Transport and concentration processes in the multidimensional zero-pressure gas dynamics model with the energy conservation law

0.27 MB·English

by  S. Albeverio

TRANSPORT AND CONCENTRATION PROCESSES IN THE
MULTIDIMENSIONAL ZERO-PRESSURE GAS DYNAMICS
MODEL WITH THE ENERGY CONSERVATION LAW
S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
1
1
0 Abstract. We introduce integral identities to define δ-shock wave type solu-
2 tionsforthemultidimensionalzero-pressuregasdynamics

n ρt+∇·(ρU) = 0,
a (ρU)t+∇·(ρU⊗U) = 0,
J ρ|U|2 ρ|U|2
0 (cid:16) 2 +H(cid:17)t+∇·(cid:16)(cid:16) 2 +H(cid:17)U(cid:17) = 0,
3 where where ρ is the density, U ∈ Rn is the velocity, H(x,t) is the internal
energy, x ∈ Rn. Using these integral identities, the Rankine-Hugoniot condi-

] tions for δ-shocks are obtained. We derive the balance laws describing mass,
h
momentum,andenergytransportfromtheareaoutsidetheδ-shockwavefront
p
onto this front. These processes are going on in such a way that the total
-
mass, momentum, and energy are conserved and at the same time mass and
h
energyofthemovingδ-shockwavefrontareincreasingquantities. Inaddition,
t
a thetotal kineticenergytransfersintothetotal internalenergy. Theprocessof
m propagation of δ-shock waves is also described. These results can be used in
modelingofmediumswhichcanbetreatedasapressureless continuum(dusty
[
gases,two-phaseflowswithsolidparticlesordroplets,granulargases).

1
v
5
1
1. Strong singular solutions and pressureless mediums
8
5 1.1. L -type solutions. Let us recallsome classicalresults. Consider the Cauchy
∞
.
1 problem for the system of conservation laws in one dimension space:
0
U + F(U) = 0, in R (0, ),
1 t x × ∞ (1.1)
1 ( (cid:0) (cid:1)U = U0, in R t=0 ,
: ×{ }
v where F :Rm Rm is called the flux-function associated with (1.1); U0 :R Rm
i → →
X are given vector-functions; U = U(x,t) = (u (x,t),...,u (x,t)) is the unknown
1 m
r function with value in Rm, and components uj(x,t), j =1,...,m; x R, t 0.
a ∈ ≥
As is well known, even in the case of smooth (and, certainly, in the case of
discontinuous) initial data U0(x), in general, does not exist any smooth and global
in time solutionofsystem(1.1). As notedin the Evans’book [9,11.1.1.],“the great
difficultyinthissubjectisdiscoveringapropernotionofweaksolutionfortheinitial
problem (1.1)”. “We must devise some way to interpret a less regular function as
somehow “solving” this initial-value problem” [9, 3.4.1.a.]. But it is a well known
that a partial differential equation may not make sense even if U is differentiable.
“However, observe that if we temporarily assume U is smooth, we can as follows
Date:
2000 Mathematics Subject Classification. Primary35L65;Secondary35L67, 76L05.
Keywords and phrases. Multidimensionalsystemofconservationlaws,δ-shocks,theRankine–
Hugoniotconditionsforδ-shocks,transportandconcentration processes.
The authors were supported by DFG Project 436 RUS 113/895. The second and third au-
thors (O.R. and V.S.)were alsosupported by the Analytical departmental special program ”The
development ofscientificpotential oftheHigherSchool”,project2.1.1/1399.
1
2 S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
rewrite, so that the resulting expression does not directly involve the derivatives of
U” [9, 3.4.1.a.]. “The idea is to multiply the partial differential equation in (1.1)
by a smooth function ϕ and then to integrate by parts, thereby transferring the
derivativesontoϕ” [9, 3.4.1.a.;11.1.1.]. Followingthis suggestionweshallderivethe
integralidentitywhichgivesthefollowingdefinitionofanL -generalized solutionof
∞
the Cauchy problem (1.1): U L R (0, );Rm is called a generalized solution
∞
∈ × ∞
of the Cauchy problem (1.1) if the integral identity
(cid:0) (cid:1)
∞ U ϕ +F(U) ϕ dxdt+ U0(x) ϕ(x,0)dx=0 (1.2)
t x
· · ·
Z0 Z (cid:16) (cid:17) Z
holdsforallcompactlysupportedsmoothtestvector-functionsϕ:R [0, ) Rm,
e e e × ∞ →
where isthescalarproductofvectors,and f(x)dxdenotestheimproperintegral
·
∞ f(x)dx. “This identity, whichwe derivedsupposing U to bee a smoothsolution
R
m−a∞kes sense if U is merely bounded” [9, 11.1.1.].
R
Theorem1.1. (see,e.g.,[9,11.1.1.]) LetΩ R (0, )bearegioncutbyasmooth
⊂ × ∞
curve Γ into a left- and right-hand parts Ω . Let us assume that the generalized
solution U of (1.1) is smooth on either side o∓f the curve Γ along which U has simple
jump discontinuities. Then the Rankine–Hugoniot condition
F(U) ν + U ν =0, (1.3)
Γ 1 Γ 2
holds along Γ, where n = (ν1(cid:2),ν2) i(cid:3)s the u(cid:2)nit(cid:3)normal to the curve Γ pointing from
Ω into Ω ,
+
−
def
[F(U)] = F(U ) F(U ),
+
− −
def
[U] = U U are the jumps in F(U) and in U across the discontinuity curve Γ,
+
respective−ly−. U are respective the left- and right-hand values of U on Γ.
If Γ= (x,t∓):x=φ(t) , where φ() C1(0,+ ), then
{ } · ∈ ∞
1
n=(ν ,ν )= 1, φ˙ (t) , (1.4)
1 2 i
1+(φ˙ (t))2 −
i
(cid:0) (cid:1)
q
and (1.3) reads
F(U) =φ˙(t) U , (1.5)
Γ Γ
where (˙)= d(). (cid:2) (cid:3) (cid:2) (cid:3)
· dt ·
It is well known that if U L R (0, );Rm is a generalized solution of the
∞
∈ × ∞
Cauchy problem (1.1) compactly supported with respect to x, then the integral of
(cid:0) (cid:1)
the solution on the whole space
U(x,t)dx= U0(x)dx, t 0 (1.6)
≥
Z Z
is independent of time. These integrals can express the conservation laws of quan-
tities like the total area, mass, momentum, energy, etc.
1.2. δ-shocks. It is well known that there are “nonclassical” situations where, in
contrastto Lax’s and Glimm’s classicalresults,the Cauchyproblem for a system of
conservationlawseitherdoesnotpossessaweakL -solutionorpossessesitforsome
∞
particular initial data. Inorder to solvethe Cauchy problemin these “nonclassical”
situations,itisnecessarytoseeksolutionsofthisCauchyprobleminclassofsingular
solutions called δ-shocks. Roughly speaking, a δ-shock is a solution such that its
components contain Dirac delta-functions.
Itiscustomarytoassumethataδ-shockwavetypesolutionwasfirstdescribedby
Korchinskiin his unpublished dissertation [13] in 1977. However,in fact, a solution
TRANSPORT PROCESSES IN THE ZERO-PRESSURE GAS DYNAMICS 3
of this type as well as the Rankine-Hugoniotcondition for the one-dimensionalcon-
tinuity equation were already derived from physical considerations in the book [33,
7, 12]in1973. Next,in1979,A.N.Kraiko[14]consideredanewtypeofdiscontinu-
§ §
ity surface which are to be introduced in certain models of media having no inherent
pressureandobtainedtheRankine-Hugoniotconditionsforthem. Thesystemunder
consideration in [14] is the zero-pressure gas dynamics described by the system of
equations:
ρ + ρu = 0,
t x
(ρu) + ρu2 = 0,
t (cid:0) (cid:1)x (1.7)
ρu2 ρu2
+ρτ + +ρ(cid:0)τ u(cid:1) = 0,
2 t 2 x
where ρ(x,t) 0 is th(cid:16)e density,(cid:17)u(x,t(cid:16))(cid:16)is the velo(cid:17)cit(cid:17)y, ρ(x,t)u(x,t) is corresponding
momentum, τ≥(x,t) is the internal energy per unit mass, x R. The lastsystemcan
∈
be derived from the Euler equations of nonisentropic gas dynamics
ρ + ρu =0, (ρu) + ρu2+p =0, (ρE) + (ρE+p)u =0, (1.8)
t x t x t x
if we set(cid:0)p=(cid:1)0, where E = ρu2(cid:0)+τ is t(cid:1)otal energy per un(cid:0)it mass. (cid:1)
2
Accordingto [14, page 502],to construct a solutionfor system(1.7) for arbitrary
initialdata,weneeddiscontinuitieswhichwouldbedifferentfromclassicalonesand
carry mass, impulse and energy. As it turned out these nonclassical discontinuities
are δ-shocks.
Thetheoryofδ-shockshasbeenintensivelydevelopedinthelastfifteenyears(for
example, see [2], [4]– [7], [17]– [20], [29]– [31] andthe references therein). Moreover,
recently, in [24], a concept of δ(n)-shock wave type solutions was introduced, n =
1,2,.... It is a new type of singular solution of a system of conservation laws such
that its components contain delta functions and their derivatives up to n-th order.
In [24], [27], the theory of δ -shocks was established. The results [24] and [27] show
′
that systems of conservation laws can develop solutions not only of the type of
Dirac measures (as in the case of δ-shocks) but also the type of derivatives of such
measures.
Theabove-mentionedsingularsolutionsdonotsatisfythestandardintegraliden-
tities of the type (1.2). To define them we use special integral identities and derive
special Rankine–Hugoniotconditions. These solutions are connected with transport
and concentration processes [2], [5], [24], [29], [28].
Inthe numerouspapers cited aboveδ-shockswere studiedfor the system of zero-
pressure gas dynamics:
ρ + (ρU)=0, (ρU) + (ρU U)=0, (1.9)
t t
∇· ∇· ⊗
where ρ=ρ(x,t) 0 is the density, U =(u (x,t),...,u (x,t)) Rn is the velocity,
1 n
≥ ∈
= ∂ ,..., ∂ , isthescalarproductofvectors, istheusualtensorproduct
∇ ∂x1 ∂xn · ⊗
of vectors.
(cid:0) (cid:1)
Thesystemofzero-pressuregasdynamics(1.9)hasaphysicalcontextandisused
in applications. This system can be considered as a model of the “sticky particle
dynamics” and was used, e.g., to describe the formation of large-scale structures of
theuniverse[25],[32],formodelingtheformationandevolutionoftrafficjams[3],for
modeling non-classical shallow water flows [8]. Nonlinear equations (in particular,
zero-pressure gas dynamics) admitting δ-shock wave type solutions are appropriate
for modeling and studying singular problems like movement of multiphase media
(dusty gases, two-phase flows with solid particles or droplets). The presence of
particles or droplets may drastically modify flow parameters. Moreover, a large
numberofphenomenathatareabsentinpuregasflowisinherentintwo-phaseflows.
4 S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
Amongthemtherearelocalaccumulationandfocusingofparticles,inter-particleand
particle-wallcollisionsresultinginparticlemixinganddispersion,surfaceerosiondue
toparticleimpacts,andparticle-turbulenceinteractionswhichgovernthedispersion
andconcentrationheterogeneitiesofinertialparticles. Thedispersedphaseisusually
treated mathematically as a pressureless continuum. Models of such media were
discussed in the papers [14]– [16], [21] –[23]. Equations admitting δ-shocks can also
be used for modeling granular gases. Granular gases are dilute assemblies of hard
sphereswhichloseenergyatcollisions. Insuchgaseslocaldensityexcessesandlocal
pressurefalls[10],[11]. In[10],[11],thefollowinghydrodynamicssystemofgranular
gas
ρ +(ρu) =0, ρ(u +uu )= (ρT) , T +uT = (γ 1)Tu ΛρT3/2,
t x t x x t x x
− − − −
was studied, where ρ is the gas density, u is the velocity, T is the temperature, γ is
the adiabaticindex, p=ρT is the pressure. Itwasshownthatfornon-zero pressure
this system admits a solution which contains a δ-function in the density ρ.
1.3. Main results. As it follows from [14]– [16], for modeling media which can be
consideredashaving nopressurewemusttakeintoaccountenergytransport. Inthe
above-cited papers zero-pressure gas dynamics was studied only in the form (1.9).
Therefore, we need to study δ-shocks in zero-pressure gas dynamics
ρ + (ρU) = 0,
t
∇·
(ρU) + (ρU U) = 0,
t ∇· ⊗ (1.10)
ρU 2 ρU 2
| | +H + | | +H U = 0,
2 t ∇· 2
where H(x,t) is th(cid:16)e internal e(cid:17)nergy, U(cid:16)(cid:16)2 = n u(cid:17)2. (cid:17)This system is obtained by
| | k=1 k
adding an energy conservation law to zero-pressure gas dynamics (1.9). As distinct
P
from(1.7)itismoreconvenientforustoconsiderasavariableH insteadofH =ρτ,
where τ is the internal energy per unit mass. The reason is that since for singular
solution τ(x,t) and ρ(x,t) must contain δ-functions, it is impossible to define the
product τ(x,t)ρ(x,t).
Under the second thermodynamics law it is natural to supplement the system
(1.7) with a state equation τ = τ(T), where T is the temperature. For (1.10) the
natural state equation is H =H(ρ,T), moreover, H(0,T)=0.
In Sec. 2, we introduce Definition 2.1 of δ-shock wave type solutions for system
(1.10). Next, using this definition, by Theorem 2.1 we derive the corresponding
Rankine-Hugoniotconditionsforδ-shocks(2.4). TheseRankine-Hugoniotconditions
are the direct analog of those that were introduced by A. N. Kraiko [14].
In Sec.3, we show that δ-shocksare relatedwith the transport processes of mass,
momentum and energy. According to Theorems 3.1, 3.2, the mass, momentum and
energy transport processes between the area outside of the moving δ-shock wave
front and this front are going on such that the total mass, momentum and energy
are independent of time. Moreover, the mass and energy concentration processes
takes place on the δ-shock wave front.
2. δ-shock type solutions and the Rankine–Hugoniot conditions
2.1. δ-shock type solutions. Throughout the paper we shall systematically use
someresultsrecalledinAppendixA. LetΓ= (x,t):S(x,t)=0 beahypersurface
of codimension 1 in the upper half-space (x,t) : x Rn, t [0, ) Rn+1, S
C (Rn [0, )), with S(x,t) =0 f{or a(cid:8)ny fixe∈d t, whe∈re (cid:9)∞= } ⊂∂ ,..., ∂ ∈.
∞ × ∞ ∇ S=0 6 ∇ ∂x1 ∂xn
Let Γ = x Rn : S(x,t) = 0 be a moving surface in Rn. Denote by ν the unit
t ∈ (cid:12) (cid:0) (cid:1)
space norm(cid:8)al to the surface Γt (cid:9)p(cid:12)ointing (in the positive direction) from Ω−t ={x∈
TRANSPORT PROCESSES IN THE ZERO-PRESSURE GAS DYNAMICS 5
Rn :S(x,t)<0 to Ω+ = x Rn :S(x,t)>0 such that ν = Sxj , j =1,...,n.
} t { ∈ } j S
The direction of the vector ν coincides with the direction in wh|i∇ch|the function S
increases, i.e., inward the domain Ω+. The time component of the normal vector
t
G= St is the velocity of the wave front Γ along the space normal ν.
− S t
For s|∇yst|em (1.10) we consider the δ-shock type initial data
U0(x),ρ0(x),H0(x), x Rn; U0(x), x Γ ,
∈ δ ∈ 0
where ρ0(x) = ρ0(x)+e0(x)δ(Γ ), (2.1)
0
(cid:0) (cid:1)
H0(x) = H0(x)+h0(x)δ(Γ ),
0
b
such that U0 L Rn;Rn , ρ0,H0 L Rn;R , e0,h0 C(Γ ), Γ = x :
∞ ∞ b 0 0
∈ ∈ ∈
S0(x) = 0 is the initial position of the δ-shock front, S0(x) = 0, U0(x),
(cid:0) (cid:1) (cid:0) (cid:1) ∇ S0=0 6 δ(cid:8)
x ∈ Γ0, is(cid:9)the initial velocity obf thbe δ-shock, δ(Γ0) (≡ δ(S0))(cid:12)is the Dirac delta
function concentrated on the surface Γ defined by (A.8): (cid:12)
0
δ(S ), ϕ(x) = ϕ(x)dΓ , ϕ (Rn),
0 0
∀ ∈D
ZΓ0
(cid:10) (cid:11)
dΓ is the surface measure on the surface Γ .
0 0
Similarly to [29, Definition 9.1.] we introduce the following definition
Definition 2.1. A triple of distributions (U,ρ,H) and a hypersurface Γ, where
ρ(x,t) and H(x,t) have the form of the sum
ρ(x,t)=ρ(x,t)+e(x,t)δ(Γ), H(x,t)=H(x,t)+h(x,t)δ(Γ),
and U L Rn (0, );Rn , ρ,H L Rn (0, );R , e,h C(Γ), is called
∈ ∞ ×b ∞ ∈ ∞ × b∞ ∈
a δ-shock wave type solution of the Cauchy problem (1.10), (2.1) if the integral
(cid:0) (cid:1) (cid:0) (cid:1)
identities b b
∞ δϕ dΓ
ρ ϕ +U ϕ dxdt+ e
t ·∇ δt √1+G2
Z0 Z (cid:16) (cid:17) ZΓ
b + ρ0(x)ϕ(x,0)dx+ e0(x)ϕ(x,0)dΓ = 0,
0
Z ZΓ0
∞ ρU ϕ +U bϕ dxdt+ eU δϕ dΓ
t ·∇ δ δt √1+G2
Z0 Z (cid:16) (cid:17) ZΓ
+ b U0(x)ρ0(x)ϕ(x,0)dx+ e0(x)U0(x)ϕ(x,0)dΓ = 0,
δ 0
Z ZΓ0 (2.2)
ρU 2 ρU 2
∞ | | b+H ϕ + | | +H U ϕ dxdt
t
2 2 ·∇
Z0 Z (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19)
+b e|Ubδ|2 +h δbϕ dΓb
2 δt √1+G2
ZΓ(cid:16) (cid:17)
ρ0(x)U0(x)2
+ | | +H0(x) ϕ(x,0)dx
2
Z (cid:16) (cid:17)
b e0(x)U0(x)2
+ | δ b| +h0(x) ϕ(x,0)dΓ = 0,
0
2
ZΓ0(cid:16) (cid:17)
hold for all ϕ (Rn [0, )). Here f(x)dx denotes the improper integral
∈ D × ∞
f(x)dx; dΓ and dΓ are the surface measures on the surfaces Γ and Γ , respec-
Rn 0 R 0
tively;
R
S S
U =νG= t∇ (2.3)
δ − S 2
|∇ |
6 S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
istheδ-shockvelocity,ν istheunitspacenormaltothesurfaceΓ introducedabove;
t
G = St , δϕ is the δ-derivative with respect to the time variable (A.5); δ(Γ) is
− S δt
|∇ |
the Dirac delta function concentrated on the surface Γ defined by (A.8):
δ(S), ϕ(x,t) = ∞ ϕ(x,t)dΓ dt= ϕ(x,t) dΓ , ϕ (Rn R).
t
√1+G2 ∀ ∈D ×
(cid:10) (cid:11) Z−∞ZΓt ZΓ
In view of (2.3), the δ-derivative in (2.2) can be rewritten as the Lagrangian
derivative:
δϕ ∂ϕ ∂ϕ ∂ϕ Dϕ
= +G = +U ϕ= .
δt ∂t ∂ν ∂t δ·∇ Dt
2.2. Rankine–Hugoniot conditions. Using Definition 2.1, we derive the δ-shock
Rankine–Hugoniot conditions for system (1.10).
Theorem 2.1. Let us assume that Ω Rn (0, ) is a region cut by a smooth
⊂ × ∞
hypersurface Γ= (x,t):S(x,t)=0 into left- and right-hand parts Ω = (x,t):
∓
{
S(x,t)>0 . Let (U,ρ,H), Γ be a δ-shock wave type solution of system (1.10) (in
∓ } (cid:8) (cid:9)
the sense of Definition 2.1), and suppose that U,ρ,H are smooth in Ω and have
±
one-sided limits U , ρ , H on Γ. Then the Rankine–Hugoniot conditions for the
± ± ±
δ-shock
bδe
+ (eU ) = [ρU] [ρ]U ν,
δt ∇Γt · δ − δ ·
δ(eU )
δ + (eU U ) = (cid:0)[ρU U] [(cid:1)ρU]U ν,
δt ∇Γt · δ⊗ δ ⊗ − δ ·
δ eU 2 (cid:0) (cid:1)
| δ| +h (2.4)
δt 2
(cid:16)+ e(cid:17)|Uδ|2 +h U = ρ|U|2 +H U
∇Γt · 2 δ 2
(cid:16)(cid:16) (cid:17) (cid:17) (cid:18)h(cid:16) (cid:17) i
ρU 2
| | +H U ν,
δ
− 2 ·
h i (cid:19)
hold on the discontinuity hypersurface Γ, where f(U,ρ,H) = f(U ,ρ ,H )
− − −
−
f(U+,ρ+,H+) is the jump of the function f(U,ρ,H) across the discontinuity hy-
(cid:2) (cid:3)
persurface Γ, δ is the δ-derivative (A.5) with respect to t, and is defined by
δt ∇Γt
(A.5), (A.6).
Proof. The first two conditions in (2.4) were proved in [29, Theorem 9.1.].
Letusprovethethirdconditionin(2.4). Foranytestfunctionϕ (Ω)wehave
∈D
ϕ(x,t) = 0 for (x,t) G, G Ω. Selecting the test function ϕ(x,t) with compact
6∈ ⊂
support in Ω , we deduce from the third identity in (2.2) that the third relation in
±
(1.10) hold in Ω , i.e.,
±
ρU 2 ρU 2
| | +H + | | +H U =0 for (x,t) Ω±. (2.5)
2 t ∇· 2 ∈
Now, if th(cid:0)e test funct(cid:1)ion ϕ(x,(cid:0)t(cid:0)) has the su(cid:1)pp(cid:1)ort in Ω, then
ρU 2 ρU 2
∞
| | +H ϕ + | | +H U ϕ dxdt
t
2 2 ·∇
Z0 Z (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19)
b b
ρUb2 ρUb2
= | | +H ϕ + | | +H U ϕ dxdt
t
ZΩ−∩G(cid:18)(cid:16) 2 (cid:17) (cid:16) 2 (cid:17) ·∇ (cid:19)
b b ρU 2 b b ρU 2
+ | | +H ϕ + | | +H U ϕ dxdt.
t
2 2 ·∇
ZΩ+∩G(cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19)
b b
b b
TRANSPORT PROCESSES IN THE ZERO-PRESSURE GAS DYNAMICS 7
Using the integrating-by-partsformula, we obtain
ρU 2 ρU 2
| | +H ϕ + | | +H U ϕ dxdt
t
2 2 ·∇
Ω±Z∩G (cid:18)(cid:16)b b(cid:17) (cid:16)b b(cid:17) (cid:19)
ρU 2 ρU 2
= | | +H + | | +H U ϕ(x,t)dxdt
− 2 t ∇· 2
Ω±ZG (cid:18)(cid:16) (cid:17) (cid:16)(cid:16) (cid:17) (cid:17)(cid:19)
∩
ρ U 2 S ρ U 2 U S
±| ±| +H± t + ±| ±| +H± ±·∇ ϕ(x,t)dΓ
∓ 2 S 2 S
ΓZG (cid:18)(cid:16) (cid:17)|∇(x,t) | (cid:16) (cid:17)|∇(x,t) |(cid:19)
∩
ρ0(x)U0(x)2
| | +H0(x) ϕ(x,0)dx,
− 2
where dΓ is the surfΩa±ce∩ZGm∩eRansu(cid:16)rbe on Γ. Next, addbing t(cid:17)he latter relations and taking
into account (2.5), we have
ρU 2 ρU 2
∞
| | +H ϕ + | | +H U ϕ dxdt
t
2 2 ·∇
Z0 Z (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19)
b b ρ0(xb)U0(x)2b
+ | | +H0(x) ϕ(x,0)dx
2
Z (cid:16) (cid:17)
ρU 2 b ρU 2 dΓ
b
= | | +H G+ | | +H U ν ϕ(x,t) . (2.6)
− 2 2 · √1+G2
ZΓ(cid:16) h i h(cid:16) (cid:17) i (cid:17)
Next, applying the integrating-by-parts formula (A.10) to the second summand in
third identity (2.2), one can see that
eU 2 δϕ dΓ e0(x)U0(x)2
| δ| +h + | δ | +h0(x) ϕ(x,0)dΓ
2 δt √1+G2 2 0
ZΓ(cid:16) (cid:17) ZΓ0(cid:16) (cid:17)
δ eU 2 dΓ
= ∗ | δ| +h ϕ ,
− δt 2 √1+G2
where the adjoint operator δ∗ is defineZdΓin ((cid:16)A.11). Thu(cid:17)s
δt
eU 2 δϕ dΓ e0(x)U0(x)2
| δ| +h + | δ | +h0(x) ϕ(x,0)dΓ
2 δt √1+G2 2 0
ZΓ(cid:16) (cid:17) ZΓ0(cid:16) (cid:17)
δ eU 2 eU 2 dΓ
= | δ| +h + | δ| +h Gν ϕ . (2.7)
− δt 2 ∇Γt · 2 √1+G2
ZΓ(cid:18) (cid:16) (cid:17) (cid:16)(cid:16) (cid:17) (cid:17)(cid:19)
Adding (2.6) and (2.7) and taking into account (2.2), (2.3), we derive
ρU 2 ρU 2
| | +H U ν+ | | +H U ν
δ
− 2 · 2 ·
ZΓ(cid:18) h i h(cid:16) (cid:17) i
δ eU 2 eU 2 dΓ
| δ| +h | δ| +h U ϕ(x,t) =0,
−δt 2 −∇Γt · 2 δ √1+G2
for all ϕ (cid:16)(Ω). Thus, t(cid:17)he third r(cid:16)e(cid:16)lation in (2.(cid:17)4) h(cid:17)o(cid:19)lds. (cid:3)
∈D
The right-hand sides of the equations in (2.4) are called the Rankine–Hugoniot
deficits in ρ, ρU, and ρ|U|2 +H, respectively.
2
Leta(x,t)beasmoothfunctiondefinedonlyonthesurfaceΓ= (x,t):S(x,t)=
0 which is the restriction of some smooth function defined in a neighborhood of Γ
(cid:8)
in Rn R. It is easy to prove that
(cid:9) ×
(aU )= 2 Ga, (2.8)
∇Γt · δ − K
8 S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
where is the mean curvature of the surface Γ (see (A.7) ). Indeed, according to
t
K
(A.5), (A.6), (A.7), (2.3), we have (aU ) = n δ(Gaνk) = n δ(Ga)ν +
∇Γt · δ k=1 δxk k=1 δxk k
Ga n δνk = 2 Ga. Here the obvious relation n δ(Ga)ν = 0 was taken
k=1 δxk − K P k=1 δxk kP
into account.
P P
Due to (2.8), the Rankine–Hugoniot conditions (2.4) can also be rewritten as
δe
2 Ge = [ρU] [ρ]U ν,
δt − K − δ ·
δ(eU )
δ 2 GeU = (cid:0)[ρU U] [(cid:1)ρU]U ν,
δt − K δ ⊗ − δ ·
δ eU 2 eU 2 ρU 2 (2.9)
| δ| +h 2 G | δ| +h = (cid:0) | | +H U (cid:1)
δt 2 − K 2 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:18)h(cid:16) ρU 2(cid:17) i
| | +H U ν.
δ
− 2 ·
(cid:19)
h i
Remark 2.1. The Rankine–Hugoniotconditions (2.4) constitute a systemofsecond-
order PDEs. According to this fact, for system (1.10) we use the initial data (2.1)
which contain the initial velocity U0(x) of a δ-shock. This is similar to the fact
δ
that in the measure-valued solution approach [4], [17], [18], [31] the velocity U is
determined on the discontinuity surface.
Inthedirectionν thecharacteristicequationofsystem(1.10)hasrepeatedeigen-
values λ=U ν. So, we assume that for the initial data (2.1) the geometric entropy
·
condition holds:
U0+(x)·ν0 Γ0 <Uδ0(x)·ν0 Γ0 <U0−(x)·ν0 Γ0, (2.10)
where ν0 = ∇SS00((xx)) is the un(cid:12)(cid:12)it space normal(cid:12)(cid:12)of Γ0, oriented f(cid:12)(cid:12)rom Ω−0 = {x ∈ Rn :
S0(x)<0 t|o∇Ω+ =| x Rn :S0(x)>0 . Similarly, we assume that for a solution
} 0 { ∈ }
of the Cauchy problem (1.10), (2.1) the geometric entropy condition holds:
U+(x,t) ν <U (x,t) ν <U (x,t) ν , (2.11)
· Γt δ · Γt − · Γt
where U is the velocity (2.3)(cid:12)of the δ-shock f(cid:12)ront Γ , U is th(cid:12)e velocity behind the
δ (cid:12) (cid:12) t ± (cid:12)
δ-shock wave front and ahead of it, respectively. Condition (2.11) implies that all
characteristics on both sides of the discontinuity Γ must overlap. For t = 0 the
t
condition (2.11) coincides with (2.10).
3. δ-shock mass, momentum and energy transport relations
Theclassicalconservationlaws(1.6)donotmakesenseforaδ-shockwavetypeso-
lution. “Generalized”analogsofconservationlaws(1.6)werederivedin[2],[24],[28]
for the one-dimensionalcase, and in [29] for the multidimensional case. Now we de-
rive these transport conservation laws for the case of system (1.10).
LetusassumethatamovingsurfaceΓ = x:S(x,t)=0 permanentlyseparates
t
Rnx intotwopartsΩ±t ={x∈Rn :±S(x,t)>(cid:8)0},andΩ±0 =(cid:9){x∈Rn :±S0(x)>0}.
Let (U,ρ,H) be compactly supported with respect to x. Denote by
M(t)= ρ(x,t)dx, m(t)= e(x,t)dΓ , (3.1)
t
ZΩ−t ∪Ω+t ZΓt
and
P(t)= ρ(x,t)U(x,t)dx, p(t)= e(x,t)U (x,t)dΓ , (3.2)
δ t
ZΩ−t ∪Ω+t ZΓt
TRANSPORT PROCESSES IN THE ZERO-PRESSURE GAS DYNAMICS 9
massesandmomentaofthe volumeΩ−t ∪Ω+t andthe movingδ-shockwavefrontΓt,
respectively, dΓ being the surface measure on Γ . Let
t t
ρ(x,t)U(x,t)2 e(x,t)U (x,t)2
W (t)= | | dx, w (t)= | δ | dΓ , (3.3)
kin kin t
2 2
Ω−tZ∪Ω+t ΓZt
and
W (t)= H(x,t)dx, w (t)= h(x,t)dΓ , (3.4)
int int t
ZΩ−t ∪Ω+t ZΓt
be the kinetic and internal energies of the volume Ω−t ∪Ω+t and the moving wave
front Γ , respectively. Here W (t)+w (t) and W (t)+w (t) are the total
t kin kin int int
kinetic and internal energies, respectively; W (t)+w (t)+W (t)+w (t) is
kin kin int int
the total energy.
Theorem 3.1. Let (U,ρ,H) together with a discontinuity hypersurface Γ= (x,t):
S(x,t) = 0 be a δ-shock wave type solution (in the sense of Definition 2.1) of the
(cid:8)
Cauchy problem (1.10), (2.1), where
(cid:9)
ρ(x,t)=ρ(x,t)+e(x,t)δ(Γ), H(x,t)=H(x,t)+h(x,t)δ(Γ).
Let this solution satisfy the entropy condition (2.11). Suppose that (U,ρ,H) is com-
pactly supported wbith respect to x, smooth in Ω = (bx,t) : S(x,t) > 0 and has
±
{ ± }
one-sidedlimitsU ,ρ ,H onΓ. Thenthefollowing massandmomentumbalance
± ± ±
relations hold:
M˙ (t)= m˙ (tb), bm˙ (t) 0, P˙(t)= p˙(t),
− ≥ − (3.5)
M(t)+m(t)=M(0)+m(0), P(t)+p(t)=P(0)+p(0).
In fact, the proof of Theorem 3.1 coincides with the proof of [29, Theorem 9.2.].
The proof of [29, Theorem 9.2.], and, consequently, the proof of Theorem 3.1 are
based onthe volume andsurface transportTheorems A.1, A.2 and use the first two
relations in (2.8).
Theorem 3.2. Let (U,ρ,H) together with a discontinuity hypersurface Γ= (x,t):
S(x,t) = 0 satisfy the same conditions as in Theorem 3.1. Then the following
(cid:8)
energy balance relations hold:
(cid:9)
w˙ (t)+w˙ (t) 0, W˙ (t) 0, W˙ (t)+w˙ (t) 0, W˙ (t) 0. (3.6)
kin int kin int int int
≥ ≤ ≥ ≤
Moreover,
W˙ (t)+w˙ (t)= W˙ (t)+w˙ (t) ,
kin kin int int
−
W (t)+w (t)+W (t)+w (t)= (3.7)
kin kin int (cid:0) int (cid:1)
=W (0)+w (0)+W (0)+w (0).
kin kin int int
Proof. 1. Let us assume that the supports of U(x,t) and ρ(x,t) with respect to x
dbeenloontge,taosabecfoomrep,athcte Kspa∈ceRnonxrmboaulntdoeΓdtbpyoi∂ntKin.g LfreotmKΩt±−t =toΩΩ±t+t .∩DKiff.ereBnytiaνtiwnge
W (t)+W (t) and using the volume transport Theorem A.1, we obtain
kin int
∂ ρ(x,t)U(x,t)2
W˙ (t)+W˙ (t)= | | +H(x,t) dx
kin int ZKt−∪Kt+ ∂t(cid:16) 2 (cid:17)
ρ(x,t)U(x,t)2
+ | | +H(x,t) V(x,t) ν˜dΓ , (3.8)
t
Z∂Kt−∪∂Kt+(cid:16) 2 (cid:17) ·
where ν˜ is the outward unit space normal to the surface ∂Kt± and V(x,t) is the
velocity of the point x in Kt±.
10 S.ALBEVERIO,O.S.ROZANOVA,ANDV.M.SHELKOVICH
Next, taking into account that for x ∈ Kt± system (1.10) has a smooth solution
(U ,ρ ,H ), i.e.,
± ± ±
ρ U 2 ρ U 2
±| ±| +H± + ±| ±| +H± U± =0,
2 t ∇· 2
(cid:16) (cid:17) (cid:16)(cid:16) (cid:17) (cid:17)
and U±, ρ±, H± are equal to zero on the hypersurface ∂Kt± except Γt, applying
Gauss’s divergence theorem to relation (3.8), we transform it to the form
ρ U 2
W˙kin(t)+W˙int(t)= −| −| +H− U− dx
−ZKt−∇·(cid:16)(cid:16) 2 (cid:17) (cid:17)
ρ+ U+ 2 ρ U 2
| | +H+ U+ dx+ | | +H U νdΓ
δ t
−ZKt+∇·(cid:16)(cid:16) 2 (cid:17) (cid:17) ZΓth 2 i ·
ρ U 2
= −| −| +H− U− νdΓt
− 2 ·
ZΓt(cid:16) (cid:17)
ρ+ U+ 2 ρ U 2
+ | | +H+ U+ ν dΓ + | | +H U νdΓ
t δ t
2 · 2 ·
ZΓt(cid:16) (cid:17) ZΓth i
ρ U 2 ρ U 2
= | | +H U | | +H U νdΓ , (3.9)
δ t
− 2 − 2 ·
ZΓt(cid:18)h(cid:16) (cid:17) i h i (cid:19)
where U = V is the velocity (2.3) of the δ-shock front Γ . Using the third
δ Γt t
Rankine–Hugoniot condition (2.4), relation (3.9) can be rewritten as
(cid:12)
(cid:12)
W˙ (t)+W˙ (t)
kin int
δ eU 2 eU 2
= | δ| +h + | δ| +h U dΓ . (3.10)
− δt 2 ∇Γt · 2 δ t
ZΓt(cid:18) (cid:16) (cid:17) (cid:16)(cid:16) (cid:17) (cid:17)(cid:19)
Applying the surface transport Theorem A.2 to the second relations in (3.3), (3.4)
one can see that the right-hand side of (3.10) coincides with w˙ (t) w˙ (t).
kin int
− −
Thus relations (3.7) hold.
Since ρ 0, H 0 and the solution (U,ρ,H) of the Cauchy problem (1.10),
± ±
≥ ≥
(2.1) satisfies the entropy condition (2.11), we have
[ρU 2U] [ρU 2]U ν
δ
| | − | | ·
(cid:0) = ρ U 2(U(cid:1) U ) ν+ρ+ U+ 2(U U+) ν 0; (3.11)
−| −| −− δ · | | δ− · Γt ≥
[HU]−[H(cid:0)]Uδ ·ν = H−(U−−Uδ)·ν+H+(Uδ−U+)·(cid:1)ν(cid:12)(cid:12) Γt ≥0. (3.12)
Form(cid:0)ulas(3.9),(3.1(cid:1)1), (3.1(cid:0)2)implythatW˙ (t)+W˙ (t) 0,i.(cid:1)e(cid:12).,dueto(3.7)the
kin int ≤ (cid:12)
first inequality in (3.6) holds.
2. In fact, the second inequality in (3.6) was proved in [26].
Let us calculate w˙(t). Taking into account formula (2.8), due to the surface
transport Theorem A.2, we obtain
1 δ
w˙ (t)= e(x,t)U (x,t)2 + (e(x,t)U (x,t)2U ) dΓ
kin 2 δt | δ | ∇Γt · | δ | δ t
ZΓt(cid:16) (cid:0) (cid:1) (cid:17)
1 δ
= e(x,t)U (x,t)2 2 Ge(x,t)U (x,t)2 dΓ
2 δt | δ | − K | δ | t
ZΓt(cid:16) (cid:0) (cid:1) (cid:17)
n
1 δ(eu ) δu
= u δk +u e δk 2 Ge(x,t)U (x,t)2 dΓ . (3.13)
2 δk δt δk δt − K | δ | t
ZΓt(cid:16)Xk=1(cid:16) (cid:17) (cid:17)