f^{\prime})^{-1}(0),(f^{\prime})^{-1}(+\infty))\rightarrow\mathbb{R}.\end{array}\right.$
(3.9)

The functions $\displaystyle f^{-1}_{+}g(\cdot)$ and $\displaystyle
g^{-1}_{-}f(\cdot)$, namely, singular maps, plays crucial role in traferring
information via the interface from left-to-right and right-to-left
respectively.

######  Lemma 3.3 (Regularity of singular maps).

Let $\displaystyle f$ and $\displaystyle g$ be the $\displaystyle
C^{1}(\mathbb{R})$ and convex functions with $\displaystyle
f(\theta_{f})=g(\theta_{g})$ with restricted flatness, i.e., satify (A1), (A2)
and (A3). Let $\displaystyle K$ is any compact set of
$\displaystyle\mathbb{R}$. Then $\displaystyle f^{-1}_{+}g(\cdot)$ and
$\displaystyle g^{-1}_{-}f(\cdot)$ are Hölder continuous functions. Moreover,
$\displaystyle f^{-1}_{+}g(\cdot)$ is a Lipschitz continuous function on
$\displaystyle K\setminus[\theta_{g}-\varepsilon,\theta_{g}+\varepsilon]$ and
$\displaystyle g^{-1}_{-}f(\cdot)$ is a Lipschitz continuous function on
$\displaystyle K\setminus[\theta_{f}-\varepsilon,\theta_{f}+\varepsilon]$ for
any fixed $\displaystyle\varepsilon>0$.

Indeed, if $\displaystyle x$ is far from $\displaystyle\theta_{g}$, then
$\displaystyle g(x)$ is far from $\displaystyle\theta_{f}$ since
$\displaystyle f(\theta_{f})=g(\theta_{g})$. Thus, $\displaystyle
f^{-1}_{+}g(\cdot)$ is Lipschitz far from $\displaystyle\theta_{g}$. The
symmetric result occurs for $\displaystyle g^{-1}_{-}f(\cdot)$ far from
$\displaystyle\theta_{f}$. Now, the detailed proof is given.

###### Proof.

Let $\displaystyle x_{1},x_{2}\in K$ then there exist $\displaystyle
y_{1},y_{2}\geq\theta_{f}$ such that $\displaystyle g(x_{1})=f(y_{1})$ and
$\displaystyle g(x_{2})=f(y_{2})$. Without loss of generality we can assume
that $\displaystyle g(x_{1})\not=g(x_{2})$ because if $\displaystyle
g(x_{1})=g(x_{2})$ then result holds anyway. As $\displaystyle f^{-1}_{+}$ is
increasing, we get $\displaystyle y_{1},y_{2}\geq\theta_{f}$. Let us consider
the following

$\displaystyle\displaystyle\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|^{p+1}}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|^{p+1}}{|g(x_{1})-g(x_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|f^{-1}_{+}f(y_{1})-f^{-1}_{+}f(y_{2})|^{p+1}}{|f(y_{1})-f(y_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|y_{1}-y_{2}|^{p+1}}{|f(y_{1})-f(y_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$

From Lemma 3.2 we have

$\displaystyle\displaystyle|f(x)-f(y)|\geq\frac{C_{2}}{p+1}|x-y|^{p+1}$
$\displaystyle\displaystyle\mbox{ for any }x\not=y,$

and $\displaystyle g$ is Lipschitz continuous function, we have
$\displaystyle\left|g(x_{1})-g(x_{2})\right|\leq
c_{1}\left|x_{1}-x_{2}\right|$ where $\displaystyle c_{1}$ depends on
$\displaystyle g$ and $\displaystyle K$. Therefore, we get a constant
$\displaystyle C$,

$\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|^{p+1}}{|x_{1}-x_{2}|}\leq C.$

Similarly, we can prove that

$\frac{|g^{-1}_{-}f(x_{1})-g^{-1}_{-}f(x_{2})|^{q+1}}{|x_{1}-x_{2}|}\leq C.$

Thus the singular maps are Hölder continuous functions.
For any $\displaystyle x_{1},x_{2}\in
K\backslash[\theta_{g}-\varepsilon,\theta_{g}+\varepsilon]$ such that
$\displaystyle g(x_{1}),g(x_{2})\geq g(\theta_{g})+\varepsilon_{1}$ for a
positive $\displaystyle\varepsilon_{1}$ only depending on
$\displaystyle\varepsilon$. Then for some $\displaystyle\varepsilon_{0}>0$
only depending on $\displaystyle\varepsilon_{1}$ there exist $\displaystyle
y_{1},y_{2}\geq\theta_{f}+\varepsilon_{0}$ such that $\displaystyle
g(x_{1})=f(y_{1})$ and $\displaystyle g(x_{2})=f(y_{2})$. Let us consider the
following

$\displaystyle\displaystyle\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|}{|g(x_{1})-g(x_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|f^{-1}_{+}f(y_{1})-f^{-1}_{+}f(y_{2})|}{|f(y_{1})-f(y_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{|y_{1}-y_{2}|}{|f(y_{1})-f(y_{2})|}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{f^{\prime}(\xi)}\cdot\frac{|g(x_{1})-g(x_{2})|}{|x_{1}-x_{2}|}$

where $\displaystyle\xi\in(y_{1},y_{2})$. Now from the Lipschitz continuous of
$\displaystyle g$ and from the convexity of $\displaystyle f$ we get

$\frac{|f^{-1}_{+}g(x_{1})-f^{-1}_{+}g(x_{2})|}{|x_{1}-x_{2}|}\leq C.$

In a similar fashion, we can prove

$\frac{|g^{-1}_{-}f(x)-g^{-1}_{-}f(y)|}{|x_{1}-x_{2}|}\leq C.$

∎

##  4 Proof of main results

In this section, we first prove the smoothing effect in fractional
$\displaystyle BV$ space with only $\displaystyle L^{\infty}$ initial data,
that is Theorem 2.1. The proof is detailed in the long subsection 4.1 with an
explicit $\displaystyle BV^{s}$ estimate, Proposition 2.1. After, similar
arguments are used in the last short subsection to prove the propagation of
fractional $\displaystyle BV$ regularity and the smoothing effect when the
initial data is already in $\displaystyle BV^{s}$.

###  4.1 Smoothing effect for compatible fluxes with $\displaystyle
L^{\infty}$ initial data

The following long proof contains the main ingredients to study the regularity
of the entropy solution through the interface. After, an explicit fractional
$\displaystyle BV$ estimate is given in a favorable case.

###### Proof of Theorem 2.1.

We assume that $\displaystyle f(\theta_{f})=g(\theta_{g})$, hence, from
Theorem 3.1 we have $\displaystyle L_{2}(t)=L_{1}(t)$ and $\displaystyle
R_{2}(t)=R_{1}(t)$, hence, it is enough to consider the following two cases.
The second case is similar so, only the first case is detailed.

Case (i):

If $\displaystyle L_{1}(t)=0$ and $\displaystyle R_{1}(t)\geq 0$.
Consider the partition
$\displaystyle\sigma=\\{-M=x_{-n}\leq\cdots<x_{-1}<x_{0}\leq
0<x_{1}<\cdots<x_{m}\leq R_{1}(t)<x_{m+1}<\cdots\leq x_{n}=M\\}$ and let
$\displaystyle s=\min\\{1/p,1/q\\}$,

$\displaystyle\displaystyle\sum_{i=-n}^{n}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{i=-n}^{-1}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}+|u(x_{0},t)-u(x_{1},t)|^{\frac{1}{s}}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\sum_{i=1}^{m-1}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}+|u(x_{m},t)-u(x_{m+1},t)|^{\frac{1}{s}}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\sum_{i=m+1}^{n-1}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}.$

By using the explicit formula from Theorem 3.1 we get,

$\displaystyle\displaystyle\sum_{i=-n}^{n}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\underbrace{\sum_{i=-n}^{-1}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}}_{\text{I}}$
(4.1) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\underbrace{\sum_{i=1}^{m-1}|f^{-1}_{+}g(u_{0}(z_{+}(x_{i},t)))-f^{-1}_{+}g(u_{0}(z_{+}(x_{i+1},t)))|^{\frac{1}{s}}}_{\text{II}}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\underbrace{\sum_{i=m+1}^{n-1}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}}_{\text{III}}+2(2||u_{0}||_{\infty})^{\frac{1}{s}}.$

Now we wish to estimate the terms I, II, and III. The simplest terms I, III
are estimated as in [7]. First taking the I and III into the account, since
$\displaystyle f,g$ are satisfying the flux non-degeneracy condition (1.5), by
Lemma 3.1, the maps $\displaystyle u\mapsto(g^{\prime})^{-1}(u)$ and
$\displaystyle u\mapsto(f^{\prime})^{-1}(u)$ are Hölder continuous with
exponents $\displaystyle 1/q$ and $\displaystyle 1/p$. From the Theorem 3.1,
for $\displaystyle x<0$, the solution of (1.4) is given by $\displaystyle
u(x,t)=(g^{\prime})^{-1}\left(\frac{x-z_{-}(x,t)}{t}\right)$. Then for
$\displaystyle-M\leq x_{i}<x_{i+1}\leq 0$ we have

$\displaystyle\displaystyle|u(x_{i},t)-u(x_{i+1},t)|^{q}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left|(g^{\prime})^{-1}\left(\frac{x_{i}-z_{-}(x_{i},t)}{t}\right)-(g^{\prime})^{-1}\bigg{(}\frac{x_{i+1}-z_{-}(x_{i+1},t)}{t}\bigg{)}\right|^{q}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\bigg{(}C_{2}^{-q^{-1}}\bigg{|}\frac{x_{i}-z_{-}(x_{i},t)}{t}-\frac{x_{i+1}-z_{-}(x_{i+1},t)}{t}\bigg{|}^{q^{-1}}\bigg{)}^{q},$

using triangle inequality we obtain,

$|u(x_{i},t)-u(x_{i+1},t)|^{q}\leq
C_{2}^{-1}\bigg{|}\frac{x_{i}-x_{i+1}}{t}\bigg{|}+C_{2}^{-1}\bigg{|}\frac{z_{-}(x_{i},t)-z_{-}(x_{i+1},t)}{t}\bigg{|}.$

Since $\displaystyle|x_{i}|,|x_{i+1}|\leq M$ and $\displaystyle
x=z_{-}(x,t)+g^{\prime}(u(x,0))t$ hence, we get the $\displaystyle
TV^{q^{-1}}u(\sigma\cap[-M,0])$,

$TV^{q^{-1}}u(\sigma\cap[-M,0])\leq\frac{4M}{C_{2}t}+\frac{1}{C_{2}}\sup\left\\{\left|g^{\prime}(v)\right|;\,\left|v\right|\leq\left|\left|u_{0}\right|\right|_{L^{\infty}(\mathbb{R})}\right\\}.$
(4.2)

In similar fashion, for the term III we have,

$TV^{p^{-1}}u(\sigma\cap[R_{1}(t),M])\leq\frac{4M}{C_{1}t}+\frac{1}{C_{1}}\sup\left\\{\left|f^{\prime}(v)\right|;\,\left|v\right|\leq\left|\left|u_{0}\right|\right|_{L^{\infty}(\mathbb{R})}\right\\}.$
(4.3)

Next we focus on computing term II. Without loss of generality we can choose a
point $\displaystyle x^{\prime}\in(0,R_{2}(t))$ to take $\displaystyle
x^{\prime}=R_{2}(t)$ such that $\displaystyle z_{+}(x^{\prime},t)<0$, suppose
there is no such point $\displaystyle x^{\prime}$, then $\displaystyle
R_{2}(t)=0$, since $\displaystyle R_{1}(t)=R_{2}(t)=0$ which implies that the
term II does not appear. Let characteristics emanating from $\displaystyle
z_{+}(x^{\prime},t)$ be hitting the interface at $\displaystyle
t_{+}(x^{\prime},t)$ then by the monotonicity of $\displaystyle
t_{+}(\cdot,t)$ we get for all $\displaystyle x\in(0,x^{\prime})$,

$0<t_{+}(x^{\prime},t)\leq t_{+}(x,t)<t,$ (4.4)

also by the monotonicity of $\displaystyle z_{+}(\cdot,t)$ we get for all
$\displaystyle x\in(0,x^{\prime})$,

$z_{+}(0+,t)\leq z_{+}(x,t)\leq z_{+}(x^{\prime},t)<0,$ (4.5)

which implies that

$\frac{z_{+}(0+,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x^{\prime},t)}{t_{+}(x,t)}<0.$
(4.6)

Hence, from (4.4) and (4.5) we get

$\displaystyle\displaystyle\frac{z_{+}(0+,t)}{t_{+}(x,t)}$
$\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\frac{z_{+}(0+,t)}{t_{+}(x^{\prime},t)},$ (4.7)
$\displaystyle\displaystyle\frac{z_{+}(x^{\prime},t)}{t}$
$\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\frac{z_{+}(x^{\prime},t)}{t_{+}(x,t)}.$ (4.8)

Therefore, from (4.6), (4.7) and (4.8) one can conclude that for
$\displaystyle x\in(0,x^{\prime})$,

$\frac{z_{+}(0+,t)}{t_{+}(x^{\prime},t)}\leq\frac{z_{+}(x,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x^{\prime},t)}{t}<0.$
(4.9)

Since $\displaystyle(g^{\prime})^{-1}$ is an increasing function and
$\displaystyle(g^{\prime})^{-1}(0)=\theta_{g}$, the above inequality (4.9)
implies that,

$(g^{\prime})^{-1}\left(-\frac{z_{+}(x,t)}{t_{+}(x,t)}\right)\geq\delta_{0}:=(g^{\prime})^{-1}\left(-\frac{z_{+}(x^{\prime},t)}{t}\right)>\theta_{g}\mbox{
for }x\in(0,x^{\prime}).$ (4.10)

Then, the map $\displaystyle a\mapsto g((g^{\prime})^{-1}(a)$ is Lipschitz
continuous for $\displaystyle a\geq g^{\prime}(\delta_{0})=:c_{0}>0$. As
$\displaystyle f(\theta_{f})=g(\theta_{g})$ we have if $\displaystyle
g((g^{\prime})^{-1}(a))$ is away from $\displaystyle g(\theta_{g})$ then also
away from $\displaystyle f(\theta_{f})$ which implies that,

$a\mapsto f_{+}^{-1}g\left((g^{\prime})^{-1}(a)\right)$ (4.11)

is a Hölder continuous function with exponent $\displaystyle 1/q$ for
$\displaystyle a\geq c_{0}>0$.

$\displaystyle x=0$$\displaystyle\bullet$$\displaystyle
z_{+}(0,t)$$\displaystyle
z_{+}(x^{\prime},t)$$\displaystyle(x^{\prime},t)$$\displaystyle
t$$\displaystyle t_{+}(x^{\prime},t)$$\displaystyle
t=0$$\displaystyle\bullet$$\displaystyle z_{-}(x,t)$$\displaystyle(x,t)$

Figure 3: An illustration of solution for the region
$\displaystyle(0,x^{\prime})$ at time $\displaystyle t$.

From Theorem 3.1 for all $\displaystyle x\in(0,x^{\prime})$ we have

$g(u_{0}(z_{+}(x,t)))=g(u(0-,t_{+}(x,t)))=(g^{\prime})^{-1}\left(-\frac{z_{+}(0,t_{+}(x,t))}{t_{+}(x,t)}\right).$
(4.12)

To get a more precise estimate with respect to t, we refine the choice of
$\displaystyle x^{\prime}$. We wish to choose a point $\displaystyle
x^{\prime}\in(0,R_{2}(t))$ such that $\displaystyle t_{+}(x^{\prime}+,t)\leq
t/2\leq t_{+}(x^{\prime}-,t)$. Since $\displaystyle t_{+}(x,t)\rightarrow t$
as $\displaystyle x\rightarrow 0+$, we can always get $\displaystyle
x^{\prime}$ satisfying $\displaystyle t_{+}(x^{\prime}-,t)\geq t/2$. If we get
an $\displaystyle x^{\prime}$ such that $\displaystyle
t_{+}(x^{\prime}+,t)\leq t/2\leq t_{+}(x^{\prime}-,t)$, we define
$\displaystyle\sigma_{1}:=\sigma\cap[0,x^{\prime})$,
$\displaystyle\sigma_{2}:=\sigma\cap[x^{\prime},R_{2}(t))$. In the other case,
definition of $\displaystyle\sigma_{1}$ remains same and
$\displaystyle\sigma_{2}$ is set to be empty. Now from (4.12) we get,

II $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{x_{i}\in\sigma_{1}}\left|f_{+}^{-1}g\bigg{(}(g^{\prime})^{-1}\left(-\frac{z_{+}(x_{i},t)}{t_{+}(x_{i},t)}\right)\bigg{)}-f_{+}^{-1}g\bigg{(}(g^{\prime})^{-1}\left(-\frac{z_{+}(x_{i+1},t)}{t_{+}(x_{i+1},t)}\right)\bigg{)}\right|^{\frac{1}{s}}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\sum_{x_{i}\in\sigma_{2}}\left|(f^{\prime})^{-1}\left(\frac{x_{i}}{t-t_{+}(x_{i},t)}\right)-(f^{\prime})^{-1}\left(\frac{x_{i+1}}{t-t_{+}(x_{i+1},t)}\right)\right|^{\frac{1}{s}}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
C(f,g,t,u_{0})\cdot\sum_{x_{i}\in\sigma_{1}}\bigg{|}-\frac{z_{+}(x_{i},t)}{t_{+}(x_{i},t)}+\frac{z_{+}(x_{i+1},t)}{t_{+}(x_{i+1},t)}\bigg{|}^{\frac{1}{sq}}$
$\displaystyle\displaystyle+$ $\displaystyle\displaystyle
C_{1}\cdot\sum_{x_{i}\in\sigma_{2}}\bigg{|}\frac{x_{i}}{t-t_{+}(x_{i},t)}-\frac{x_{i+1}}{t-t_{+}(x_{i+1},t)}\bigg{|}^{\frac{1}{sp}}.$

Note that
$\displaystyle\left|t-t_{+}(x,t)\right|\geq\left|t-t_{+}(x^{\prime}+,t)\right|\geq
t/2$ for $\displaystyle x\in(x^{\prime},R_{1}(t))$. Since $\displaystyle
s\,p<1$, we obtain

$\displaystyle\displaystyle\left|\frac{x_{i}}{t-t_{+}(x_{i},t)}-\frac{x_{i+1}}{t-t_{+}(x_{i+1},t)}\right|^{\frac{1}{sp}}$
$\displaystyle\displaystyle\leq\frac{2^{\frac{1}{sp}}\left|x_{i}-x_{i+1}\right|^{\frac{1}{sp}}}{\left|t-t_{+}(x_{i},t)\right|^{\frac{1}{sp}}}+2^{\frac{1}{sp}}\left|\frac{x_{i+1}}{t-t_{+}(x_{i},t)}-\frac{x_{i+1}}{t-t_{+}(x_{i+1},t)}\right|^{\frac{1}{sp}}$
$\displaystyle\displaystyle\leq\frac{2^{\frac{1}{sp}}}{\left|t-t_{+}(x^{\prime},t)\right|^{\frac{1}{sp}}}\left|x_{i}-x_{i+1}\right|^{\frac{1}{sp}}+\frac{(2M)^{\frac{1}{sp}}}{\left|t-t_{+}(x^{\prime},t)\right|^{\frac{2}{sp}}}\left|t_{+}(x_{i},t)-t_{+}(x_{i+1},t)\right|^{\frac{1}{sp}}$
$\displaystyle\displaystyle\leq\frac{2^{\frac{2}{sp}}}{t^{\frac{1}{sp}}}\left|x_{i}-x_{i+1}\right|^{\frac{1}{sp}}+\frac{(8M)^{\frac{1}{sp}}}{t^{\frac{2}{sp}}}\left|t_{+}(x_{i},t)-t_{+}(x_{i+1},t)\right|^{\frac{1}{sp}}.$

By a similar argument, we have

$\displaystyle\displaystyle\left|\frac{z_{+}(x_{i},t)}{t_{+}(x_{i},t)}-\frac{z_{+}(x_{i+1},t)}{t_{+}(x_{i+1},t)}\right|$
$\displaystyle\displaystyle\leq\frac{1}{t_{+}(x_{i},t)}\left|z_{+}(x_{i},t)-z_{+}(x_{i+1},t)\right|+\frac{\left|z_{+}(x_{i+1},t)\right|}{t_{+}(x_{i},t)t_{+}(x_{i+1},t)}\left|t_{+}(x_{i},t)-t_{+}(x_{i+1},t)\right|.$

Due to the choice of $\displaystyle x^{\prime}$, $\displaystyle t_{+}(x,t)\geq
t/2$ for $\displaystyle x\in(0,x^{\prime})$. Therefore,

$\displaystyle\displaystyle\left|\frac{z_{+}(x_{i},t)}{t_{+}(x_{i},t)}-\frac{z_{+}(x_{i+1},t)}{t_{+}(x_{i+1},t)}\right|$
$\displaystyle\displaystyle\leq\frac{2}{t}\left|z_{+}(x_{i},t)-z_{+}(x_{i+1},t)\right|+\frac{4\left|z_{+}(x_{i+1},t)\right|}{t^{2}}\left|t_{+}(x_{i},t)-t_{+}(x_{i+1},t)\right|.$
(4.13)

Hence, by the monotonicity of $\displaystyle x\mapsto t_{+}(x,t)$ and
$\displaystyle x\mapsto z_{+}(x,t)$ we estimate,

II
$\displaystyle\displaystyle\leq\frac{2^{\frac{1}{s}}C(f,g,t,u_{0})}{t^{\frac{1}{sq}}}\left|z_{+}(0+,t)-z_{+}(x^{\prime}-,t)\right|^{1/sq}$
$\displaystyle\displaystyle+\frac{4^{\frac{1}{sq}}\left|z_{+}(0+,t)\right|^{\frac{1}{sq}}C(f,g,t,u_{0})}{t^{\frac{2}{sq}}}\left|t_{+}(0+,t)-t_{+}(x^{\prime},t)\right|^{\frac{1}{sq}}$
$\displaystyle\displaystyle+\frac{2^{\frac{2}{sp}}C_{1}}{t^{\frac{1}{sp}}}\left|x_{1}-x_{m+1}\right|^{\frac{1}{sp}}+\frac{(8M)^{\frac{1}{sp}}C_{1}}{t^{\frac{2}{sp}}}\left|t_{+}(x_{1},t)-t_{+}(x_{m+1},t)\right|^{\frac{1}{sp}}.$
(4.14)

Note that $\displaystyle\left|t_{+}(0+,t)-t_{+}(x^{\prime},t)\right|\leq t/2$
and $\displaystyle\left|t_{+}(x_{1},t)-t_{+}(x_{m+1},t)\right|\leq t/2$. Hence
we get

$\textup{II}\leq C(f,g,t,u_{0},M)\frac{1}{t^{\frac{1}{sp}}}.$ (4.15)

Thus, combining I, II and III we obtain,

$\displaystyle\displaystyle\sum_{-n}^{n}|u(x_{i},t)-u(x_{i+1},t)|^{\frac{1}{s}}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
C(f,g,t,u_{0},M)+2(2||u_{0}||_{\infty})^{\frac{1}{s}}.$ (4.16)

Case (ii):

$\displaystyle R_{1}(t)=0$, $\displaystyle L_{1}(t)<0$. This case follows by a
similar argument as in Case (i).

This proves Theorem 2.1. ∎

Next, we prove Proposition 2.1 which concerns the precise estimate of
$\displaystyle TV^{s}(u(\cdot,t))$ when initial data satisfies condition
(2.1). This condition means when the initial data are far from critical points
of fluxes. In general, without this condition, the situation is quite
intricate and no estimates are proposed.

###### Proof of Proposition 2.1:.

We prove an explicit estimate of term II as in (4.1) and note that for terms I
and III we can use explicit estimates (4.2), (4.3). To this end, it is enough
to get an explicit estimate of $\displaystyle C(f,g,M,t,u_{0})$ in (4.15). We
note that $\displaystyle C(f,g,M,t,u_{0})$ comes from estimation of Lipschitz
constant of $\displaystyle f^{-1}_{+}g$ which depends on domain
$\displaystyle[a,m]$ of $\displaystyle f_{+}^{-1}g$ for
$\displaystyle\theta_{g}<a<m<\infty$. Observe that as $\displaystyle
a\rightarrow\theta_{g}$ the Lipschitz constant of $\displaystyle f_{+}^{-1}g$
blows up. We prove that when initial data is satisfying (2.1), we can estimate
the effective domain of $\displaystyle f_{+}^{-1}g$ in terms of
$\displaystyle\delta,r,t$. Since $\displaystyle u_{0}$ satisfies (2.1), we can
re-write (4.9) as

$\frac{z_{+}(0+,t)}{t_{+}(x^{\prime},t)}\leq\frac{z_{+}(x,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x^{\prime},t)}{t}\leq\max\left\\{\frac{-\delta}{t},-g^{\prime}(\theta_{g}+r)\right\\}.$
(4.17)

We observe the following if $\displaystyle\varphi\in C(\mathbb{R})$ is an
increasing function satisfying
$\displaystyle\left|\varphi(a_{1})-\varphi(a_{2})\right|\leq
C_{1}\left|a_{1}-a_{2}\right|^{\alpha}$ with $\displaystyle\alpha\in(0,1]$
then we have $\displaystyle a_{1}\geq
a_{2}+C_{1}^{-1/\alpha}\left|\varphi(a_{1})-\varphi(a_{2})\right|^{1/\alpha}$
for $\displaystyle a_{1}\geq a_{2}$. Since $\displaystyle g$ is a
$\displaystyle C^{2}$ function we have

$(g^{\prime})^{-1}\left(-\frac{z_{+}(x,t)}{t_{+}(x,t)}\right)\geq\delta_{0}:=\theta_{g}+C_{g}\min\left\\{{\delta}/{t},r\right\\}\mbox{
for }x\in(0,x^{\prime}).$ (4.18)

By Lemma 3.2, $\displaystyle g_{+}$ satisfies the non-degeneracy condition
(1.5) with exponent $\displaystyle q+1$, therefore, we obtain for
$\displaystyle a\geq\theta_{g}+C_{g}\min\left\\{{\delta}/{t},r\right\\}$,

$\displaystyle\displaystyle g(g^{\prime})^{-1}(a)$
$\displaystyle\displaystyle\geq
g(\theta_{g})+C_{g,q}\min\left\\{{\delta^{q+1}}/{t^{q+1}},r^{q+1}\right\\}$
$\displaystyle\displaystyle=f(\theta_{f})+C_{g,q}\min\left\\{{\delta^{q+1}}/{t^{q+1}},r^{q+1}\right\\}.$

Since $\displaystyle f\in C^{1}(\mathbb{R})$ we get

$f_{+}^{-1}(g(g^{\prime})^{-1}(a))\geq\theta_{f}+C_{f,g,q}\min\left\\{{\delta^{q+1}}/{t^{q+1}},r^{q+1}\right\\},$

for $\displaystyle a\geq\theta_{g}+C_{g}\min\left\\{{\delta}/{t},r\right\\}$.
Let $\displaystyle h$ be defined as $\displaystyle h=f_{+}^{-1}\circ g$. Then,
$\displaystyle
h^{\prime}(b)=\frac{g^{\prime}(b)}{f^{\prime}(f_{+}^{-1}(g(b)))}$. Since
$\displaystyle f$ satisfies (1.5),
$\displaystyle\left|h^{\prime}(b)\right|\leq C_{g,q,m}\lambda^{-p}$ for
$\displaystyle b$ satisfying $\displaystyle\left|b\right|\leq m$ and
$\displaystyle f_{+}^{-1}g(b)\geq\theta_{f}+\lambda$. Hence, we have for
$\displaystyle
a,b\geq\min\left\\{(g^{\prime})^{-1}(\delta/t),\theta_{g}+r\right\\}$,

$\left|f_{+}^{-1}g(g^{\prime})^{-1}(a)-f_{+}^{-1}g(g^{\prime})^{-1}(b)\right|\leq
C_{f,g}\max\left\\{{t^{pq+p}}/{\delta^{pq+p}},{r^{-pq-p}}\right\\}\left|a-b\right|^{\frac{1}{q}}.$

As $\displaystyle 1/s\geq\max\\{p,q\\}$ we get $\displaystyle a^{pq+p}\leq
a^{pq+\frac{1}{s}}$ for $\displaystyle a\geq 1$. Applying the above
observations in estimate (4.15) of term II we obtain (2.2). ∎

###  4.2 Propagation of the regularity and smoothing for compatible fluxes

Now Theorem 2.2 is proved and then its Corollary 2.1. Many arguments given in
the previous subsection are used and shortly recalled in the following proof
to prove that the initial fractional regularity is propagated as for a smooth
flux.

###### Proof of Theorem 2.2.

Due to the assumption $\displaystyle f(\theta_{f})=g(\theta_{g})$, it is
enough to consider the following two cases: (i) $\displaystyle
L_{1}(t)=0,R_{1}(t)\geq 0$, (ii) $\displaystyle R_{1}(t)=0,L_{1}(t)\leq 0$.
Likewise Theorem 2.1, we give detailed proof only for case (i).

Case (i):

If $\displaystyle L_{1}(t)=0$, $\displaystyle R_{1}(t)\geq 0$.
Consider the partition $\displaystyle\\{\cdots<x_{-1}<x_{0}\leq
0=L_{2}(t)=L_{1}(t)<x_{1}<\cdots<x_{m}\leq
R_{2}(t)=R_{1}(t)<x_{m+1}<\cdots\\}$ and fix $\displaystyle s\in(0,1)$,

$\displaystyle\displaystyle\sum_{i=-\infty}^{\infty}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{i=-\infty}^{-1}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}+|u(x_{0},t)-u(x_{1},t)|^{1/s}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\sum_{i=1}^{m-1}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}+|u(x_{m},t)-u(x_{m+1},t)|^{1/s}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\sum_{i=m+1}^{\infty}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}.$

From the Theorem 3.1 we get,

$\displaystyle\displaystyle\sum_{i=-\infty}^{\infty}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\underbrace{\sum_{i=-\infty}^{-1}|u_{0}(z_{-}(x_{i},t))-u_{0}(z_{-}(x_{i+1},t))|^{1/s}}_{\text{I}}+2(2||u_{0}||_{\infty})^{1/s}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\underbrace{\sum_{i=1}^{m-1}|f^{-1}_{+}g(u_{0}(z_{+}(x_{i},t)))-f^{-1}_{+}g(u_{0}(z_{+}(x_{i+1},t)))|^{1/s}}_{\text{II}}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\underbrace{\sum_{i=m+1}^{\infty}|u_{0}(z_{+}(x_{i},t))-u_{0}(z_{+}(x_{i+1},t))|^{1/s}}_{\text{III}}.$

Since the initial data $\displaystyle u_{0}\in BV^{s}(\mathbb{R})$, hence, I
and III

$\textup{I}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+}\textup{III}\leq
TV^{s}(u_{0}).$

Hence, from (4.6), (4.7) and (4.8) one can conclude that for $\displaystyle
x\in(0,x^{\prime})$,

$\frac{z_{+}(0,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x,t)}{t_{+}(x,t)}\leq\frac{z_{+}(x^{\prime},t)}{t}.$

$\displaystyle g$ is a convex function which implies that
$\displaystyle(g^{\prime})^{-1}$ is an increasing function and
$\displaystyle(g^{\prime})^{-1}(0)=\theta_{g}$ hence,
$\displaystyle(g^{\prime})^{-1}\Big{(}-\frac{z_{+}(x,t)}{t_{+}(x,t)}\Big{)}$,
for $\displaystyle x\in(0,x^{\prime})$, is always away from
$\displaystyle\theta_{g}$. Therefore, similarly as in the proof of Theorem 2.1
we get

$\textup{II}=\sum_{i=1}^{l-1}|f^{-1}_{+}g(u_{0}(z_{+}(x_{i},t)))-f^{-1}_{+}g(u_{0}(z_{+}(x_{i+1},t)))|^{1/s}\leq
C(f,g,t,u_{0}).$

Hence,

$\sum_{i=-\infty}^{\infty}|u(x_{i},t)-u(x_{i+1},t)|^{1/s}\leq
2TV^{s}(u_{0})+C(f,g,t)+2(2||u_{0}||_{\infty})^{1/s}.$

Case (ii):

$\displaystyle R_{1}(t)=0$, $\displaystyle L_{1}(t)<0$.

This case follows by a similar argument as in Case (i). Hence, the theorem.

∎

Now the last main result is proven. It gives the regularity of the solution
due to the fractional regularity of the initial data and the smoothing effect.
Indeed Corollary 2.1 is a direct consequence of Theorem 2.2 and 2.1.

###### Proof of Corollary 2.1.

We assume $\displaystyle f(\theta_{f})=g(\theta_{g})$ and fluxes satisfies the
non-degeneracy condition (1.5). Hence, Theorem 2.1 and Theorem 2.2 together
imply that the solution is in fractional BV space with exponent $\displaystyle
s_{1}=\min\\{\max\\{s,r_{1}\\},$$\displaystyle\max\\{s,r_{2}\\}\\}$. ∎

######  Remark 4.1.

With appropriate sufficient conditions, we can obtain estimates of type (2.2)
for Theorem 2.2 and Corollary 2.1 as well.

##  5 Acknowledgement

The authors thank the IFCAM project: ”Conservation laws: $\displaystyle
BV^{s}$, interface, and control”. The second author would like to thank TIFR-
CAM for hospitality.

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11institutetext:  Robert W. Easton 22institutetext: Applied Mathematics,
University of Colorado at Boulder, Boulder, CO 80309
22email<EMAIL_ADDRESS>33institutetext: Rodney L. Anderson
44institutetext: Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, M/S 301-121, Pasadena, CA 91109
44email<EMAIL_ADDRESS>55institutetext: Martin W. Lo
66institutetext: Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, M/S 301-121, Pasadena, CA 91109
66email<EMAIL_ADDRESS>
© 2021 California Institute of Technology. Government sponsorship
acknowledged.

# An Elementary Solution To Lambert’s Problem

Robert W. Easton  Rodney L. Anderson  Martin W. Lo

(Received: date / Accepted: date)

###### Abstract

A fundamental problem in spacecraft mission design is to find a free-flight
path from one place to another with a given transfer time. This problem for
paths in a central force field is known as Lambert’s problem. Although this is
an old problem, we take a new approach. Given two points in the plane, we
produce the conic parameters $(p,e,\omega)$ for all conic paths between these
points. For a given central force gravitational parameter, the travel time
between the launch and destination points is computed along with the initial
and final velocities for each transfer conic. For a given travel time, we
calculate the parameters for a transfer conic having that travel time.

††journal: Celestial Mechanics and Dynamical Astronomy

##  1 Introduction

Lagrange showed in 1778 that the time required to traverse an elliptic arc
between specified endpoints for Kepler’s problem depends only on the semi-
major axis of the ellipse, the distance between the two points, and the sum of
their distances from the focus of the ellipse. A wide range of methods for
solving Lambert’s problem have subsequently been proposed over the years with
various areas of applicability for different problems. These methods often
focus on the use of various parameters such as F & G series (Bate et al.,
1971), universal variables (Vallado, 2013), and semimajor axis (Prussing and
Conway, 2013) just to mention a few. Other standard solutions to Lambert’s
problem can be found in Pollard (1966), Battin (1999), and Gooding (1990). A
more complete description of various Lambert problem solutions may be found in
Russell (2019). Many of these approaches focus on the proper selection of
variables to use in combination with the time-of-flight to optimize subsequent
convergence on a desired time-of-flight using various iteration methods.

The textbook solution to Lambert’s problem typically derives the transfer time
using Kepler’s equation and the eccentric anomalies of the endpoints. To
calculate travel times between points on a transfer conic arc we use
conservation of angular momentum and numerical integration. This avoids using
Kepler’s equation and eccentric anomalies. In our approach, we first focus on
determining the set of all elliptical orbit solutions that connect two points.
We use a geometric approach based on standard orbital elements that eliminates
the need for using various intermediate parameters that may not always have an
obvious physical meaning. In particular, the conic parameters $p$ and $e$ (the
scale parameter and eccentricity) form a triangle in the parameter plane for
the elliptical transfer orbit case, which is the basis of our approach.

In the final section, the results presented here are used to construct a
sequence of conic arcs between a sequence of patch points  as an application.
The method is general and can be extended to construct conic arcs between
patch points in three dimensions. The method can be used as a preliminary
mission design tool producing a sequence of conic arcs between a sequence of
patch points which could include maneuvers or flybys.

##  2 A Geometry Problem

For two points with polar coordinates $(R,\alpha)$ and $(R\gamma,\alpha+\phi)$
with $\gamma>1$, the first step of the problem is to find the parameters of
all conic arcs between them. The equation of a general conic has the form

$r(\theta)=\frac{p}{1+e\cos(\theta-\omega)}$ (1)

where $\nu=\theta-\omega$ is the true anomaly and $r$ is the radius. A general
conic is defined by its parameters $(p,e,\omega)$. The scale parameter $p$ and
the elliptic parameter, or eccentricity, $e$ are non-negative. A standard
conic as we define it here has the argument of periapse $\omega=0$. Coordinate
rotation by $\omega$ takes the conic with parameters $(p,e,0)$ to the conic
with parameters $(p,e,\omega)$. Suppose a standard conic with parameters
$(p,e,0)$ contains the points with polar coordinates $(1,\nu_{1})$ and
$(\gamma,\nu_{1}+\Delta\nu)$ as shown in Figure 1. Here, the initial true
anomaly $\nu_{1}$ is also referred to as the inside angle, and the change in
the true anomaly $\Delta\nu$ is called the transfer angle.

Figure 1: Schematic showing the standard conic with $\nu_{1}$ and
$\nu_{2}=\nu_{1}+\Delta\nu$.

Then the rotated and rescaled conic with parameters $(Rp,e,\alpha-\nu_{1})$
contains the points $(R,\alpha)$ and $(R\gamma,\alpha+\Delta\nu)$. We will
rotate and rescale to produce transfer conics from standard conics.

Given circles of radius $1$ and radius $\gamma>1$ centered at the origin of a
plane, we first find all standard ellipses that intersect both circles. The
minimum and maximum radii of an ellipse with parameters $(p,e,0)$ are equal to
$p/(1+e)$ and $p/(1-e)$, respectively. Necessary and sufficient conditions for
intersections with the circles are $p/(1+e)\leq 1\leq\gamma\leq p/(1+e)$. The
set of points in the $(p,e)$ plane which satisfy these conditions forms a
triangle $T(\gamma)$ as shown in Figure 2 for $\gamma=1.524$.  The value 1.524
was used because this is the roughly the distance of the orbit of Mars from
the Sun in astronomical units. The triangle is bounded by the lines
$e=1,p=1+e$, and $p=\gamma(1-e)$. For conics with $e\geq 1$ the condition for
intersection is $p/(1+e)\leq 1$.

Figure 2:  Allowed elliptic parameters

A standard conic with parameters $(p,e,0)$ generates two equations for its
intersections with the two circles:

$\frac{p}{1+e\cos(\nu_{1})}=1$ (2) $\frac{p}{1+e\cos(\nu_{2})}=\gamma$ (3)

The cosines $(c_{1},c_{2})$ of the angles $(\nu_{1},\nu_{2})$ are found by
solving the equations

$p=1+ec_{1}$ (4) $\frac{p}{\gamma}=1+ec_{2}$ (5)

The solution defines a map

$f:T(\gamma)\to[-1,1]\times[-1,1]$ (6)
$f(p,e)=(c_{1},c_{2})=((p-1)/e,(p-\gamma)/e\gamma)$ (7)

The inverse of this map has the formula

$f^{-1}(c_{1},c_{2})=(1+c_{1}(\gamma-1)/(c_{1}-\gamma
c_{2}),(\gamma-1)/(c_{1}-\gamma c_{2}))$ (8)

The set $T^{*}(\gamma)=f(T(\gamma))$ is also a triangle contained in the
square $[-1,1]\times[-1,1]$. The top boundary of $T^{*}(\gamma)$ is the image
of the boundary of $T(\gamma)$ with $e=1$. By using the formula for $f^{-1}$
one shows that this boundary is the line segment

$c_{2}=\gamma^{-1}(c_{1}+1)-1,\,-1\leq c_{1}\leq 1.$ (9)

The triangle $T^{*}(1.524)$ is shown in Figure 3. The cosine oval
$Oval(\Delta\nu)$ shown in the plot is the parameterized curve
$Oval(x)=(\cos(x),\cos(x+\Delta\nu))$ for $0\leq x\leq 2\pi$. The intersection
of the oval with the triangle will be used to solve for the parameters of all
standard ellipses that intersect the circles at points with coordinates of the
form $(1,\nu_{1})$, and $(\gamma,\nu_{1}+\Delta\nu)$.

Figure 3:  The set $Oval(\Delta\nu)$ and $T^{*}(\gamma)$ for $\gamma=1.524$
and $\Delta\nu=\pi/3$

An example of a standard ellipse intersecting two circles is shown in Figure
4.

Figure 4:  Circles of radius 1 and radius 1.5. Elliptic parameters
$p=.9,e=.5,\omega=0$. Angles $\nu$ and $\Delta\nu$ are shown.

For a given transfer angle draw the oval and intersect this oval with the
image $T^{*}(\gamma)$ of the map $f$. The intersection determines an interval
of allowed inside angles
$\nu_{1}(\Delta\nu,\gamma)\leq\nu\leq\nu_{2}(\Delta\nu,\gamma)$ . The
intersection points of the cosine oval $Oval(\Delta\nu)$ with the upper
boundary of the triangle $T^{*}(\gamma)$ are found by finding the solutions
$\nu_{1}$ and $\nu_{2}$ of the equation

$\cos(\nu_{2})=\gamma^{-1}(\cos(\nu_{1})+1)-1$ (10)

where $\nu_{2}=\nu_{1}+\Delta\nu$. A typical intersection is shown in Figure 3
for $\Delta\nu=\pi/3$. The preimage of the intersection $Oval(\Delta\nu)\cap
T^{*}(\gamma)$ of the cosine oval as viewed in the $p$-$e$ plane is shown in
Figure 5. This shows the elliptic parameters of the transfer ellipses with
transfer angle $\Delta\nu$.

Figure 5:  The preimage of the set $T^{*}(\gamma)\cap Oval(\Delta\nu)$ viewed
in the e-p plane.

To compute the set of allowed elliptic parameters $f^{-1}[T^{*}(\gamma)\cap
Oval(\Delta\nu)]$ one uses the map

$g:[\nu_{1}(\Delta\nu,\gamma),\nu_{2}(\Delta\nu,\gamma)]\to R^{2}$ (11)
$g(\nu_{1})=f^{-1}(\cos(\nu_{1}),\cos(\nu_{1}+\Delta\nu))$ (12)
$g(\nu_{1})=(p(\gamma,\Delta\nu,\nu_{1}),e(\gamma,\Delta\nu,\nu_{1}))$ (13)
$e(\gamma,\Delta\nu,\nu_{1})=[\gamma-1]/[\cos(\nu_{1})-\gamma\cos(\nu_{1}+\Delta\nu)]$
(14) $p(\gamma,\Delta\nu,\nu_{1})=e(\gamma,\Delta\nu,\nu_{1})\cos(\nu_{1})+1$
(15)

The function $g$ determines the set of allowed elliptic parameters of transfer
ellipses as a function of the outer radius and the transfer and inner angles.
The function$f^{-1}$ applied to the set $Oval(\Delta\nu)$ also determines the
set of all conic transfer parameters. To create elliptic arcs joining the
points $(R_{1},\alpha)$ and $(R_{2},\alpha+\Delta\nu)$, choose
$\gamma=R_{2}/R_{1}$. Choose elliptic
parameters$(p(\gamma,\Delta\nu,\nu_{1}),e(\gamma,\Delta\nu,\nu_{1}))$. These
are the parameters of a standard ellipse that intersects the inner and outer
circles with the transfer angle $\Delta\nu$ and inside angle $\nu_{1}$. Set
$\omega=\alpha-\nu_{1}$. The angle is the argument of the periapse of a
rotated standard ellipse. Then the transfer ellipse between the points
$(R_{1},\alpha)$ and $(R_{2},\alpha+\Delta\nu)$ is the general conic

$r(\theta)=R_{1}p(\gamma,\Delta\nu,\nu_{1})/[1+e(\gamma,\Delta\nu,\nu_{1})\cos(\theta-\omega)]$
(16)

When the inside angle $\nu_{1}$ is unconstrained, the formula (16) produces
parabolas and hyperbolas.

##  3 Travel Times

As a first step to determining the transfer orbit between two points for a
specific time, the transfer orbits for all times may be computed.  Consider
travel times for a given gravitational parameter $\mu$, from an inner radius
$R$ to an outer radius $\gamma R$ . The travel times depend on the values of
$(\mu,R,\gamma,\Delta\nu,\nu_{1})$. Consider travel times for standard conics
as a function of these variables. Use equations (14) and (15) to produce
parameters $p(\gamma,\Delta\nu,\nu_{1})$, $e(\gamma,\Delta\nu,\nu_{1})$ for a
standard conic with the chosen transfer and inner angles. Then compute travel
times using the following algorithm.

Algorithm: travel times as a function of $(\mu,R,\gamma,\Delta\nu,\nu_{1})$.

1\. Calculate
$(e(\gamma,\Delta\nu,\nu_{1}),p(\gamma,\Delta\nu,\nu_{1}))=(e,p)$ using
equations (14) and (15).

2\. Set $P=Rp$ (rescale), $\sigma=\sqrt{\mu p}$ (angular momentum)

3\. Integrate using the formula $r^{2}(\theta)d\theta=\sigma dt$

4\.
$T=\sigma^{-1}R^{2}p^{2}\int_{\nu_{1}}^{\nu_{1}+\Delta\nu}(1+ecos(\theta))^{-2}d\theta$
(travel time)

5\. For fixed parameters $(\mu,R,\gamma,\Delta\nu)$ and for a given travel
time $\tau$, solve the equation $T(\mu,R,\gamma,\Delta\nu,\nu_{1})=\tau$ for
$\nu_{1}^{*}$ .

6\. Calculate the parameters
$(e(\gamma,\Delta\nu,\nu_{1}^{*}),p(\gamma,\Delta\nu,\nu_{1}^{*}))=(e^{*},p^{*})$
of the transfer ellipse.

The time to travel from $(R,\,\alpha)$ to $(\gamma R,\alpha+\Delta\nu)$ on the
transfer ellipse

$r(\theta)=Rp(\gamma,\Delta\nu,\nu_{1})/[1+e(\gamma,\Delta\nu,\nu_{1})cos(\theta-\omega)]$
(17)

with $\omega=\alpha-\nu_{1}$ is the same as the time to travel on the
corresponding standard ellipse with $\omega=0$ . Thus the travel time does not
depend on $\omega$. Steps 5 and 6 solve Lambert’s problem. Travel times for
$\mu=1.327*10^{11}\,km^{3}/s^{2},\,\gamma=1.524,\,\Delta\nu_{1}=\pi/6$ are
displayed in Figure 6 as a function of the admissible inside angles.

Figure 6:  Travel times for allowed inside angles $\nu_{1}$ for $\gamma=1.524$
and $\Delta\nu=\pi/6$

##  4 Specific Travel Time Solution

Once the possible transfer orbits have been computed as a function of the
inside angle $\nu_{1}$, the possible transfer orbits with a specific transfer
time may be computed. A standard method for finding the zero of a function may
then be used, and some sample results were computed next using Matlab’s
_fzero_ function. As an example, we consider the Mars 2020 transfer trajectory
which launched on July 30, 2020 and arrived on February 18, 2021. The angle
between launch and landing points is about 143.2 degrees, and the travel time
is about 203 days. The allowable range of inside angles $(\nu_{1},\nu_{2})$
may first be computed by finding the intersections of the oval with the
triangle as shown in Figure 7. The permissible values of $\cos(\nu_{1})$ and
$\cos(\nu_{2})$ are given by the non-shaded region.

Figure 7: The set $Oval(\Delta\nu)$ and $T^{*}(\gamma)$ for $\gamma=1.524$ and
$\Delta\nu=143.2$ degrees.

By plotting the transfer time as a function of $\nu_{1}$, the different
possible solutions between the two desired points may now be clearly seen as
shown in Figure 8. By using different initial guesses for $\nu_{1}$, the
algorithm can converge on possible different zeros along the curve.

Figure 8:  Travel time as a function of $\nu_{1}$ for the Mars 2020
trajectory.

By limiting the search space to the allowable $\nu_{1}$ values and using the
algorithm, the solution converged to $(\nu_{1},e,p)$ = (0.302347076950009,
0.21911558915832, 1.20917656075465). The geometry of this ellipse is shown in
Figure 9. While these results were computed for the dimensionless orbits in
the plane, the actual transfer orbits may be computed in three dimensions by
simply scaling and rotating the resulting orbits in combination with using
states from the ephemeris.

(a) Dimensionless ellipse

(b) Dimensional transfer

Figure 9:  A transfer ellipse with elliptic parameters $(e,p)$ =
(0.21911558915832, 1.20917656075465) is plotted along with circles of radius 1
and radius 1.524. The inside angle $\nu_{1}$ was approximately
0.302347076950009.

##  5 Conclusion

A new elementary approach to Lambert’s problem based on computing all conic
transfer orbits between two points is derived as a function of the polar
coordinate of the points and the initial true anomaly. This method was
successfully used to compute these conic transfers across a range of possible
transfer times for the direct case using elliptic orbits. By implementing a
straightforward zero finding method, it is possible to compute specific orbits
with desired transfer times using a standard conic. This approach solves
Lambert’s problem once subsequent standard rotations are performed to orient
the orbit correctly in three-dimensional space. The presented approach also
allows the computation of various constraints on the possible astrodynamic
parameters for each transfer, which reduces the search space and speeds up the
computation of the desired solution.

##  6 Acknowledgements

Part of the research presented in this paper has been carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration. The authors would like
to thank James Meiss and David Sterling for valuable comments and suggestions.

## References

  * Bate et al. (1971) Bate RR, Mueller DD, White JE (1971) Fundamentals of Astrodynamics. Dover Publications, Inc., New York
  * Battin (1999) Battin RH (1999) An Introduction to the Mathematics and Methods of Astrodynamics, Revised edn. AIAA Education Series
  * Gooding (1990) Gooding RH (1990) A procedure for the solution of lambert’s orbital boundary-value problem. Celestial Mechanics and Dynamical Astronomy 48(2):145–165
  * Pollard (1966) Pollard H (1966) Mathematical Introduction to Celestial Mechanics. Prentice-Hall Inc., Englewood Cliffs, New Jersey
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  * Vallado (2013) Vallado DA (2013) Fundamentals of Astrodynamics and Applications, 4th edn. Space Technology Library, Microcosm Press

##  Appendix A Appendix: Velocity Changes at Patch Points

In this section the formulas for transfer conics and transfer times are
applied to design patched conic orbits. To transfer from one arc to the next
on a patched conic path, one must know the angle between the tangent lines to
the arcs at the patch point. For a general conic with equation

$r(\theta)=\frac{p}{1+e\cos(\theta-\omega)}$ (18)

The angle $\Phi$ of a unit tangent vector to the conic at a point
$(r(\theta),\theta)$ on the conic is required for this calculation. To find
this angle one can calculate using complex variables. For this calculation the
notation for eccentricity conflicts with exponential notation. Just for this
calculation the symbol for eccentricity will be $\epsilon$.

$q(\theta)=r(\theta)e^{i\theta},\dot{q}(\theta)=\dot{r}(\theta)e^{i\theta}+r(\theta)ie^{i\theta}=s(\theta)e^{i\Phi}$
(19) $s(\theta)=\sqrt{\dot{r}(\theta)^{2}+(r(\theta)^{2}}$ (20)
$\dot{r}(\theta)+r(\theta)i=s(\theta)e^{i(\Phi-\theta)}$ (21)
$sin(\Delta\nu-\theta)=r(\theta)/s(\theta)$ (22)
$\Phi=\theta+asin(r(\theta)/s(\theta))$ (23)
$\dot{r}(\theta)=(\epsilon/p)r^{2}(\theta)sin(\theta-\omega)$ (24)
$\dot{r}(\theta)/r(\theta)=[1+\epsilon cos(\theta-\omega)]^{-1}\epsilon
sin(\theta-\omega)$ (25)
$r(\theta)/s(\theta)=1/\sqrt{\epsilon^{2}sin^{2}(\theta-\omega)/{[1+\epsilon
cos(\theta-\omega)]}^{2}+1}$ (26)

Now replace $\epsilon$ by $e$ as the symbol for eccentricity. The result is

$\Phi=\theta+asin(1/\sqrt{\epsilon^{2}sin^{2}(\theta-\omega)/r^{2}(\theta)+1}\,)$
(27)

The energy of the orbit in terms of the elliptic parameters $(p,\,e,\,\omega)$
is $E=\mu(e^{2}-1)/2p$. For an elliptic orbit of the Kepler problem
$\ddot{q}=-\mu r^{{}_{3}}\,\,with\,r=|q|$ The Cartesian velocity
$v(r,\theta)=\dot{q}$ is the velocity at the point $(r,\theta)$ on the conic
and the speed is

$s(r,\theta)=|\dot{q}|=\sqrt{2(E+\mu
r^{-1})}=\sqrt{2\mu}\sqrt{[(e^{2}-1)/p]+r^{-1}}$ (28)

The velocity in polar coordinates is
$(s(r,\mu,e,p),\Phi(r,\theta,e,p,\omega))$. To transfer from an incoming
elliptic orbit to an outgoing elliptic orbit at a common patch point
$(r,\theta)$ the velocity change needed is the difference between the outgoing
and incoming velocities at this point. These velocities are determined for
polar coordinates by the above formulas.

To patch elliptic arcs in an outward spiral one can go back to geometry and
choose conic arcs between circles of radii $1,\,\gamma,\gamma^{2},...$ with a
fixed transfer angle $\Delta\nu$ between patch points and choose allowed
elliptic parameters $(1\,,e_{0},\omega_{0})$. Then choose an initial scale
parameter and generate a sequence of elliptic parameters $p_{1}=\gamma
p_{0},\,e_{1}=e_{0},\omega_{1}=\omega_{0}+\Delta\nu,\,...$ Elliptic arcs,
travel times, and velocity changes can now be computed from this data. A plot
for the choice $\gamma=1.524,\,\Delta\nu=2\pi/3,\,p_{0}=1.202,\,e_{0}=.2385$
is shown in Figure 10.

Figure 10:  Four conic arcs with
$\gamma=1.524,\,\Delta\nu=2\pi/3\,\,p_{0}=1.202,\,e_{0}=.2385$


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