Tong Yang- Singular Behavior of Vacuum States for Compressible Fluids

8/3/2019 Tong Yang- Singular Behavior of Vacuum States for Compressible Fluids

1/21

Singular Behavior of Vacuum States for Compressible Fluids

Tong Yang

Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Received and accepted

Dedicated to Roderick S.C. Wong on the occasion of his 60th birthday

Abstract

In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by

illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations,

Euler-Poisson equations and Navier-Stokes equations. The main concern is the well-posedness of the problem

when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the

gas. Furthermore, the relation of the solutions for gas dynamics with vacuum to those of Boltzmann equation will

also be discussed. In fact, the results obtained so far for vacuum states are far from complete and satisfactory.

Therefore, this paper can only be served as an introduction to this interesting field which has many open and

challenging mathematical problems. Moreover, the problems considered here are limited to the authors interestand knowledge in this area.

Key words: Vacuum singularity, Euler-Poisson equations, Euler equations, Navier-Stokes equations, Boltzmann equation.PACS:XXXXX

1. Introduction

The study on the singularity associated with vacuum state in fluid dynamics can be traced back at

least to the collected work of von Neumann, [69]. However, there is still no satisfactory answer so far tothis important physical problem. One of the reasons could be the existing equations or systems provideeither trivial solutions in this region or contain the singularity which prevents the use of present theory.

In this paper, we will survey some recent work in this direction on some fluid dynamics systems andthe Boltzmann equation. One can see that there are more open problems than those which have been

Email address: [email protected].

Preprint submitted to Elsevier Science 5th January 1970


2/21

solved. Therefore, we expect this paper can provide the readers some updated information on the studyof the existence, stability, uniqueness and interface evolution of the interface separating the gas andvacuum, called vacuum boundary, to the solutions with vacuum states. Since we are mainly interestedin the singular behavior of the solutions with vacuum, the solutions need to have some regularity for

this purpose. For this, those work on weak solutions obtained by compensated compactness will not bediscussed here.

In the following sections, we will first discuss the behavior of the solutions with vacuum states to somefluid dynamics systems, which include the Euler equations, Euler-Poisson equations and Navier-Stokesequations. Finally, we will also present some special solutions to the Boltzmann equation related to thefluid dynamics in this aspect. To be more precise, we will give some brief discussions on each topic asfollows.

It is well known that the system of Euler equations becomes non-strictly hyperbolic when vacuumappears. Therefore, many existence theory for strictly hyperbolic system can not be applied here andricher nonlinear phenomena are expected. In fact, whether the vacuum state appears in finite time ifinitial data contain no vacuum states is a main step to obtain the global existence of entropy solutions forlarge data. This is a long standing open problem which we will not discuss it in this paper. What we willshow in Section 2 is that the solution near vacuum boundary has a canonical singular behavior, called thephysical boundary condition. This singularity prevents the application of the existing theory on existencefor hyperbolic systems even locally in time. And the linearized version of this singularity is of the Keldyshtype degenerate hyperbolic equation. This is different from the Tricomi type singularity which is betterunderstood. By looking at the system of Euler equations with the linear frictional damping, it is shownthat it has some similar behavior on the evolution of the vacuum boundary to the one for porous mediaequation. The latter has been well investigated. However, the former is almost open because it comesfrom a system instead of a single equation.

The system of Euler-Poisson equations consisting of a hyperbolic system coupled with a Poisson equa-tion. It is a model system for the time evolution of the self-gravitating gas coming from astrophysics.There has been a lot of study on this system when the initial data has compact support starting from1800s. Since this system can be viewed as a model for the time evolution of gaseous stars, the existenceand stability of the stationary solutions are among the important research topics. In the linear theory,

there are classical Chandrasekhars principle and Eddingtons variational principles for the stability of thestationary solutions with spherical symmetry. By assuming the gas to be polytropic, according to thesetwo principles, it is shown that the linear stability depends on the adiabatic constant . When < 4

3, the

stationary solution is instable, while it is stable when > 43

in three dimensional space. However, how tojustify this for the fully nonlinear system is not complete yet. What we can prove mathematically for thenonlinear system is merely on the stability part. The instability which is more important in physics andinteresting in mathematics is now still based on the linearized model.

When the evolution of the gas or fluid becomes viscous and heat conductive, one of the typical systemsis the system of Navier-Stokes equations. When the initial density is of compact support, a lot of workhas been done on the existence, stability and uniqueness of these solutions. And the time evolution ofthe vacuum boundary is also investigated under some special conditions. Most of these works are on thecase when both the viscosity coefficient and the heat conductivity are constants. However, by deriving the

Navier-Stokes equations from the Boltzmann equation, the viscosity and heat conductivity are functionsof temperature. Since there is no a priori estimate on the temperature for vacuum states, it is not clearhow to pose the problem when this physical dependence on temperature is included. Hence, there isalmost no result on regular solutions to the full Navier-stokes equations when the viscosity and heatconductivity are functions of temperature and the density function is of compact support. However, ifone considers only the isentropic gas, the dependence on temperature becomes the dependence on thedensity. In this direction, there are some work on the existence and uniqueness. But even in this case,

2


3/21

there is no description of the large time behavior of the solution because most of the estimates are timedependent coming from Granwall inequality. To overcome this, new techniques in analysis are needed andnew estimates are also required. We should mention that the large time behavior of the solutions whenthe viscosity is constant can be given in details including the expansion rate of the gas region. And the

non-appearance of vacuum states is proved if initial data contains no vacuum. However, this result cannot be directly generalized to the case when the viscosity depends on the density function.

Finally, Boltzmann equation is a fundamental equation for the statistical time evolution of a largenumber of particles in rarefied gas interacting through various physical assumptions on collisions. Whenthe Knudsen number ( or the mean free path) tends to zero, one can formally obtain the systems ofEuler equations and Navier-Stokes equations in different orders of approximations. These limit procedureshave been justified in some mathematical settings. In fact, the study on the relations between these fluiddynamic systems and the Boltzmann equation has been one of the main topics in this field since Hilbertraised his sixth problem, Mathematical treatment of the axioms of physics in his famous lectureMathematical Problems at ICM in 1900.

To illustrate the relation between Boltzmann equation and the systems for fluid dynamics with vacuumstates, we will present some particular solutions to the Boltzmann equation when the initial data connectto vacuum at

|x|

=

with the gravitational force between the particles in the gas and a fixed regionof finite mass. We will focus on the hard sphere model of collision. The main objective is to study thesolution behavior when tends to zero which should be related to some known results on the solutionsto the corresponding fluid dynamic systems, such as Euler equations and Navier-Stokes equations withexternal force. These problems have physical significance besides its mathematical interest. For example,when we consider the atmosphere around a star, such as our earth, in the outer space the density ofthe atmosphere is rarefied and the particle experiences the gravitational force from the earth. If we firstignore the radiation from the sun and from the reflection on the surface of the earth, then this situationcan be modelled by the above equation. The detailed structure of the solution in this case will be helpfulto understand atmosphere distribution in this simplified setting. Notice that the general existence, largetime behavior, stability and the asymptotic phenomena when 0 for the Boltzmann equation withforce and non-trivial time asymptotic nonlinear profile connecting to vacuum are still not known. Whenthe solution tends to a trivial state, i.e., f(x,t,)

0, the existence results can be found in [32], [25] for

both the cases with and without force using the method introduced in [31].As mentioned earlier, there are a lot of open questions on vacuum problems for systems of compressible

fluids and the Boltzmann equation. Vacuum states have its unique singularity that have challengingmathematical difficulty and mistry. The above four models only cover part of the works on this subject.Similar problems can also be found in the study of the magnetohydrodynamics where Alfen wave is relatedto vacuum, and some particle systems.

2. Euler Equations

The system of Euler equations consisting of conservation of mass, momentum and energy is one of the

typical hyperbolic systems in fluid dynamics. When the solution contains vacuum, the system becomesdegenerate in the sense that some of its characteristics coincide. If there is no external force, a stateconnects to vacuum through rarefaction wave, not shock wave, cf. [69]. And it is also believed that theappearance of vacuum in finite time does not happen if there is no vacuum initially even though a rigorousproof is not known. The intuitive thinking is that the appearance of vacuum state comes only from theinteraction of two large rarefaction waves of different families. And numerical computation suggests thatin this case the vacuum does not appear in the interaction zone in finite time, [42].

3


4/21

In this section, we will try to illustrate the time evolution of the vacuum boundary under the phys-ical boundary condition for some simple physical model. More precisely, we assume that the governedequations for the gas dynamics are Euler equations with linear frictional damping. For physical interpre-tation of the linear damping term, please refer to [60]. For the previous works on Euler equations related

to vacuum, please see [24,46,47,51,54,55,56,73,77,78] and references therein. For the case with frictionaldamping, the canonical vacuum boundary behavior corresponds to the case when the space derivative ofthe enthalpy is bounded but not zero which is different from the one for centered rarefaction wave withoutforce, cf. [41]. That is, the pressure has a non-zero finite effect on the evolution of the vacuum boundary.However, for this canonical (physical) case, the system is singular in the sense that it is not symmetrizablewith regular coefficients so that the local existence thoery for the classical hyperbolic systems can notbe applied. Moreover, the linearized equation at the boundary gives a Keyldish type equation for whichgeneral local existence thoery is still not known. Notice that this linearized equation is quite differentfrom the one considered in [65] for weakly hyperbolic equation which is of Tricomi type.

To capture this singularity in the nonlinear settting, a transformation was introduced in [47] and somelocal existence results for bounded domain were also discussed. The transformed equation is a secondorder nonlinear wave equation of an unknown function (y, t) with coefficients as functions of y1(y, t)and (0, t)

0. Along the vacuum boundary y = 0 in this setting, the physical boundary condition implies

that the coefficients along the vacuum boundary are functions of y(0, t) which are bounded and awayfrom zero. Hence, the wave equation has no singularity or degeneracy. However, its coefficients have theabove special form so that the local existence thoery developed for the classical nonlinear wave equationcan not be applied directly, [29,35].

Through the transformation capturing the singularity at the vacuum boundary under the physicalboundary condition at the vacuum interface, the coefficients in the reduced wave equation which arepower functions of y1 correspond to the fractional differentiations of . Thus, one can apply theLittlewood-Paley theory based on Fourier theory to obtain some local existence results. Indeed, this wasdone when the initial data is a small perturbation of a planar wave solution where the enthalpy is linearin the space variable, [72].

To appreciate this analysis, consider the one dimensional compressible Euler equations for the isentropicflow with frictional damping in Eulerian coordinates

t + (u)x = 0,

ut + uux + p()x = u, (2.1)

where , u and p() are density, velocity and pressure respectively. And the linear frictional coefficient isnormalized to 1. When the initial density function contains vacuum, the vacuum boundary is definedas

= cl{(x, t) | (x, t) > 0} cl{(x, t) | (x, t) = 0}.Since the second equation in (2.1) can be rewritten as

ut + uux + ix = u,

where i is the enthalpy. One can see that the term ix represents the effect of the pressure on the particlepath, especially the vacuum boundary. It is shown in [46,54,56,77,78] that there is no global existence ofregular solutions satisfying ix 0 along the vacuum boundary. That is, in general, i is not C1 crossing thevacuum boundary. Hence, the canonical behavior of the vacuum boundary should satisfy the conditionix = 0 and is bounded. This special feature of the solution can be illustrated by the stationary solutionsand some self-similar solutions, also for different physical systems, such as Euler-Possion equations forgaseous stars and Navier-Stokes equations discussed in the next two sections.

4


5/21

When ix 0 at the vacuum boundary, the proof on the non-global existence of regular solutions isbased on the estimation of the support of the density function which should be sub-linear, and the timeevolution of the second inertia of the gas w.r.t. the origin, i.e., H(t) =

|x|2 dx. For general system,

a non-global existence of regular solution violating the physical boundary condition is given in [78] by

considering n + 2 entropy and entropy flux pairs for a symmetrizable system in n-dimensional space.Notice that the charateristics of Euler equations is u c. And for isentropic polytropy gas when

p() = k for a positive constant k and the adiabatic constant > 1, we have i = c2

1. Hence the

characteristics are singular with infinite space derivatives at the vacuum boundary under the physicalboundary condition. This kind of singularity yields the smooth reflection of the charactertics on thevacuum boundary and then thus causes analytical difficulty.

There is an interesting way to view the canonical boundary condition from the study of porous mediaequation. It is known that the Euler equations with linear damping behave like the porous media equation,t = p()xx, at least away from vacuum when t , cf.[30]. Notice also that there are some correspondingresults with vacuum in the weak sense by using compensated compactness, cf. [ ?]. Since the vacuumboundary is not well-defined in the case, we will not discuss them here.

For porous media equation, there is a well established theory on the waiting time problem and boundary

singularities, [2,37]. It can be briefly explained as follows. Consider a parabolic equation

Vt(x, t) = (Vm(x, t))xx, m > 1. (2.2)

For initial data of compact support, one can show that the canonical boundary behavior is the one with(Vm1)x being bounded and non-zero. That is, if initially, (V)x is bounded and non-zero at the edge ofthe support, then we have the following cases on the value of :

1. When 0 < m12

, there exists a solution with the same behavior(i.e. unchanged) up to a finitetime, and then it behaves like = m 1 afterwards.

2. When m12 < < m 1 or > m 1, then the solution has the behavior with = m 1 for t > 0.

3. When = m 1, the solution has the same behavior for all time.

If we compare this kind of waiting time behavior to the system (2.1), then one can see that they have alot of similarity. Then the canonical behavior corresponds to (1)x being bounded and non-zero at thevacuum boundary. It turn out that the conjecture on the behavior of ()x being bounded and non-zeroat the vacuum boundary for (2.1) can be stated by introducing V by and m by respectively. However,the rigorous proof on this waiting time problem for (2.1) is almost open because it is for a system notjust for a scalar equation. The techniques used in proving the theorem for (2.2) may not work here.

We now come back to the local existence of solutions with physical vacuum boundary for Euler equationswith linear damping. Since our concern is on the behavior of the solution related to vacuum and any shockwave vanishes at vacuum [48], it is reasonable to consider the problem without shock waves. In fact, anyshock wave appears initially or in finite time will decay to zero in time because of the dissipation from

the linear damping. By using the special property of the one dimensional gas dynamics, we can rewritethe system (2.1) by using Lagrangian coordinates to make all the particle paths, in particular the vacuumboundary, as straight lines. (2.1) in Lagrangian coordinates takes the form

vt u = 0,ut + p(v) = u, (2.3)

5


6/21

where v = 1 is the specific volume and =x0

(y, t)dy. Notice that the physical sigularity, ix = 0 butbounded, along the vacuum boundary in Eulerian coordinates corresponds to 0 < |p(v)| < in theLagrangian coordinates.

In order to symmetrize the system (2.3) with the physical boundary condition, the following coordinate

transformation was introduced in [47], = y

21 .

Here for simplicity, we assume that the initial density function satisfies 0(x) = 0 for x < 0 in the Euleriancoordinates. Then the system (2.3) can be rewritten as

(v)t + uy = 0,

ut + (v)y = u, y > 0, t > 0, (2.4)

where (v) =2

k

1 v12 , and

=( 1)k

(vy

21 )

+12 = (y1)

+11 ,

for some positive constant . Notice that near the vacuum boundary, both (v)y and are bounded andnon-zero under the physical boundary condition. And this allows the application of the Hardy-Littlewood-Paley theory in the proof of local existence, cf. [8,13].

In fact, there are two families of global solutions connecting to vacuum constructed in [51]. One hasp(v) as a linear function in with constant velocity, while the velocity in the other solution is linearfunction in space variable and p(v) is a polynomial of second order. It was shown that the solutions inthe second family converge to the Barenblatt solutions to the porous media equation (2.2) as time tendsto infinity. The local existence of solutions for a small perturbation to the solutions of the first family wasproved in [72].

3. Euler-Poisson Equations

In this section, we will consider the Euler-Poisson equations which can be viewed as a model for thetime evolution of self-gravitating gaseous stars. The system is

t+x (u) = 0,

(u)

t+x (u u) + xp = x ,

(s)

t+x (su) = 0, (3.1)

x = 4g, (3.2)

where t 0, x R3

, is the density, u R3

the velocity, s is the entropy, p(, s) the pressure of the gas, the potential function of the self-gravitational force, and g the gravitational constant. In the following,we consider the equation of state for a polytropic gas:

p = kes.

We are interested in solutions representing the gaseous stars in that the density (x, t) has compactsupport in the space x R3. (3.1) is the system of compressible Euler equations with gravitational force

6


7/21

provided by the gravitational potential , while is determined by the density distribution of the starthrough the Poisson equation (3.2). Notice that the adiabatic exponent = 5/3 for monatomic gas, 7/5for a diatomic gas and 1 + 0 for heavier molecules. Other values of have significance of their own,cf. [12,40]. For instance, and the value = 4/3 is important because of the quantum effects, [6]. In the

following discussions, we will see that plays an important role in the existence, stability, and uniquenessof the stationary solutions. For instance, the linear theory by the astrophysicists shows that sphericalsymmetric stationary solutions are unstable when 4

3, while stable when > 4

3.

The above stability criterion was believed long time ago since the work of A. Ritter in nineteenthcentury. And the justification by linear theory was obtained by Chandrasekhar and Eddington under theassumption of spherical symmetry stated as variational principles. For completeness of the presentation,we briefly include their arguments as following, cf. [6].

Consider the system (3.1)-(3.2), and assume that there is a spherically symmetric stationary profile(0, u0) satisfying

u0 = 0,d0(r)

dr< 0, 0(rm) = 0,

where r is the radius and rm is a positive constant. By linearizing the system around (3.1)-(3.2) as

= 0 + 1 , p = p0 + p1, u = u1, = 0 + 1,the leading term yields

1t

+ (0u1) = 0, (3.3)u1t

= h1 1,1 = 4g1,

where h1 = (dPd )0

10

. Here ()0 means the value in the bracket is evaluated at the stationary state (u0, 0).By taking the normal modes as

1(r, t) = Re{a(r)eiwat}, u1(r, t) = Re{ua(r)eiwat},1(r, t) = Re{a(r)eiwat}, h1(r, t) = Re{ha(r)eiwat},

it is straightforward to derive the following identity:

w2a|wa|2

0|ua|2d3r =

|( d

d)0||a|2d3r g

d3rd3r

|r r| a(r)a(r), (3.4)

where r and r are vector in R3, and represents the complex conjugate. Notice that if w2a 0, thenthe normal mode is stable and oscillates with frequency wa. Otherwise, it is unstable. This is the followingChandrasekhars variational principle.

Chandrasekhars Variational Principle: A barotropic star with d0dr < 0 and 0(rm) = 0 is table ifthe quantity

E[1] =|( dd )0||a|2d3r g d

3rd3r

|r r| a(r)a(r

),

is non-negative for all real functions a(r) that conserve the total mass of star, i.e.,

1d3r = 0.

According to the Antonov-Lebovitz Theorem, any spherical symmetric barotropic star with d0dr < 0and 0(rm) = 0 is stable to all perturbations that are not spherically symmetric. Therefore, one can onlylook at the spherically symmetric perturbation to study the stability. Set

7


8/21

ua = vaer, ha = ( dhadr

)er, etc.,

where er is the unit vector in radius direction. Then by defining the local adiabatic index

(r) = (

d lnp

d ln )0,

and the amplitude of the fractional displacement in the radius of a fluid element

a =ivawar

,

(3.3) gives the following identity obtained by Eddington in 1918:

d

dr(p0r

4 dadr

) + {w2a0r4 + r3d

dr[(3 4)p0]}a = 0.

And then

w2a

rm0

0r4|a|2dr =

rm0

{p0r4|dadr|2 r3|a|2 d

dr[(3 4)p0]}dr,

which yields the Eddingtons variational principle.

Eddingtons Variational Principle: A barotropic star with d0dr < 0 and 0(rm) = 0 is stable toradial perturbations if the quantity

F[] =

rm0

{p0r4|dadr|2 r3|a|2 d

dr[(3 4)p0]}dr,

is non-negative for all function a(r).

Since for isentropic and polytropic star, i.e. p = k, F is reduced to

F[] =

rm0

p0r4|da

dr|2 (3 4)

rm0

r3|a|2 dp0dr

dr,

the fact that dp0

dr

< 0 implies that the star is stable for

4

3

and unstable otherwise.Now we turn to the nonlinear theory, consider again first the stationary solutions of non-moving gas

u = 0, to (3.1) and(3.2)

p() = , = 4g,p = kes(x). (3.5)

Let the gas is confined to a spatially bounded domain R3, that is,

= 0, > 0 in . We now

look at the existence, uniqueness and stability of the stationary solutions to (3.5). Through a nonlineartransformation,

Q(, x) =

1

1es(x),

(3.5) can be reduced to a semi-linear elliptic equation with a nonlinear function of Q(, x) as a source,

es Q

1

(S)Qe s = 4g

1

Qes 11

es . (3.6)

for which the classical theory of the elliptic equation leads us to existence theorems under some assump-tions on and s(x). In fact, it was proved in [15] that non-trivial positive solutions of (3.6) exist for

8


9/21

65

< < 2. Moreover, for isentropic gases, the behavior of the solutions, such as uniqueness, multiplicityand radial symmetry can be discussed more explicitly, cf. [15].

There are extensive studies of weak solutions with shocks for conservation laws and also for Euler-Poisson equations. On the other hand, the regularizing effect of the Poisson equation indicates that the

interesting singular behaviour of the solutions is not mainly with shocks. Same with the Euler equationswith damping, there is the interesting singular behavior of solutions for the Euler-Poisson equations atthe interface separating the gas and vacuum, [15] and [46], that is, for isentropic fluid,

< p , n >

=< c2,n >

< 0, and bounded,

where c is the sound speed. This singular behavior of the solution at the boundary again prevents thesymmetrization of the system and the classical local existence theory for symmetrizable hyperbolic sys-tems can not be applied directly. In the study of the stationary solutions to the Euler-Poisson equationsin any bounded region, the physical boundary singularity is guaranteed by Hopfs maximal principle.Moreover, one can show that the integral of the quantity (c2,n) along the boundary gives exactly thetotal mass up to a constant factor. As for the nonlinear stability of the above stationary solutions can besummarized as follows.

Consider a solution with no shocks to the isentropic Euler-Poisson equations which has finite totalenergy E and total mass M. When > 43 , there is no blowup phenomena where part of the solutioncollapses to a point with finite mass. This also holds when = 4

3and the total mass is less than the

critical mass Mc = (3K2

)32 (M 4

3)2. Here M 4

3is the Marcinkiewicz interpolation constant.

This statement can be proved by the Fourier analysis using the Hardy-Littlewood-Paley inequality,cf. [67], from which the Marcinkiewicz interpolation constant arises naturally. It is consistent with theaforementioned conjecture and linear theory, in that, for > 4

3there is no collapse of the gas to a single

point, called core collapse. It remains open to construct blowup solutions for the case < 43 . For thesystem (3.1) and (3.2), the local existence was studied in [5,19,55,56,57]. The stability of linearized systemaround a stationary solution with spherical symmetry was also discussed in [40].

For the core collapse at finite time for =43 , the following example was constructed in [54] for isen-

tropic flow and is later generalized to non-isentropic case in [16].

Assume = 43

and p = k

43 y

343(1) ( ra(t))

43 , there exists a family of isentropic solutions (, v)(t, r)

with spherical symmetry in the form of

(t, r) =

a(t)Ny

11

ra(t)

, r < a(t)z,

0, r a(t)z,

v(t, r) =a(t)

a(t)r r

a(t)

.

Here r = |x|, =12(1)g

K

12

, > 1, 0 is a sufficiently small positive constant, z is a positiveconstant depending only on , is a smooth, cut-off function such that (z) = 1 for 0 z z and(z) = 0 for 2z z < +. Here a = a(t) and y = y(z) satisfy the following two ordinary differentialequations respectively,

d2a(t)

dt2=

a2, a(0) = a0 > 0, a(0) = a1

9


10/21

and

d2y

dz2+

2

z

dy

dz+ y

11 = , y(0) = 1, y(0) = 0

where = 4g.If a1 0, , u and p() have the same meaning as those in the previous section, 0 isthe viscosity coefficient. For simplicity of presentation, we again consider only the polytropic gas wherep() = k .

We will consider this hyperbolic-parabolic system when the initial data is of compact support, i.e.,connecting to vacuum state. Our main concern here is the global existence of solutions and the evolutionof the vacuum boundary. Notice that one of the important features of this problem is that the interfaceseparating the gas and vacuum propagates with finite speed if the initial data are of compact support.It is interesting to note that the proof of this finite speed propagation is obtained after the lower boundof the density function is given, for example, in Lagrangian coordinates. In other words, this finite speedpropagation property is difficult to be justified without the estimate on the density function.

Lets first review some of the works in this direction. When the viscosity coefficient is a constant,the study in [27] shows that there is no continuous dependence on the initial data of the solutions tothe Navier-Stokes equations (1.1) with vacuum. The main reason for this non-continuous dependence atthe vacuum comes from the kinetic viscosity coefficient being independent of the density. It is motivatedby the physical consideration that in the derivation of the Navier-Stokes equations from the Boltzmannequation, cf. [23,?], the viscosity is not constant but depends on the temperature. For isentropic flow,this dependence is reduced to the dependence on the density by the laws of Boyle and Gay-Lussac forideal gas. In particular, the viscosity of gas is proportional to the square root of the temperature for hard

sphere collision model. Under this hypothesis, the temperature is of the order of 1 for the perfect gaswhere the pressure is proportional to the product of the density and the temperature. Therefore, for thehard sphere model with = 5

3for monotomic gas, the viscosity is proportional to

13 .

The above non-continuous dependence on the initial data for constant viscosity with vacuum is themotivation for the works on the case when the viscosity function is a function of density, such as = c,where c and are positive constants. Notice that now the viscosity coefficient vanishes at vacuum andthis property yields the well-posedness of the Cauchy problem when the initial density is of compactsupport. In this situation, the local existence of weak solutions to Navier-Stokes equations with vacuumwas studied in [49], where the initial density was assumed to be connected to vacuum with discontinuities.This property can be maintained for some positive finite time. And the authors in [64] obtained the globalexistence of weak solutions when 0 < < 1/3 which was later generalized to the cases when 0 < < 1

2and 0 < < 1 in [74] and [33] respectively.

It is noticed that the above analysis is based on the uniform positive lower bound of the density withrespect to the construction of the approximate solutions, for example, by line method. This estimate iscrucial because the other estimates for the convergence of a subsequence of the approximate solutionsand the uniqueness of the solution thus obtained will follow from the estimation by standard techniquesas long as the vacuum does not appear in the solutions in finite time. And this uniform positive lowerbound on the density function can only be obtained when the density function connects to vacuum withdiscontinuities. In this case, the density function is positive for any finite time and thus the viscosity

11


12/21

coefficient never vanishes. This good property of the solution was obtained and used to prove globalexistence of solutions to (1.1) when the initial data is of compact support, cf. [33,64,74].

If the density function connects to vacuum continuously, there is no positive lower bound for the densityand the viscosity coefficient vanishes at vacuum. This degeneracy in the viscosity coefficient gives arise

to new analysis difficulty because of the weaker regularizing effect on the solutions. For this problem, alocal existence result was obtained in [75], and global existence result in [76] for 0 < < 2

9and in [70]

for 0 < < 13

. Notice also that the singularity arises at the vacuum boundary when the density functionconnects to vacuum continuously. This can be seen from the analysis in [71] on the non-global existenceof regular solution to Navier-Stokes equations when the density function is of compact support when theviscosity coefficient is constant. The proof there is based on the estimation on the growth rate of thesupport of the density function in time t. If the growth rate is sub-linear, then the nonlinear functionalintroduced in [71] gives the non-global existence of regular solutions. The intuitive explanation of thisphenomena comes from the consideration of the pressure in the gas. No matter how smooth the initialdata is, the pressure of the gas will build up at the vacuum boundary in finite time and it will push thegas into the vacuum region. This effort can not be compensated by the dissipation from the viscosity sothat the support of the gas stays unchanged. This is different from the system of Euler-Possion equationsfor gaseous stars where the pressure and the gravitational force can become balanced to have a stationarysolutions. In the case of compressible Navier-Stokes equations, the pressure will have the effort on theevolution of the vacuum boundary in finite time so that the density function at the interface will not besmooth. This was overcome by introducing some appropriate weights in the energy estimates as in [70,76]which vanish at the vacuum boundary.

For the case when the density function of compact support in both Eulerian and Lagrangian coordinates,the restriction on the solution coming from the boundedness of the support is1

0

1

(x, t)dx < , (1.3)

in Lagrangian coordinates (x, t). This as a consequence of the boundedness of velocity in L norm isjustified after the a prioriestimates on the density function are obtained. If the density function of infinitesupport in Eulerian coordinates, then even though the total mass is assumed to be finite, no restriction

like (1.3) is needed. Some new a priori estimates were established in [76] for this case so that the globalexistence of weak solution is also obtained when 13 < 0, c > 0 and s are constants, and the pressure P = ces with > 1 being a constant.It is easy to check that the local Maxwellian corresponding to these solutions are not solutions to theBoltzmann equation (5.14). But whether they can be viewed as asymptotic states when the Knudsennumber tends to zero is not clear. However, the answer to this question may be no if we considerthe problem as follows. When we derive the Navier-Stokes equations from the Boltzmann equation, theviscosity coefficient and the heat conductivity are not constant but functions of temperature. For the hard

shpere model, they are proportional to the squareroot of the temperature. Under this consideration, it isstraightforward to calculate that the corresponding stationary solutions to the Navier-Stokes equationsand to find out that they behave quite differently from (5.19). That is, as for the hard sphere model, thedensity does not have compact support and the temperature tends to infinity as the gas tends to vacuumin the following way:

(ax + b)2/3e(ax+b)1/3 , (ax + b)2/3, (5.19)where a and b are positive constants. Notice that the local Maxwellian corresponding to all these are notsolutions to the Boltzmann equation (5.14). However, it would be interesting to find out whether there isany relation between the above solutions and those for Boltzmann equation connecting to vacuum. Noticethat here we illustate the problem with one dimemsional macroscopic force and one can also study thesimilar problems in three dimensional space with general force x.

There are a lot of important research on global existence of small perturbation of a global Maxwellian

and the renomalized solutions which are not directly related to our study so that we wont present themhere, cf. [17,23,68] and reference therein. In the constructional proof of Kaniel and Shinbrot on the localexistence of solutions, the local existence and uniqueness of solutions was proved by constructing twosequences of approximate solutions, one sequence is decreasing and the other is increasing. The solutionsthus obtained are local because the beginning condition for the sequences was proved to hold only locallyin time in general [34]. For a rare gas in an infinite vacuum, this method was used to prove globalexistence of solutions later by Illner and Shinbrot [31]. The advantage of this approach is that it gives aconstructional picture of the solution so that one can understand its behavior more clearly. Furthermore,the approach is nonlinear because it does not use any linearization because of the singularity of thevacuum states.

There are also many results on Boltzmann equation about its relation to compressible or incompressibleEuler equations and Navier-Stokes equations. The most recent ones are the series work by P.-L. Lions and

N. Masmoudi [45], and by Bardos, Glose and Levermore [?], and by Bardos and Ukai [4] and by Nishida[61], etc. Please refer to these works and the reference therein.Since the analysis may not go through if we linearize around any of the above background solutions(local

Maxwellians) and the standard energy type of estimate doesnt not hold because the linearized operatorwith the decay density factor can not be used to control the non-fluid part in the solutions of theBoltzmann equation, a full nonlinear approach is needed. One of these approaches was given by Kanieland Shinbrot [34] for local existence of solutions to Boltzmann equation for general L data and was

17


18/21

generalized by Illner and Shinbrot [31] for the global existence of a rare gas in an infinite vacuum withoutexternal force. Recently, Guo applied this method to the case with external force, but the solution tendsto zero as time goes to infinity.

So far the external force is assumed to be the gravitational force between the particle and a fixed region

of mass. It will be more interesting to look at the case when radiation is also taken into account. Then,one needs to include the Maxwell equations and there will be electro-magnetic fields giving other externalforces, complexity and phenomena.

Acknowledgment: The research of the author was supported by the Competitive Earmarked ResearchGrant of Hong Kong CityU 102703.

References

a1 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and application, J. Funct. Anal.,14(1973), 349381.

a2 [2] D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initial stationary interface begins to move in porous media flow,SIAM J. Math., Anal., 14(1983), 639-658.

b2 [3] C. Bardos, F. Glose and D. Levermore, Fluid dynamic limits of Kinetic equations, I. Formal derivations, J. Statist.Phys. 63, 323-344, 1991; II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46, 667-753,1993.

b3 [4] C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann eqaution, Math. Models andMethods in Appl. Sci., Vol. 1 No.2, 1991, 235-257.

b4 [5] M. Beard, Existence locale pour les equations dEuler-Poisson, Japan J. Indust. Appl. Math., 10(1993) 431-450.

b5 [6] J. Binney and S. Tremaine, Galactic Dynamics, Princeton Univ. Press, 1994.

b6 [7] L. Boltzmann, Weitere studien uber das Warme gleichgenicht unfer Gasmolekular. Sitzungsberichte der Akademie derWissenschaften, Wien, 66(1872), 275-370.

b7 [8] J.-M. Bony, Calcul symbolique et propagation des singularites pour les equations aux derivees partielles non lineaires,Annales de l Ecole Normale Superieure, 14, 1981, pages 209246.

b8 [9] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm.on Pure and Applied Mathematics, 36(1983), 437-477.

c2 [10] S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases. 2nd ed., Cambridge Univ. Press, 1952.

c3 [11] C. Cercignani, The Boltzmann equation and its application. Springer Berlin, 1988.

c4 [12] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Univ. of Chicago Press, 1938.

c5 [13] J.-Y. Chemin, Fluides parfaits incompressibles, Asterisque, 230 (1995).

c8 [14] G.Q. Chen and M. Kratka, Global solutions to the Navier-Stokes equations for compressible heat-conducting flow withsymmetry and free boundary, Comm. Partial Diff. Equations, 27(2002), no. 5-6, 907-943.

d2 [15] Y.B. Deng, T.-P. Liu, T. Yang and Z.A. Yao, Solutions with vacuum of Euler-Poisson equations, Arch. Rat. Mech.Anal., 164(2002), 3, 261-285.

d2-1 [16] Y.B. Deng, J.L. Xiang and T. Yang, Blowup phenomena of the Euler-Poisson equations for gaseous stars, Jour. Math.Anal. Appl., 286(2003), 295-306.

d4 [17] R.J. DiPerna and P.L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability.Ann. Math., 130(1989), 321-366.

f1 [18] Chun-Chieh Fu, Song-Sun Lin, On the critical mass of the collapse of a gaseous star in spherically symmetric andisentropic motion, Japan J. Indust Appl Math., 15(1998), 461-469.

g1 [19] P. Gamblin, Solution reguliere a temps petit pour lequation dEuler-Poisson, Commun. PDE., 18(1993), 731-745.

18


19/21

g3 [20] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math.Phys., 68(1979), 209243.

g4 [21] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 1983.

g5 [22] H. Grad, Principles of the kinetic theory of gases, in Flugges Handbuch der Physik, XII, Springer, Berlin, (1958),

205-294.g6 [23] H. Grad, Asymptotic theory of the Boltzmann equation II. In: Rarefied gas dynamics, 1(ed. J. Laurmann), New York

Academic Press(1963), 26-59.

g7 [24] M. Grassin and D. Serre, Global smooth solutions to Euler equations for an isentropic perfect gas, C.R. Acad. Sci. ParisSer. I Math, 325(1997), No. 7, 721-726.

g8 [25] Y. Guo, The Vlasov-Poisson-Boltzmann system bear vacuum, Comm. Math. Phys. 218(2001), no.2, 293-313.

h1 [26] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropicequations of state and discontinuous initial data, Arch. Rat. Mech. Anal., 132(1995), 1-14.

h3 [27] D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressibleflow, SIAM J. Appl. Math., 51(1991), 887-898.

h4 [28] D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys.,216(2001), 255276.

h4 [29] L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations, 1997, Springer-Verlag.h5 [30] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation

laws with damping, Comm. Math. Phys.143 (1992), 599-605.

i1 [31] Illner R. and M. Shinbrot, The Boltzmann equation: global existence for a rare gas in an infinite vacuum, Comm. Math.Phys., 95, 217-226(1984).

i2 [32] R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gases invacuum, Comm. Math. Phys. 105, 189-203(1986).

j2 [33] S. Jiang, Z.P. Xin and P. Zhang, Global weak solutions to 1 D compressible isentropic Navier-Stokes equations withdensity-dependent viscosity, preprint.

k1 [34] S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence. Comm. Math. Phys. 59, (1978),65-84.

k2 [35] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational. Mech. Anay., 58(1945),181-205.

k3 [36] S. Kawashima; A. Matsumura; T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at thelevel of the Navier-Stokes equation. Comm. Math. Phys. 70 (1979), no. 2, 97124.

k4 [37] B. Knerr, The porous medium equation in one dimension, Transactions of AMS, Vol. 234, No. 2, 1977.

l1 [38] O. Lanford III, Time evolution of large classical system, Lecture Notes in Physics 38, E.J. Moser(ed.), 1-111, Springer-Verlag(1975).

l2 [39] P. Ledowx and T. Welrevein, Variable stars, Handbuch der Physic, Bd. LI(1958), Springer, 353-604.

l2 [40] S. S. Lin, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28(3)(1997), 539-569.

l3 [41] L.W. Lin, On the vacuum state for the Equations of Isentropic Gas Dynamics, J. of Math. Anal. Appl., Vol. 121, No.2 (1987) pp. 406425.

lin1 [42] L.W. Lin and X.P. Ye, Conjecture on the non-appearance of vacuum states in isentropic gas flow with shocks, Proc.5th Inter. Colloquium Diff. Equations, Provdiv(1994).

l4 [43] P.L. Lions, On Boltzmann and Landau equations. Phil. Trans. R. Soc. London, A346(1994), 191-204.l5 [44] P.L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, 2, (1998), Clarendon Press-Oxford.

l6 [45] P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics, I,Arch. Rational Mech. Anal. 158(2001), 173-193.Boltzmann sans troncature angulaire, C.R. Acad. Sci. Paris Ser. I Math., 326(1998), 37-42.

l9 [46] T.-P. Liu and T. Yang, Compressible Euler equations with vacuum, Journal of Differential Equations, Vol. 140, No. 2,1997, 223-237.

19


20/21

l10 [47] T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods and Applications of Analysis,Vol. 7, No. 3, 495-510, 2000.

l11 [48] T.-P. Liu and J. Smoller, On the vacuum state for isentropic gas dynamics equations, Advances in Math., 1(1980),345-359.

l12 [49] T.-P. Liu, Z. Xin and T. Yang, Vacuum states of compressible flow, Discrete and Continuous Dynamical Systems,4(1998), 1-32.

l13 [50] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Physica D., 188(2004), 178-192.

l14 [51] T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Appl. Math., Vol. 13, No. 1, 25-32, 1996.

l16 [52] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math.Anal., 31(2000), 1175-1191.

l17 [53] T. Luo and J. Smoller, Rotating fluids with self-gravitation in bounded domains, Arch. Rational Mech. Anal.(2004).

m1 [54] T. Makino, Blowing up solutions of the Euler-Poisson equation for the evolution of gaseous stars, Transport Theoryand Statistical Physics 21 (1992), 615624.

m2 [55] T. Makino et B. Perthame, Sur les Solutions a symmetric spherique de lequation dEuler-Poisson Pour levolutiondetoiles gazeuses, Japan J. Appl. Math., 7(1990), 165-170.

m3 [56] T. Makino et S. Ukai, Sur lexistence des solutions locales de lequation dEuler-Poisson pour levolution gazeuses, J.Math. Kyoto Univ., 27(1987), 387-399.

m4 [57] T. Makino, On a local existence theorem for the evolution equations of gaseous stars, Patterns and wave-qualitativeanalysis of nonlinear differential equations, Ed. T. Nishida, M. Mimura and H. Fujii, North-Holland, 1986, 459-479.

m5 [58] J.C. Maxwell, Scientific papers. Vol. 2, Cambridge, Cambridge Univ. Press, 1890.(Reprinted by Dover Publications,New York, 1965)

n1 [59] W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of u = f(u, r), Comm.Pure Appl. Math., 38(1985), 67-108.

n2 [60] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. DOrsay, (1978), 46-53.

n6 [61] T. Nishida, Asymptotic behavior of solutions of the Boltzmann equation. Trends in applications of pure mathematicsto mechanics, Vol. III (Edinburgh, 1979), 190203, Monographs Stud. Math., 11, Pitman, Boston, Mass.-London, 1981.

o1 [62] M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl.Math., 6(1989), 161-177.

o2 [63] M. Okada and T. Makino, Free boundary problem for the equations of spherically symmetrical motion of viscous gas,Japan J. Indust. Appl. Math., 10(1993), 219-235.

o4 [64] M. Okada, S. Matusu-Necasova and T. Makino, Free boundary problem for the equation of one-dimensional motion ofcompressible gas with density-dependent viscosity, Ann. Univ. Ferrara Sez. VII(N.S.) 48 (2002), 1-20.

o5 [65] O.A. Oleinik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math., Vol., XXIII, (1970),569-586.

s1 [66] D. Serre, Sur lequation mondimensionnelle dun fluide visqueux, compressible et conducteur de chaleur, Comptes rendusAcad. des Sciences. 303(1986), 703-706.

s2 [67] E.M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton UniversityPress, 1993.

u2 [68] S. Ukai, Les solutions globales de lequation non lineaire de Boltzmann dans lespace tout entier et dans demi-espace,C.R. Acad. Sci.(paris) A, 282(1996), 317-320.

v2 [69] J. von Neumann, Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations,17, August, 1949, Collected works of J. von Neumann.

v3 [70] S.W. Vong, T. Yang and C.J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient andvaccum (II), Jour. Diff. Equations, 192, 475-501, 2003.

x1 [71] Z. Xin, Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density, Comm. PureAppl. Math., 51(1998), 229-240.

20


21/21

x2 [72] C.J. Xu and T. Yang, Local existence with physical vacuum boundary condition to Euler equations with damping, toappear in Jour. Diff. Equations.

y1 [73] T. Yang, Some recent results on compressible flow with vacuum, Taiwanese J. Math., 4(2000), 33-44.

y2 [74] T. Yang, Z.A. Yao and C.J. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum,

Comm. Partial Differential Equations, 26(2001), 965-981.

y3 [75] T. Yang and H.J. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, Jour. Diff. Equations, 184, 163-184, 2002.

y4 [76] T. Yang and C.J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,Commun. Math. Phys., 230, 329-363(2002).

y6 [77] T. Yang and C.J. Zhu, Existence and non-existence of global smooth solutions for p-system with relaxation. Jour. Diff.Equations, 161(2000), 321-336.

y7 [78] T. Yang and C.J. Zhu, Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems,Proceedings of the Royal Society of Edinburgh, 133A, 719-728, 2003.

21

Tong Yang- Singular Behavior of Vacuum States for Compressible Fluids

Documents

Transcript of Tong Yang- Singular Behavior of Vacuum States for Compressible Fluids