doi:10.1155/2009/279818
Research Article
Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function
H. Holden,
1, 2K. H. Karlsen,
3and D. Mitrovic
11Department of Mathematical Sciences, Norwegian University of Science and Technology, Alfred Getz vei 1, 7491 Trondheim, Norway
2Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
3Department of Mathematics, Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Correspondence should be addressed to H. Holden,holden@math.ntnu.no Received 2 April 2009; Revised 24 August 2009; Accepted 24 September 2009 Recommended by Philippe G. LeFloch
We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach ofH-measures to investigate the zero diffusion-dispersion-smoothing limit.
Copyrightq2009 H. Holden et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider the convergence of smooth solutionsu uεt, xwith t, x ∈ R×Rd of the nonlinear partial differential equation
∂tudivxft, x, u εdivxb∇u δ d
j1
∂3xjxjxju 1.1
asε → 0 andδδε, ε → 0. Heref ∈CR;BVRt ×Rdxis the Caratheodory flux vector such that
max|u|≤lft, x, u−ft, x, u−→0, −→0, inLploc
R×Rd
, 1.2
for p > 2 and every l > 0. The aim is to show convergence to a weak solution of the corresponding hyperbolic conservation law:
∂tudivxft, x, u 0, uut, x, x∈Rd, t≥0. 1.3
We refer to this problem as the zero diffusion-dispersion-smoothing limit.
In the case when the fluxfis at least Lipschitz continuous, it is well known that the Cauchy problem corresponding to1.3has a unique admissible entropy solution in the sense of Kruˇzhkov1 or measure valued solution in the sense of DiPerna 2. The situation is more complicated when the flux is discontinuous and it has been the subject of intensive investigations in the recent yearssee, e.g.,3and references therein. The one-dimensional case of the problem is widely investigated using several approachesnumerical techniques 3,4, compensated compactness5,6, and kinetic approach7,8. In the multidimensional case there are only a few results concerning existence of a weak solution. In 9 existence is obtained by a two-dimensional variant of compensated compactness, while in 10 the approach of H-measures 11,12 is used for the case of arbitrary space dimensions. Still, many open questions remain such as the uniqueness and stability of solutions.
A problem that has not yet been studied in the context of conservation laws with discontinuous flux, and which is the topic of the present paper, is that of zero diffusion- dispersion limits. When the flux is independent of the spatial and temporal positions, the study of zero diffusion-dispersion limits was initiated in 13 and further addressed in numerous works by LeFloch et al. e.g., 14–17. The compensated compactness method is the basic tool used in the one-dimensional situation for the so-called limiting case in which the diffusion and dispersion parameters are in an appropriate balance. On the other hand, when diffusion dominates dispersion, the notion of measure valued solutions2,18is used.
More recently, in19the limiting case has also been analyzed using the kinetic approach and velocity averaging20.
The remaining part of this paper is organized as follows. InSection 2we collect some basic a priori estimates for smooth solutions of1.1. InSection 3we look into the diffusion- dispersion-smoothing limit for multidimensional conservation laws with a flux vector which is discontinuous with respect to spatial variable. In doing so we rely on the a priori estimates from the previous section in combination with Panov’s H-measures approach10. Finally, inSection 4we restrict ourselves to the one-dimensional case for which we obtain slightly stronger results using the compensated compactness method.
2. A priori Inequalities
Assume that the flux f in 1.1is smooth in all variables. Consider a sequence uε,δε,δ of solutions of
∂tudivxft, x, u εdivxb∇u δ d j1
∂3xjxjxju,
ux,0 u0x, x∈Rd.
2.1
We assume thatuε,δε,δhas enough regularity so that all formal computations below are correct. So, following Schonbek13, we assume that for everyε, δ > 0 we haveuε,δ ∈ L∞0, T;H4Rd.
Later on, we will assume that the initial datau0depends onε. In this section, we will determine a priori inequalities for the solutions of problem2.1.
To simplify the notation we will writeuεinstead ofuε,δ.
We will need the following assumptions on the diffusion termbλ b1λ, . . . , bnλ.
H1For some positive constantsC1, C2we have
C1|λ|2 ≤λ·bλ≤C2|λ|2 ∀ λ∈Rd. 2.2 H2The gradient matrixDbλis a positive definite matrix, uniformly inλ∈Rd, that is, for everyλ, ∈Rd, there exists a positive constantC3such that we have
TDbλ≥C32. 2.3
We use the following notation:
D2u2 d
i,k1
∂2xixku2. 2.4
In the sequel, for a vector valued functiong g1, . . . , gddefined onR×Rd×R, we denote g2 d
i1
gi2. 2.5
The partial derivative∂xi in the pointt, x, u, whereupossibly depends ont, x, is defined by the formula
∂xigt, x, ut, x
Dxigt, x, λ
|λut,x. 2.6
In particular, the total derivative Dxi and the partial derivative ∂xi are connected by the identity
Dxigt, x, u ∂xigt, x, u ∂ugt, x, u∂xiu. 2.7 Finally we use
divxgt, x, u d
i1
Dxigit, x, u, g
g1, . . . , gd ,
Δxqt, x, u d
i1
D2xixiqt, x, u, q∈C2
R×Rd×R .
2.8
With the previous conventions, we introduce the following assumption on the flux vectorf.
H3The growth of the velocity variableuand the spatial derivative of the fluxfare such that for someC, α >0,p≥1, and everyl >0, we have
max|λ|<lfit, x, λ∈Lp
R×Rd
, i1, . . . , d, d
i1
∂ufit, x, u≤C,
d i,j1
∂xifjt, x, u≤ μt, x 1|u|1α,
2.9
where μ ∈ MR ×Rd is a bounded measure and, accordingly, the above inequality is understood in the sense of measures.
Now, we can prove the following theorem.
Theorem 2.1. Suppose that the flux function f ft, x, usatisfies (H3) and that it is Lipschitz continuous onR×Rd×R. Assume also that initial datau0belongs toL2Rd. Under conditions (H1)- (H2) the sequence of solutionsuεε>0of2.1for everyt∈0, Tsatisfies the following inequalities:
Rd|uεt, x|2dxε t
0
Rd
∇uε
t, x2dxdt
≤C4
Rd|u0x|2dx− t
0
Rd
uεt,x
0
divxf t, x, v
dvdxdt
,
2.10
ε2
Rd
∇uεt, x|2dxε3 t
0
Rd
D2uε
t, x
|2dxdt
≤C5 ε2
Rd|∇u0x|2dxε t
0
Rd
d k1
∂xkf
t, x, uε
t, x2dxdt ∂uf2
L∞R×Rd×R
, 2.11
for some constantsC4andC5.
Proof. We follow the procedure from19. Given a smooth functionηηu,u∈R, we define
qit, x, u u
0
ηv∂vfit, x, vdv, i1, . . . , d. 2.12
If we multiply2.1byηu, it becomes
∂tηuε
d i1
∂xiqit, x, uε−d
i1
uε
0
∂2xivfit, x, vηvdvd
i1
ηuε∂xifit, x, uε
ε d
i1
∂xi
ηuεbi∇uε
−εη uεd
i1
bi∇uε∂xiuεδ d
i1
∂xi
ηuε∂2xixiuε
−δ
2η uεd
i1
∂xi∂xiuε2.
2.13
Choosing hereηu u2/2 and integrating over0, t×Rd, we get
Rd|uεt, x|2dxε t
0
Rd∇uε
t, x
·b
∇uε
t, x dxdt
Rd|u0x|2dxd
j1
t
0
Rd
uεt,x
0
vD2xjvfj
t, x, v dvdxdt
−d
i1
t
0
Rduε t, x
∂xifi
t, x, uε t, x
dxdt
Rd|u0x|2dx−d
i1
t
0
Rd
uεt,x
0
∂xifi t, x, v
dvdxdt,
2.14
where the second equality sign is justified by the following partial integration:
t
0
Rd
uε
0
vD2xjvfj t, x, v
dvdxdt
t
0
Rduε∂xifi
t, x, uε dxdt −
t
0
Rd
uε
0
∂xifi t, x, v
dvdxdt.
2.15
Now inequality2.10follows from2.14, usingH1.
As for inequality2.11, we start by using2.14, namely,
Rd|uεt, x|2dxε t
0
Rd∇uε
t, x
·b
∇uε
t, x dxdt
Rd|u0x|2dx−d
i1
t
0
Rd
uεt,x
0
∂xifi t, x, v
dvdxdt
≤
Rd|u0x|2dxd
i1
t
0
Rd
uεt,x
0
∂xifi
t, x, v dvdxdt
≤
Rd|u0x|2dx t
0
Rd
R
μt, x
1|v|1αdvdxdt
≤
Rd|u0x|2dxC t
0
Rdμ t, x
dxdt,
2.16
whereC
Rdv/1|v|1α.
From here, usingH3, we conclude in particular that
ε t
0
Rd
∇uε
t, x2dxdt ≤C11, 2.17
for some constantC11independent ofε.
Next, we differentiate 2.1with respect toxk and multiply the expression by∂xku.
Integrating overRd, using integration by parts and then summing overk1, . . . , d,we get:
1 2
Rd∂t|∇uε|2dx−d
k1
Rd∇∂xkuε·
∂xkfkt, x, uε ∂ufk∂xkuε
dx
−εd
k1
Rd∇∂xkuεTDb∇uε∇∂xkuεdx.
2.18
Integrating this over0, tand using the Cauchy-Schwarz inequality and conditionH2, we find
1 2
Rd|∇uεt,·|2dxεC3 d k1
t
0
Rd|∇∂xkuε|2dxdt
≤ 1 2
Rd|∇u0|2dxd
k1
∇∂xkuεL2R×Rd∂xkfk·,·, uε ∂ufk∂xkuε
L2R×Rd,
2.19
whereC3is independent ofε. Then, using Young’s inequalitythe constantC3is the same as previously mentioned
ab≤ C3ε
2 a2 1
2C3εb2, a, b∈R, 2.20
we obtain
1 2
Rd|∇uεt,·|2dxεC3 d k1
t
0
Rd|∇∂xkuε|2dxdt
≤ 1 2
Rd|∇u0|2dxC3ε 2
d k1
t
0
Rd|∇∂xkuε|2dxdt 1
2C3ε t
0
Rd
d k1
∂xkfk
t, x, uε
∂ufk∂xkuε2dxdt.
2.21
Multiplying this byε2, usingab2≤2a22b2, and applying2.17, we conclude
ε2 2
Rd|∇uεt, · |2dxC3ε3 2
Rd
t
0
D2uε2dxdt
≤ ε2 2
Rd|∇u0|2dxdt ε C3
t
0
Rd
d k1
∂xkfk
t, x, uε
t, x2dxdt C11
C3
∂ufk2
L∞R×Rd×R. 2.22
This inequality is actually inequality 2.11 when we take C5 2 max{1,1/C3, C11/ C3}/min{1, C3}.
3. The Multidimensional Case
Consider the following initial-value problem. Finduut, xsuch that
∂tudivxft, x, u 0, ux,0 u0x, x∈Rd,
3.1
whereu0∈L2Rdis a given initial data.
For the fluxf f1, . . . , fdwe need the following assumption, denotedH4.
H4aFor the fluxfft, x, u,t, x, u∈R×Rd×R, we assume thatf ∈CR;BVR× Rdand that for everyl∈Rwe have maxu∈−l,l|ft, x, u| ∈LpR×Rd,p >2 .
H4bThere exists a sequencef f1, . . . , fd,∈0,1, such thatfft, x, u∈ C1R×Rd×R, satisfying for somep >2 and everyl∈R:
z∈−l,lmaxf·,·, z−f·,·, z −→
→00 in Lp
R×Rd
0, 3.2a d
i1
R×Rd
∂xifit, x, udxdt≤ C1
1|u|1α, 3.2b
d i1,k
R×Rd
∂xkfit, x, u2dxdt≤C2, 3.2c
d i1
∂ufit, x, u≤ C β
, 3.2d
d i1
R×Rd
∂2xiufit, x, udxdt≤ C3
1|u|1α, 3.2e
where Ci, i 1,2,3, and C are constants, while the function β : R → R is such that limρ→0βρ 0.
In the case when we have only vanishing diffusion, it is usually possible to obtain uniform L∞ bound for the corresponding sequence of solutions under relatively mild assumptions on the flux and initial datasee, e.g.,9,10. In the case when we have both vanishing diffusion and vanishing dispersion, we must assume more on the flux in order to obtain even much weaker boundsseeTheorem 3.2. We remark that demand on controlling the flux at infinity is rather usual in the case of conservation laws with vanishing diffusion and dispersionsee, e.g.,16,17,19.
Remark 3.1. For an arbitrary compactly supported, nonnegativeϕ1 ∈C∞0 R×Rdandϕ2 ∈ C∞0Rwith total mass one denote
ϕz, u 1 d1ϕ1
z
1 β
ϕ2 u β
, 3.3
z ∈ R ×Rd and u ∈ R, where β is a positive function tending to zero as → 0. In the case when the fluxf ∈CR;BVR×Rd ∩BVR×R×Rdis bounded, straightforward computation shows that the sequenceff ϕ f1, . . . , fdsatisfiesH4bwithβ . We also need to assume that the fluxfis genuinely nonlinear, that is, for everyt, x∈ R×Rdand everyξ∈Rd\ {0}, the mapping
Rλ−→d
i1
fit, x, λξi
|ξ| 3.4
is nonconstant on every nondegenerate interval of the real line.
We will analyze the vanishing diffusion-dispersion-smoothing limit of the problem
∂tudivxft, x, u εdivxb∇u δ d j1
∂3xjxjxju, 3.5
ux,0 u0,εx, x∈Rd, 3.6
where the fluxfsatisfies the conditionsH4b. We denote the solution of3.5-3.6byuε uεt, x. We assume that
u0,ε−u0L2Rd−→0, u0,εL2Rd εu0,εH1Rd≤C. 3.7
We also assume thatε → 0 andδδε → 0 asε → 0. We want to prove that under certain conditions, a sequence of solutionsuεε>0of3.5-3.6converges to a weak solution of problem3.1asε → 0. To do this in the multidimensional case we use the approach of H-measures, introduced in11and further developed in10,21. In the one-dimensional case, we use the compensated compactness method, following13.
In order to accomplish the plan we need the following a priori estimates.
Theorem 3.2a priori inequalities. Suppose that the flux ft, x, usatisfies (H4). Also assume that the initial datau0satisfies3.7. Under these conditions the sequence of smooth solutionsuεε>0 of 3.5-3.6satisfies the following inequalities for everyt∈0, T:
Rd|uεt, x|2dxε t
0
Rd|∇uεx, s|2dxds≤C4
Rd|u0,εx|2dxC10
, 3.8
ε2
Rd|∇uεt, x|2dxε3 t
0
Rd
D2uε
t, x2dxdt ≤C5 ε2
Rd|∇u0,εx|2dx ε
C11 C12
β 2
, 3.9
for some constantsC10, C11, C12(the constantsC4, C5are introduced inTheorem 2.1).
Proof. For every fixed, the functionf f1, . . . , fdis smooth, and, due toH4, we see thatfsatisfiesH3. This means that we can applyTheorem 2.1.
Replacing the fluxfbyffrom3.5andu0byu0,ε from3.6in2.10and2.11, we get
Rd|uεt, x|2dxε t
0
Rd|∇uεx, s|2dxds
≤C3
Rd|u0,εx|2dx− t
0
Rd
uεt,x
0
divxf t, x, v
dvdxdt
,
3.10
ε2
Rd|∇uεt, x|2dxε3 t
0
Rd
D2uε
t, x2dxdt
≤C4 ε2
Rd|∇u0,εx|2dx∂uf2
L∞R×Rd×R ε
t
0
Rd
d k1
d i1
∂xkfi
t, x, uε t, x2
dxdt
.
3.11
To proceed, we use assumptionH4. We have
t
0
Rd
uεt,x
0
divfi
t, x, v
dvdxdt ≤ t
0
Rd
R
d i1
∂xifi
t, x, vdvdxdt
≤
R
C1
1|v|1αdv≤C10,
3.12
which together with3.10immediately gives3.8.
Similarly, combiningH4and3.11, and arguing as in3.12, we get3.9.
In this section, we will inspect the convergence of a familyuεε>0of solutions to3.5- 3.6in the case when
bλ1, . . . , λd λ1, . . . , λd 3.13
for the functionbappearing in the right-hand side of3.5. This is not an essential restriction, but we will use it in order to simplify the presentation.
Thus, we use the following theorem which can be proved using the H-measures approachsee, e.g.,10, Corollary 2 and Remark 3. We letθdenote the Heaviside function.
Theorem 3.3 see10. Assume that the vectorft, x, u is genuinely nonlinear in the sense of 3.4. Then each familyvεt, xε>0⊂L∞R×Rdsuch that for everyc∈R the distribution
∂tθvε−cvε−c divx
θvε−c
ft, x, vε−ft, x, c
3.14 is precompact inHloc−1contains a subsequence convergent inL1locR×Rd.
We can now prove the following theorem.
Theorem 3.4. Assume that the flux vectorf is genuinely nonlinear in the sense of 3.4and that it satisfies (H4). Furthermore, assume that
ε, δε2ρ2ε with ρε O
βε
, 3.15
and that u0,ε satisfies 3.7. Then, there exists a subsequence of the familyuεε>0 of solutions to 3.5–3.6that converges to a weak solution of problem3.1.
Proof. We will useTheorem 3.3. Since it is well known that the familyuεε>0of solutions of problem3.5–3.6 is not uniformly bounded, we cannot directly apply the conditions of Theorem 3.3.
Take an arbitraryC2functionS Su,u∈R, and multiply the regularized equation 3.5bySuε. As usual, put
qt, x, u u
0
Sv∂uf dv, q
q1, . . . , qd
. 3.16
We easily find that
∂tSuε divxqt, x, uε−divxqt, x, v|vuε Suεdivxft, x, v|vuε εdivx
Suε∇uε
−εS uε|∇uε|2δ d
j1
Dxj
Suε∂2xjxjuε
−δ d
j1
S uε∂xjuε∂2xjxjuε. 3.17 We will apply this formula repeatedly with different choices forSu.
In order to apply Theorem 3.3, we will consider a truncated sequence Tluεε>0, where the truncation functionTlis defined for every fixedl∈N as
Tlu
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
−l, u≤ −l, u, −l≤u≤l, l, u≥l.
3.18
We will prove that the sequenceTluεε>0 is precompact for every fixedl. Denote byul a subsequential limitin L1locof the family Tluεε>0, which gives raise to a new sequence ull>1that we prove converges to a weak solution of3.1.
To carry out this plan, we must replaceTl by aC2 regularization Tl,σ : R → R. We defineTl,σ:R → R byTl,σ0 0 and
Tl,σu
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1, |u|< l, l− |u|σ
σ , l <|u|< lσ, 0, |u|> lσ.
3.19
Next, we want to estimate Tl,σ uε∇uεL2R×Rd. To accomplish this, we insert the functionsTl,σ± forSin3.17whereTl,σ± are defined byTl,σ± 0 0 and
Tl,σ
u
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1, u < l, lσ−u
σ , l < u < lσ, 0, u > lσ,
3.20
Tl,σ−
u
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1, u >−l, lσu
σ , −l−σ < u <−l, 0, u <−l−σ.
3.21
Notice that
Tl,σ±u
≤1, Tl,σ±u≤ |u|σ 2, Tl,σu Tl,σ− u for −l≤u≤l.
3.22
By insertingSu −Tl,σu,q qt, x, u −u
0Tl,σ v∂ufdvin3.17and integrating overΠt 0, t×Rd, we get
−
RdTl,σuεdx
RdTl,σu0dx ε σ
Πt∩{l<uε<lσ}|∇uε|2dxdt
Πt
divxqt, x, v|vuεdxdt
Πt
Tl,σ
uεdivxft, x, v|vuεdxdt
−δ σ
Πt∩{l<uε<lσ}
d j1
∂xjuε∂2xjxjuεdxdt.
3.23
Similarly, forSu Tl,σ−u,qq−t, x, u u
0Tl,σ− v∂ufdv, we have from3.17
RdTl,σ−uεdx−
RdTl,σ− u0dx ε σ
Πt∩{−l−σ<uε<−l}|∇uε|2dxdt
Πt
divxq−t, x, v|vuεdxdt−
Πt
Tl,σ−
uεdivxft, x, v|vuεdxdt
δ σ
Πt∩{−l−σ<uε<−l}
d j1
∂xjuε∂2xjxjuεdxdt.
3.24
Adding3.23to3.24, we get
ε σ
Πt∩{l<|uε|<lσ}|∇uε|2dxdt −
Rd
Tl,σ− uε−Tl,σuε dx
Rd
Tl,σ−u0−Tl,σ u0 dx
Πt
divxq−t, x, v|vuεdxdt
Πt
divxqt, x, v|vuεdxdt
−
Πt
Tl,σ−
uεdivxft, x, v|vuεdxdt
Πt
Tl,σ
uεdivxft, x, v|vuεdxdt
δ σ
Πt∩{−l−σ<uε<−l}
d j1
∂xjuε∂2xjxjuεdxdt− δ σ
Πt∩{l<uε<lσ}
d j1
∂xjuε∂2xjxjuεdxdt.
3.25
From3.22and the definition ofq−andq, it follows
ε σ
Πt∩{l<|uε|<lσ}|∇uε|2dxdt≤
|uε|>l2|uε|dx
|u0|>l2|u0|dx 2
Πt
R
d i1
D2xivfit, x, vdvdxdt 2
Πt
d i1
∂xifit, x, uεdxdt 2δ
σ
Πt∩{l−σ<|uε|<l}
d j1
∂xjuε∂2xjxjuεdxdt.
3.26
Without loss of generality, we can assume thatl >1. Having this in mind, we get fromH4 and3.26
ε σ
Πt∩{l<|uε|<lσ}|∇uε|2dxdt
≤
|uε|>l2|uε|2dx
|u0|>l2|u0|2dx2
R
d i1
C3 1|v|1αdv 2
Πt
d i1
∂xifit, x, uεdxdt2δ σ
Πt∩{l<|uε|<lσ}
d j1
∂xjuε∂2xjxjuεdxdt
≤
Rd2
|uεx, t|2|u0x, t|2
dxK1K22 δ σε2
d i1
ε1/2∂xiuε
L2R×Rd
×ε3/2∂2xixiuε
L2R×Rd
≤K5 δ2 σ2ε4
β
2 δ2 σ2ε4
1/2 K3K4,
3.27
whereKi,i1, . . . ,5, are constants such thatcf.3.8and3.9
2
R
d i1
C3
1|v|1αdv≤K1, 2
Πt
d i1
∂xifit, x, uεdxdt≤K2,
d i1
ε1/2∂xiuε
L2R×Rd≤K3, d
i1
ε3/2∂2xixiuε
L2R×Rd≤ 1 β
2 ε
1/2
K4,
Rd2
|uεx, t|2|u0x, t|2
dxK1K2≤K5.
3.28
These estimates follow fromH4and the a priori estimates3.8,3.9. If in addition we use the assumptionεfrom3.15, we conclude
δ
σε2ε1/2∇uεL2R×Rd d
i1
ε3/2∂2xixiuεL2R×Rd≤ δ2
σ2ε4β2ε δ2 σ2ε4
1/2
K3K4. 3.29
Thus, in view of3.27,
ε σ
Πt∩{l<|uε|<lσ}|∇uε|2dxdt≤K5 δ2
σ2ε4β2ε δ2 σ2ε4
1/2
K3K4, 3.30
which is the sought for estimate forTl,σ uε∇uεL2R×Rd. Next, take a functionUρzsatisfyingUρ0 0 and
Uρz
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
0, z <0, z
ρ, 0< z < ρ, 1, z > ρ.
3.31
Clearly,Uρis convex, andUρz → θzinLplocRasρ → 0, for anyp < ∞; as before,θ denotes the Heaviside function.
InsertingSuε UρTl,σuε−cin3.17, we get
∂tUρTl,σuε−c divx uε
UρTl,σv−cTl,σv∂vft, x, vdv
uε
UρTl,σv−cTl,σvdivx∂vft, x, vdv
−UρTl,σuε−cTl,σuεdivxft, x, v|vuε εΔxUρTl,σuε−c−εDuu2
UρTl,σuε−c
|∇uε|2 δ
d i1
Dxi Du
UρTl,σuε−c
∂2xixiuε
−δ d i1
Duu2
UρTl,σuε−c
∂xiuε∂2xixiuε.
3.32
We rewrite the previous expression in the following manner:
∂tθTluε−cTluε−c divx
θTluε−c
ft, x, Tluε−ft, x, c
Γ1,ε Γ2,ε Γ3,ε Γ4,ε Γ5,ε Γ6,ε Γ7,ε, 3.33
