数理解析研究所講究録
1495 巻 2006 年 154-161

154

Continuous Dependence on Initial Data for Multidimensional,
Viscous, Compressible Flows
David Hoff’

We prove uniqueness and continuous dependence on initial data of weak solutions of
the Navier-Stokes equations of compressible flow$\rho_{t}+\mathrm{d}\mathrm{i}\mathrm{v}(\rho u)=0$

(1)

$\{$

$(\rho u^{j})_{t}+\mathrm{d}\mathrm{i}\mathrm{v}(\rho u^{j}u)+P(\rho)_{x_{j}}=\mu\Delta u^{j}+\lambda \mathrm{d}\mathrm{i}\mathrm{v}u_{x_{j}}+\rho f^{j}$$n=2,3$ , and $t>0$ with initial datafor$(\rho, u)|_{t=0}=(\rho_{0}, u_{0})$ .
(2)
Here and $u=(u^{1}, \ldots, u^{n})$ are unknown functions of and representing density andvelocity, $P=P(\rho)$ is the pressure, is a given external force, $\mu>0$ andare viscosity
constants, andand are the usual spatial divergence and Laplace operators.
Our purpose is to compare solutions with only minimal regularity, including the genericsingular solutions of the Navier-Stokes system-solutions with codimension-one discontinuities in density, pressure, and velocity gradient (see Hoff [4]). Our result extends that ofDanchin [1], which applies to solutions in certain Besov spaces of continuous functions.
Danchin makes a direct comparison by subtracting the equations satisfied by different solutions and controlling the linearization errors by the assumed regularity. This regularity
is absent for solutions in the class considered here, however, and a different approach is
required.
Ideally, one would prove uniqueness and continuous dependence for solutions in thelargest class for which existence is known, that is, solutions with finite energies and nonnegative densities (see Feireisl [2] and [3] and Lions [9], for example). Such a result may
be unattainable, however, and may be in fact be false: it is known that certain anomalies
can arise in solutions of (1) when densities are zero on nonnegligible sets, and there areexamples of nonphysical solutions (see Hoff and Serre [7]). Moreover, it appears likely,
although it is not presently known, that large energies can force positive densities to zero,so that the aforementioned anomalies may occur even when initial densities are strictly
positive. It thus remains an interesting open problem to select physical solutions fromthe most general class of known weak solutions and to establish their uniqueness and continuous dependence on initial data. We shall discuss this issue in greater detail below.
For the time being we avoid these issues and difficulties by assuming that the solutions
being compared have strictly positive densities, which is known to hold for a large class ofsolutions with positive initial densities and small initial energies (Hoff [5]).
To be more specific, the solutions we consider are weak solutions in a regularity classwhich we shall call . Properties of solutions in are rather technical and are listedin Hoff [6], $(1.4)-(1.12)$ . Rather than repeating these conditions here, we instead list the
hypotheses on the system parameters, initial data, and external force under which solutionsin the class are known to exist (Hoff [5]): if$x\in \mathbb{R}^{n},$$x$$\rho$$f$$t$$\lambda\geq 0$$\Delta$$\mathrm{d}\mathrm{i}\mathrm{v}$

$C$

$C$

$C$

Department of Mathematics, Indiana University, Bloomington, IN 47405. This research was supported in part by the NSF under Grant No. DMS-0305072.
2000 Mathematics Subject Classiflcation:
Key Words: uniqueness, continuous dependence, Navier-Stokes equations, compressible
flow.
t$76\mathrm{N}10,35\mathrm{Q}30$

155

(3)

$P(\rho)=K\rho,$ $K$

(4)

$1.4$

(5)

$1.5$

$C^{-1}<\rho 0<C\mathrm{a}.\mathrm{e}$$\int\rho_{0}|u_{0}|^{q}dx<\infty$(6). for some constant$\tilde{\rho}$

$1.8$

,

,

is a constant, positive reference density,$u_{0}\in H^{\alpha}(\mathbb{R}^{n})$

(8)

$C$

for some $q>6$ for $n=3$ and $q>2$ for $n=2$ ,$\int[\rho_{0}|u_{0}|^{2}+(\rho_{0}-\rho)^{2}\sim]dx<<1$

where
(7)

constant, $\mu>0$ , and $0<\lambda<5\mu/4$ ,where $\alpha>0$ for $n=2$ and $\alpha>1/2$ for $n=3$ ,$f\in L^{\infty}([0, T];L^{2})$and$\int_{0}^{T}(t^{7}|\nabla f|_{L^{4}}^{2}+t^{5}|f_{t}|_{L^{2}}^{2})dt<\infty$,

then there is a weak solution
to the initial-value problem (1) $-(2)$ in the class$T<\infty$
defined on. The condition in (3) on can be relaxed considerably
and the strict positivity of in (4) and the condition (7) can be dropped entirely for thegeneral existence theory of [5]; all three are required here, however, for reasons which will
be explained later.
$C$$(\rho, u)$$P$$\mathbb{R}^{n}\mathrm{x}[0, T],$$\rho_{0}$The following is our main result:

Theorem: Let $(\rho, u, f)$ andbe weak solutions of (1) in the regularity classcorresponding to respec tive external forces and satisfying (8). Assume also that,
, where and are H\"older conjugates inandand that$C$$(\overline{\rho},\overline{u}, \overline{f})$$f$$L^{1}((0, T);L^{2r}(\mathbb{R}^{n}))$

(9)

where

$\rho_{0}-\overline{\rho}0\in L^{2\mathit{0}}(\mathbb{R}^{n})$$\nabla\Gamma*(\rho-\tilde{\rho}),$$\Gamma$$\overline{f}\in$$\overline{f}$$r$$[1, \infty]$$s$$\nabla\Gamma*(\overline{\rho}-\tilde{\rho})\in L^{1}((0, T);W^{1,\infty}(\mathbb{R}^{n}))$is the fundamental solution of the Laplace operator on$\mathbb{R}^{\mathrm{n}}$,. Then$( \int_{0}^{T}\int_{\mathrm{R}^{n}}|u-\overline{u}|^{2}dxdt)^{1/2}+\sup_{0\leq t\leq T}||(\rho-\overline{\rho})(\cdot, t)||_{H^{-1}(\mathbb{R}^{n})}$(10)$\leq C[||\rho_{0}-\overline{\rho}_{0}||_{L^{2}\cap L^{2\epsilon}}+||\rho_{0}u_{0}-\overline{\rho}_{0}\overline{u}_{0}||_{L^{2}}+(\int_{0}^{T}\int_{\mathbb{R}^{n}}|f-\overline{f}\circ S|^{2}dxdt)^{1/2}]$

156

is explained below in (12)). The constant $C$ depends on the system(the notation
parameters $P,$ , and , on , and on upper bounds for the norms of solutions and forces$\overline{f}\circ S$$\lambda$

$\mu$

$T$

occurring in the above hypotheses. If
by$\overline{f}$then$\int_{0}^{T}t||\nabla\overline{f}(\cdot, t)||_{L\infty}dt<\infty,$$\overline{f}\circ S$

may be replacedin (10).

Details of the proof are given in Hoff [6]. We describe here the main ideas by giving
an overview of the analysis and remarking on several of the hypotheses, particular thecondition in (3) that be the pressure of an ideal isothermal fluid and the condition (9)concerning the Newtonian potentials of the densities. We first note that the smallness
condition in (6), while required for the existence of solutions in the class , plays no role
$P$

$C$

in the proof of uniqueness and continuous dependence.
The key point of the analysis is that solutions with minimal regularity are best compared in a Lagrangean, rather than Eulerian framework. To describe this we first notethat (7) and (9) insure the existence of particle trajectories $X(y, t, s)$ satisfying$\frac{\partial X}{\partial t}(y, t, s)=u(X(y, t, s), t)$

(11)

$\{$

$X(y, s, s)=y$and similarly for . $X(y, t, s)$ is therefore the position at time of the fluid particle whose
position at time is . In particular, the fluid particle in $(\rho, u, f)$ at $(x, t)$ was at $X(x, 0, t)$initially and so corresponds to the particle inwhich at time is at the point$\overline{X}$

$s$

$t$

$y$

$t$

$(\overline{\rho}, \overline{u}, \overline{f})$(12) $1.3.4$$\overline{X}(X(x, 0, t), t, \mathrm{O})\equiv S(x, t)$(this is the appearing in (10)). The “Lagrangean” comparison therefore consists of an$t)=$, where we abuse notation slightly and abbreviate
estimate for. The advantages of the Lagrangean formulation may be understood briefly asFirst,.
from its definition, $S(X(y, t, 0), t)=\overline{X}(y, t, 0)$ , so that
follows.
,
It then follows easily that, if$S$$u-\overline{u}\circ S$$\overline{u}(S(x, t),$$(\overline{u}\circ S)(x, t)$$S_{t}+\nabla Su=\overline{u}\circ S$$w=\overline{u}\circ S$,$(\overline{u}_{t}+\nabla\overline{u}\overline{u})\circ S=w_{t}+\nabla wu$

will bewhich shows that the convective time derivative in the evolution equation for.
equation,comparison
the
form
the
of
Given
of the momentum
the same as that forwiththerefore appears quite natural and proceeds in the usual way: bounds forare obtained by duality from estimates for solutions of the adjoint of the weak
. A bound for the Eulerian difference
can then beequation satisfied byderived via the regularity assumptions and a bound for $I-S$ .
We now discuss three of the estimates that go into the proof, beginning with a boundand integrating in time,
for $I-S$ . Subtracting the equations (11) satisfied by $X$ andwe obtain
$w$

$u$

$u$

$w=\overline{u}\circ S$$u-\overline{u}\circ S$$u-\overline{u}\circ S$$u-\overline{u}$$\overline{X}$$|X(y, t, 0)- \overline{X}(y, t, 0)|\leq\int_{0}^{t}|(u\circ X)(y, s)-(\overline{u}\circ\overline{X})(y, s)|ds$

157

so that$|X( \cdot, t, 0)-\overline{X}(\cdot, t, 0)|_{L^{2}}^{2}\leq t\int_{0}^{t}|(u\mathrm{o}X)(\cdot, s)-(\overline{u}\circ\overline{X})(\cdot, s)|_{L^{2}}^{2}ds$.

Assuming for the time being that $X$ is Lipschitz continuous with respect to , we can thenmake the change of variables $x=X(y, t, 0)$ to obtain the bound$y$(13)2.2$|I-S( \cdot, t)|_{L^{2}}^{2}\leq Ct\int_{0}^{t}|(u-\overline{u}\mathrm{o}S)(\cdot, s)|_{L^{2}}^{2}ds$

Next we apply this estimate to obtain a bound for
as a consequence of the conservation of mass,(14)$\mathrm{a}.\mathrm{e}$.

$||(\rho-\overline{\rho})(\cdot, t)||_{H^{-1}}$. First note that,$\rho(X(y, t, s), t)\det\nabla_{y}X(y, t, s)=\rho(y, s)$. (this is the precise formulation of the more intuitive statement that$\rho(\cdot, s)dy.)$$\rho(\cdot, t)dX=$

Thus by (12) and (14),$\det\nabla S(x, t)=\det(\nabla_{y}\overline{X}(X(x, 0, t), t, 0))\det(\nabla_{y}X(x, 0, t), t, 0))$$=\rho(x, t)/\overline{\rho}(S(x, t),$ $t)$

Then for$\varphi\in D(\mathbb{R}^{n})$.

and fixed,$t$$| \int(\rho-\overline{\rho})\varphi dx|=|\int[\rho\varphi-(\overline{\rho}\circ S)(\varphi\circ S)\det\nabla S]dx|$$=| \int\rho[\varphi-\varphi\circ S]dx|$$\leq C||\nabla\varphi||_{L^{2}}||I-S||_{L^{2}}$.

It then follows from (13) that(15)$\mathrm{z}.3$$\sup_{0\leq t\leq T}||(\rho-\overline{\rho})(\cdot, t)||_{H}^{2}-1\leq CT\int_{0}^{T}\int|u-\overline{u}\circ S|^{2}dxdt$.

To complete the proof we need to obtain a bound for the integral on the right sideof (15) in terms of the left side and differences in the initial data and forces. This is themost difficult and technical part of the analysis, and, as indicated earlier, proceeds by
duality by subtracting the weak forms of the equations satisfied by the different solutions
and choosing the test function to satisfy a suitably mollified adjoint equation. Complete
details are given in Hoff [6]. We shall content ourselves here with the observation that the
momentum equation, which may be written$[ \rho(\frac{\partial}{\partial t}+u\cdot\nabla)-\mu\Delta-\lambda \mathrm{d}\mathrm{i}\mathrm{v}]u=-\nabla P(\rho)+\rho f$

,

158

shows thatdifferences in velocities are driven bydifferences in pressuresanddifferences in forces . We therefore anticipate an estimate of the form$L^{2}$$H^{-1}$$u$$L^{2}$$P(\rho)$$f$$\int_{0}^{T}\int|u-\overline{u}\circ S|^{2}dxdt\leq\ldots+CT.\sup_{0\leq t\leq T}||P(\rho(\cdot, t))-P(\overline{\rho}(\cdot, t)||_{H}^{2}-1$,

where the ellipsis indicates norms of differences in forces and initial data. This together
with (15) then completes the proof provided that $P=K\rho$ , which is the assumption required
to insure thatdifferences in pressures are equivalent todifferences in densities.
We remark that this is the only point in the analysis where this condition is required. Moregeneral pressures could be included in this analysis, but only at the expense of having toimpose unreasonably strong conditions on the regularity of the solutions being compared.
We now discuss briefly the requirement (9) concerning the Newtonian potentials of
the densities. As indicated above, we require at several points of the analysis that theparticle trajectories $X(y, t, s)$ be Lipschitz continuous functions of their initial positions. This follows from (11) when the Lipschitz constants with respect to space of andare integrable are time, that is, when
. Now by adding andsubtracting ternis we obtain that$H^{-1}$$H^{-1}$$u$$y$$\overline{u}$$u,\overline{u}\in L^{1}((0, T);W^{1,\infty}(\mathbb{R}^{n}))$$(\mu+\lambda)\Delta u^{j}=((\mu+\lambda)\mathrm{d}\mathrm{i}\mathrm{v}u-P(\rho))_{x_{j}}+(\mu+\lambda)(u_{x_{k}}^{j}-u_{x_{j}}^{k})_{x_{k}}+(P(\rho)-P(\tilde{\rho}))_{x_{J}}$ ,$\equiv(\mu+\lambda)\Delta u_{F,\omega}^{j}+(\mu+\lambda)\Delta u_{P}^{j}$where
and summation over repeated indices is understood.
(The standard notation is thatand
) For thesolutions constructed in [5], and are more regular than the summands which define
them, this fact being a consequence of the cancellation of singularities predicted by the
Rankine-Hugoniot conditions applied to (1) (see the introduction to [4], for example). A
straightforward argument based on standard elliptic theory and Sobolev estimates ([6],, as required. (The hypotheses (7)section 1) then shows thatand (8) are needed here to insure thatis integrable in the initial layer
near $t=0.$ ) On the other hand, the relation
implies that
only for $1<p<\infty$ , not for $p=1$ or $p=\infty$ . We must
therefore impose (9) as an posteriori condition on the solutions being compared. We donote, however, that a large class of piecewise smooth solutions exhibiting codimension-one
singularities and satisfying the condition (9) is constructed in Hoff [4].
$(\mu+\lambda)u_{P}^{j}=\Gamma_{x_{j}}*(P(\rho)-P(\tilde{\rho}))$$F=(\mu+\lambda)\mathrm{d}\mathrm{i}\mathrm{v}u-P(\rho)$$F$$\omega^{j,k}=u_{x_{k}}^{j}-u_{x_{\mathrm{j}}}^{k}.$$\omega$$\nabla u_{F,\omega}\in L^{1}((0, T);L^{\infty}(\mathbb{R}^{n}))$$||\nabla u_{F,\omega}(\cdot, t)||_{L\infty}$$(\mu+\lambda)\Delta u_{P}^{j}=(P(\rho)-P(\tilde{\rho}))_{x_{j}}$$||\nabla u(\cdot, t)||_{L^{\mathrm{p}}}\leq C_{p}||P(\rho)-P(\tilde{\rho})||_{L^{\mathrm{p}}}$

We now turn to the question of uniqueness and continuous dependence for more general, large-energy solutions (again see Feireisl [2] and [3] and Lions [9] for the corresponding
existence theory.) These solutions are much less regular than those in the class and consequently are not amenable to the analysis outlined above. Additionally, it is known thatcertain anomalies can arise in solutions in which densities vanish on nonnegligible sets,
and it appears likely, although it is not presently known, that large initial energies cancause spontaneous cavitation even when the initial density is strictly positive. This raises
the possibility that the known set of large-energy weak solutions includes nonphysicalsolutions, even when initial densities are strictly positive.
$C$

159

To be more specific, we say that a solution
is locally momentum conserving if,$E$wheneverandare bounded open sets in -space withand with.
in $(V-E)\cross[t_{1}, t_{2}]$ , then the change in the momentum of the fluid in $E$ from time to$(\rho, u)$$V$$\overline{E}\subset V$$x$$\rho=0\mathrm{a}.\mathrm{e}$$t_{1}$time$E$$t_{2}$

should be the impulse$\int_{t_{1}}^{t_{2}}\int_{E}\rho fdxdt$

applied by the external force to the fluid in. Weak solutions violating this condition do in fact exist, one such being constructedin Hoff and Serre [7]: initial data
is given corresponding to two fluids initiallyseparated by a vacuum; it is shown that, if, then the limit asis replaced byof the corresponding perturbed solutions exists and is a weak solution in the entire
physical space, but fails to be locally momentum conserving. This failure is a consequenceof the fact that the momentum equation in (1) determines a velocity even in open setsin which $\rho=0$ , or, stated differently, the model posits nonzero viscous forces generated
from the motion of nonexistent fluid particles. It is also shown in [8] for the above example
that there is a different solution corresponding to the same initial data which locallymomentum conserving, but this solution is defined only on the support of the density andcannot be extended as a weak solution to the entire physical space.
These observations suggest a remedy: the viscous force terms in (1) should be madeto vanish when the density vanishes and the momentum equation should be imposed only
on the support of the density. We recall that kinetic theory predicts that, to first order,
and are functions of temperature . If it is assumed that the entropy of each fluid
particle is fixed, that is, that the flow is isentropic, then will be a function of , andunder reasonable assumptions on the functional relationships, and will be zero whenis zero. Whether or not the isentropic assumption is consistent with the presence ofviscosity is in some doubt, however. If we instead regard and as functions of 9 but
assume that is constant, that is, that the flow is isothermal, then and are effectively
constant. This is the case we consider, and for this case we propose that the second
equation in (1) should be modified by replacing the termson the right, whereside byis one or zero according as is positive or zero.
To be more specific, we propose that, for viscous, isothermal flow governed by the
Navier-Stokes system (1), there should be established an existence, uniqueness, continuous
dependence theory as follows: given initial density
whichand momentum density
vanishes when $\rho_{0}=0$ ( $m$ is a surrogate for ), both defined, let us say, on , there should
exist a unique pair $(\rho, m)$ withandsatisfying the weak form of the mass equation$(\rho 0, u_{0})$$\rho 0+\delta$$\rho_{0}$$6\downarrow 0$

$u$

$is$

$\lambda$$\theta$$\mu$$\theta$$\rho$$\lambda$$\mu$$\rho$$\lambda$$\mu$$\theta$$\lambda$$\mu$$\mu u_{x_{k}x_{k}}^{j}+\lambda u_{x_{j}x_{k}}^{k}$$\chi(\rho)$$(\chi\mu u_{x_{k}}^{j})_{x_{k}}+(\chi\lambda u_{x_{j}}^{k})_{x_{k}}$$\rho$$m_{0}$$\rho_{0}$$\mathbb{R}^{3}$$\rho u$$\rho,$$m\in C([0, \infty);L_{loc}^{1}(\mathbb{R}^{n}))$$m^{2}/\rho\in L_{loc}^{1}(\mathbb{R}^{n}\mathrm{x}[0, \infty))$$\int\rho\varphi dx|_{t_{1}}^{t_{2}}=\int_{t_{1}}^{t_{2}}\int_{\mathrm{R}^{n}}(\rho\varphi_{t}+m\cdot\nabla\varphi)dxdt$

for Lipschitz test functions having compact support; for each $t>0$ there should be an
open set. on the complement of ;. on andsuch that, so
the set $G=\{(x, t):x\in G_{t}, t>0\}$ should be open and $u\equiv m/\rho$ should be in$\nablau\inD’(G)$;
equation
finally, the weak form of the momentum
thatshould hold on in$\varphi$$G_{t}$$\rho(\cdot, t)>0\mathrm{a}.\mathrm{e}$$G_{t}$$G_{t}$$\rho(\cdot, t)=0\mathrm{a}.\mathrm{e}$$L_{loc}^{1}(G)$

$G$

160

the sense that(16)2.4$\int_{\mathrm{R}^{n}}m\cdot\psi dx|_{t_{1}}^{t_{2}}=\int_{t_{1}}^{t_{2}}\int_{\mathrm{N}^{n}}[m\cdot(\psi_{t}+\nabla\psi u)+(P(\rho)-\tilde{P})\mathrm{d}\mathrm{i}\mathrm{v}\psi$$-\chi c\mu\nabla u^{j}\cdot\nabla\psi^{j}-\chi_{G}\lambda(\mathrm{d}\mathrm{i}\mathrm{v}u)(\mathrm{d}\mathrm{i}\mathrm{v}\psi)+\rho f\cdot\psi]dxdt$

is the characteristic function of and. It is easy towherecheck that solutions in this sense are locally momentum conserving.
A version of this program has been carried out in the simplest case thatwith the obvious modifications required to accommodate no-slip boundary conditions. The
is nonnegative and bounded, that there is an
hypotheses on the initial data are that. on the complement of ,. on andopen set $G_{0}\subset(0,1)$ with$G$$\chi_{G}$$\psi\in D(\mathbb{R}^{n}\mathrm{x}(-\infty, \infty))^{n}$$\Omega=(0,1)\subset \mathbb{R}^{1}$$\rho 0$$\rho_{0}>0\mathrm{a}.\mathrm{e}$that the initial energy$\int_{G_{\mathrm{O}}}\rho_{0}u_{0}^{2}dx$$G_{0}$$G_{0}$$\rho 0=0\mathrm{a}.\mathrm{e}$is finite, and that there exists a constant$\overline{u}^{2}$such that$\int_{E}[\frac{1}{2}\rho_{0}u_{0}^{2}+(\rho_{0}-\tilde{\rho})^{2}]dx\leq\frac{1}{2}\overline{u}^{2}\int_{E}\rho_{0}dx$. The latter condition insures that the there are no fluidfor all measureable sets
particles of fixed energy and vanishingly small mass, and the second condition disallows
data with mass concentrated on Cantor-like sets.
Complete details of this work will be presented elsewhere. We do note, however, that
locally momentum conserving solutions in this sense are in fact not unique: an additional
condition, which might be called “maximal Lagrangean structure” is required. Rather thangiving precise technical details here, we instead describe an example which illustrates theissue in a fairly simple way. First recall the result of Serre [10] showing that, given a fluid
initially occupying a one-dimensional interval
surrounded entirely by vacuum,of (1) defined on the set $\{(x, t) : a(t)<x<b(t), t>0\}$ ,there is a weak solution
where $a(t)$ and $b(t)$ are fluid particle paths satisfying $\dot{a}=u(a, t)$ and $\dot{b}=u(b, t)$ defining
the free boundaries of the fluid, subject to the weak form of momentum conservation
boundary conditions $(\mu+\lambda)u_{x}-P=0$ at $x=a(t)$ and $b(t)$ . It is easily checked that
this solution is locally momentum conserving in the sense described above. We can apply
this result to demonstrate the nonuniqueness of locally momentum conserving solutions
as follows. First apply Serre’s result to a fluid with strictly positive density initially, perform a Galilean translation of the -axis to transform
occupying
the resulting solution to one for which $a(t)>0$ for all $t>0$ , then reflect about the t-axis
to obtain a locally momentum conserving solution
witheven in
odd in, and $G_{t}=(-b(t), -a(t))\cup(a(t), b(t))$ and $G_{0}=(-1,1)$ . A second solution
can be obtained more simply by applying Serre’s result directly to the same initial dataon $(-1,1)$ . For this second solution, $G_{t}=(-c(t), c(t))$ for a free boundaryfunction , and since
is strictly positive on. Our theory picksoutas the unique momentum conserving solution having the property that there isuniqueparticle
trajectory emanating from each point of . Observe that this conditionafails forsince there are two distinct particle trajectories for this solution emanatingfrom $x=0$ .
$E\subset G_{0}$$(a(\mathrm{O}), b(\mathrm{O}))$$(\rho, u)$$(a(\mathrm{O}), b(\mathrm{O}))=(0,1)$$x$$(\rho_{1}, u_{1})$$\rho_{1}$$x,$$u_{1}$$(\rho_{2}, u_{2})$$x$$(\rho_{1}(0), u_{1}(0))$$c$$\rho_{2}(\cdot, t)$$G_{t},$$(\rho_{1}, u_{1})\neq(\rho_{2}, u_{2})$$(\rho_{2}, u_{2})$$\overline{G}_{0}$$(\rho_{1}, u_{1})$161

We conclude by remarking that the extension of this program to three space dimensions promises to be an especially interesting and challenging mathematical project, owing
to the increased complexity of the geometry of the set and to the fact that spontaneous
cavitation may be possible after the initial time, a phenomenon which cannot occur for
one-dimensional flows (Hoff and Smoller [8]).
$G$

References[1] Raphael Danchin, Local theory in critical spaces for compressible viscous and heatconductive gases. Comm. Partial Differential Equations 26 (2001), no. 7-8, 1183-

1233.

[2] Eduard Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in
Math. and Appl. 26, 2004.
[3] Eduard Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,
Indiana Univ. Math. J. 53, number 6 (2004), 1707-1740.
[4] David Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two
space dimensions, Comm. Pure Appl. Math. LV (2002), 1365-1407.
[5] David Hoff, Compressible Flow in a Half-Space with Navier Boundary Conditions, toappear in J. Math. Fluid Mech.

[6] David Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow, to appear in SIAM J. Math. Ana.

[7] David Hoff and Denis Serre, The failure of continuous dependence on initial data for
the Navier-Stokes equations of compressible flow, SIAM J. Appl. Math. 51 (1991),
887-898.
[8] David Hoff and Joel Smoller, Non-formation of vacuum states for compressible NavierStokes equations, Comm. Math. Physics 216 (2001), 255-276.
[9]. P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, Oxford Lecture Series inMathematics, 10 (1998).

[10] Denis Serre, Solutions faible globales des \’equations de Navier-Stokes pour un fluidecompressible, C.R. Acad. Sci. Paris t. 303, S\’erie I, 13, 1986.