# Weak–strong uniqueness property for models of compressible viscous fluids
near vacuum

Eduard Feireisl The work of E.F. was supported by the Czech Sciences
Foundation (GAČR), Grant Agreement 21–02411S. Antonín Novotný The work of A.N.
was partially supported by the Eduard Čech visiting program at the
Mathematical Institute of the Academy of Sciences of the Czech Republic.

###### Abstract

We extend the weak–strong uniqueness principle to general models of
compressible viscous fluids near/on the vacuum. In particular, the physically
relevant case of positive density with polynomial decay at infinity is
considered.

Institute of Mathematics of the Academy of Sciences of the Czech Republic;

Žitná 25, CZ-115 67 Praha 1, Czech Republic

<EMAIL_ADDRESS>
and

IMATH, EA 2134, Université de Toulon,

BP 20132, 83957 La Garde, France

<EMAIL_ADDRESS>
Keywords: Compressible Navier–Stokes system, weak strong uniqueness, vacuum

##  1 Introduction

The motion of a compressible viscous fluid in the barotropic regime is
described by the time evolution of the mass density $\varrho=\varrho(t,x)$ and
the velocity field ${\bf u}={\bf u}(t,x)$, $t>0$, $x\in\Omega\subset R^{d}$,
$d=1,2,3$, satisfying the equation of continuity

$\partial_{t}\varrho+{\rm div}_{x}(\varrho{\bf u})=0,$ (1.1)

and the momentum balance

$\partial_{t}(\varrho{\bf u})+{\rm div}_{x}(\varrho{\bf u}\otimes{\bf
u})+\nabla_{x}p(\varrho)={\rm div}_{x}\mathbb{S}+\varrho{\bf f}.$ (1.2)

Here, $p=p(\varrho)$ is the pressure, ${\bf f}$ the external driving force,
and $\mathbb{S}$ the viscous stress. To close the system, we suppose that
$\mathbb{S}$ is related to the symmetric part of the velocity gradient

$\mathbb{D}_{x}{\bf u}\equiv\frac{1}{2}\left(\nabla_{x}{\bf
u}+\nabla_{x}^{t}{\bf u}\right)$

through a general rheological law

$F(\mathbb{D}_{x}{\bf u})+F^{*}(\mathbb{S})=\mathbb{S}:\nabla_{x}{\bf u}\
\Leftrightarrow\ \mathbb{S}\in\partial F(\mathbb{D}_{x}{\bf u})\
\Leftrightarrow\ \mathbb{D}_{x}{\bf u}\in\partial F^{*}(\mathbb{S}),$ (1.3)

where

$F:R^{d\times d}_{\rm sym}\to[0,\infty]\ \mbox{is a convex l.s.c. function.}$

The best known example is the isentropic Navier–Stokes system, where

$p(\varrho)=a\varrho^{\gamma},\ a>0,\ \gamma>1,\
F=\frac{\mu}{2}\left|\nabla_{x}{\bf u}+\nabla_{x}^{t}{\bf u}-\frac{2}{d}{\rm
div}_{x}{\bf u}\mathbb{I}\right|^{2}+\frac{\lambda}{2}|{\rm div}_{x}{\bf
u}|^{2},\mu>0,\ \lambda\geq 0.$ (1.4)

The “implicit” rheological law (1.3) covers the class of power law fluids as
well as other non-Newtonian fluids, see e.g. Bulíček et al. [2].

Our goal is to study stability of strong solutions of the problem in the
regime when the fluid density either vanishes or approaches asymptotically the
vacuum state. To begin, we point out that the model has been derived for
non–dilute fluids out of vacuum. In particular, as strong solutions satisfy
the equation of continuity

$\partial_{t}\varrho+{\bf u}\cdot\nabla_{x}{\bf u}+\varrho{\rm div}_{x}{\bf
u}=0,$

the density will always remain positive unless the vacuum state is
artificially imposed through the initial or boundary data. However, allowing
$\varrho\approx 0$ is still physically relevant at least in certain asymptotic
regimes:

  * •

Solutions of (1.1)–(1.3) with ${\bf f}=\nabla_{x}G$, $G=G(x)$ approach an
equilibrium state $\varrho=\tilde{\varrho}$, ${\bf u}=0$ as $t\to\infty$, see
[7], where

$\nabla_{x}p(\tilde{\varrho})=\tilde{\varrho}\nabla_{x}G.$ (1.5)

It is a simple observation that $\tilde{\varrho}$ may vanish on a non–void
subset of the physical domain, in particular if the total mass of the fluid is
small enough.

  * •

In models of gaseous stars in astrophysics, the physical domain $\Omega$ is
exterior to a rigid object, while

$\varrho\to 0,\ {\bf u}\to 0\ \mbox{as}\ |x|\to\infty.$ (1.6)

Thus the density is close to zero at least in the far field. Note that, on the
one hand, the isentropic Navier–Stokes system (1.1), (1.2), (1.4) is globally
well posed in this regime if the total energy is small enough, see Huang, Li,
and Xin [9], and also Fang, Zhu, and Guo [5] for a similar result in the non-
Newtonian case and $d=1$. On the other hand, Rozanova [15] and Xin [17] showed
that fast decay of solutions as $|x|\to\infty$ is not compatible with global
existence. More recently, Merle et al. [14] obtained blow–up results in a
similar regime for radially symmetric solutions with certain profiles and
certain values of the adiabatic exponent $\gamma$.

  * •

Last but not least, the problem with vanishing initial density is
mathematically challenging and has been considered in a large number of recent
studies, see Gong et al. [8], Li and Xin [11], Liang [12], to name only a few.

Our aim is to clarify in which way the strong solutions obtained in the above
references can coexist/coincide with the weak solutions in the sense of
P.-L.Lions [13] or even more general dissipative solutions introduced in [1].
More specifically, we establish the weak–strong strong uniqueness principle: A
weak solution of the problem (1.1)–(1.3) coincides with the strong one with
the same initial/boundary data as long as the strong solution exists. As shown
in [1, Theorem 6.3], the weak–strong uniqueness principle holds for the
problem (1.1)–(1.3) in the class of strong solutions away from vacuum, and for
$\Omega\subset R^{d}$ a bounded domain with general in/out flow boundary
conditions:

${\bf u}|_{\partial\Omega}={\bf u}_{B},\ \varrho|_{\Gamma_{\rm
in}}=\varrho_{B},\ \Gamma_{\rm in}\equiv\left\\{x\in\partial\Omega\ \Big{|}\
{\bf u}_{B}\cdot{\bf n}<0\right\\},$ (1.7)

where ${\bf n}$ is the outer normal vector to $\partial\Omega$. We show that
this result can be extended to the class of strong solutions near/with the
vacuum as long as:

  1. 1.

The potential $F$ is uniformly strictly convex in $R^{d\times d}_{{\rm
sym},0}$.

  2. 2.

The pressure $p$ is strictly convex in $[0,\infty)_{\rm loc}$.

  3. 3.

The density component $\tilde{\varrho}$ of the strong solution satisfies

$\nabla_{x}P^{\prime}(\tilde{\varrho})\in L^{2}(0,T;L^{q}(\Omega)),\ q=q(d),$

where $P$ is the pressure potential,

$P^{\prime}(\varrho)\varrho-P(\varrho)=p(\varrho).$

  4. 4.

If $\Omega$ is unbouded, the initial density must decay sufficiently fast as
$|x|\to\infty$.

Conditions 1, 2 are constitutive. Condition 1 holds in particular for the
Newtonian stress in the Navier–Stokes system. Condition 2 is satisfied for
$p=a\varrho^{\gamma}$ as long as $1<\gamma\leq 2$, which is physically
relevant in the isentropic regime. Condition 3 is associated to the flow. If
the velocity field ${\tilde{\bf u}}$ is smooth enough, the Sobolev regularity
of $P^{\prime}(\tilde{\varrho})$ is propagated along streamlines. It is
therefore enough to impose 3 on the data only. We point out that

$\nabla_{x}P^{\prime}(\tilde{\varrho})\approx\tilde{\varrho}^{\gamma-2}\nabla_{x}\tilde{\varrho}$

becomes singular near the vacuum if $\gamma<2$. Condition 4 is relevant if the
physical domain is unbounded. Here again, the decay may be inherited from the
initial/boundary data if the velocity field is sufficiently regular.

Our method is based on a general relative energy inequality obtained in [1]
that must be adapted to the vanishing density regime. The most delicate issue
is the behavior of the fluid velocity on or near the vacuum region. As a
matter of fact, the concept of velocity in the absence of matter is
meaningless, however, the velocity can be recovered as a solution of an
elliptic equation in the whole physical space. This paradox can be seen as a
consequence of the infinite speed of propagation due to the viscous stress
with constant viscosity coefficients. By the same token, the decay of the
velocity for $|x|\to\infty$ is not expected to be faster than that of the
Dirichlet kernel at least in the linearly viscous case. This fact makes the
analysis on unbounded domains rather delicate and requires certain decay
properties of the initial density profile. Seen from this perspective, the
hypothesis ${\bf u}(t,\cdot)\in W^{1,2}$ assumed sometimes in the literature
is not very realistic.

The paper is organized as follows. We begin by recalling the basic concepts of
strong as well as weak (dissipative) solution in Section 2. In Section 3, we
introduce the relative energy functional. The main results are stated and
proved in Section 4. We distinguish three cases: Bounded domain, general
unbounded domain, and the whole space $R^{3}$ with compactly supported density
or rapidly decaying density.

##  2 Strong and weak solutions

We recall the definition of strong and weak (dissipative) solutions to the
problem (1.1)–(1.3), with the boundary conditions (1.7), and with the far
field conditions (1.6) if the domain $\Omega$ is unbounded. For the sake of
simplicity, we assume that the boundary data ${\bf u}_{B}$, $\varrho_{B}$ are
independent of $t$. First, extend the function ${\bf u}_{B}$ to be smooth in
$\overline{\Omega}$ and

${\bf u}_{B}(x)=0\ \mbox{for}\ |x|>R$ (2.1)

if $\Omega$ is unbounded. To avoid technical difficulties, we suppose that the
outer normal is defined whenever ${\bf u}_{B}\cdot{\bf n}\neq 0$. Moreover, we
consider the pressure in the isentropic form

$p(\varrho)=a\varrho^{\gamma},\ a>0,\ \gamma>1.$ (2.2)

Finally, we impose certain growth restrictions on the dissipation potential
$F$,

$0\leq F(\mathbb{D})\stackrel{{\scriptstyle<}}{{\sim}}|\mathbb{D}|^{2},$ (2.3)
$F(\mathbb{D}+\mathbb{Q})-F(\mathbb{D})-\mathbb{S}:\mathbb{Q}\stackrel{{\scriptstyle>}}{{\sim}}\left|\mathbb{Q}-\beta{\rm
tr}[\mathbb{Q}]\mathbb{I}\right|^{2},\ \beta=\frac{1}{d}\ \mbox{if}\ d=2,3,\
\beta=0\ \mbox{if}\ d=1,$ (2.4)

for any $\mathbb{D},\mathbb{Q}\in R^{d\times d}_{\rm sym}$,
$\mathbb{S}\in\partial F(\mathbb{D})$. Note that both (2.3) and (2.4) are
compatible with the Newtonian potential (1.4). The condition (2.3) can be
relaxed at the expense of several technical difficulties in the proofs. The
coercivity assumption (2.4), reflecting non-degeneracy of the viscous stress,
is essential.

###  2.1 Strong solutions

The goal is to identify the largest possible class of strong solutions.
Inspired by the existence result by Cho, Choe, Kim [3] we consider the
velocity ${\tilde{\bf u}}$ in the class

$({\tilde{\bf u}}-{\bf u}_{B})\in C([0,T];D^{1,2}_{0}(\Omega;R^{d})\cap
L^{2}(0,T;D^{2,q}(\Omega;R^{d})),\partial_{t}{\tilde{\bf u}}\in
L^{2}(0,T;L^{q}(\Omega;R^{d}))$ (2.5)

for some $q>d$. Here, $D^{k,q}$ denotes the space of locally integrable
functions $v$ with $D^{k}_{x}v\in L^{q}(\Omega)$, $D^{1,2}_{0}$ is the closure
of $C^{\infty}_{c}(\Omega)$ in $D^{1,2}(\Omega)$. If $\Omega$ is bounded,
$D^{k,p}$ coincide with the standard Sobolev spaces $W^{k,p}$. Unbounded
domains will be considered only for $d=3$. Note that the class (2.5) includes
the case $q=6$, $d=3$ considered in [3]. In particular, by interpolation and
the standard embedding $D^{1,q}\hookrightarrow C$, $q>d$,

${\tilde{\bf u}}\in C([0,T];D^{1,q}\cap
D^{1,2}_{0}(\Omega;R^{d}))\hookrightarrow
BC([0,T]\times\overline{\Omega};R^{d}),$

where $BC(Q)$ is the Banach space of bounded continuous functions on $Q$.
Similarly, we deduce from (2.5)

$\int_{0}^{T}\|{\tilde{\bf u}}\|_{W^{1,\infty}(\Omega;R^{d})}^{2}\,{\rm
d}t\stackrel{{\scriptstyle<}}{{\sim}}1$ (2.6)

if either $\Omega$ is bounded or $d=3$.

In particular, given the velocity field ${\tilde{\bf u}}$, the initial state
and the boundary data

$\tilde{\varrho}(0,\cdot)=\varrho_{0}\in L^{1}_{\rm loc}(\Omega),\
\tilde{\varrho}|_{\Gamma_{\rm in}}=\varrho_{B}\in L^{1}(\partial\Omega),$

the density $\tilde{\varrho}$ is uniquely determined by the equation of
continuity (1.1) and can be computed explicitly via the method of
characteristics. More specifically, supposing $\Omega$ has a compact Lipschitz
boundary, we may extend ${\tilde{\bf u}}$ outside $\Omega$ to remain in the
class

${\tilde{\bf u}}\in L^{2}(0,T;D^{2,q}(R^{d};R^{d}))\cap BC([0,T]\times
R^{d};R^{d}).$

Then we define the characteristic curves

$\frac{{\rm d}}{\,{\rm d}t}{\bf X}(t,x)={\tilde{\bf u}}(t,{\bf X}(t,x)),\ {\bf
X}(0,x)=x.$

For any $t>0$, $x\in\Omega$, there is a unique characteristic satisfying

${\bf X}(t,x_{0})=x.$

Let $\tau=\inf\left\\{s\in[0,t]\ \Big{|}{\bf X}(z,x_{0})\in\Omega\ \mbox{for
all}\ z\in(s,t)\right\\}$. If $\tau=0$, we set

$\tilde{\varrho}(t,x)=\varrho_{0}(x_{0})\exp\left(-\int_{0}^{t}{\rm
div}_{x}{\tilde{\bf u}}(s,{\bf X}(s,x_{0})){\rm d}s\right).$ (2.7)

If $\tau>0$, then necessarily

$x_{\tau}={\bf X}(\tau,x_{0})\in\overline{\Gamma}_{\rm in},$

and we set

$\tilde{\varrho}(t,x)=\varrho_{B}(x_{\tau})\exp\left(-\int_{\tau}^{t}{\rm
div}_{x}{\tilde{\bf u}}(s,{\bf X}(s,x_{\tau})){\rm d}s\right).$ (2.8)

To avoid jumps in the resulting density, the above procedure requires an
obvious compatibility condition

$\varrho_{0}\in BC(\overline{\Omega}),\ \varrho_{0}|_{\Gamma_{\rm
in}}=\varrho_{B}.$

Still the construction via characteristic lines may give rise to ambiguous
results as soon as part of the boundary is tangent to the characteristic
curves. To avoid this problem, several non–degeneracy conditions on
$\Gamma_{\rm in}$ are usually imposed in the literature to eliminate ingoing
characteristics emanating from $\partial\Gamma_{\rm in}$, cf. e.g. Valli and
Zajaczkowski [16]. The density $\tilde{\varrho}\in
BC([0,T]\times\overline{\Omega})$ associated to ${\tilde{\bf u}}$ is then well
defined through formulae (2.7), (2.8). As a direct consequence, we obtain the
following standard result concerning propagation of Sobolev regularity. To
avoid the afore mentioned technical problems and also because the available
regularity of $\partial_{t}{\tilde{\bf u}}$ is rather limited, we restrict
ourselves to the case $\Gamma_{\rm in}=\emptyset$.

######  Lemma 2.1.

Let $\Omega\subset R^{d}$ be a domain with compact (possibly empty) Lipschitz
boundary. Let a vector field ${\tilde{\bf u}}$ be given in the class

${\tilde{\bf u}}\in L^{2}(0,T;D^{2,q}(\Omega;R^{d}))\cap
BC([0,T]\times\overline{\Omega};R^{d}),\ d<q\leq\infty$

and such that ${\tilde{\bf u}}\cdot{\bf n}|_{\partial\Omega}\geq 0$. Suppose
that

$\varrho_{0}\in BC(\overline{\Omega}),\ \varrho_{0}\geq 0,\
\nabla_{x}(\varrho_{0}^{\gamma-1})\in L^{q}(\Omega;R^{d})\ \mbox{for some}\ \
1<\gamma\leq 2.$

Then the unique solution $\tilde{\varrho}$ of the equation of continuity (1.1)
determined by (2.7) satisfies

$\|\nabla_{x}(\tilde{\varrho}^{\gamma-1})(t,\cdot)\|_{L^{q}(\Omega;R^{d})}\leq
c(T,\varrho_{0},{\tilde{\bf u}})\ \mbox{for any}\ t\in[0,T].$

###### Proof.

Extending ${\tilde{\bf u}}$ and $\varrho_{0}$ outside $\Omega$ as the case may
be, we may assume $\Omega=R^{d}$. As $\Gamma_{\rm in}=\emptyset$, the solution
$\tilde{\varrho}$ is given by formula (2.7). In particular,

$\tilde{\varrho}(t,x)=\varrho_{0}({\bf
X}^{-1}(t,x))\exp\left(-\int_{0}^{t}{\rm div}_{x}{\tilde{\bf u}}(s,{\bf
X}(s,{\bf X}^{-1}(t,x))){\rm d}s\right),$
$\tilde{\varrho}^{\gamma-1}(t,x)=\varrho^{\gamma-1}_{0}({\bf
X}^{-1}(t,x))\exp\left((1-\gamma)\int_{0}^{t}{\rm div}_{x}{\tilde{\bf
u}}(s,{\bf X}(s,{\bf X}^{-1}(t,x))){\rm d}s\right).$

Thus the desired result follows by direct differentiation of the above
formula.

∎

If the exponent $\gamma$ remains in the range $1<\gamma\leq 2$,
differentiablity of $\tilde{\varrho}^{\gamma-1}$ implies differentiablity of
$\tilde{\varrho}$ and of $\tilde{\varrho}^{\gamma}$. In particular, the
equation of continuity (1.1) can be interpreted in the strong sense.
Similarly, the left hand side of the momentum equation can be written in the
form

$\tilde{\varrho}\partial_{t}{\tilde{\bf u}}+\tilde{\varrho}{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\tilde{\varrho}\nabla_{x}\tilde{\varrho}^{\gamma-1}$

Thus for ${\tilde{\bf u}}$ belonging to the class (2.5), the system
(1.1)–(1.3) can be written as

$\begin{split}\partial_{t}\tilde{\varrho}+{\tilde{\bf
u}}\cdot\nabla_{x}\tilde{\varrho}+\tilde{\varrho}{\rm div}_{x}{\tilde{\bf
u}}&=0,\\\ \tilde{\varrho}\partial_{t}{\tilde{\bf
u}}+\tilde{\varrho}{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\tilde{\varrho}\nabla_{x}\tilde{\varrho}^{\gamma-1}&={\rm
div}_{x}\widetilde{\mathbb{S}}+\tilde{\varrho}{\bf f},\end{split}$ (2.9)

with

$\widetilde{\mathbb{S}}\in\partial F(\mathbb{D}_{x}{\tilde{\bf u}})\ \mbox{for
a.a.}\ t\in(0,T).$ (2.10)

###  2.2 Dissipative (weak) solutions

The dissipative solutions to the system (1.1)–(1.3) were introduced in [1]. We
denote $\mathcal{M}^{+}(\overline{\Omega};R^{d\times d}_{\rm sym})$ the set of
positively semi–definite finite tensorial measures on $\overline{\Omega}$.
Specifically, $\mathfrak{R}\in\mathcal{M}^{+}(\overline{\Omega};R^{d\times
d}_{\rm sym})$ if $\mathfrak{R}$ is a tensor valued finite measure satisfying

$\mathfrak{R}:(\xi\otimes\xi)\geq 0\ \mbox{for any}\ \xi\in R^{d}.$

######  Definition 2.2 (Dissipative solution).

The functions $(\varrho,{\bf u})$ represent _dissipative solution_ of the
problem (1.1)–(1.3), with the boundary conditions (1.7), the far field
conditions (1.6), and the initial conditions

$\varrho(0,\cdot)=\varrho_{0},\ (\varrho{\bf u})(0,\cdot)={\bf m}_{0},$

if the following holds:

  * •

Integrability.

$\begin{split}\varrho\geq 0,\ \varrho&\in C_{\rm
weak}([0,T];L^{\gamma}(\Omega))\cap
L^{\gamma}((0,T);L^{\gamma}(\partial\Omega,|{\bf u}_{B}\cdot{\bf n}|{\rm
d}\sigma)),\\\ ({\bf u}-{\bf u}_{B})&\in
L^{2}(0,T;D^{1,2}_{0}(\Omega;R^{d})),\ \varrho{\bf u}\in C_{\rm
weak}([0,T];L^{\frac{2\gamma}{\gamma+1}}(\Omega;R^{d})),\\\ \mathbb{S}&\in
L^{2}(0,T;L^{2}(\Omega;R^{d\times d}_{\rm sym})).\end{split}$

  * •

Equation of continuity. The integral identity

$\begin{split}\left[\int_{\Omega}\varrho\varphi\ \,{\rm
d}{x}\right]_{t=0}^{t=\tau}&+\int_{0}^{\tau}\int_{\partial\Omega}\varphi\varrho[{\bf
u}_{B}\cdot{\bf n}]^{+}\ {\rm d}\
S_{x}+\int_{0}^{\tau}\int_{\partial\Omega}\varphi\varrho_{B}[{\bf
u}_{B}\cdot{\bf n}]^{-}\ {\rm d}\ S_{x}\\\
&=\int_{0}^{\tau}\int_{\Omega}\Big{[}\varrho\partial_{t}\varphi+\varrho{\bf
u}\cdot\nabla_{x}\varphi\Big{]}\ \,{\rm d}{x}\,{\rm d}t,\
\varrho(0,\cdot)=\varrho_{0}\end{split}$ (2.11)

holds for any $0\leq\tau\leq T$, and any test function $\varphi\in
C^{1}_{c}([0,T]\times\overline{\Omega})$.

  * •

Momentum equation. There exists

$\mathfrak{R}\in L^{\infty}_{\rm
weak(*)}(0,T;\mathcal{M}^{+}(\overline{\Omega};R^{d\times d}_{\rm sym}))$

such that

$\begin{split}\left[\int_{\Omega}\varrho{\bf u}\cdot\boldsymbol{\varphi}\
\,{\rm
d}{x}\right]_{t=0}^{t=\tau}&=\int_{0}^{\tau}\int_{\Omega}\Big{[}\varrho{\bf
u}\cdot\partial_{t}\boldsymbol{\varphi}+\varrho{\bf u}\otimes{\bf
u}:\nabla_{x}\boldsymbol{\varphi}+p(\varrho){\rm
div}_{x}\boldsymbol{\varphi}-\mathbb{S}:\nabla_{x}\boldsymbol{\varphi}+\varrho{\bf
f}\cdot\boldsymbol{\varphi}\Big{]}\ \,{\rm d}{x}\\\
&+\int_{0}^{\tau}\int_{{\Omega}}\nabla_{x}\boldsymbol{\varphi}:{\rm d}\
\mathfrak{R}(t)\ \,{\rm d}t,\ \varrho{\bf u}(0,\cdot)={\bf m}_{0}\end{split}$
(2.12)

holds for any $0\leq\tau\leq T$ and any test function $\boldsymbol{\varphi}\in
C^{1}_{c}([0,T]\times\overline{\Omega};R^{d})$,
$\boldsymbol{\varphi}|_{\partial\Omega}=0$.

  * •

Energy inequality. There exists

$\mathfrak{E}\in L^{\infty}_{\rm
weak(*)}(0,T;\mathcal{M}^{+}(\overline{\Omega}))$

such that

$\begin{split}&\left[\int_{\Omega}\left[\frac{1}{2}\varrho|{\bf u}-{\bf
u}_{B}|^{2}+P(\varrho)\right]\ \,{\rm
d}{x}\right]_{t=0}^{t=\tau}+\int_{0}^{\tau}\int_{\Omega}\Big{[}F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})\Big{]}\ \,{\rm d}{x}\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\partial\Omega}P(\varrho)[{\bf u}_{B}\cdot{\bf n}]^{+}\
{\rm d}S_{x}\,{\rm d}t+\int_{0}^{\tau}\int_{\partial\Omega}P(\varrho_{B})[{\bf
u}_{B}\cdot{\bf n}]^{-}\ {\rm d}S_{x}\,{\rm d}t+\int_{\overline{\Omega}}{\rm
d}\ \mathfrak{E}(\tau)\\\ \leq&-\int_{0}^{\tau}\int_{\Omega}\left[\varrho{\bf
u}\otimes{\bf u}+p(\varrho)\mathbb{I}\right]:\nabla_{x}{\bf u}_{B}\ \,{\rm
d}{x}\,{\rm d}t+\int_{0}^{\tau}{\frac{1}{2}\int_{\Omega}{\varrho}{\bf
u}\cdot\nabla_{x}|{\bf u}_{B}|^{2}\ \,{\rm d}{x}}\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}\Big{[}\mathbb{S}:\nabla_{x}{\bf
u}_{B}+\varrho{\bf f}\cdot{({\bf u}-{\bf u}_{B})}\Big{]}\ \,{\rm d}{x}\,{\rm
d}t\\\ &-\int_{0}^{\tau}\int_{\overline{\Omega}}\nabla_{x}{\bf u}_{B}:{\rm d}\
\mathfrak{R}(t)\,{\rm d}t,\
P(\varrho)\equiv\frac{a}{\gamma-1}\varrho^{\gamma}\end{split}$ (2.13)

for a.a. $0\leq\tau\leq T$.

  * •

Defect compatibility.

$\underline{d}\mathfrak{E}\leq{\rm
tr}[\mathfrak{R}]\leq\overline{d}\mathfrak{E},\ \ \mbox{for certain
constants}\ 0<\underline{d}\leq\overline{d}.$ (2.14)

######  Remark 2.3.

The hypothesis

$\mathbb{S}\in L^{2}(0,T;L^{2}(\Omega;R^{d\times d}_{\rm sym}))$

is pertinent to the class of the dissipative potentials $F$ with at most
quadratic growth (2.3).

##  3 Relative energy

The relative energy reads:

$E\left(\varrho,{\bf u}\ \Big{|}\ \tilde{\varrho},{\tilde{\bf
u}}\right)=\frac{1}{2}\varrho|{\bf u}-{\tilde{\bf
u}}|^{2}+P(\varrho)-P^{\prime}(\tilde{\varrho})(\varrho-\tilde{\varrho})-P(\tilde{\varrho}).$

The relative energy inequality associated to the problem (1.1)–(1.3), (1.7)
was derived in [1, Section 5, formula (5.5)]. Its generalization to the case
of unbounded domain is straightforward (cf. also [6]):

$\begin{split}&\left[\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)\ \,{\rm
d}{x}\right]_{t=0}^{t=\tau}+\int_{0}^{\tau}\int_{\Omega}\Big{[}F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}\mathbb{S}:\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\
&{+\int_{0}^{\tau}\int_{\partial\Omega}\left[P(\varrho)-P^{\prime}(\tilde{\varrho})(\varrho-\tilde{\varrho})-P(\tilde{\varrho})\right][{\bf
u}_{B}\cdot{\bf n}]^{+}\ {\rm d}S_{x}\,{\rm d}t}\\\
&{+\int_{0}^{\tau}\int_{\partial\Omega}\left[P(\varrho_{B})-P^{\prime}(\tilde{\varrho})(\varrho_{B}-\tilde{\varrho})-P(\tilde{\varrho})\right][{\bf
u}_{B}\cdot{\bf n}]^{-}\ {\rm d}S_{x}\,{\rm d}t}+\int_{\overline{\Omega}}1{\rm
d}\ \mathfrak{E}(\tau)\\\
\leq&-\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot({\tilde{\bf u}}-{\bf u})\cdot\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\
&-\int_{0}^{\tau}\int_{\Omega}\Big{[}p(\varrho)-p^{\prime}(\tilde{\varrho})(\varrho-\tilde{\varrho})-p(\tilde{\varrho})\Big{]}{\rm
div}_{x}{\tilde{\bf u}}\ \,{\rm d}{x}\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t{-\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf u})\cdot{\bf f}\
\,{\rm d}{x}\,{\rm d}t}\\\
&+\int_{0}^{\tau}\int_{\Omega}p^{\prime}(\tilde{\varrho})\left(1-\frac{\varrho}{\tilde{\varrho}}\right)\Big{[}\partial_{t}\tilde{\varrho}+{\rm
div}_{x}(\tilde{\varrho}{\tilde{\bf u}})\Big{]}\ \,{\rm d}{x}\,{\rm d}t\\\
&-\int_{0}^{\tau}\int_{\overline{\Omega}}\nabla_{x}{\tilde{\bf u}}:{\rm d}\
\mathfrak{R}(t)\ \,{\rm d}t\ \mbox{for a.a.}\ \tau\in(0,T),\end{split}$ (3.1)

for a.a. $0<\tau<T$, and any pair of “test functions”

$\begin{split}{\tilde{\bf u}}&\in
C^{1}_{c}([0,T]\times\overline{\Omega};R^{d}),{\tilde{\bf
u}}|_{\partial\Omega}={\bf u}_{B},\\\ \tilde{\varrho}&\in
C^{1}([0,T]\times\overline{\Omega}),\
\tilde{\varrho}>0,(\tilde{\varrho}-\varepsilon)\in
C_{c}([0,T]\times\overline{\Omega})\ \mbox{for some}\
\varepsilon>0.\end{split}$ (3.2)

Our goal is to extend validity of (3.1) to the class of test functions
containing $(\tilde{\varrho}+\varepsilon,{\tilde{\bf u}})$, where
$(\tilde{\varrho},{\tilde{\bf u}})$ is a strong solution. Using the standard
regularization by a convolution kernel in time, we first observe that the
class of test functions can be extended to

$\begin{split}{\tilde{\bf u}}&\in
C_{c}([0,T]\times\overline{\Omega};R^{d}),{\tilde{\bf
u}}|_{\partial\Omega}={\bf u}_{B},\ \partial_{t}{\tilde{\bf u}}\in
L^{2}(0,T;C_{c}(\overline{\Omega})),\ \nabla_{x}{\tilde{\bf u}}\in
L^{2}(0,T;C_{c}(\overline{\Omega};R^{d\times d}))\\\
(\tilde{\varrho}-\varepsilon)&\in C_{c}([0,T]\times\overline{\Omega}),\
\tilde{\varrho}>0,\ \partial_{t}\tilde{\varrho}\in L^{\infty}_{\rm
weak-(*)}(0,T;C_{c}(\overline{\Omega})),\ \nabla_{x}\tilde{\varrho}\in
L^{\infty}_{\rm weak-(*)}(0,T;C_{c}(\overline{\Omega};R^{d}))\end{split}$
(3.3)

for some $\varepsilon>0$.

The class (3.3) covers all interesting cases if $\Omega$ is bounded. If
$\Omega$ is an exterior domain or $\Omega=R^{d}$, integrability for
$|x|\to\infty$ is relevant. In this case, we consider $d=3$ only, where we
consider the class:

$\begin{split}{\tilde{\bf u}}&\in
BC([0,T]\times\overline{\Omega};R^{3}),{\tilde{\bf u}}|_{\partial\Omega}={\bf
u}_{B},\ \nabla_{x}{\tilde{\bf u}}\in L^{2}(0,T;BC\cap
L^{2}(\overline{\Omega};R^{d\times d})),\\\ \partial_{t}{\tilde{\bf u}}&\in
L^{2}(0,T;BC\cap L^{6}(\overline{\Omega};R^{3})),\\\ \tilde{\varrho}&\in
BC([0,T]\times\overline{\Omega}),\ \inf\tilde{\varrho}>0,\
\nabla_{x}\tilde{\varrho}\in L^{\infty}_{\rm weak-(*)}(0,T;BC\cap
L^{6}(\overline{\Omega};R^{3}))\\\ \partial_{t}\tilde{\varrho}&\in
L^{\infty}_{\rm weak-(*)}(0,T;BC\cap L^{6}(\overline{\Omega})).\end{split}$
(3.4)

The extension to this class can be justified by the standard cut-off procedure
as long as all integrals in (3.1) are finite. Extensions to other classes of
test functions can be obtained in a similar manner as the case may be.

##  4 Weak strong uniqueness

We are ready to establish the main results. We distinguish the cases of
$\Omega$ bounded, unbounded, or $\Omega=R^{3}$. For the sake of simplicity, we
suppose here and hereafter that

${\bf f}=0.$

###  4.1 Weak–strong uniqueness–bounded domain

We start with our main result concerning bounded domains, where inflow is
absent.

######  Theorem 4.1 (Weak–strong uniqueness, bounded domain, no inflow).

Suppose that $\Omega\subset R^{d}$, $d=1,2,3$ is a bounded Lipschitz domain.
Let the boundary velocity ${\bf u}_{B}$ be a twice continuously differentiable
function satisfying

${\bf u}_{B}\cdot{\bf n}\geq 0\ \mbox{on}\ \partial\Omega.$

Let $(\tilde{\varrho},{\tilde{\bf u}})$ be a strong solution of the problem
(1.1), (1.2), (1.7) in $(0,T)\times\Omega$ in the sense specified in Section
2.1 (see (2.9),(2.10)) belonging to the class

$\begin{split}{\tilde{\bf u}}&\in C([0,T]\times\overline{\Omega};R^{d}),\
\tilde{\varrho}\in C([0,T]\times{\Omega}),\ \tilde{\varrho}\geq 0,\\\
\nabla^{2}_{x}{\tilde{\bf u}}&\in L^{2}(0,T;L^{q}(\Omega;R^{d\times d\times
d})),\ \partial_{t}{\tilde{\bf u}}\in
L^{2}(0,T;L^{q}(\Omega;R^{d})),\end{split}$ (4.1)

where

$1<\gamma\leq 2,\ q\geq\frac{2\gamma}{\gamma-1}.$ (4.2)

Let $(\varrho,{\bf u})$ be a weak solution of the same problem in the sense of
Definition 2.2 such that

$\varrho(0,\cdot)=\tilde{\varrho}(0,\cdot)=\varrho_{0},\ (\varrho{\bf
u})(0,\cdot)=(\tilde{\varrho}{\tilde{\bf u}})(0,\cdot)={\bf m}_{0},{\bf
u}|_{\partial\Omega}={\tilde{\bf u}}|_{\partial\Omega}={\bf u}_{B},$

where

$(\varrho_{0})^{\gamma-1}\in W^{1,q}(\Omega;R^{d}).$ (4.3)

Then

$\varrho=\tilde{\varrho},\ {\bf u}={\tilde{\bf u}}\ \mbox{in}\
(0,T)\times\Omega.$

######  Remark 4.2.

In the Newtonian case, the local in time existence of strong solutions with
vacuum in the case ${\bf u}_{B}=0$ (no in/outflow) on smooth bounded domains,
say $\Omega\in C^{3}$, with the pressure law (2.2) was proved in Cho, Kim [4,
Theorem 3]. The solutions belong to a regularity class included in (2.4) if
$q=6$. Local existence for general non-Newtonian viscous stress was recently
established by Kalousek, Mácha, and Nečasová [10] on periodic spatial domains.

###### Proof.

The strategy is to apply the relative energy inequality (3.1) to
$(\tilde{\varrho},{\tilde{\bf u}})$. This cannot be done in a direct manner as
the latter requires $\inf\tilde{\varrho}>0$. Thus we start with the choice
$(\tilde{\varrho}+\varepsilon,{\tilde{\bf u}})$, $\varepsilon>0$, obtaining

$\begin{split}&\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho}+\varepsilon,{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(\tau)\\\
&+\int_{0}^{\tau}\int_{\Omega}\Big{[}F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}\mathbb{S}:\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\ &\leq\int_{\Omega}P(\varrho_{0})+\varepsilon
P^{\prime}(\varrho_{0})-P(\varrho_{0}+\varepsilon)\ \,{\rm d}{x}\\\
&-\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot({\tilde{\bf u}}-{\bf u})\cdot\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\
&-\int_{0}^{\tau}\int_{\Omega}\Big{[}p(\varrho)-p^{\prime}(\tilde{\varrho}+\varepsilon)(\varrho-\tilde{\varrho}-\varepsilon)-p(\tilde{\varrho}+\varepsilon)\Big{]}{\rm
div}_{x}{\tilde{\bf u}}\ \,{\rm d}{x}\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho}+\varepsilon)\Big{]}\ \,{\rm
d}{x}\,{\rm d}t\\\
&+\varepsilon\int_{0}^{\tau}\int_{\Omega}p^{\prime}(\tilde{\varrho}+\varepsilon)\left(1-\frac{\varrho}{\tilde{\varrho}+\varepsilon}\right){\rm
div}_{x}{\tilde{\bf u}}\ \,{\rm d}{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\overline{\Omega}}\nabla_{x}{\tilde{\bf u}}:{\rm d}\
\mathfrak{R}(t)\ \,{\rm d}t\ \mbox{for a.a.}\ \tau\in(0,T),\end{split}$

Moreover, as ${\tilde{\bf u}}$ belongs to the class (4.1) and $q\geq 4$, we
get $\|\nabla_{x}{\tilde{\bf u}}\|_{L^{\infty}(\Omega;R^{d\times d})}\in
L^{2}(0,T)$, and the above inequality reduces to

$\begin{split}&\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho}+\varepsilon,{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(\tau)\\\
&+\int_{0}^{\tau}\int_{\Omega}\Big{[}F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}\mathbb{S}:\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\ &\leq\int_{\Omega}P(\varrho_{0})+\varepsilon
P^{\prime}(\varrho_{0})-P(\varrho_{0}+\varepsilon)\ \,{\rm
d}{x}+\int_{0}^{\tau}\chi(t)\left[\int_{\Omega}E\left(\varrho,{\bf u}\
\Big{|}\ \tilde{\varrho}+\varepsilon,{\tilde{\bf u}}\right)(t,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(t)\right]\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho}+\varepsilon)\Big{]}\ \,{\rm
d}{x}\,{\rm d}t\\\
&+\varepsilon\int_{0}^{\tau}\int_{\Omega}p^{\prime}(\tilde{\varrho}+\varepsilon)\left(1-\frac{\varrho}{\tilde{\varrho}+\varepsilon}\right){\rm
div}_{x}{\tilde{\bf u}}\ \,{\rm d}{x}\,{\rm d}t,\ \mbox{with}\ \chi\in
L^{2}(0,T).\end{split}$ (4.4)

Next, we let $\varepsilon\to 0$ in (4.4). Obviously,

$\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho}+\varepsilon,{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm
d}{x}\to\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm d}{x}\ \mbox{as}\
\varepsilon\to 0\ \mbox{uniformly in}\ \tau\in(0,T),$
$\int_{\Omega}P(\varrho_{0})+\varepsilon
P^{\prime}(\varrho_{0})-P(\varrho_{0}+\varepsilon)\ \,{\rm d}{x}\to 0\
\mbox{as}\ \varepsilon\to 0,$

and

$\varepsilon\int_{0}^{\tau}\int_{\Omega}p^{\prime}(\tilde{\varrho}+\varepsilon){\rm
div}_{x}{\tilde{\bf u}}\ \,{\rm d}{x}\to 0\ \mbox{as}\ \varepsilon\to 0.$

In addition,

$\varepsilon\varrho\frac{p^{\prime}(\tilde{\varrho}+\varepsilon)}{\tilde{\varrho}+\varepsilon}\stackrel{{\scriptstyle<}}{{\sim}}\varepsilon\varrho(\tilde{\varrho}+\varepsilon)^{\gamma-2}\leq\varepsilon^{\gamma-1}\varrho\to
0\ \mbox{in}\ L^{\gamma}(\Omega)\ \mbox{as}\ \varepsilon\to 0.$

Finally, by virtue of Lemma 2.1 and hypotheses (4.1), (4.3), we have

$\nabla_{x}(\tilde{\varrho})^{\gamma-1}\in
L^{\infty}(0,T;L^{q}(\Omega;R^{d})).$ (4.5)

Seeing that

$\varrho{\bf u}\in L^{\infty}(0,T;L^{\frac{2\gamma}{\gamma+1}}(\Omega;R^{3}))$

we may use hypothesis (4.2) to conclude

$\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\nabla_{x}P^{\prime}(\tilde{\varrho}+\varepsilon)\ \,{\rm d}{x}\,{\rm
d}t\to\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\nabla_{x}P^{\prime}(\tilde{\varrho})\ \,{\rm d}{x}\,{\rm d}t\
\mbox{as}\ \varepsilon\to 0.$

Performing the limit $\varepsilon\to 0$ in (4.1) we may infer that

$\begin{split}&\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(\tau)\\\
&+\int_{0}^{\tau}\int_{\Omega}\Big{[}F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}\mathbb{S}:\nabla_{x}{\tilde{\bf u}}\ \,{\rm
d}{x}\,{\rm d}t\\\
&\leq\int_{0}^{\tau}\chi(t)\left[\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)(t,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(t)\right]\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}\varrho({\tilde{\bf u}}-{\bf
u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t.\end{split}$ (4.6)

Now, as $(\tilde{\varrho},{\tilde{\bf u}})$ is a strong solution, we have

$\tilde{\varrho}\partial_{t}{\tilde{\bf u}}+\tilde{\varrho}{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\tilde{\varrho}\nabla_{x}P^{\prime}(\tilde{\varrho})={\rm
div}_{x}\widetilde{\mathbb{S}}.$ (4.7)

Consequently, we may add the integral

$\int_{0}^{\tau}\int_{\Omega}({\bf u}-{\tilde{\bf u}})\cdot{\rm
div}_{x}\widetilde{\mathbb{S}}\ \,{\rm d}{x}\,{\rm
d}t=\int_{0}^{\tau}\int_{\Omega}\mathbb{D}_{x}({\tilde{\bf u}}-{\bf
u}):\widetilde{\mathbb{S}}\ \,{\rm d}{x}\,{\rm d}t$

to both sides of the inequality (4.6). Regrouping terms on the left–hand side
we get

$\begin{split}\int_{0}^{\tau}&\int_{\Omega}\left[F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})-\mathbb{S}:\mathbb{D}_{x}{\tilde{\bf
u}}+\widetilde{\mathbb{S}}:(\mathbb{D}_{x}{\tilde{\bf u}}-\mathbb{D}_{x}{\bf
u})\right]\ \,{\rm d}{x}\,{\rm d}t\\\
&=\int_{0}^{\tau}\int_{\Omega}\left[F(\mathbb{D}_{x}{\bf
u})-F(\mathbb{D}_{x}{\tilde{\bf
u}})-\widetilde{\mathbb{S}}:(\mathbb{D}_{x}{\bf u}-\mathbb{D}_{x}{\tilde{\bf
u}})\right]\ \,{\rm d}{x},\end{split}$

where we have used Fenchel–Young inequality

$F(\mathbb{D}_{x}{\tilde{\bf
u}})+F^{*}(\mathbb{S})\geq\mathbb{S}:\mathbb{D}_{x}{\tilde{\bf u}}.$

As $\widetilde{\mathbb{S}}\in\partial F(\mathbb{D}_{x}{\tilde{\bf u}})$ we can
apply hypothesis (2.4) for $\mathbb{D}=\mathbb{D}_{x}{\tilde{\bf u}}$,
$\mathbb{S}=\widetilde{\mathbb{S}}$, $\mathbb{Q}=\mathbb{D}_{x}{\bf
u}-\mathbb{D}_{x}{\tilde{\bf u}}$ to obtain

$\begin{split}\int_{0}^{\tau}&\int_{\Omega}\left|(\mathbb{D}_{x}{\bf
u}-\mathbb{D}_{x}{\tilde{\bf u}})-\frac{1}{d}{\rm tr}[(\mathbb{D}_{x}{\bf
u}-\mathbb{D}_{x}{\tilde{\bf u}})]\mathbb{I}\right|^{2}\ \,{\rm d}{x}\\\
&\stackrel{{\scriptstyle<}}{{\sim}}\int_{0}^{\tau}\int_{\Omega}\left[F(\mathbb{D}_{x}{\bf
u})+F^{*}(\mathbb{S})-\mathbb{S}:\mathbb{D}_{x}{\tilde{\bf
u}}+\widetilde{\mathbb{S}}:(\mathbb{D}_{x}{\tilde{\bf u}}-\mathbb{D}_{x}{\bf
u})\right]\ \,{\rm d}{x}\,{\rm d}t.\end{split}$

Finally, using the traceless version of Korn’s inequality, we may rewrite
(4.6) in the form

$\begin{split}\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)(\tau,\cdot)\ \,{\rm
d}{x}&+\int_{\overline{\Omega}}1{\rm d}\
\mathfrak{E}(\tau)+\int_{0}^{\tau}\int_{\Omega}|\nabla_{x}{\bf
u}-\nabla_{x}{\tilde{\bf u}}|^{2}\ \,{\rm d}{x}\,{\rm d}t\\\
&\stackrel{{\scriptstyle<}}{{\sim}}\int_{0}^{\tau}\chi(t)\left[\int_{\Omega}E\left(\varrho,{\bf
u}\ \Big{|}\ \tilde{\varrho},{\tilde{\bf u}}\right)(t,\cdot)\ \,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(t)\right]\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}(\varrho-\tilde{\varrho})({\tilde{\bf u}}-{\bf
u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{]}\ \,{\rm d}{x}\,{\rm
d}t.\end{split}$ (4.8)

Thus it remains to control the last integral in (4.8). Let
$\overline{\varrho}$ be a positive constant chosen so that
$\tilde{\varrho}\leq\frac{1}{2}\overline{\varrho}$ in $(0,T)\times\Omega$. We
consider two complementary cases. Suppose first that
$\varrho\geq\overline{\varrho}$. By virtue of hypotheses (4.1) – (4.3), we get

$\begin{split}\int_{\Omega}&1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{]}\,{\rm d}{x}\\\
&\leq\left\|1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})^{1/2}\right\|_{L^{2\gamma}(\Omega)}\left\|1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})^{1/2}({\bf
u}-{\tilde{\bf
u}})\right\|_{L^{2}(\Omega;R^{d})}\left\|\partial_{t}{\tilde{\bf
u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\right\|_{L^{q}(\Omega;R^{d})}\\\
&\stackrel{{\scriptstyle<}}{{\sim}}\left\|\partial_{t}{\tilde{\bf
u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\right\|_{L^{q}(\Omega;R^{d})}\int_{\Omega}E\left(\varrho,{\bf
u}\Big{|}\tilde{\varrho},{\tilde{\bf u}}\right)\ \,{\rm d}{x}.\end{split}$
(4.9)

If $0\leq\varrho\leq\overline{\varrho}$, then

$\begin{split}\int_{\Omega}&1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot\Big{[}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{]}\,{\rm d}{x}\\\
&\leq\left\|1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})\right\|_{L^{2}(\Omega)}\left\|{\bf
u}-{\tilde{\bf
u}}\right\|_{L^{2\gamma}(\Omega;R^{d})}\left\|\partial_{t}{\tilde{\bf
u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\right\|_{L^{q}(\Omega;R^{d})}\\\
&\leq\delta\|{\bf u}-{\tilde{\bf
u}}\|^{2}_{L^{2\gamma}(\Omega;R^{d})}+c(\delta)\left\|1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})\right\|_{L^{2}(\Omega)}^{2}\left\|\partial_{t}{\tilde{\bf
u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\right\|_{L^{q}(\Omega;R^{d})}^{2}.\end{split}$
(4.10)

for any $\delta>0$. As $\gamma\leq 2$, the standard Poincaré–Sobolev
inequality yields

$\delta\|{\bf u}-{\tilde{\bf
u}}\|^{2}_{L^{2\gamma}(\Omega;R^{d})}\leq\frac{1}{2}\int_{\Omega}|\nabla_{x}{\bf
u}-\nabla_{x}{\tilde{\bf u}}|^{2}\ \,{\rm d}{x}$ (4.11)

for a suitably small $\delta>0$. By the same token $P$ is strictly convex on
the compact interval $[0,\overline{\varrho}]$; whence

$\left\|1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})\right\|_{L^{2}(\Omega)}^{2}\stackrel{{\scriptstyle<}}{{\sim}}\int_{\Omega}E\left(\varrho,{\bf
u}\Big{|}\tilde{\varrho},{\tilde{\bf u}}\right)\ \,{\rm d}{x}.$ (4.12)

In view of hypothesis (4.1), and (4.5), we have

$\left\|\partial_{t}{\tilde{\bf u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\right\|_{L^{q}(\Omega;R^{d})}\in
L^{2}(0,T).$

Thus plugging (4.9)–(4.12) in (4.8) we may use the standard Gronwall argument
to conclude

$\int_{\Omega}E\left(\varrho,{\bf u}\Big{|}\tilde{\varrho},{\tilde{\bf
u}}\right)(\tau,\cdot)\ \,{\rm d}{x}=0,\ \tau\in(0,T),\
\int_{0}^{T}\int_{\Omega}|\nabla_{x}{\bf u}-\nabla_{x}{\tilde{\bf u}}|^{2}\
\,{\rm d}{x}\,{\rm d}t=0.$

∎

The extension of the above result to more general viscous potentials still
satisfying the coercivity condition (2.4) is straightforward, see [1]. The
major drawback of Theorem 4.1 is the absence of inflow. If $\Gamma_{\rm
in}\neq\emptyset$, a slight modification of the above arguments yields a
positive result provided the boundary density $\varrho_{B}$ is bounded below
away from vacuum.

######  Theorem 4.3 (Weak–strong uniqueness, bounded domain, int/out flow).

Suppose that $\Omega\subset R^{d}$, $d=1,2,3$ is a bounded Lipschitz domain.
Let the boundary conditions be determined by $\varrho_{B}\in
C^{1}_{c}(R^{d})$, ${\bf u}_{B}\in C^{2}_{c}(R^{d};R^{d})$, where

$\varrho_{B}(x)\geq\underline{\varrho}>0\ \mbox{for any}\ x\in\Gamma_{\rm
in}.$ (4.13)

Let $(\tilde{\varrho},{\tilde{\bf u}})$ be a strong solution of the problem
(1.1), (1.2), (1.7) in $(0,T)\times\Omega$ in the sense specified in Section
2.1 belonging to the class

$\begin{split}{\tilde{\bf u}}&\in C([0,T]\times\overline{\Omega};R^{d}),\
\tilde{\varrho}\in C([0,T];W^{1,q}(\Omega)),\ \tilde{\varrho}\geq 0,\\\
\nabla^{2}_{x}{\tilde{\bf u}}&\in L^{2}(0,T;L^{q}(\Omega;R^{d\times d\times
d})),\ \partial_{t}{\tilde{\bf u}}\in
L^{2}(0,T;L^{q}(\Omega;R^{d})),\end{split}$ (4.14)

where

$1<\gamma\leq 2,\ q\geq\frac{2\gamma}{\gamma-1}.$

Let $(\varrho,{\bf u})$ be a weak solution of the same problem in the sense of
Definition 2.2 such that

$\begin{split}\varrho(0,\cdot)&=\tilde{\varrho}(0,\cdot)=\varrho_{0},\
(\varrho{\bf u})(0,\cdot)=(\tilde{\varrho}{\tilde{\bf u}})(0,\cdot)={\bf
m}_{0},\\\ {\bf u}|_{\partial\Omega}&={\tilde{\bf u}}|_{\partial\Omega}={\bf
u}_{B},\ (\varrho{\bf u})\cdot{\bf n}|_{\Gamma_{\rm
in}}=(\tilde{\varrho}{\tilde{\bf u}})\cdot{\bf n}|_{\Gamma{\rm
in}}=\varrho_{B}{\bf u}_{B}\cdot{\bf n},\end{split}$

where

$(\varrho_{0})^{\gamma-1}\in W^{1,q}(\Omega;R^{d}).$ (4.15)

Then

$\varrho=\tilde{\varrho},\ {\bf u}={\tilde{\bf u}}\ \mbox{in}\
(0,T)\times\Omega.$

######  Remark 4.4.

The class (4.14) is slightly smaller than (4.1) in Theorem 4.1 but still large
enough to accommodate the strong solutions obtained in the Newtonian
non–degenerate case by Valli and Zajaczkowski [16]. Note, however, that quite
severe restrictions are imposed on $\Gamma_{\rm in}$ in [16].

###### Proof.

The proof can be done following the same arguments as in Theorem 4.1 as soon
as we observe that

$\tilde{\varrho}^{\gamma-1}\in L^{\infty}(0,T;W^{1,q}(\Omega)).$

To see this, write

$\tilde{\varrho}^{\gamma-1}=r_{1}+r_{2},\
r_{1}=\delta+[\tilde{\varrho}^{\gamma-1}-\delta]^{+},\
r_{2}=[\tilde{\varrho}^{\gamma-1}-\delta]^{-}\ \mbox{for a suitable}\
\delta>0,$

where $a^{+}=\max\\{a,0\\}$, $a^{-}=\min\\{a,0\\}$. Consequently, it is enough
to show

$\nabla_{x}r_{i}\in L^{\infty}(0,T;L^{q}(\Omega;R^{d})),\ i=1,2.$

As for $r_{1}$, we have

$\nabla_{x}r_{1}=\nabla_{x}[\tilde{\varrho}^{\gamma-1}-\delta]^{+}=(\gamma-1){\rm
sgn}^{+}[\tilde{\varrho}^{\gamma-1}-\delta]\tilde{\varrho}^{\gamma-2}\nabla_{x}\tilde{\varrho},$

so the desired conclusion follows from the hypothesis (4.14) as long as
$\delta>0$.

To obtain a similar estimate for $r_{2}$, we first extend the velocity field
${\tilde{\bf u}}$ and the initial density $\varrho_{0}$ outside $\Omega$ so
that they satisfy the hypotheses (4.14), (4.15) on $R^{d}$. Accordingly, the
equation of continuity (1.1) endowed with the extended velocity field
${\tilde{\bf u}}$ admits a solution $r$ determined uniquely through formula
(2.7). By virtue of Lemma 2.1, we have

$r^{\gamma-1}\in L^{\infty}(0,T;W^{1,q}(R^{d})).$ (4.16)

We say that a point $(t,x)\in[0,T]\times\Omega$ is regular, if the value of
$\tilde{\varrho}(t,x)$ is determined by formula (2.7), otherwise $(t,x)$ is
singular. Clearly the backward characteristic curve emanating from a singular
point reaches the boundary $\partial\Omega$ at a positive time, and
$\tilde{\varrho}(t,x)$ is then given by (2.8). Obviously,

$\tilde{\varrho}(t,x)=r(t,x)\ \mbox{whenever}\ (t,x)\ \mbox{is regular}.$

As stated in (4.13), $\varrho_{B}$ is bounded below away from zero.
Consequently, there exists $\varepsilon=\varepsilon(T,{\tilde{\bf u}})>0$ such
that

$\tilde{\varrho}^{\gamma-1}(t,x)\geq\varepsilon>0\ \mbox{whenever}\ (t,x)\
\mbox{is singular}.$

Consequently, for $0<\delta<\frac{\varepsilon}{2}$ and any fixed $t\in[0,T]$,
the set $\\{x\in\Omega\ \Big{|}\ \tilde{\varrho}^{\gamma-1}(t,x)\leq\delta\\}$
admits and open neighbourhood in $\Omega$ that consists of regular points.
Consequently,

$\nabla_{x}r_{2}=\nabla_{x}[\tilde{\varrho}^{\gamma-1}-\delta]^{-}=(\gamma-1){\rm
sgn}^{-}[\tilde{\varrho}^{\gamma-1}-\delta]\nabla_{x}r^{\gamma-1},$

and the desired conclusion follows from (4.16).

∎

##  5 Weak–strong uniqueness, unbounded domains

The main difficulty when extending the previous results to unbounded domains
is the lack of Poincaré–Sobolev inequality. For this reason, we consider only
the case $d=3$, where Sobolev’s inequality is available, namely

$\|v\|_{L^{6}(\Omega)}\stackrel{{\scriptstyle<}}{{\sim}}\|\nabla_{x}v\|_{L^{2}(\Omega,R^{3})}\
\mbox{for any}\ v\in D^{1,2}_{0}(\Omega).$ (5.1)

For simplicity, we restrict ourselves to the case ${\bf u}_{B}=0$.

###  5.1 General exterior domain

We look for strong solutions $(\tilde{\varrho},{\tilde{\bf u}})$ belonging to
the class introduced by Huang et al. [9]. In particular,

$\begin{split}{\tilde{\bf u}}&\in C([0,T];D^{1,2}_{0}\cap
D^{3,2}(\Omega;R^{3}))\cap L^{2}(0,T;D^{4,2}(\Omega;R^{3})),\\\
\partial_{t}{\tilde{\bf u}}&\in L^{\infty}(0,T;D^{1,2}_{0}(\Omega,R^{3}))\cap
L^{2}(0,T;D^{2,2}(\Omega;R^{3})),\\\ \tilde{\varrho},\ p(\tilde{\varrho})&\in
C([0,T];W^{3,2}(\Omega)).\end{split}$ (5.2)

We point out that the class is pertinent to the Newtonian case, where the
momentum equation can be differentiated with respect to time to obtain higher
order estimates on $\partial_{t}{\tilde{\bf u}}$.

In addition, similarly to Theorem 4.1, we impose the condition

$(\varrho_{0})^{\gamma-1}\in W^{1,6}\cap W^{1,\infty}(\Omega;R^{3}).$ (5.3)

As shown in Lemma 2.1, this regularity will propagate in time; whence we get

$\sup_{t\in(0,T)}\|\nabla_{x}(\tilde{\varrho})^{\gamma-1}(t,\cdot)\|_{L^{6}\cap
L^{\infty}(\Omega;R^{3})}\leq c.$ (5.4)

Now, following step by step the proof of Theorem 4.1 we arrive at the
inequality (4.8), specifically,

$\begin{split}\int_{\Omega}E\left(\varrho,{\bf u}\ \Big{|}\
\tilde{\varrho},{\tilde{\bf u}}\right)(\tau,\cdot)\,{\rm
d}{x}&+\int_{\Omega}1{\rm d}\
\mathfrak{E}(\tau)+\int_{0}^{\tau}\int_{\Omega}{|\nabla_{x}{\bf
u}-\nabla_{x}{\tilde{\bf u}}|^{2}}\,{\rm d}{x}\,{\rm d}t\\\
&\stackrel{{\scriptstyle<}}{{\sim}}\int_{0}^{\tau}\chi(t)\left[\int_{\Omega}{E\left(\varrho,{\bf
u}\ \Big{|}\ \tilde{\varrho},{\tilde{\bf u}}\right)(t,\cdot)}\,{\rm
d}{x}+\int_{\overline{\Omega}}1{\rm d}\ \mathfrak{E}(t)\right]\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Omega}{(\varrho-\tilde{\varrho})({\tilde{\bf u}}-{\bf
u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t,\end{split}$ (5.5)

where, by virtue of (5.2), (5.4),

${\bf b}=\partial_{t}{\tilde{\bf u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\in L^{2}(0,T;L^{6}\cap
L^{\infty}(\Omega;R^{3})).$ (5.6)

Similarly to the preceding section, we write

$\begin{split}\int_{0}^{\tau}&\int_{\Omega}{(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t\\\
&=\int_{0}^{\tau}\int_{\Omega}{1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}{1_{\varrho<\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t,\end{split}$

where, exactly as in (4.9),

$\begin{split}\int_{0}^{\tau}&\int_{\Omega}{1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t\\\
&\leq\left\|1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})^{1/2}\right\|_{L^{2}(\Omega)}\left\|1_{\varrho\geq\overline{\varrho}}(\varrho-\tilde{\varrho})^{1/2}({\bf
u}-{\tilde{\bf u}})\right\|_{L^{2}(\Omega;R^{3})}\left\|{\bf
b}\right\|_{L^{\infty}(\Omega;R^{3})}\\\
&\stackrel{{\scriptstyle<}}{{\sim}}\left\|{\bf
b}\right\|_{L^{\infty}(\Omega;R^{3})}\int_{\Omega}E\left(\varrho,{\bf
u}\Big{|}\tilde{\varrho},{\tilde{\bf u}}\right)\ \,{\rm d}{x}.\end{split}$
(5.7)

The integral

$\int_{0}^{\tau}\int_{\Omega}{1_{\varrho<\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t$ (5.8)

is more difficult to handle. Indeed, in order to mimick the arguments leading
to (4.10), we would need

${\bf b}\ \mbox{bounded in}\ L^{2}(0,T;L^{3}(\Omega;R^{3})),$

which does not follow from (5.2) if $\Omega$ is an exterior domain and must be
imposed as an extra hypothesis.

######  Theorem 5.1 (Weak–strong uniqueness, unbounded domains).

Suppose that $\Omega\subset R^{3}$, $d=1,2,3$ is an exterior domain with
compact Lipschitz boundary. Let $(\tilde{\varrho},{\tilde{\bf u}})$ be a
strong solution of the problem (1.1), (1.2), (1.7) with ${\bf u}_{B}=0$ in
$(0,T)\times\Omega$ in the sense specified in Section 2.1 belonging to the
class (5.2). In addition, suppose that

$\partial_{t}{\tilde{\bf u}}\in L^{2}(0,T;L^{3}(\Omega;R^{3})).$ (5.9)

Let $(\varrho,{\bf u})$ be a weak solution of the same problem in the sense of
Definition 2.2 such that

$\varrho(0,\cdot)=\tilde{\varrho}(0,\cdot)=\varrho_{0},\ (\varrho{\bf
u})(0,\cdot)=(\tilde{\varrho}{\tilde{\bf u}})(0,\cdot)={\bf m}_{0},$

where

$(\varrho_{0})^{\gamma-1}\in W^{1,3}\cap W^{1,\infty}(\Omega;R^{3}).$ (5.10)

Then

$\varrho=\tilde{\varrho},\ {\bf u}={\tilde{\bf u}}\ \mbox{in}\
(0,T)\times\Omega.$

The result extends easily to general in/out flow boundary conditions exactly
as in Theorem 4.3.

###  5.2 Compactly supported density

In view of the existence class (5.2) identified in [9], the hypothesis (5.9)
does not seem very realistic, in particular in the presence of vacuum. To
handle the general case, we restrict ourselves to the initial data with
_compactly supported_ initial density $\varrho_{0}$ considered in [9] or [17].
Moreover, we restrict ourselves to the class of Newtonian fluids, where

$\mathbb{S}(\nabla_{x}{\tilde{\bf u}})=\mu\left(\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}^{t}{\tilde{\bf u}}-\frac{2}{3}{\rm div}_{x}{\tilde{\bf
u}}\mathbb{I}\right)+\lambda{\rm div}_{x}{\tilde{\bf u}}\mathbb{I},\ \mu>0,\
\lambda\geq 0.$

In view of Lemma 2.1, there exists $R>0$ sufficiently large such that

$\tilde{\varrho}(t,x)=0\ \mbox{for all}\ t\in[0,T],\ |x|\geq R.$ (5.11)

Going back to the integral (5.8), we get

$\begin{split}\int_{0}^{\tau}&\int_{\Omega}{1_{\varrho<\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t\\\
&=\int_{0}^{\tau}\int_{x\in\Omega,|x|\leq
R}{1_{\varrho<\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{|x|>R}{1_{\varrho<\overline{\varrho}}\varrho({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm d}{x}\,{\rm d}t,\end{split}$

where

$\int_{x\in\Omega,|x|\leq
R}{1_{\varrho<\overline{\varrho}}(\varrho-\tilde{\varrho})({\tilde{\bf
u}}-{\bf u})\cdot{\bf b}}\,{\rm
d}{x}\leq\left\|1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})\right\|_{L^{2}(\Omega)}\|{\bf
u}-{\tilde{\bf u}}\|_{L^{6}(\Omega;R^{3})}\|{\bf
b}\|_{L^{3}(\Omega\cap\\{|x|\leq R\\})}.$

Seeing that

$\|{\bf b}\|_{L^{3}(\Omega\cap\\{|x|\leq R;R^{3}\\})}\leq c(R)\|{\bf
b}\|_{L^{6}(\Omega;R^{3})}$

the above integral is controlled by (5.6).

Finally,

$\int_{|x|>R}{1_{\varrho<\overline{\varrho}}\varrho({\tilde{\bf u}}-{\bf
u})\cdot{\bf b}}\,{\rm d}{x}\leq\delta\|{\bf u}-{\tilde{\bf
u}}\|^{2}_{L^{6}(\Omega;R^{3})}+c(\delta)\|1_{\varrho\leq\overline{\varrho}}\varrho\|^{2}_{L^{\gamma}(\Omega)}\|{\bf
b}\|_{L^{q}(|x|\geq R)}^{2}$ (5.12)

for any $\delta>0$, where

$\frac{1}{q}=\frac{5}{6}-\frac{1}{\gamma}.$

Suppose that $1<\gamma\leq 2$. Consequently,

$\|1_{\varrho\leq\overline{\varrho}}\varrho\|^{2}_{L^{\gamma}(\Omega)}\leq
c(\overline{\varrho})\|1_{\varrho\leq\overline{\varrho}}\varrho\|^{\gamma}_{L^{\gamma}(\Omega)}\stackrel{{\scriptstyle<}}{{\sim}}\int_{\Omega}E\left(\varrho,{\bf
u}\ \Big{|}\tilde{\varrho},{\tilde{\bf u}}\right)\,{\rm d}{x}.$

Thus, to close the estimates, we need to control

$\int_{0}^{T}\|{\bf b}\|_{L^{q}(|x|\geq R)}^{2}\,{\rm d}t\leq c(q)\ \mbox{for
any}\ q>3.$ (5.13)

To see (5.13), we realize that

${\bf b}=\partial_{t}{\tilde{\bf u}}+{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}\ \mbox{if}\ |x|>R.$

If the fluid is Newtonian, we get immediately from (4.7) that

$\mu{\rm div}_{x}\left(\nabla_{x}\partial_{t}{\tilde{\bf
u}}+\nabla_{x}^{t}\partial_{t}{\tilde{\bf u}}-\frac{2}{3}{\rm
div}_{x}\partial_{t}{\tilde{\bf u}}\mathbb{I}\right)+\lambda\nabla_{x}{\rm
div}_{x}\partial_{t}{\tilde{\bf u}}=0\ \mbox{for}\ |x|>R,$

where, in view of (5.2)

$\partial_{t}{\tilde{\bf u}}\in D^{1,2}\cap L^{6}(|x|>R),\
\partial_{t}{\tilde{\bf u}}\in C^{\alpha}(|x|=R)\ \mbox{for some}\ \alpha>0\
\mbox{and a.a.}\ t\in(0,T).$

Using the standard elliptic estimates, we get

$|\partial_{t}{\tilde{\bf
u}}(t,x)|\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{|x|}\|\partial_{t}{\tilde{\bf
u}}(t,\cdot)\|_{C(|x|=R)}\ \mbox{for all}\ |x|\geq R,$

which, together with (5.2), yields (5.13). Thus we are able to control the
integral (5.12) as soon as $1<\gamma<2$.

We have shown the following result.

######  Theorem 5.2 (Weak–strong uniqueness, compactly supported density).

Suppose that $\Omega\subset R^{3}$ is a Lipschitz exterior domain, ${\bf
u}_{B}=0$, and $1<\gamma<2$. In addition, let $\mathbb{S}$ be Newtonian,

$\mathbb{S}=\mu\left(\nabla_{x}{\bf u}+\nabla_{x}^{t}{\bf u}-\frac{2}{3}{\rm
div}_{x}{\bf u}\mathbb{I}\right)+\lambda{\rm div}_{x}{\bf u}\mathbb{I},\
\mu>0,\ \lambda\geq 0.$ (5.14)

Let $(\tilde{\varrho},{\tilde{\bf u}})$ be a strong solution of the problem
(1.1), (1.2) in $(0,T)\times\Omega$ in the sense specified in Section 2.1
belonging to the class (5.2). Let $(\varrho,{\bf u})$ be a weak solution of
the same problem in the sense of Definition 2.2 such that

$\varrho(0,\cdot)=\tilde{\varrho}(0,\cdot)=\varrho_{0},\ (\varrho{\bf
u})(0,\cdot)=(\tilde{\varrho}{\tilde{\bf u}})(0,\cdot)={\bf m}_{0},$

where

$\varrho_{0}\in C_{c}(\overline{\Omega}),\ (\varrho_{0})^{\gamma-1}\in
W^{1,6}\cap W^{1,\infty}(\Omega;R^{3}).$

Then

$\varrho=\tilde{\varrho},\ {\bf u}={\tilde{\bf u}}\ \mbox{in}\
(0,T)\times\Omega.$

######  Remark 5.3.

Local existence of strong solutions as well as global existence for small
initial data in the class included in (5.2) was proved by Huang et al. [9,
Lemma 2.1], see also Cho and Kim [4, Theorem 3] for local existence in the
class of more regular solution.

The result can be extended to the case of general in/out flow boundary
conditions in the spirit of Theorem 4.3. The hypothesis of Newtonian viscous
stress is however necessary.

###  5.3 Positive density

Finally, we consider the physically relevant case $\Omega=R^{3}$,

$\begin{split}0<\varrho_{0}(x)&\leq\overline{\varrho},\\\
|\nabla_{x}(\varrho_{0})^{\gamma-1}(x)|&\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha}},\
\alpha=\alpha(\gamma)>0,\end{split}$ (5.15)

meaning vacuum is not present but $\varrho_{0}$ decays to zero as
$|x|\to\infty$. We consider the Newtonian viscous stress (5.14) and restrict
slightly the class (5.2) of strong solutions

$\begin{split}{\tilde{\bf u}}&\in C([0,T];D^{1,2}_{0}\cap
D^{3,2}(R^{3};R^{3}))\cap L^{2}(0,T;D^{4,2}(R^{3};R^{3})),\\\
\partial_{t}{\tilde{\bf u}}&\in L^{\infty}(0,T;D^{1,2}_{0}(R^{3},R^{3}))\cap
L^{2}(0,T;D^{2,2}(R^{3};R^{3})),\\\ \tilde{\varrho},\ p(\tilde{\varrho})&\in
C([0,T];W^{3,2}(R^{3})),\\\ \sqrt{\varrho}\partial^{2}_{t,t}{\tilde{\bf
u}}&\in L^{2}(0,T;L^{2}(R^{3};R^{3})).\end{split}$ (5.16)

introduced by Cho and Kim [4].

The first observation is that any solution belonging to (5.16) satisfies

${\tilde{\bf u}}\in C([0,T];BC^{1}(R^{3};R^{3})),\ \nabla^{2}{\tilde{\bf
u}}\in L^{2}(0,T;BC(R^{3};R^{9})).$

Thus a direct inspection of formula (2.7) reveals that the decay properties of
the initial density propagate in time. More specifically, we have

$\begin{split}|\nabla_{x}(\varrho_{0})^{\gamma-1}(x)|\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha}}\
&\Rightarrow\
(\varrho_{0})^{\gamma-1}(x)\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha-1}}\
\Rightarrow\varrho^{\gamma-1}(t,x)\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha-1}}\\\
&\Rightarrow\
|\nabla_{x}(\varrho(t,x))^{\gamma-1}|\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha-1}}\end{split}$

In particular, we get

$\nabla_{x}P^{\prime}(\tilde{\varrho})\in L^{\infty}(0,T;L^{1}\cap
L^{\infty}(R^{3};R^{3}))\ \mbox{provided}\ \alpha>2,$ (5.17)

and

$\tilde{\varrho}(t,x)\approx\frac{1}{1+|x|^{\beta}},\
|\nabla_{x}\tilde{\varrho}(t,x)|\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\beta}}\
\mbox{for all}\ t\in[0,T],\ x\in R^{3},\ \beta=\frac{\alpha-1}{\gamma-1}.$
(5.18)

Consequently, all steps in Section 5.1 up to formula (5.7) can be performed
and we are left with the integral (5.8)

$\int_{0}^{\tau}\int_{R^{3}}1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot\Big{(}\partial_{t}{\tilde{\bf u}}+{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{)}\,{\rm d}{x}\,{\rm d}t.$
(5.19)

We proceed in several steps:

Step 1.

As ${\tilde{\bf u}}\in L^{\infty}((0,T)\times R^{3};R^{3})$,
$\nabla_{x}{\tilde{\bf u}}\in L^{\infty}\cap L^{2}((0,T)\times R^{3};R^{9})$,
we get, by interpolation,

${\bf u}\cdot\nabla_{x}{\tilde{\bf u}}\in L^{2}(0,T;L^{3}(R^{3},R^{3}))$

Similarly, it follows from (5.18) that

$\nabla_{x}P^{\prime}(\tilde{\varrho})\in L^{\infty}(0,T;L^{3}\cap
L^{\infty}(R^{3};R^{3})).$ (5.20)

Consequently,

$\begin{split}\int_{R^{3}}&1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot\Big{(}{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\Big{)}\,{\rm d}{x}\,{\rm d}t\\\
&\leq\|\varrho-\tilde{\varrho}\|_{L^{2}(R^{3})}\|{\bf u}-{\tilde{\bf
u}}\|_{L^{6}(R^{3};R^{3})}\|{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\nabla_{x}P^{\prime}(\tilde{\varrho})\|_{L^{3}(R^{3};R^{3})}\end{split},$

and we may use the arguments of Section 5.1 to control the expression on the
right–side. Thus (5.19) reduces to

$\int_{0}^{\tau}\int_{R^{3}}1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot\partial_{t}{\tilde{\bf u}}\,{\rm d}{x}\,{\rm d}t.$
(5.21)

Step 2. To control $\partial_{t}{\tilde{\bf u}}$, we differentiate the
momentum equation with respect to time obtaining

$\mathcal{L}[\partial_{t}{\tilde{\bf
u}}]=\partial^{2}_{t,t}(\tilde{\varrho}{\tilde{\bf u}})+{\rm
div}_{x}(\partial_{t}(\tilde{\varrho}{\tilde{\bf u}}\otimes{\tilde{\bf
u}}))+\nabla_{x}\partial_{t}p(\tilde{\varrho}),$ (5.22)

where $\mathcal{L}$ denotes the Lamé elliptic operator

$\mathcal{L}[{\bf v}]=\mu{\rm div}_{x}\left(\nabla_{x}{\bf v}+\nabla_{x}{\bf
v}^{t}-\frac{2}{3}{\rm div}_{x}{\bf v}\mathbb{I}\right)+\lambda\nabla_{x}{\rm
div}_{x}{\bf v}.$ (5.23)

Now,

$\begin{split}\partial_{t}(\tilde{\varrho}{\tilde{\bf u}}\otimes{\tilde{\bf
u}})&=\partial_{t}\tilde{\varrho}({\tilde{\bf u}}\otimes{\tilde{\bf
u}})+\tilde{\varrho}(\partial_{t}{\tilde{\bf u}}\otimes{\tilde{\bf
u}})+\tilde{\varrho}({\tilde{\bf u}}\otimes\partial_{t}{\tilde{\bf u}})\\\
&=-{\tilde{\bf u}}\cdot\nabla_{x}\tilde{\varrho}({\tilde{\bf
u}}\otimes{\tilde{\bf u}})-\tilde{\varrho}{\rm div}_{x}{\tilde{\bf
u}}({\tilde{\bf u}}\otimes{\tilde{\bf
u}})+\tilde{\varrho}(\partial_{t}{\tilde{\bf u}}\otimes{\tilde{\bf
u}})+\tilde{\varrho}({\tilde{\bf u}}\otimes\partial_{t}{\tilde{\bf
u}}).\end{split}$

By Hölder’s inequality

$\|{\tilde{\bf u}}\cdot\nabla_{x}\tilde{\varrho}({\tilde{\bf
u}}\otimes{\tilde{\bf u}})\|_{L^{\frac{3}{2}}(R^{3};R^{3})}\leq\|{\tilde{\bf
u}}\|_{L^{6}(R^{3};R^{3})}^{3}\|\nabla_{x}\tilde{\varrho}\|_{L^{6}(R^{3};R^{3})}.$

Similarly

$\|\tilde{\varrho}{\rm div}_{x}{\tilde{\bf u}}({\tilde{\bf
u}}\otimes{\tilde{\bf
u}})\|_{L^{\frac{3}{2}}(R^{3};R^{3})}\leq\|\tilde{\varrho}\|_{L^{\infty}(R^{3})}\|{\rm
div}_{x}{\tilde{\bf u}}\|_{L^{2}(R^{3})}\|{\tilde{\bf
u}}\|^{2}_{L^{6}(R^{3})},$

and

$\|\tilde{\varrho}(\partial_{t}{\tilde{\bf u}}\otimes{\tilde{\bf
u}})+\tilde{\varrho}({\tilde{\bf u}}\otimes\partial_{t}{\tilde{\bf
u}})\|_{L^{\frac{3}{2}}(R^{3};R^{3})}\leq
2\|\tilde{\varrho}\|_{L^{3}(R^{3})}\|{\tilde{\bf
u}}\|_{L^{6}(R^{3};R^{3})}\|\partial_{t}{\tilde{\bf
u}}\|_{L^{6}(R^{3};R^{3})}.$

As $(\tilde{\varrho},{\tilde{\bf u}})$ belongs to the class (5.16) and
$\tilde{\varrho}$ satisfies (5.20), it is easy to check that

$\partial_{t}(\tilde{\varrho}{\tilde{\bf u}}\otimes{\tilde{\bf u}})\in
L^{2}(0,T;L^{\frac{3}{2}}(R^{3};R^{9})).$ (5.24)

Next,

$\partial_{t}p(\tilde{\varrho})=-p^{\prime}(\tilde{\varrho})\Big{(}\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf
u}}-\tilde{\varrho}{\rm div}_{x}{\tilde{\bf u}}\Big{)},$

where

$\|p^{\prime}(\tilde{\varrho})\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf
u}}\|_{L^{\frac{3}{2}}(\Omega)}\stackrel{{\scriptstyle<}}{{\sim}}\|\tilde{\varrho}\|_{L^{6}(R^{3})}\|\nabla_{x}P^{\prime}(\tilde{\varrho})\|_{L^{3}(R^{3})}\|{\tilde{\bf
u}}\|_{L^{6}(R^{3})}$

Similarly to the above, we conclude

$\partial_{t}p(\tilde{\varrho})\in L^{\infty}(0,T;L^{\frac{3}{2}}(R^{3})).$
(5.25)

Going back to the equation (5.22), we get

$\mathcal{L}[\partial_{t}{\tilde{\bf
u}}]=\partial^{2}_{t,t}(\tilde{\varrho}{\tilde{\bf u}})+{\rm
div}_{x}\mathbb{B}$

where

$\mathbb{B}\in L^{2}(0,T;L^{\frac{3}{2}}(R^{3};R^{9})).$

In view of the standard elliptic estimates, we may write

$\partial_{t}{\tilde{\bf u}}={\bf z}+{\bf v},$

where

$\mathcal{L}[{\bf z}]=\partial^{2}_{t,t}(\tilde{\varrho}{\tilde{\bf u}}),$
(5.26)

while

${\bf v}\in L^{2}(0,T;D^{1,\frac{3}{2}}_{0}(R^{3};R^{3}))\hookrightarrow
L^{2}(0,T;L^{3}(R^{3};R^{3})).$

In view of the arguments employed in Step 1, our task reduces to controlling
the integral

$\int_{0}^{\tau}\int_{R^{3}}1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot{\bf z}\,{\rm d}{x}\,{\rm d}t.$ (5.27)

Step 3.

First, we write the right–hand side of the elliptic equation (5.26) in the
form

$\partial^{2}_{t,t}(\tilde{\varrho}{\tilde{\bf
u}})=\partial^{2}_{t,t}\tilde{\varrho}{\tilde{\bf
u}}+\tilde{\varrho}\partial^{2}_{t,t}{\tilde{\bf u}}-2{\rm
div}_{x}(\tilde{\varrho}{\tilde{\bf u}})\partial_{t}{\tilde{\bf
u}}=-\partial_{t}{\rm div}_{x}(\tilde{\varrho}{\tilde{\bf u}}){\tilde{\bf
u}}+\tilde{\varrho}\partial^{2}_{t,t}{\tilde{\bf u}}-2\tilde{\varrho}{\rm
div}_{x}({\tilde{\bf u}})\partial_{t}{\tilde{\bf
u}}-2\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf u}}\partial_{t}{\tilde{\bf u}}$

where, furthermore,

$\partial_{t}{\rm div}_{x}(\tilde{\varrho}{\tilde{\bf u}}){\tilde{\bf u}}={\rm
div}_{x}\left(\partial_{t}(\tilde{\varrho}{\tilde{\bf u}})\otimes{\tilde{\bf
u}}\right)-\partial_{t}(\tilde{\varrho}{\tilde{\bf
u}})\cdot\nabla_{x}{\tilde{\bf u}}.$

As the term ${\rm div}_{x}\left(\partial_{t}(\tilde{\varrho}{\tilde{\bf
u}})\otimes{\tilde{\bf u}}\right)$ can be handled exactly as in Step 2, our
task reduces to estimating the integral

$\int_{0}^{\tau}\int_{R^{3}}1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot{\bf w}\,{\rm d}{x}\,{\rm d}t,$ (5.28)

where ${\bf w}$ solves the elliptic system

$\mathcal{L}[{\bf w}]={\bf g},$ (5.29)

with the right–hand side

${\bf g}=\tilde{\varrho}\partial_{t}{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}-\tilde{\varrho}{\rm div}_{x}{\tilde{\bf u}}({\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf u}})+{\tilde{\bf
u}}\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}+\tilde{\varrho}\partial^{2}_{t,t}{\tilde{\bf u}}-2\tilde{\varrho}{\rm
div}_{x}({\tilde{\bf u}})\partial_{t}{\tilde{\bf
u}}-2\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf u}}\partial_{t}{\tilde{\bf
u}}.$ (5.30)

Our goal is to show that that function ${\bf w}$ decays to zero for large $x$,
more specifically, we shall see that

$|{\bf w}(t,x)|\leq\chi(t)\frac{1}{|x|}\ \mbox{with}\ \chi\in L^{2}(0,T).$
(5.31)

Taking (5.31) for granted, we use Hardy’s inequality to estimate the integral
(5.28),

$\begin{split}\int_{R^{3}}1_{\varrho\leq\overline{\varrho}}(\varrho-\tilde{\varrho})({\bf
u}-{\tilde{\bf u}})\cdot{\bf w}\,{\rm d}{x}\leq\delta\int_{R^{3}}\frac{|{\bf
u}-{\tilde{\bf u}}|^{2}}{|x|^{2}}\,{\rm
d}{x}+c(\delta)\int_{R^{3}}\chi^{2}1_{\varrho\leq\overline{\varrho}}|\varrho-\tilde{\varrho}|^{2}\\\
\stackrel{{\scriptstyle<}}{{\sim}}\delta\int_{R^{3}}|\nabla_{x}{\bf
u}-\nabla_{x}{\tilde{\bf u}}|^{2}\,{\rm
d}{x}+c(\delta)\int_{R^{3}}\chi^{2}\mathcal{E}\left(\varrho,{\bf
u}\Big{|}\tilde{\varrho},{\tilde{\bf u}}\right),\ \delta>0\
\mbox{arbitrary.}\end{split}$

Thus the proof of weak strong uniqueness can be completed via Gronwall’s
argument exactly as in Section 4.1.

It remains to show (5.31). As ${\bf w}$ solves the elliptic problem (5.29), we
get

${\bf w}(t,x)=\int_{R^{3}}\mathbb{G}(x,y)\cdot{\bf g}(t,y){\rm d}y,$

where $\mathbb{G}$ is the Green kernel associated to $\mathcal{L}$. Writing

$\begin{split}\int_{R^{3}}\mathbb{G}(x,y)\cdot{\bf g}(t,y){\rm
d}y=\frac{1}{|x|}\int_{R^{3}}(|x|-|y|)\mathbb{G}(x,y)\cdot{\bf g}(t,y){\rm
d}y\\\ +\frac{1}{|x|}\int_{R^{3}}|y|\mathbb{G}(x,y)\cdot{\bf g}(t,y){\rm
d}y\end{split}$

and using the fact

$|\mathbb{G}(x,y)|\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{|x-y|},$

we observe that (5.31) follows provided

$\|{\bf g}(t,\cdot)\|_{L^{1}(R^{3};R^{3})}+\||x|{\bf
g}(t,\cdot)\|_{L^{q}(R^{3};R^{3})}\leq\chi(t)\ \mbox{for some}\ \ \chi\in
L^{2}(0,T),\
q\in(\frac{3}{2}-\varepsilon,\frac{3}{2}+\varepsilon),\varepsilon>0.$ (5.32)

Thus our ultimate goal is to check (5.32) for all terms appearing on the
right–hand side of (5.30). First, we have

$\|\tilde{\varrho}\partial_{t}{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}\|_{L^{1}(R^{3})}\leq\|\tilde{\varrho}\|_{L^{3}(R^{3})}\|\partial_{t}{\tilde{\bf
u}}\|_{L^{6}(R^{3})}\|\nabla_{x}{\tilde{\bf u}}\|_{L^{2}(R^{3})},$

and

$\||x|\tilde{\varrho}\partial_{t}{\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf
u}}\|_{L^{q}(R^{3})}\leq\|\ |x|\tilde{\varrho}\
\|_{L^{q}(R^{3})}\|\partial_{t}{\tilde{\bf
u}}\|_{L^{\infty}(R^{3})}\|\nabla_{x}{\tilde{\bf u}}\|_{L^{\infty}(R^{3})}.$

In view of (5.16), the bound (5.32) holds as soon as $\beta>3$. The same
argument applies to the terms $\tilde{\varrho}{\rm div}_{x}{\tilde{\bf
u}}\partial_{t}{\tilde{\bf u}}$, $\tilde{\varrho}{\rm div}_{x}{\tilde{\bf
u}}({\tilde{\bf u}}\cdot\nabla_{x}{\tilde{\bf u}})$. Moreover, seeing that
$\nabla_{x}\tilde{\varrho}$ decays at least as fast as $\tilde{\varrho}$, we
can handle ${\tilde{\bf u}}\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf
u}}\cdot\nabla_{x}{\tilde{\bf u}}$ and
$\nabla_{x}\tilde{\varrho}\cdot{\tilde{\bf u}}\partial_{t}{\tilde{\bf u}}$ in
the same manner. Here, we have used the fact that ${\tilde{\bf u}}$ is
uniformly bounded.

Finally,

$\|\tilde{\varrho}\partial^{2}_{t,t}{\tilde{\bf
u}}\|_{L^{1}(R^{3})}\leq\|\sqrt{\tilde{\varrho}}\|_{L^{2}(R^{3})}\|\sqrt{\tilde{\varrho}}\partial^{2}_{t,t}{\tilde{\bf
u}}\|_{L^{2}(R^{3})},$ $\||x|\tilde{\varrho}\partial^{2}_{t,t}{\tilde{\bf
u}}\|_{L^{q}(R^{3})}\leq\||x|\sqrt{\tilde{\varrho}}\|_{L^{p}(R^{3})}\|\sqrt{\tilde{\varrho}}\partial^{2}_{t,t}{\tilde{\bf
u}}\|_{L^{2}(R^{3})},\ \frac{1}{p}=\frac{1}{q}-\frac{1}{2}.$

Thus the choice $\beta>3$ yields (5.32).

We have shown the following result:

######  Theorem 5.4 (Weak–strong uniqueness, positive density).

Suppose that $\Omega=R^{3}$ and $1<\gamma\leq 2$. In addition, let
$\mathbb{S}$ be Newtonian,

$\mathbb{S}=\mu\left(\nabla_{x}{\bf u}+\nabla_{x}^{t}{\bf u}-\frac{2}{3}{\rm
div}_{x}{\bf u}\mathbb{I}\right)+\lambda{\rm div}_{x}{\bf u}\mathbb{I},\
\mu>0,\ \lambda\geq 0.$

Let $(\tilde{\varrho},{\tilde{\bf u}})$ be a strong solution of the problem
(1.1), (1.2) in $(0,T)\times R^{3}$ in the sense specified in Section 2.1
belonging to the class (5.16). Let $(\varrho,{\bf u})$ be a weak solution of
the same problem in the sense of Definition 2.2 such that

$\varrho(0,\cdot)=\tilde{\varrho}(0,\cdot)=\varrho_{0},\ (\varrho{\bf
u})(0,\cdot)=(\tilde{\varrho}{\tilde{\bf u}})(0,\cdot)={\bf m}_{0},$

where

$\begin{split}0<\varrho_{0}(x)&\leq\overline{\varrho},\\\
|\nabla_{x}(\varrho_{0})^{\gamma-1}(x)|&\stackrel{{\scriptstyle<}}{{\sim}}\frac{1}{1+|x|^{\alpha}},\
\alpha>\max\left\\{2;3\gamma-2\right\\}.\end{split}$

Then

$\varrho=\tilde{\varrho},\ {\bf u}={\tilde{\bf u}}\ \mbox{in}\ (0,T)\times
R^{3}.$

######  Remark 5.5.

Local existence of strong solutions in the class (5.16) was proved by Cho and
Kim [4, Theorem 3], see also Huang et al. [9, Lemma 2.1].

Similarly to the preceding results, Theorem (5.4) can be extended to exterior
domains with general in/out flow boundary conditions. Vacuum can be
accommodated at the expense of various technical difficulties. Finally, it is
worth noting that the condition $\alpha>3\gamma-2$ in Theorem 5.4 is critical
for the fluid to have finite mass, namely

$\int_{R^{3}}\tilde{\varrho}\,{\rm d}{x}<\infty.$

## References

  * [1] A. Abbatiello, E. Feireisl, and A. Novotný.  Generalized solutions to models of compressible viscous fluids.  Discrete Contin. Dyn. Syst., 41(1):1–28, 2021.
  * [2] M. Bulíček, P. Gwiazda, J. Málek, and A. Świerczewska Gwiazda.  On unsteady flows of implicitly constituted incompressible fluids.  SIAM J. Math. Anal., 44(4):2756–2801, 2012.
  * [3] Y. Cho, H.J. Choe, and H. Kim.  Unique solvability of the initial boundary value problems for compressible viscous fluids.  J. Math. Pures. Appl., 83:243–275, 2004.
  * [4] Y. Cho and H. Kim.  On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities.  Manuscripta Math., 120(1):91–129, 2006.
  * [5] L. Fang, H. Zhu, and Z. Guo.  Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum.  Nonlinear Anal., 174:189–208, 2018.
  * [6] E. Feireisl, Bum Ja Jin, and A. Novotný.  Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system.  J. Math. Fluid Mech., 14:712–730, 2012.
  * [7] E. Feireisl, Y.-S. Kwon, and A. Novotný.  On the long–time behavior of dissipative solutions to models of non-Newtonian compressible fluids.  Archive Preprint Series, 2020.  arxiv preprint No. 2001.03313.
  * [8] H. Gong, J. Li, X.-Gao Liu, and X. Zhang.  Local well-posedness of isentropic compressible Navier-Stokes equations with vacuum.  Commun. Math. Sci., 18(7):1891–1909, 2020.
  * [9] X. Huang, J. Li, and Z. Xin.  Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations.  Comm. Pure Appl. Math., 65(4):549–585, 2012.
  * [10] M. Kalousek, V. Mácha, and Š. Nečasová.  Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations.  Archive Preprint Series, 2020.  arxiv preprint No. 2012.01795v1.
  * [11] J. Li and Z. Xin.  Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum.  Ann. PDE, 5(1):Paper No. 7, 37, 2019.
  * [12] Z. Liang.  Global Strong Solutions of Navier-Stokes Equations for Heat-Conducting Compressible Fluids with Vacuum at Infinity.  J. Math. Fluid Mech., 23(1):17, 2021.
  * [13] P.-L. Lions.  Mathematical topics in fluid dynamics, Vol.2, Compressible models.  Oxford Science Publication, Oxford, 1998.
  * [14] F Merle, P. Raphael, I. Rodnianski, and J. Szeftel.  On the implosion of a three dimensional compressible fluid.  Arxive Preprint Series, arxiv preprint No. 1912.11009, 2019\.
  * [15] O. Rozanova.  Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations.  J. Differential Equations, 245(7):1762–1774, 2008.
  * [16] A. Valli and M. Zajaczkowski.  Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case.  Commun. Math. Phys., 103:259–296, 1986.
  * [17] Z. Xin.  Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density.  Commun. Pure Appl. Math., 51:229–240, 1998.


11institutetext:  DIÉNS, ÉNS, CNRS, PSL University, Paris, France
45 rue d’Ulm, 75230, Paris cedex 05, France
11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>
# A Note on the Ramanujan Machine

Éric Brier  David Naccache 11 Ofer Yifrach-Stav 11

###### Abstract

The Ramanujan Machine project detects new expressions related to constants of
interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to
name a few).

In particular the project lists a number of conjectures involving even and odd
$\zeta$ function values, logarithms etc.

We show that many relations detected by the Ramanujan Machine Project stem
from a specific algebraic observation and show how to generate infinitely
many.

This provides an automated proof and/or an explanation of many of the
relations listed as conjectures by the project (although not all of them).

##  1 Introduction

The Ramanujan Machine project [2, 1, 3] detects new expressions related to
constants of interests, such as $\zeta$ function values, $\gamma$ and various
algebraic numbers (to name a few).

In particular the project lists several of
conjectures111http://www.ramanujanmachine.com/wp-
content/uploads/2022/07/results_different_zeta_orders.pdf concerning values of
the $\zeta$ function.

We show that many of the relations detected by the Ramanujan Machine Project
stem from a specific algebraic observation and show how to generate and
machine-prove infinitely many.

This provides an automated proof and/or an explanation of many of the
relations listed as conjectures by the project (although not all of them).

##  2 Theoretical Preamble

Consider continued fractions defined by the formulae

$\displaystyle\left\\{\begin{aligned} p_{n}&=b_{n}p_{n-1}+a_{n}p_{n-2}\\\
q_{n}&=b_{n}q_{n-1}+a_{n}q_{n-2}\\\ \end{aligned}\right.\mbox{~{}~{}with the
initial conditions:}$ $\displaystyle\left\\{\begin{aligned} p_{-1}&=1\\\
p_{0}&=b_{0}\\\ \end{aligned}\right.\mbox{~{}~{}and~{}~{}}$
$\displaystyle\left\\{\begin{aligned} q_{-1}&=0\\\ q_{0}&=1\\\
\end{aligned}\right.$

Let $f$ and $g$ be two functions from which we build $\forall n\geq 1$:

$\displaystyle\left\\{\begin{aligned} a_{n}&=-f(n)^{2}\\\
b_{n}&=\frac{f(n+1)g(n+1)+f(n)g(n-1)}{g(n)}\\\
\end{aligned}\right.\mbox{~{}~{}with the initial
condition:~{}~{}}b_{0}=f(1)g(1)$

We further require $f$ and $g$ to be nonzero for positive integers.

Define the sequence $\mathbf{F}_{n}$ by:

$F_{n}=\prod_{i=1}^{n+1}f(i)\mbox{~{}~{}with the initial
conditions:~{}~{}}\mathbf{F}_{-1}=1\mbox{~{}~{}and~{}~{}}\mathbf{F}_{0}=f(1)$

We will now prove by induction that

$\forall n\geq-1,p_{n}=\mathbf{F}_{n}g(n+1)$

The above initial conditions ensure this equality for $n=-1$ and $n=0$. Assume
that the hypothesis holds true for all values below $n$ and let us compute:

$\displaystyle\begin{aligned} p_{n}&=b_{n}p_{n-1}+a_{n}p_{n-2}\\\
&=b_{n}\mathbf{F}_{n-1}g(n)-f(n)^{2}\mathbf{F}_{n-2}g(n-1)\\\
&=\mathbf{F}_{n-1}(f(n+1)g(n+1)+f(n)g(n-1))-f(n)\mathbf{F}_{n-1}g(n-1)\\\
&=\mathbf{F}_{n-1}f(n+1)g(n+1)\\\ &=\mathbf{F}_{n}g(n+1)\\\ \end{aligned}$

as desired, thereby proving a closed form for the convergents $p_{n}$.

We will now handle $q_{n}$ by first remarking that:

$\frac{q_{n}}{p_{n}}-\frac{q_{n-1}}{p_{n-1}}=\frac{q_{n}p_{n-1}-q_{n-1}p_{n}}{p_{n}p_{n-1}}$

The recurrence conditions for $p_{n}$ and $q_{n}$ ensure that:

$q_{n}p_{n-1}-q_{n-1}p_{n}=-a_{n}(q_{n-1}p_{n-2}-q_{n-2}p_{n-1})$

Induction yields:

$q_{n}p_{n-1}-q_{n-1}p_{n}=(\prod_{i=1}^{n}-a_{n})(q_{-1}p_{-2}-q_{-2}p_{-1})$

The last term is equal to 1 given the initial conditions. The $n$-term product
is, by the definition of the sequence $a_{n}$, equal to
$(\mathbf{F}_{n-1})^{2}$.

We hence get:

$\displaystyle\begin{aligned}
\frac{q_{n}}{p_{n}}-\frac{q_{n-1}}{p_{n-1}}&=\frac{(\mathbf{F}_{n-1})^{2}}{p_{n}p_{n-1}}\\\
&=\frac{(\mathbf{F}_{n-1})^{2}}{\mathbf{F}_{n}g(n+1)\mathbf{F}_{n-1}g(n)}\\\
&=\frac{1}{f(n+1)g(n)g(n+1)}\\\ \end{aligned}$

Once again, by induction (taking into account the initial values), we get:

$\frac{q_{n}}{p_{n}}=\sum_{i=0}^{n}\frac{1}{f(i+1)g(i)g(i+1)}$

The limit $L$ of the continued fraction is thus given by the equation:

$\dfrac{1}{L}=\sum_{n=0}^{\infty}\dfrac{1}{f(n+1)g(n)g(n+1)}$

It suffices now to resort to standard partial fraction decomposition to get
relations such as those given by the Ramanujan Machine Project. This is done
automatically by symbolic computation software such as Mathematica to avoid
tedious yet standard formula manipulation by hand.

Consider the Ramanujan Project identity222Note that our notations of $a_{n}$
and $b_{n}$ are reversed with respect to theirs where $a_{n}=-n^{8}$ and
$b_{n}=n^{4}+(n+1)^{4}+2(n^{2}+(n+1)^{2})$. Posing $g(n)=\eta_{1}n+\eta_{0}$
and identifying we get:
$b_{n}g(n)-((n+1)^{4}g(n+1)+n^{4}g(n-1))=(1+2n+2n^{2})(2\eta_{0}-\eta_{1})=0$
Which gives the solution $\\{\eta_{1},\eta_{0}\\}=\\{2,1\\}$ (the first
example processed by our code).

The last step connecting the observation and the Ramanujan Machine Project is
somewhat technical we hence break it down into successive steps:

### Step 1:

The Ramanujan Machine Project considers that the $a_{n},b_{n}$ defining the
continued fraction are polynomials. Because nothing in our analysis imposes
that the $a_{n},b_{n}$ are polynomials we can write:

$\dfrac{1}{L}=\frac{f(1)g(1)+f(0)g(-1)}{g(0)}-{\cfrac{f(1)^{2}}{\frac{f(2)g(2)+f(1)g(0)}{g(1)}-{\cfrac{f(2)^{2}}{\frac{f(3)g(3)+f(2)g(1)}{g(2)}-{\cfrac{f(3)^{2}}{\frac{f(4)g(4)+f(3)g(2)}{g(3)}-{\cfrac{f(4)^{2}}{\frac{f(5)g(5)+f(4)g(3)}{g(4)}-\cdots}}}}}}}}$

$\dfrac{1}{L}=\frac{f(1)g(1)+f(0)g(-1)}{g(0)}-{\cfrac{g(1)f(1)^{2}}{\frac{f(2)g(2)+f(1)g(0)}{1}-{\cfrac{g(2)g(1)f(2)^{2}}{\frac{f(3)g(3)+f(2)g(1)}{1}-{\cfrac{g(3)g(2)f(3)^{2}}{\frac{f(4)g(4)+f(3)g(2)}{1}-{\cfrac{g(4)g(3)f(4)^{2}}{\frac{f(5)g(5)+f(4)g(3)}{1}-\cdots}}}}}}}}$

$\dfrac{g(0)}{L}=\frac{f(1)g(1)+f(0)g(-1)}{1}-{\cfrac{g(1)g(0)f(1)^{2}}{\frac{f(2)g(2)+f(1)g(0)}{1}-{\cfrac{g(2)g(1)f(2)^{2}}{\frac{f(3)g(3)+f(2)g(1)}{1}-{\cfrac{g(3)g(2)f(3)^{2}}{\frac{f(4)g(4)+f(3)g(2)}{1}-{\cfrac{g(4)g(3)f(4)^{2}}{\frac{f(5)g(5)+f(4)g(3)}{1}-\cdots}}}}}}}}$

$\dfrac{g(0)}{L}={f(1)g(1)+f(0)g(-1)}-{\cfrac{g(1)g(0)f(1)^{2}}{{f(2)g(2)+f(1)g(0)}-{\cfrac{g(2)g(1)f(2)^{2}}{{f(3)g(3)+f(2)g(1)}-{\cfrac{g(3)g(2)f(3)^{2}}{{f(4)g(4)+f(3)g(2)}-\cdots}}}}}}$

### Step 2:

In the Ramanujan Project many formulae have an $a_{n}$ (in our notations, i.e.
$b_{n}$ in theirs) of the form $Q=(n+\alpha)^{2}(n+\beta)(n+\gamma)$, i.e.
with a total degree of 4 and a term having a power of two. In many other cases
the $a_{n}$ can be written as $\phi_{1}(n)^{2}\phi_{2}(n)\phi_{3}(n)$ for some
functions $\phi_{1},\phi_{2},\phi_{3}$. The form ($f(n)^{2}g(n-1)g(n)$) at the
numerator of the continued fraction obtained in Step 1 explains why:

### Step 3:

Operate the change of variable $n_{0}=n+\alpha$ to get:

$Q={n_{0}}^{2}(n_{0}-\alpha+\beta)(n_{0}-\alpha+\gamma)$

### Step 4:

Operate the change of variables $\beta_{1}=-\alpha+\beta$,
$\gamma_{1}=-\alpha+\gamma$ to get:

$Q={n_{0}}^{2}(n_{0}+\beta_{1})(n_{0}+\gamma_{1})$

### Step 5:

Operate the change of variable $n_{0}=n_{1}(\gamma_{1}-\beta_{1})$ to get:

$Q={(n_{1}(\gamma_{1}-\beta_{1}))}^{2}(n_{1}(\gamma_{1}-\beta_{1})+\beta_{1})(n_{1}(\gamma_{1}-\beta_{1})+\gamma_{1})$

### Step 6:

Kick the parasite factor $(\gamma_{1}-\beta_{1})^{2}$ out of the continued
fraction and integrate it in the limit. We get:

$Q^{\prime}=\frac{Q}{(\gamma_{1}-\beta_{1})^{2}}={n_{1}}^{2}(n_{1}(\gamma_{1}-\beta_{1})+\beta_{1})(n_{1}(\gamma_{1}-\beta_{1})+\gamma_{1})$

### Step 5:

Declare $f(n_{1})=\mbox{ID}(n_{1})=n_{1}$ and
$g(n_{1})=n_{1}(\gamma_{1}-\beta_{1})+\beta_{1}$ to get:

$Q^{\prime}=f(n_{1})^{2}g(n_{1})(n_{1}(\gamma_{1}-\beta_{1})+\gamma_{1})$

### Step 7:

Observe that:

$g(n_{1}-1)=(n_{1}-1)(\gamma_{1}-\beta_{1})+\gamma_{1}=(\gamma_{1}-\beta_{1})n_{1}+\beta_{1}$

and hence:

$Q^{\prime}=f(n_{1})^{2}g(n_{1})g(n_{1}-1)$

### Step 8:

Replace in the simplified continued fraction the terms of the form:

$f(n)g(n)+f(n-1)g(n-2)$

by their variable-changed expression in $n_{1}$.

### Step 9:

Light an altar candle hoping that the variable changes did not alter the
initial conditions. If so a new relation of the form
$a_{n}=(n+\alpha)^{2}(n+\beta)(n+\gamma)$ and $b_{n}=\mbox{polynomial}$ was
found.

###### Remark 1

Non Polynomial Rational Fractions:  Nothing in the preamble assumed that $f$
and $g$ are polynomials or rational fractions, hence any functions satisfying
the few properties announced can be used to derive “magic” continued fractions
provided that there is a way to write the infinite sum under a fancy closed
form. We give a few examples in the next section.

##  3 Implementation

The implementation assumes that $f(0)=0$ to enforce the initial
conditions333It is also possible to tweak the code to work with other initial
conditions provided that $q_{-1}p_{-2}-q_{-2}p_{-1}=1$, we did not do this
here., sets:

$a_{n}=-f(n)^{2}\mbox{~{}~{}and~{}~{}}b_{n}=\frac{f(n+1)g(n+1)+f(n)g(n-1)}{g(n)}$

and prints:

$L=g(0)^{2}\sum_{i=0}^{\infty}\frac{1}{f(i+1)g(i)g(i+1)}$

As well as the approximate numerical values of both the exact expression and
the continued fraction (to visually compare both).

For the sake of compactness we display $L$ and not its inverse.

The code takes $f$ and $g$ from an example list Ex into which the reader can
plug any desirable function to generate new relations at wish. In the listing
above we changed the order of the printed formulae to fit the longest examples
in a landscape layout. In the formulae $C$ stands for Catalan’s constant.

⬇

1

2Ex={{z^4,1+2z},{z^3,z+1},{z^7,z+1},{z^4/(z+2),(z+3)},{z^5(z+1),z+1},{z^6/(z+2),(z+1)/(z^2+1)},{z^4(z+1)/(z+2),(z+1)/(z^2+1)},{z^2,(1+Sqrt[2](z^2+z))},{z^4,(1-Sqrt[2](z^2+z))},{z^6,(1+29(z^2+z))},{z^5,(1+4(z^2+z))},{E^(-8-2z)z^3,E^z},{E^(-8-2z)z(2+5z+2z^2)^2,E^z},{E^(-2-2z)z(1+z)^2,E^z},{E^(-2z)z(2+z)^2,E^z},{E^(-2-2z)z(1+4z)^2,E^z},{E^(-2z)z(z+z^2)^2,E^z},{E^(-2z)z(1+2z+z^2)^2,E^z},{E^(-2-2z)z(z+2z^2)^2,E^z},{E^(-2z)z(3z+4z^2)^2,E^z},{9z/Exp[z],Exp[z]},{(-2-3z)z,(2+z)^10},{(-2-2z)z,(2+z)^10},{(-3-z)z,(2+z)^10},{-2z,(2+z)^10},{z(-2+3z),(2+z)^10},{3z^2,(2+z)^10},{z(1+3z),(2+z)^10},{(2z+1)/E^z,E^z},{(2z+3)/E^z,E^z},{(2z^2+z)/E^z,E^z},{(2z^2+3z+1)/E^z,E^z},{z^2,1+z},{z(1+2z),1+z},{z(1+3z),1+z},{z(2+z^2),2+z},{z(2+z+z^2),3+z},{z(2+2z+z^2),4+z},{z(2+4z+z^2),3+z},{z(4+2z^2),2+z},{z(z+3z^2),1+z},{z(2z+3z^2),1+z},{z(4z+3z^2),1+z},{z(3z+4z^2),1+z},{z(3z+4z^2),2+z}};

3

4F:=Function[{w,x},x[[1]]/.{z->w}];

5G:=Function[{w,x},x[[2]]/.{z->w}];

6For[i=1,i<=Length[Ex],

7 v=.;

8 CF=(F[v+1,Ex[[i]]] G[v+1,Ex[[i]]]+F[v,Ex[[i]]] G[v-1,Ex[[i]]])/G[v,Ex[[i]]];

9 st=N[{1,0,CF/.v->0,1},10000];

10 For[n=1,n<= 1000,n++,

11 bn=CF/.v->n;

12 an=-F[n,Ex[[i]]]^2;

13 st={{0,0,1,0},{0,0,0,1},{an,0,bn,0},{0,an,0,bn}}.st];

14 formalexpr=G[0,Ex[[i]]]^2/F[t+1,Ex[[i]]]/G[t,Ex[[i]]]/G[t+1,Ex[[i]]];

15 closedform=Simplify[Sum[formalexpr, {t,0,Infinity}]];

16 approxform=N[st[[4]]/st[[3]],1000];

17 Print[{closedform,N[closedform,100],N[approxform,100]}];

18 i++]

$\left\\{v^{4},2v+1,8-\frac{2\pi^{2}}{3}-\frac{\pi^{4}}{90}\right\\}$

$\left\\{v^{3},v+1,-\zeta(3)+\frac{\pi^{4}}{90}+\frac{\pi^{2}}{6}-1\right\\}$

$\left\\{v^{7},v+1,-\zeta(3)-\zeta(5)-\zeta(7)+\frac{\pi^{8}}{9450}+\frac{\pi^{6}}{945}+\frac{\pi^{4}}{90}+\frac{\pi^{2}}{6}-1\right\\}$

$\left\\{\frac{v^{4}}{v+2},v+3,\frac{1}{270}\left(-270\zeta(3)+9\pi^{4}+15\pi^{2}-55\right)\right\\}$

$\left\\{v^{5}(v+1),v+1,-4\zeta(3)-2\zeta(5)+\frac{\pi^{6}}{945}+\frac{\pi^{4}}{30}+\pi^{2}-7\right\\}$

$\left\\{\frac{v^{6}}{v+2},\frac{v+1}{v^{2}+1},11\zeta(3)+10\zeta(5)+4\zeta(7)-\frac{5\pi^{2}}{3}-\frac{11\pi^{4}}{90}-\frac{2\pi^{6}}{315}+10\right\\}$

$\left\\{\frac{v^{4}(v+1)}{v+2},\frac{v+1}{v^{2}+1},4(5\zeta(3)+\zeta(5)+13)-\frac{41\pi^{2}}{6}-\frac{\pi^{4}}{9}\right\\}$

$\left\\{v^{2},\sqrt{2}\left(v^{2}+v\right)+1,\frac{\pi^{2}\left(\sqrt{2}-4\right)-30\sqrt{2}+36+3\pi\left(3\sqrt{2}-2\right)\sqrt{2\sqrt{2}-1}\tanh\left(\frac{1}{2}\sqrt{2\sqrt{2}-1}\pi\right)}{6\left(\sqrt{2}-4\right)}\right\\}$

$\left\\{v^{4},1-\sqrt{2}\left(v^{2}+v\right),-4\sqrt{2}+\frac{\pi^{4}}{90}+\frac{\pi^{2}\left(3\sqrt{2}+5\right)}{6\sqrt{2}+3}-5-\frac{\left(27\sqrt{2}+38\right)\pi\tan\left(\frac{1}{2}\sqrt{2\sqrt{2}+1}\pi\right)}{2\left(2\sqrt{2}+1\right)^{3/2}}\right\\}$

$\left\\{v\left(v^{2}+2v+2\right),v+4,\frac{49}{18}-\frac{4}{5}\pi\coth(\pi)\right\\}$

$\left\\{v\left(v^{2}+4v+2\right),v+3,\frac{1}{8}\left(9\sqrt{2}\pi\cot\left(\sqrt{2}\pi\right)-10\right)\right\\}$

$\left\\{v\left(2v^{2}+4\right),v+2,\frac{1}{12}\left(6-\sqrt{2}\pi\coth\left(\sqrt{2}\pi\right)\right)\right\\}$

$\left\\{v\left(3v^{2}+v\right),v+1,\zeta(3)-\frac{2\pi^{2}}{3}-\frac{9\sqrt{3}\pi}{4}+40+\log\left(\frac{1}{3486784401\sqrt[4]{3}}\right)\right\\}$

$\left\\{v\left(3v^{2}+2v\right),v+1,\frac{1}{48}\left(-3(-8\zeta(3)-65+\log(3)+16\log(243))+27\pi\sqrt{3}-10\pi^{2}\right)\right\\}$

$\left\\{v\left(3v^{2}+4v\right),v+1,\frac{1}{768}\left(192\zeta(3)-56\pi^{2}+54\pi\sqrt{3}-447-324\log(2)+162\log(3)+324\log(6)\right)\right\\}$

$\left\\{v\left(4v^{2}+3v\right),v+1,\frac{1}{162}\left(54\zeta(3)-21\pi^{2}+192\pi+350-384\log(8)\right)\right\\}$

$\left\\{v\left(4v^{2}+3v\right),v+2,\frac{1}{270}\left(30\pi^{2}-768\pi-1049+1536\log(8)\right)\right\\}$

$\left\\{e^{-v}\left(2v^{2}+3v+1\right),e^{v},e\left(-1+e(\log(e-1)-1)+2\sqrt{e}\tanh^{-1}\left(\frac{1}{\sqrt{e}}\right)\right)\right\\}$

$\left\\{v^{2},v+1,\zeta(3)-\frac{\pi^{2}}{6}+1\right\\}$

$\left\\{v(2v+1),v+1,\frac{\pi^{2}}{6}-7+\log(256)\right\\}$

$\left\\{v(3v+1),v+1,\frac{1}{12}\left(2\pi^{2}+9\pi\sqrt{3}-156+81\log(3)\right)\right\\}$

$\left\\{v\left(v^{2}+2\right),v+2,1-\frac{\pi\coth\left(\sqrt{2}\pi\right)}{3\sqrt{2}}\right\\}$

$\left\\{v\left(v^{2}+v+2\right),v+3,\frac{25}{16}-\frac{9\pi\tanh\left(\frac{\sqrt{7}\pi}{2}\right)}{8\sqrt{7}}\right\\}$

$\left\\{e^{-v}(2v+1),e^{v},e\left(\sqrt{e}\tanh^{-1}\left(\frac{1}{\sqrt{e}}\right)-1\right)\right\\}$

$\left\\{e^{-v}(2v+3),e^{v},\frac{1}{3}e\left(-3e-1+3e^{3/2}\tanh^{-1}\left(\frac{1}{\sqrt{e}}\right)\right)\right\\}$

$\left\\{e^{-v}\left(2v^{2}+v\right),e^{v},-e\left(-3+\log(e-1)+2\sqrt{e}\tanh^{-1}\left(\frac{1}{\sqrt{e}}\right)\right)\right\\}$

$\left\\{v^{6},29\left(v^{2}+v\right)+1,\frac{\pi^{6}}{945}+\frac{87\pi^{4}}{10}+\frac{306124\pi^{2}}{3}-\frac{478731681}{2}+\frac{15895741}{10}\pi\sqrt{29}\tan\left(\frac{5\pi}{2\sqrt{29}}\right)\right\\}$

$\left\\{v^{5},4\left(v^{2}+v\right)+1,8\zeta(3)+\zeta(5)+56-48\log(4)\right\\}$

$\left\\{e^{-2v-8}v^{3},e^{v},e^{9}\zeta(3)\right\\}$

$\left\\{e^{-2v-8}v\left(2v^{2}+5v+2\right)^{2},e^{v},-\frac{1}{108}e^{9}\left(13\pi^{2}+8(\log(16)-19)\right)\right\\}$

$\left\\{e^{-2v-2}v(v+1)^{2},e^{v},-\frac{1}{6}e^{3}\left(\pi^{2}-12\right)\right\\}$

$\left\\{e^{-2v}v(v+2)^{2},e^{v},e-\frac{e\pi^{2}}{12}\right\\}$

$\left\\{e^{-2v-2}v(4v+1)^{2},e^{v},-\frac{1}{4}e^{3}\left(8C+\pi^{2}+2\pi+4(\log(8)-8)\right)\right\\}$

$\left\\{e^{-2v}v\left(v^{2}+v\right)^{2},e^{v},e\left(\zeta(3)-\frac{\pi^{2}}{2}+4\right)\right\\}$

$\left\\{e^{-2v}v\left(v^{2}+2v+1\right)^{2},e^{v},-\frac{1}{90}e\left(90(\zeta(3)-4)+\pi^{4}+15\pi^{2}\right)\right\\}$

$\left\\{e^{-2v-2}v\left(2v^{2}+v\right)^{2},e^{v},-\frac{1}{3}e^{3}\left(-3(\zeta(3)+32)+5\pi^{2}+36\log(4)\right)\right\\}$

$\left\\{e^{-2v}v\left(4v^{2}+3v\right)^{2},e^{v},-\frac{1}{243}e\left(-288C-27\zeta(3)+48\pi^{2}-72\pi-256+144\log(8)\right)\right\\}$

$\left\\{9e^{-v}v,e^{v},\frac{1}{9}(e-e\log(e-1))\right\\}$

$\left\\{v^{4},\sqrt{2}\left(v^{2}+v\right)+2,\frac{4\pi^{4}\left(\sqrt{2}-8\right)+30\pi^{2}\left(17\sqrt{2}-12\right)+45\left(64-39\sqrt{2}\right)+90\pi\left(5\sqrt{2}-11\right)\sqrt{4\sqrt{2}-1}\tanh\left(\frac{1}{2}\sqrt{4\sqrt{2}-1}\pi\right)}{360\left(\sqrt{2}-8\right)}\right\\}$

$\left\\{v^{6},\sqrt{2}\left(v^{2}+v\right)+2,\frac{16\pi^{6}\left(\sqrt{2}-8\right)+84\pi^{4}\left(17\sqrt{2}-12\right)-6615\left(19\sqrt{2}-28\right)+630\pi^{2}\left(39\sqrt{2}-64\right)+1890\pi\left(21\sqrt{2}-29\right)\sqrt{4\sqrt{2}-1}\tanh\left(\frac{1}{2}\sqrt{4\sqrt{2}-1}\pi\right)}{15120\left(\sqrt{2}-8\right)}\right\\}$

$\left\\{v^{8},\sqrt{2}\left(v^{2}+v\right)+2,\frac{16\pi^{8}\left(\sqrt{2}-8\right)+80\pi^{6}\left(17\sqrt{2}-12\right)+22050\pi^{2}\left(19\sqrt{2}-28\right)+420\pi^{4}\left(39\sqrt{2}-64\right)-14175\left(149\sqrt{2}-200\right)+9450\pi\left(69\sqrt{2}-91\right)\sqrt{4\sqrt{2}-1}\tanh\left(\frac{1}{2}\sqrt{4\sqrt{2}-1}\pi\right)}{151200\left(\sqrt{2}-8\right)}\right\\}$

$\left\\{v^{10},1-\sqrt{2}\left(v^{2}+v\right),\frac{2\pi^{6}\left(8\sqrt{2}+11\right)}{135\left(4\sqrt{2}+9\right)}+\frac{\pi^{10}}{93555}+\frac{\pi^{8}\left(13\sqrt{2}+17\right)}{4725\left(4\sqrt{2}+9\right)}+\frac{8\pi^{4}\left(30\sqrt{2}+43\right)}{45\left(4\sqrt{2}+9\right)}-4\left(79\sqrt{2}+111\right)+\frac{2\pi^{2}\left(176\sqrt{2}+249\right)}{4\sqrt{2}+9}-\frac{2\left(2159\sqrt{2}+3057\right)\pi\tan\left(\frac{1}{2}\sqrt{2\sqrt{2}+1}\pi\right)}{\left(2\sqrt{2}+1\right)^{5/2}}\right\\}$

$\left\\{v^{12},\sqrt{2}\left(v^{2}+v\right)+2,\frac{88448\pi^{12}\left(\sqrt{2}-8\right)+436800\pi^{10}\left(17\sqrt{2}-12\right)+75675600\pi^{6}\left(19\sqrt{2}-28\right)+2162160\pi^{8}\left(39\sqrt{2}-64\right)+170270100\pi^{4}\left(149\sqrt{2}-200\right)+425675250\pi^{2}\left(1381\sqrt{2}-1996\right)-638512875\left(4479\sqrt{2}-6320\right)+1277025750\pi\left(689\sqrt{2}-963\right)\sqrt{4\sqrt{2}-1}\tanh\left(\frac{1}{2}\sqrt{4\sqrt{2}-1}\pi\right)}{81729648000\left(\sqrt{2}-8\right)}\right\\}$

$\left\\{(-3v-2)v,(v+2)^{10},\frac{3}{4}(654616480\zeta(3)+342927872\zeta(5)+73113600\zeta(7)+6422528\zeta(9)+31737050334-2324522934\log(2)+1162261467\log(3)+2324522934\log(6))-\frac{1}{4}1162261467\pi\sqrt{3}-2279750473\pi^{2}-\frac{156065456\pi^{4}}{5}-\frac{913408\pi^{8}}{225}-\frac{141308416\pi^{6}}{315}-\frac{131072\pi^{10}}{10395}\right\\}$

$\left\\{(-2v-2)v,(v+2)^{10},-\frac{512\left(-467775(4341764\zeta(3)+1794064\zeta(5)+389184\zeta(7)+41216\zeta(9)+1024\zeta(11)+110374897)+48640\pi^{10}+10638144\pi^{8}+1035033120\pi^{6}+68103341460\pi^{4}+4803747223275\pi^{2}\right)}{467775}\right\\}$

$\left\\{(-v-3)v,(v+2)^{10},\frac{512\left(-39106972899225+3428161436700\pi^{2}+47204776080\pi^{4}+649503360\pi^{6}+5144832\pi^{8}+10240\pi^{10}\right)}{467775}\right\\}$

$\left\\{v(3v-2),(v+2)^{10},-\frac{3(2240882559523414400\zeta(3)+902645353815040000\zeta(5)+178178187264000000\zeta(7)+13985382400000000\zeta(9)+126999855214834440651+2324522934\log(2)-1162261467\log(3)-2324522934\log(6))}{40000000000}+\frac{1162261467\pi\sqrt{3}}{40000000000}+\frac{65536\pi^{10}}{22275}+\frac{88928768\pi^{8}}{65625}+c\right\\}$

$\mbox{where:~{}~{}}c=\frac{162966967472\pi^{6}}{984375}+\frac{111360024325867\pi^{4}}{9375000}+\frac{859444691048496267\pi^{2}}{1000000000}$

$\left\\{3v^{2},(v+2)^{10},\frac{256\left(-467775(3325728\zeta(3)+1342656\zeta(5)+267264\zeta(7)+21504\zeta(9)+159629723)+25600\pi^{10}+11037312\pi^{8}+1318680000\pi^{6}+93801403080\pi^{4}+6769898635500\pi^{2}\right)}{1403325}\right\\}$

$\left\\{v(3v+1),(v+2)^{10},-\frac{512\left(-18711(-2(1307738292800\zeta(3)+529342480000\zeta(5)+106368000000\zeta(7)+8800000000\zeta(9)+55119052506237+1162261467\log(6))+2324522934\log(2)-1162261467\log(3))+7249024769679\pi\sqrt{3}+c^{\prime}\right)}{60908203125}\right\\}$

$\mbox{where:~{}~{}}c^{\prime}=-800000000\pi^{10}-317671200000\pi^{8}-37028146320000\pi^{6}-2609348019339600\pi^{4}-187811501659121340\pi^{2}$

$\left\\{-2v,(v+2)^{10},-3072(106154\zeta(3)+42664\zeta(5)+8352\zeta(7)+640\zeta(9)+10269357)+\frac{262144\pi^{10}}{31185}+\frac{950272\pi^{8}}{225}+\frac{18661376\pi^{6}}{35}+\frac{581261312\pi^{4}}{15}+2814203392\pi^{2}\right\\}$

##  4 Using Formal Manipulations

Mathematica is powerful enough to take into account parametric summations. If
we remove from our code the numerical part and add parameters to the input we
can get general forms in which we later set parameters at wish.

In the following example we arbitrarily broke the result into several pieces
to fit in the page:

⬇

1Ex={{((a z)^4+(c z)^3)u,d+b(2+2z)}};

2F:=Function[{w,x},x[[1]]/.{z->w}];

3G:=Function[{w,x},x[[2]]/.{z->w}];

4For[i=1,i<=Length[Ex],v=.;

5CF=(F[v+1,Ex[[i]]]G[v+1,Ex[[i]]]+

6F[v,Ex[[i]]]G[v-1,Ex[[i]]])/G[v,Ex[[i]]];

7formalexpr=

8G[0,Ex[[i]]]^2/F[t+1,Ex[[i]]]/G[t,Ex[[i]]]/G[t+1,Ex[[i]]];

9closedform=Simplify[Sum[formalexpr,{t,0,Infinity}]];

10Print[{closedform}];

11i++]

$\mbox{result}=\frac{\ell_{1}+\ell_{2}+\ell_{3}+\ell_{4}+\ell_{5}+\ell_{7}+\ell_{6}}{6c^{9}d^{3}u(2b+d)^{2}\left(2bc^{3}-a^{4}d\right)\left(2bc^{3}-a^{4}(2b+d)\right)}$

$\ell_{1}=96b^{3}c^{9}(2b+d)\psi^{(0)}\left(\frac{d}{2b}+1\right)\left(bc^{3}\left(4b^{2}+6bd+3d^{2}\right)-2a^{4}\left(2b^{3}+4b^{2}d+3bd^{2}+d^{3}\right)\right)$

$\ell_{2}=+6a^{16}d^{3}(2b+d)^{4}\psi^{(0)}\left(\frac{c^{3}}{a^{4}}+1\right)+\left(2bc^{3}-a^{4}d\right)$

$\ell_{3}=384\gamma
b^{6}c^{6}\left(c^{3}-a^{4}\right)-32b^{5}c^{3}d\left(6\gamma
a^{8}+\left(24\gamma-\pi^{2}\right)a^{4}c^{3}+\left(\pi^{2}-24\gamma\right)c^{6}\right)+a^{4}d^{6}\left(-6\gamma
a^{8}+\pi^{2}a^{4}c^{3}-6c^{6}\zeta(3)\right)$

$\ell_{4}=-16b^{4}d^{2}\left(6\gamma
a^{12}+\left(24\gamma-\pi^{2}\right)a^{8}c^{3}+2a^{4}c^{6}\left(3\zeta(3)-2\pi^{2}+18\gamma\right)-2c^{9}\left(3\zeta(3)-2\pi^{2}+18\gamma\right)\right)$

$\ell_{5}=8b^{3}d^{3}\left(-24\gamma
a^{12}+4\left(\pi^{2}-9\gamma\right)a^{8}c^{3}-6a^{4}c^{6}\left(4\zeta(3)-\pi^{2}+4\gamma\right)+c^{9}\left(18\zeta(3)-5\pi^{2}+18\gamma-6\right)\right)$

$\ell_{6}=-2bd^{5}\left(24\gamma
a^{12}+2\left(3\gamma-2\pi^{2}\right)a^{8}c^{3}-a^{4}c^{6}\left(\pi^{2}-24\zeta(3)\right)-6c^{9}\zeta(3)\right)$

$\ell_{7}=-8b^{2}d^{4}\left(18\gamma
a^{12}+3\left(4\gamma-\pi^{2}\right)a^{8}c^{3}+a^{4}c^{6}\left(18\zeta(3)-2\pi^{2}+3\gamma\right)+c^{9}\left(\pi^{2}-9\zeta(3)\right)\right)$

This is much more practical than the blind exploration of values. Whilst the
code does nothing with CF, we left it in the program to explicit the continued
fraction being calculated. The chosen example has nothing fundamental or
conceptual in it and was chosen at random for illustrative purposes.

##  5 Conclusion & Further Challenges

Given the above it appears that several of the Ramanujan Project
conjectures444We did not exhaust all the relations listed online. can be
explained and/or automatically machine-proved with virtually no effort.

We did not machine-prove the online conjectures one by one as this implies
retyping their polynomials, a retro-solving for $f$ and $g$ by identification
and computing the closed forms using formal summation to infinity.
Nonetheless, many such relations can be generated automatically quasi-
instantaneously on a simple PC.

An interesting question is that of reversing continued fractions from a target
constant. For instance, determine $a_{n}$ and $b_{n}$ such that the continued
fraction converges to a constant chosen a priori, e.g.:

$\frac{1}{L}=\sum_{i=2}^{100}\zeta(i)$

We did not research this challenge and leave it to readers interesting in
pursuing this line of investigation.

## References

  * [1] N. B. David, G. Nimri, U. Mendlovic, Y. Manor, and I. Kaminer.  On the Connection Between Irrationality Measures and Polynomial Continued Fractions, 2021.
  * [2] G. Raayoni, S. Gottlieb, Y. Manor, G. Pisha, Y. Harris, U. Mendlovic, D. Haviv, Y. Hadad, and I. Kaminer.  Generating Conjectures on Fundamental Constants with the Ramanujan Machine.  Nature, 590(7844):67–73, Feb 2021.
  * [3] G. Raayoni, G. Pisha, Y. Manor, U. Mendlovic, D. Haviv, Y. Hadad, and I. Kaminer.  The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants.  CoRR, abs/1907.00205, 2019.