c 2004 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL.
Vol. 36, No. 2, pp. 405–422
RENORMALIZED ENTROPY SOLUTIONS FOR QUASI-LINEAR
ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS∗
MOSTAFA BENDAHMANE† AND KENNETH H. KARLSEN† ‡
Abstract. We prove the well-posedness (existence and uniqueness) of renormalized entropy
solutions to the Cauchy problem for quasi-linear anisotropic degenerate parabolic equations with L1
data. This paper complements the work by Chen and Perthame [Ann. Inst. H. Poincaré Anal. Non
Linéaire, 20 (2003), pp. 645–668], who developed a pure L1 theory based on the notion of kinetic
solutions.
Key words. degenerate parabolic equation, quasi-linear, anisotropic diffusion, entropy solution,
renormalized solution, uniqueness, existence
AMS subject classifications. 35K65, 35L65
DOI. 10.1137/S0036141003428937
1. Introduction. We consider the Cauchy problem for quasi-linear anisotropic
degenerate parabolic equations with L1 data. This convection–diffusion-type problem
is of the form
(1.1)
∂t u + divf (u) = ∇ · (a(u)∇u) + F,
u(0, x) = u0 (x),
where (t, x) ∈ (0, T ) × Rd ; T > 0 is fixed; div and ∇ are with respect to x ∈ Rd ; and
u = u(t, x) is the scalar unknown function that is sought. The (initial and source)
data u0 (x) and F (t, x) satisfy
u0 ∈ L1 (Rd ),
(1.2)
F ∈ L1 ((0, T ) × Rd ).
The diffusion function a(u) = (aij (u)) is a symmetric d × d matrix of the form
(1.3)
a(u) = σ(u)σ(u) ≥ 0,
σ ∈ (L∞
loc (R))
d×K
1 ≤ K ≤ d,
,
and hence has entries
aij (u) =
K
σik (u)σjk (u),
i, j = 1, . . . , d.
k=1
The inequality in (1.3) means that for all u ∈ R
d
aij (u)λi λj ≥ 0
∀λ = (λ1 , . . . , λd ) ∈ Rd .
i,j=1
Finally, the convection flux f (u) is a vector-valued function that satisfies
(1.4)
d
f (u) = (f1 (u), . . . , fd (u)) ∈ (Liploc (R)) .
∗ Received by the editors June 11, 2003; accepted for publication (in revised form) March 5,
2004; published electronically July 29, 2004. This work was supported by the BeMatA program of
the Research Council of Norway and the European network HYKE, funded by the EC as contract
HPRN-CT-2002-00282.
http://www.siam.org/journals/sima/36-2/42893.html
† Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N–5008 Bergen, Norway
([email protected]).
‡ Centre of Mathematics for Applications, Department of Mathematics, University of Oslo,
P.O. Box 1053, Blindern, N–0316 Oslo, Norway ([email protected], http://www.mi.uib.no/
˜kennethk/).
405
406
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
It is well known that (1.1) possesses discontinuous solutions and that weak solutions are not uniquely determined by their initial data (the scalar conservation law
is a special case of (1.1)). Hence (1.1) must be interpreted in the sense of entropy
solutions [16, 20, 21]. In recent years the isotropic diffusion case, for example, the
equation
u
∂t u + divf (u) = ∆A(u), A(u) =
(1.5)
a(ξ) dξ, 0 ≤ a ∈ L∞
loc (R),
0
has received much attention, at least when the data are regular enough (say L1 ∩ L∞ )
to ensure ∇A(u) ∈ L2 . Various existence results for entropy solutions of (1.5) (and
(1.1)) can be derived from the work by Vol’pert and Hudjaev [21]. Some general
uniqueness results for entropy solutions have been proved in the one-dimensional context by Wu and Yin [22] and Bénilan and Touré [2]. In the multidimensional context
a general uniqueness result is more recent and was proved by Carrillo [6, 5] using
Kružkov’s doubling-of-variables device. Various extensions of his result can be found
in [4, 13, 14, 15, 17, 18, 19]; see also [7] for a different approach. Explicit “continuous
dependence on the nonlinearities” estimates were proved in [10]. In the literature
just cited it is essential that the solutions u possess the regularity ∇A(u) ∈ L2 . This
excludes the possibility of imposing general L1 data, since it is well known that in
this case one cannot expect that much integrability.
The general anisotropic diffusion case (1.1) is more delicate and was successfully
solved only recently by Chen and Perthame [9]. Chen and Perthame introduced the
notion of kinetic solutions and provided a well-posedness theory for (1.1) with L1 data.
Using their kinetic framework, explicit continuous dependence and error estimates for
L1 ∩ L∞ entropy solutions were obtained in [8]. With the only assumption that
the data belong to L1 , we cannot expect a solution of (1.1) to be more than L1 .
Hence it is in general impossible to make distributional sense to (1.1) (or its
entropy
formulation). In addition, as already mentioned above, we cannot expect a(u)∇u
to be square-integrable, which seems to be an essential condition for uniqueness. Both
these problems were elegantly dealt with in [9] using the kinetic approach.
The purpose of the present paper is to offer an alternative “pure” L1 wellposedness theory for (1.1) based on a notion of renormalized entropy solutions and
the classical Kružkov method [16]. The notion of renormalized solutions was introduced by DiPerna and Lions in the context of Boltzmann equations [11]. This notion
(and a similar one called entropy solutions) was then adapted to nonlinear elliptic and
parabolic equations with L1 (or measure) data by various authors. We refer to [3] for
some recent results in this context and a list of relevant references. Bénilan, Carrillo,
and Wittbold [1] introduced a notion of renormalized Kružkov entropy solutions for
scalar conservation laws with L1 data and proved the existence and uniqueness of
such solutions. Their theory generalizes the Kružkov well-posedness theory for L∞
entropy solutions [16].
Motivated by [1, 3] and [9], we introduce herein a notion of renormalized entropy
solutions for (1.1) and prove its well-posedness. Let us illustrate our notion of an L1
solution on the isotropic diffusion equation (1.5) with initial data u|t=0 = u0 ∈ L1 .
To this end, letTl : R → R denote the truncation function at height l > 0 and
z
let ζ(z) = 0 a(ξ) dξ. A renormalized entropy solution of (1.5) is a function
u ∈ L∞ (0, T ; L1 (Rd )) such that (i) ∇ζ(Tl (u)) is square-integrable on (0, T ) × Rd for
any l > 0; (ii) for any convex C 2 entropy-entropy flux triple (η, q, r), with η bounded
and q = η f , r = η a, there exists for any l > 0 a nonnegative bounded Radon
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
407
measure µl on (0, T ) × Rd , whose total mass tends to zero as l ↑ ∞, such that
(1.6)
∂t η(Tl (u)) + divq(Tl (u)) − ∆r(Tl (u))
≤ −η (Tl (u)) |∇ζ(Tl (u))| + µl (t, x)
2
in D ((0, T ) × Rd ).
Roughly speaking, (1.6) expresses the entropy condition satisfied by the truncated
function Tl (u). Of course, if u is bounded by M , choosing l > M in (1.6) yields the
usual entropy formulation for u, i.e., a bounded renormalized entropy solution is an
entropy solution. However, in contrast to the usual entropy formulation, (1.6) makes
sense also when u is merely L1 and possibly unbounded. Intuitively the measure µl
should be supported on {|u| = l} and carry information about the behavior of the
“energy” on the set where |u| is large. The requirement is that the energy should be
small for large values of |u|, that is, the total mass of the renormalization measure
µl should vanish as l ↑ ∞. This is essential for proving uniqueness of a renormalized
entropy solution. Being explicit, the existence proof reveals that
|u0 | dx → 0 as l ↑ ∞.
µl ((0, T ) × Rd ) ≤
{|u0 |>l}
We prove existence of a renormalized entropy solution to (1.1) using an approximation procedure based on artificial viscosity [21] and bounded data. We derive a
priori estimates and pass to the limit in the approximations.
Uniqueness of renormalized entropy solutions is proved by adapting the doublingof-variables device due to Kružkov [16]. In the first order case, the uniqueness proof
of Kružkov depends crucially on the fact that
∇x Φ(x − y) + ∇y Φ(x − y) = 0,
Φ smooth function on Rd ,
which allows for a cancellation of certain singular terms. The proof herein for the
second order case relies in addition crucially on the following identity involving the
Hessian matrices of Φ(x − y):
∇xx Φ(x − y) + 2∇xy Φ(x − y) + ∇yy Φ(x − y) = 0,
which, when used together with the parabolic dissipation terms (like the one found in
(1.6)), allows for a cancellation of certain singular terms involving the second order
operator in (1.1). Compared to [9], our uniqueness proof is new even in the case of
bounded entropy solutions.
The remaining part of this paper is organized as follows: In section 2 we introduce the notion of a renormalized entropy solution for (1.1) and state our main
well-posedness theorem. The proof of this theorem is given in section 3 (uniqueness)
and section 4 (existence).
2. Definitions and statement of main result. We start by defining an
entropy-entropy flux triple.
Definition 2.1 (entropy-entropy flux triple). For any convex C 2 entropy function η : R → R, the corresponding entropy fluxes
q = (q1 , . . . , qd ) : R → Rd
and
r = (rij ) : R → Rd×d
are defined by q (u) = η (u)f (u) and r (u) = η (u)a(u). We will refer to (η, q, r) as
an entropy-entropy flux triple.
408
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
For 1 ≤ k ≤ K and 1 ≤ i ≤ d, we let
u
ζik (u) =
σik (ξ) dξ,
ζk (u) = (ζ1k (u), . . . , ζdk (u)) ,
0
and for any ψ ∈ C(R)
ψ
ζik
(u) =
ψ
ψ
(u) .
(u), . . . , ζdk
ζkψ (u) = ζ1k
u
ψ(ξ)σik (ξ) dξ,
0
Let us introduce the following set of vector fields:
L2 (0, T ; L2 (div; Rd ))
d
= w = (w1 , . . . , wd ) ∈ L2 ((0, T ) × Rd ) : divw ∈ L2 ((0, T ) × Rd ) .
Following [9] we define an entropy solution as follows.
Definition 2.2 (entropy solution). An entropy solution of (1.1) is a measurable
function u : (0, T ) × Rd → R satisfying the following conditions:
(D.1) u ∈ L∞ (0, T ; L1 (Rd )) ∩ L∞ ((0, T ) × Rd ).
(D.2) For any k = 1, . . . , K, ζk (u) ∈ L2 (0, T ; L2 (div; Rd )).
(D.3) (chain rule) For any k = 1, . . . , K and ψ ∈ C(R),
divζkψ (u) = ψ(u)divζk (u)
a.e. in (0, T ) × Rd and in L2 ((0, T ) × Rd ).
(t, x) by
(D.4) Define the parabolic dissipation measure nu,ψ
l
nu,ψ (t, x) = ψ(u(t, x))
K 2
divζk (u(t, x)) .
k=1
For any entropy-entropy flux triple (η, q, r),
(2.1)
∂t η(u) +
d
∂xi qi (u) −
i=1
− η (u)F ≤ −n
d
∂x2i xj rij (u)
i,j=1
u,η in D ((0, T ) × Rd ).
(D.5) ess limt↓0 u(t, ·) − u0 L1 (Rd ) = 0.
An important contribution of Chen and Perthame [9] is to make explicit the point
that the chain rule (D.3) should be included in the definition of an entropy solution
in the anisotropic diffusion case. They also note that (D.3) is automatically fulfilled
when a(u) is a diagonal matrix, and can then be deleted from Definition 2.2. This
applies to the isotropic case (1.5).
Uniqueness of an entropy solution in the sense of Definition 2.2 was proved in
[9] using a kinetic formulation and regularization by convolution. The present paper
offers an alternative proof based on the more classical Kružkov method of doubling
the variables [16].
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
409
Let us mention that (D.4) implies that the following Kružkov-type entropy condition holds for all c ∈ R (here Aij (·) = aij (·)):
∂t |u − c| +
d
∂xi sign (u − c) (fi (u) − fi (c))
i=1
(2.2)
−
d
∂x2i xj sign (u − c) (Aij (u) − Aij (c)) − sign (u − c) F ≤ 0.
i,j=1
In the isotropic case (1.5), (2.2) simplifies to
(2.3)
∂t |u − c| + div sign (u − c) (f (u) − f (c)) − ∆ |A(u) − A(c)| ≤ 0.
After Carrillo’s work [6, 5], it is known that (2.3) implies uniqueness in the isotropic
case (1.5). In the anisotropic case (1.1), (2.2) is not sufficient for uniqueness. Indeed,
it is necessary to explicitly include the parabolic dissipation measure in the entropy
condition, as is done in (D.4).
As we discussed in section 1, for unbounded L1 solutions Definition 2.2 is in
general not meaningful. In [9] the authors use a notion of kinetic solutions to handle
this problem. It is the purpose of this paper to use instead a notion of renormalized
entropy solutions. Before we can introduce this notion, let us recall the definition of
the (Lipschitz continuous) truncation function Tl : R → R at height l > 0:
⎧
⎪
⎨−l, u < −l,
Tl (u) = u,
(2.4)
|u| ≤ l,
⎪
⎩
l,
u > l.
We then suggest the following notion of an L1 solution.
Definition 2.3 (renormalized entropy solution). A renormalized entropy solution of (1.1) is a measurable function u : (0, T ) × Rd → R satisfying the following
conditions:
(D.1) u ∈ L∞ (0, T ; L1 (Rd )).
(D.2) For any k = 1, . . . , K, ζk (Tl (u)) ∈ L2 (0, T ; L2 (div; Rd )) for all l > 0.
(D.3) (renormalized chain rule) For any k = 1, . . . , K and ψ ∈ C(R),
divζkψ (Tl (u)) = ψ(Tl (u))divζk (Tl (u))
a.e. in (0, T ) × Rd and in L2 ((0, T ) × Rd ) for all l > 0.
(D.4) For l > 0, introduce the renormalized parabolic dissipation measure
(t, x) = ψ(Tl (u(t, x)))
nu,ψ
l
K 2
divζk (Tl (u(t, x))) .
k=1
For any l > 0 and any entropy-entropy flux triple (η, q, r), with |η | bounded
by K (for some given K), there exists a nonnegative bounded Radon measure
µu,K
on (0, T ) × Rd such that
l
(2.5)
∂t η(Tl (u)) +
d
∂xi qi (Tl (u)) −
i=1
− η (Tl (u))F ≤
d
∂x2i xj rij (Tl (u))
i,j=1
−nu,η
l
+ µu,K
l
in D ((0, T ) × Rd ).
410
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
(D.5) The total mass of the renormalization measure µu,K
vanishes as l ↑ ∞:
l
((0, T ) × Rd ) = 0.
lim µu,K
l
l↑∞
(D.6) ess limt↓0 u(t, ·) − u0 L1 (Rd ) = 0.
Note that since Tl (u) ∈ L∞ ((0, T ) × Rd ), the terms in (2.5) are all well defined.
Moreover, if a renormalized entropy solution u belongs to L∞ ((0, T ) × Rd ), then it
is also an entropy solution in the sense of Definition 2.2 (let l ↑ ∞ in Definition
2.3).
Our well-posedness result is contained in the following theorem, which is proved
in section 3 (uniqueness) and section 4 (existence).
Theorem 2.1 (well-posedness). Suppose that (1.2), (1.3), and (1.4) hold. Then
there exists a unique renormalized entropy solution u of (1.1).
It is worthwhile mentioning that Theorem 2.1 holds under merely local regularity
assumptions on f (u), a(u). Moreover, a(u) can be discontinuous, which is of interest
in some applications [4].
Remark 2.1. In the isotropic case it is not necessary to include the chain rule
(D.3) as a part of Definition 2.3, since it is then automatically fulfilled. Indeed, let
∞
0 ≤ σ ∈ L∞
loc (R) and ψ ∈ Lloc (R). Set
β(z) =
z
σ(ξ) dξ,
0
β ψ (z) =
z
ψ(ξ)σ(ξ) dξ.
0
Then, for any measurable function u(x) such that ∂xi β(Tl (u)) ∈ L1loc (Rd ), for some
fixed i = 1, . . . , d, there holds
(2.6)
∂xi β ψ (Tl (u(x))) = ψ(Tl (u(x)))∂xi β(Tl (u(x)))
for a.e. x ∈ Rd and in L2loc (Rd ) for all l > 0. To establish (2.6) we can apply the
proof in [9] to the function v := β(Tl (u)) ∈ L∞ (R), which satisfies ∂xi v ∈ L1loc (Rd ).
Remark 2.2 (the initial condition). In Definition 2.3 of a renormalized entropy
solution we require that the initial condition at t = 0 be satisfied in the strong L1
sense. When proving convergence of approximate solution sequences without having
BV estimates at our disposal, it can be difficult to verify condition (D.6) for a limit
function. To have a more flexible framework, the initial condition can be included
into the renormalized entropy formulation, that is, delete condition (D.6) and require
instead that the renormalized entropy inequality in (2.5) hold in D ([0, T ) × Rd ).
should be bounded on [0, T ) × Rd and satisfy
In this case, the Radon measure µu,K
l
u,K
d
liml↑∞ µl ([0, T )×R ) = 0. Such a weak formulation of the initial condition is much
easier to verify for limits of certain approximate solution sequences. This point was
made explicit in [12]; see also [13] for degenerate parabolic equations. To prove Theorem 2.1 with a weak formulation of the initial condition we simply have to combine
the proof of Theorem 3.1 below with a straightforward adaptation of the arguments
in [12, 13] (we leave the details to the reader). Of course, the comments above also
apply to Definition 2.3 of an entropy solution.
3. Uniqueness of renormalized entropy solution. For the uniqueness proof,
we need a C 1 approximation of sign (·) and a corresponding C 2 approximation of the
Kružkov entropy flux |· − c|, c ∈ R.
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
For ε > 0, set
(3.1)
411
⎧
⎪
ξ < −ε,
⎨−1,
π
signε (ξ) = sin 2ε ξ , |ξ| ≤ ε,
⎪
⎩
1,
ξ > ε.
For each c ∈ R, the corresponding entropy function
u
u → ηε (u, c) =
signε (ξ − c) dξ
c
ηε
is convex and belongs to C (R) with
∈ Cc (R) and |ηε | ≤ 1 (so that the constant
K appearing in Definition 2.3 is 1). Moreover, ηε is symmetric in the sense that
ηε (u, c) = ηε (c, u) and
2
ηε (u, c) → η(u, c) := |u − c|
as ε ↓ 0.
For each c ∈ R and 1 ≤ i, j ≤ d, we define the entropy flux functions
u
ε
signε (ξ − c) fi (ξ) dξ,
u → qi (u, c) =
c
(3.2)
u
ε
u → rij (u, c) =
signε (ξ − c) Aij (ξ) dξ,
c
where
Aij (·)
= aij (·) for 1 ≤ i, j ≤ d. Then as ε ↓ 0
qiε (u, c) → qi (u, c) := sign (u − c) (fi (u) − fi (c)) ,
ε
rij
(u, c) → rij (u, c) := sign (u − c) (Aij (u) − Aij (c))
ε
for 1 ≤ i, j ≤ d. Let q ε = (q1ε , . . . , qdε ), rε = rij
, and similarly for q, r.
We are now ready to prove uniqueness of renormalized entropy solutions.
Theorem 3.1 (uniqueness). Suppose that (1.3) and (1.4) hold. Let u and v be
renormalized entropy solutions of (1.1) with data F ∈ L1 ((0, T ) × Rd ), u0 ∈ L1 (Rd )
and G ∈ L1 ((0, T ) × Rd ), v0 ∈ L1 (Rd ), respectively. Then for a.e. t ∈ (0, T ),
(3.3)
u(·, t) − v(·, t)
L1 (Rd )
(3.4)
≤ u0 − v0 L1 (Rd ) +
0
t
F (s, ·) − G(s, ·)
L1 (Rd ) ds.
In particular, (1.1) admits at most one renormalized entropy solution.
Proof. We shall prove (3.4) using Kružkov’s doubling-of-variables method [16].
When it is notationally convenient we drop the domain of integration.
ε
Let (ηε , qiε , rij
) be the entropy flux triple defined above, and denote by µul , µvl the
corresponding renormalization measures.
From the definition of a renormalized entropy solution for u = u(t, x),
⎛
⎞
d
d
ε
⎝ηε (Tl (u), c)∂t φ +
qiε (Tl (u), c)∂xi φ +
rij
(Tl (u), c)∂x2i xj φ⎠ dx dt
(3.5)
−
i=1
i,j=1
signε (Tl (u) − c) F (t, x)φ dx dt
≥ nu,signε (·−c) (t, x)φ dx dt − φ dµul (t, x)
412
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
for all c ∈ R, for all l > 0, and for every 0 ≤ φ = φ(t, x) ∈ D((0, T ) × Rd ).
From the definition of a renormalized entropy solution for u = u(s, y),
⎛
⎞
d
d
ε
⎝ηε (Tl (v), c)∂s φ +
qiε (Tl (v), c)∂yj φ +
rij
(Tl (v), c)∂y2i yj φ⎠ dy ds
(3.6)
i=1
−
i,j=1
signε (Tl (v) − c) G(s, y)φ dy ds
≥ nv,signε (c−·) (s, y)φ dy ds − φ dµvl (s, y)
for all c ∈ R, for all l > 0, and for every 0 ≤ φ = φ(s, y) ∈ D((0, T ) × Rd ).
Choose c = Tl (v(s, y)) in (3.5) and integrate over (s, y). Choose c = Tl (u(t, x))
in (3.6) and integrate over (t, x). Then adding the two resulting inequalities yields
ηε (Tl (u), Tl (v)) (∂t + ∂s ) φ
+
d
[qiε (Tl (u), Tl (v))∂xi φ + qiε (Tl (v), Tl (u))∂yi φ]
i=1
+
d i,j=1
(3.7)
−
ε
rij
(Tl (u), Tl (v))∂x2i xj φ
+
ε
rij
(Tl (v), Tl (u))∂y2i yj φ
dx dt dy ds
signε (Tl (u) − Tl (v)) (F (t, x) − G(s, y)) dx dt dy ds
≥
nu,signε (·−c) (t, x) + nv,signε (·−c) (s, y) φ dx dt dy ds
−
φ(t, x, s, y) dµul (t, x) dy ds
−
φ(t, x, s, y) dµvl (s, y) dx dt,
where φ = φ(t, x, s, y) is any nonnegative function in D(((0, T ) × Rd )2 ).
We introduce next a standard mollifier sequence ωρ : R × Rd → R, ρ > 0, and
take our test function φ = φ(t, x, s, y) to be of the form
t−s x−y
t+s x+y
,
ωρ
,
,
ϕ ∈ D((0, T )×Rd ),
0 ≤ ϕ ≤ 1.
φ(t, x, s, y) = ϕ
2
2
2
2
t−s x−y ωρ 2 , 2 and
With this choice, we have (∂t + ∂s ) φ = (∂t + ∂s ) ϕ t+s
, x+y
2
2
t+s x+y t−s x−y (∇x + ∇y ) φ = (∇x + ∇y ) ϕ 2 , 2 ωρ 2 , 2 .
Introduce the Hessian matrices
∇xx φ = ∂x2i xj φ , ∇xy φ = ∂x2i yj φ , ∇yy φ = ∂y2i yj φ .
Then one can check that the following crucial matrix equality holds:
t−s x−y
t+s x+y
,
ωρ
,
.
(∇xx + 2∇xy + ∇yy ) φ = (∇xx + 2∇xy + ∇yy ) ϕ
2
2
2
2
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
413
Note that the two latter properties imply that for 1 ≤ i ≤ d,
(3.8)
qiε (Tl (u), Tl (v))∂xi φ + qiε (Tl (v), Tl (u))∂yi φ
t−s x−y
t+s x+y
ε
ωρ
,
= qi (Tl (u), Tl (v)) (∂xi + ∂yi ) ϕ
,
2
2
2
2
+ [qiε (Tl (v), Tl (u)) − qiε (Tl (u), Tl (v))] ∂yi φ,
and for 1 ≤ i, j ≤ d,
(3.9)
ε
ε
rij
(Tl (v), Tl (u))∂y2i yj φ
(Tl (u), Tl (v))∂x2i xj φ + rij
t + s x + y t − s x − y ε
2
2
2
,
ωρ
,
= rij (Tl (u), Tl (v)) ∂xi xj + 2∂xi yj + ∂yi yj ϕ
2
2
2
2
ε
− 2rij
(Tl (u), Tl (v))∂x2i yj φ
ε
ε
+ rij
(Tl (v), Tl (u)) − rij
(Tl (u), Tl (v)) ∂y2i yj φ.
We also have
−
(3.10)
φ(t, x, s, y) dµul (t, x) dy ds
t−s x−y
≥ − ωρ
,
dy ds dµul (t, x) ≥ −µul ((0, T ) × Rd )
2
2
and similarly
(3.11)
−
φ(t, x, s, y) dµvl (s, y) dx dt ≥ −µvl ((0, T ) × Rd ).
Insertion of (3.8)–(3.11) into (3.7) gives
(3.12)
t+s x+y
ηε (Tl (u), Tl (v)) (∂t + ∂s ) ϕ
,
2
2
d
t+s x+y
+
,
qiε (Tl (u), Tl (v)) (∂xi + ∂yi ) ϕ
2
2
i=1
d
t + s x + y ε
2
2
2
+
rij (Tl (u), Tl (v)) ∂xi xj + 2∂xi yj + ∂yi yj ϕ
,
2
2
i,j=1
t−s x−y
× ωρ
,
dx dt dy ds
2
2
− signε (Tl (u) − Tl (v)) (F (t, x) − G(s, y)) dx dt dy ds
≥ E1 (ε) + E2 (ε) + E3 (ε) − µvl ((0, T ) × Rd ) − µvl ((0, T ) × Rd ),
414
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
Ij (ε) dx dt dy ds, j = 1, 2, 3, with
where Ej (ε) =
I1 (ε) = nu,signε (·−c) (t, x) + nv,signε (·−c) (s, y) φ,
d
I2 (ε) = 2
ε
rij
(Tl (u), Tl (v))∂x2i yj φ,
i,j=1
d
I3 (ε) =
[qiε (Tl (u), Tl (v)) − qiε (Tl (v), Tl (u))] ∂yi φ
i=1
d
ε
ε
(Tl (v), Tl (u)) ∂y2i yj φ.
rij (Tl (u), Tl (v)) − rij
+
i,j=1
Clearly, we have limε↓0 E3 (ε) = 0 and
(3.13)
lim E2 (ε) =
2
ε↓0
d
rij (Tl (u), Tl (v))∂x2i yj φ dx dt dy ds.
i,j=1
Our goal now is to show that
lim E1 (ε) + lim E2 (ε) ≥ 0.
(3.14)
ε↓0
ε↓0
To this end, note first that, since signε (·) ≥ 0,
I1 (ε) ≥ 2
K
signε (Tl (u) − Tl (v)) divx ζk (Tl (u))divy ζk (Tl (v))φ,
k=1
so that
E1 (ε) ≥
2
K
signε (Tl (u) − Tl (v)) divx ζk (Tl (u))
k=1
× divy ζk (Tl (v))φ dx dt dy ds.
Invoking the chain rule (D.3) in Definition 2.3 (we can do this since signε (·)
belongs to C(R)), we have for 1 ≤ k ≤ K
(3.15)
signε (Tl (u)−·)
signε (Tl (u) − Tl (v)) divy ζk (Tl (v)) = divy ζk
(Tl (v)).
If we now use (3.15), then we have
E1 (ε) ≥
2
K
k=1
signε (Tl (u)−·)
divx ζk (Tl (u))divy ζk
(Tl (v))φ dx dt dy ds.
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
415
Integration by parts in y yields
E1 (ε) ≥
2
K d
∂xi ζik (Tl (u))
k=1 i,j=1
Tl (v)
× ∂yj
signε
(Tl (u) − ξ) σjk (ξ) dξ
φ dx dt dy ds
Tl (u)
=−
2
d
K ∂xi ζik (Tl (u))
k=1 i,j=1
×
Tl (v)
signε
(Tl (u) − ξ) σjk (ξ) dξ
∂yj φ dx dt dy ds.
Tl (u)
ε
For 1 ≤ k ≤ K and 1 ≤ j ≤ d, define the function ψjk
: R → R by
ε
ψjk
(η)
Tl (v)
=
signε (η − ξ) σjk (ξ) dξ.
η
ε
Since signε (·) ∈ C(R) and σjk (·) ∈ L∞
loc (R), we have ψjk (·) ∈ C(R) and the chain
rule can therefore be used.
Using the chain rule (D.3) in Definition 2.3 and then doing integration by parts
in x, we derive
E1 (ε) ≥ −
2
K d
ε
∂xi ζik (Tl (u))ψjk
(Tl (u))∂yj φ dx dt dy ds
k=1 i,j=1
=−
2
K d
ψε
∂xi ζikjk (Tl (u))∂yj φ dx dt dy ds
k=1 i,j=1
=−
2
K d
=
2
K d
∂yj φ dx dt dy ds
Tl (v)
k=1 i,j=1
ε
ψjk
(ξ)σik (ξ) dξ
∂xi
k=1 i,j=1
Tl (u)
Tl (u)
ε
ψjk
(ξ)σik (ξ) dξ
Tl (v)
∂x2i yj φ dx dt dy ds.
Observe that for a.e. η ∈ R
ε
lim ψjk
(η) = −sign (η − Tl (v)) σjk (η),
ε↓0
so that by the dominated convergence theorem we have for a.e. (t, x, s, y)
ε
ψjk
(ξ)σik (ξ) dξ
lim
ε↓0
Tl (u)
Tl (v)
=−
Tl (u)
Tl (v)
sign (ξ − Tl (v)) σjk (ξ)σik (ξ) dξ.
416
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
Hence, after another application of the dominated convergence theorem,
d
K Tl (u)
sign (ξ − Tl (v)) σjk (ξ)σik (ξ) dξ
lim E1 (ε) ≥ − 2
ε↓0
Tl (v)
k=1 i,j=1
× ∂x2i yj φ dx dt dy ds
(3.16)
=−
2
d
rij (Tl (u), Tl (v))∂x2i yj φ dx dt dy ds.
i,j=1
Finally, adding (3.16) to (3.13) yields (3.14).
Summing up, sending ε ↓ 0 in (3.12) gives
t−s x−y
,
dx dt dy ds
Itime + Iconv + Idiff (t, x, s, y)ωρ
2
2
− sign (Tl (u) − Tl (v)) (F (t, x) − G(s, y))
(3.17)
t+s x+y
t−s x−y
ϕ
dx dt dyds
× ωρ
,
,
2
2
2
2
≥ −µul ((0, T ) × Rd ) − µvl ((0, T ) × Rd ),
where
t+s x+y
,
,
2
2
t+s x+y
,
+ ∂yi ) ϕ
,
2
2
Itime (t, x, s, y) = |Tl (u(t, x)) − Tl (v(s, y))| (∂t + ∂s ) ϕ
Iconv (t, x, s, y) =
d
qi (Tl (u(t, x)), Tl (v(s, y))) (∂xi
i=1
Idiff (t, x, s, y) =
d
rij (Tl (u), Tl (v))
∂x2i xj
+
2∂x2i yj
+
∂y2i yj
i,j=1
t + s x + y ,
.
ϕ
2
2
Let us introduce the change of variables
x̃ =
x+y
,
2
t̃ =
t+s
,
2
z=
x−y
,
2
which maps (0, T ) × Rd × (0, T ) × Rd into
Ω = Rd × Rd × t̃, τ 0 ≤ t̃ + τ ≤ T,
τ=
t−s
,
2
0 ≤ t̃ − τ ≤ T .
Observe that
(∂t + ∂s ) ϕ
t+s x+y
,
2
2
= ϕt̃ (t̃, x̃),
(∇x + ∇y ) ϕ(t, x, s, y) = ∇x̃ ϕ(t̃, x̃).
This change of variables diagonalizes also the operator ∇xx + 2∇xy + ∇yy :
t+s x+y
,
= ∇x̃x̃ ϕ(t̃, x̃).
(∇xx + 2∇xy + ∇yy ) ϕ
2
2
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
417
Keeping in mind that
x = x̃ + z,
y = x̃ − z,
s = t̃ − τ,
t = t̃ + τ,
we may now estimate (3.17) as
Itime + Iconv − Idiff (t̃, x̃, τ, z)ωρ (τ, z) dt̃ dx̃ dτ dz
Ω
(3.18)
F (t̃ + τ, x̃ + z) − G(t̃ − τ, x̃ − z) ωρ (τ, z) dx̃ dt̃ dτ dz − o(1/l),
≥−
Ω
where
Itime (t̃, x̃, τ, z) = Tl (u(t̃ + τ, x̃ + z)) − Tl (v(t̃ − τ, x̃ − z)) ϕt̃ (t̃, x̃),
Iconv (t̃, x̃, τ, z) =
d
qi Tl (u(t̃ + τ, x̃ + z)), Tl (v(t̃ − τ, x̃ − z)) ∂x̃i ϕ(t̃, x̃),
i=1
Idiff (t̃, x̃, τ, z) =
d
rij Tl (u(t̃ + τ, x̃ + z)), Tl (v(t̃ − τ, x̃ − z)) ∂x̃2i x̃j ϕ.
i,j=1
Sending ρ ↓ 0 in (3.18) yields
|Tl (u) − Tl (v)| ∂t ϕ +
d
qi (Tl (u), Tl (v)) ∂xi ϕ
i=1
(3.19)
+
d
rij (Tl (u), Tl (v)) ∂x2i xj ϕ
i,j=1
≥−
dx dt
|F (t, x) − G(t, x)| ϕ dx dt − o(1/l).
By standard arguments (choosing a sequence of functions 0 ≤ ϕ ≤ 1 from
D((0, T ) × Rd ) that converges to 1(0,t)×Rd and using the initial conditions for u, v in
the sense of, say, (D.6) in Definition 2.3, it follows from (3.19) that for a.e. t ∈ (0, T )
|Tl (u(t, x)) − Tl (v(t, x))| dx
Rd
(3.20)
t
≤
|Tl (u0 ) − Tl (v0 )| dx +
|F (s, x) − G(s, x)| dx ds + o(1/l).
Rd
0
Rd
Equipped with (D.5) in Definition 2.3 for u and v, sending l ↑ ∞ in (3.20) finally
yields the L1 stability property (3.4).
4. Existence of renormalized entropy solution. The purpose of this section
is to prove the following theorem.
Theorem 4.1 (existence). Suppose that (1.2), (1.3), and (1.4) hold. Then there
exists at least one renormalized entropy solution u of (1.1).
We divide the proof into two steps.
Step 1 (bounded data). Suppose the data u0 and F are bounded and integrable
functions. Repeating the proof in [9] we find that there exists a unique entropy solution
u to (1.1) (interpreted in the sense of Definition 2.2), and this entropy solution can
418
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
be constructed by the vanishing viscosity method [21]. For us it remains to prove
that this entropy solution is also a renormalized entropy solution in the sense of
Definition 2.3. To this end, let uρ be the unique classical (say C 1,2 ) solution to the
uniformly parabolic problem
(4.1)
∂t uρ + divf (uρ ) = ∇ · (a(uρ )∇uρ ) + ρ∆uρ + F,
uρ (x, 0) = u0 (x).
ρ > 0,
Equipped with the a priori estimates in [21], Chen and Perthame [9] proved
uρ → u
(4.2)
a.e. and in C(0, T ; L1 (Rd )) as ρ ↓ 0,
where u is the unique entropy solution to (1.1).
S For any C 2 function S and qiS = S fi , rij
= S aij for 1 ≤ i, j ≤ d, multiplying the equation in (4.1) by S (uρ ) yields
∂t S(uρ ) +
(4.3)
d
∂xi qiS (uρ ) −
i=1
d
S
∂x2i xj rij
(uρ ) − ρ∆S(uρ )
i,j=1
− S (uρ )F (uρ ) = − nSρ + mSρ (t, x),
where the parabolic dissipation measure nSn,ρ (t, x) is defined by
nSρ (t, x) =
d
K
k=1
2
S
∂xi ζik
(u(t, x))
,
i=1
and the entropy dissipation measure mSρ (t, x) is defined by
mSρ (t, x) = ρS (uρ ) |∇uρ | .
2
An easy approximation argument reveals that (4.3) continues to hold for any function
S ∈ W 2,∞ (R).
u
Inserting S(u) = 1l 0 Tl (ξ)ξ into (4.3) and then sending l ↓ 0, we get the wellknown estimate
(4.4)
uρ L∞ (0,T ;L1 (Rd )) ≤ u0 L1 (Rd ) + F L1 ((0,T )×Rd ) .
We need to derive some additional a priori estimates (involving (2.4)) that are independent of ρ and u0 L∞ (Rd ) , F L∞ ((0,T )×Rd ) .
Lemma 4.1. For any l > 0, we have
⎛
⎞
d
2
K
2
⎝
∂xi ζik (Tl (uρ )) + ρ |∇Tl (uρ )| ⎠ dx dt ≤ Cl
(0,T )×Rd
k=1
i=1
for some constant Cl that is independent of ρ but not l. More precisely,
Cl = l u0 L1 (Rd ) + F L1 ((0,T )×Rd ) .
Proof. Introduce the function
2
u
|u|
2 ,
S(u) =
Tl (ξ) dξ =
l |u| −
0
l2
2,
|u| ≤ l,
|u| > l.
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
419
The lemma follows by choosing this S(·) in (4.3).
Lemma 4.2. For any l > 0 and any δ > 0,
⎛
⎞
d
2
K
1
2
⎝
(4.5)
∂xi ζik (uρ ) + ρ |∇uρ | ⎠ dx dt ≤ E(l)
δ {l<|uρ |<l+δ}
i=1
k=1
for some bounded function E(·) on R+ that is independent of ρ, δ and satisfies
liml↑∞ E(l) = 0.
If the data u0 , F are bounded and l > M := u0 L∞ (Rd ) + F L∞ ((0,T )×Rd ) , then
E(l) = 0.
Proof. Let us define the function S(·) by S(0) = 0 and
⎧
⎪
−1,
u < −l − δ,
⎪
⎪
⎪
−u−l
⎪
⎪
,
−l
− δ < u < −l,
⎨ δ
1
S (u) = (Tl+δ (u) − Tl (u)) = 0,
−l < u < l,
⎪
δ
⎪
⎪ u−l ,
l < u < l + δ,
⎪
δ
⎪
⎪
⎩1,
u > l + δ.
Inserting this S into (4.3) gives
⎛
⎞
d
2
K
1
2
⎝
∂xi ζik (uρ ) + ρ |∇uρ | ⎠ dx dt
δ {l<|uρ |<l+δ}
k=1 i=1
(4.6)
≤
|F | dx dt := E(l).
|u0 | dx +
{|uρ |>l}
{|u0 |>l}
Since u0 ∈ L1 (Rd ), F ∈ L1 ((0, T ) × Rd ), and, thanks to (4.4), uρ is uniformly (in ρ)
bounded in L1 ((0, T ) × Rd ), we have E(l) → 0 as l ↑ ∞.
If the data u0 , F are bounded, then it is well known that uρ L∞ ((0,T )×Rd ) ≤ M ,
where M is defined in the lemma. We observe that if l > M , then S(u0 ) = 0 and
S (uρ ) = 0. Hence we deduce E(l) = 0.
Let us choose a particular S = Sη,h in (4.3) of the form
Sη,h
= η h ,
Sη,h (0) = 0,
η ∈ C 2 (R),
η ≥ 0,
|η | ≤ K,
h ∈ C 2 (R),
supp(h ) ⊂ [−l, l].
This gives
∂t Sη,h (uρ ) +
(4.7)
d
i=1
S
∂xi qi η,h (uρ ) −
S
∂x2i xj rijη,h (uρ ) − ρ∆Sη,h (uρ )
i,j=1
(uρ )F (uρ ) = − nηρ
− Sη,h
where
d
h
+ µηρ h
(t, x),
µηρ h (t, x) := − nηρ h + mηρ h (t, x)
⎞
⎛
d
2
K
2
= −η (uρ )h (uρ ) ⎝
∂xi ζik (uρ ) + ρ |∇uρ | ⎠ .
k=1
i=1
420
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
Let hl,δ : R → R denote the function defined by hl,δ (0) = 0 and
hl,δ (u) =
⎧
⎪
⎨1,
|u| < l,
l < |u| < l + δ,
|u| > l + δ.
l+δ−|u|
,
⎪ δ
⎩
0,
Clearly,
hl,δ (u) → Tl (u),
(4.8)
hl,δ (u) → 1{|u|<l}
for any u ∈ R. The idea is to choose h = hn,l in (4.7) and then let δ ↓ 0. To this end,
d
let us first define the Radon measure µK
l,ρ,δ on (0, T ) × R by
⎛
dµK
l,ρ,δ (t, x) :=
K
1{l<|uρ |<l+δ} ⎝
δ
d
K
k=1
⎞
2
∂xi ζik (uρ )
2
+ ρ |∇uρ | ⎠ dx dt,
i=1
that is, for any Borel set E ⊂ (0, T ) × Rd ,
K
µK
l,ρ,δ (E) =
δ
⎛
⎝
E∩{l<|uρ |<l+δ}
d
K
k=1
⎞
2
∂xi ζik (uρ )
2⎠
+ ρ |∇uρ |
dx dt.
i=1
d
Then, by Lemma 4.2, µK
l,ρ,δ ((0, T ) × R ) ≤ E(l). Consequently, we may assume that
(4.9)
in the sense of measures on (0, T ) × Rd as δ ↓ 0,
K
µK
l,ρ,δ µl,ρ
K
µK
l,ρ µl
in the sense of measures on (0, T ) × Rd ρ ↓ 0
for some nonnegative bounded Radon measure µK
l satisfying
d
µK
l ((0, T ) × R ) ≤ E(l) → 0
(4.10)
as l ↑ ∞.
For any 0 ≤ φ ∈ D((0, T ) × Rd ), thanks to (4.8) and the convexity of η,
η hl,δ
lim
δ↓0
nρ
(4.11)
(t, x)φ dx dt
(0,T )×Rd
≥
η (Tl (uρ ))
(0,T )×Rd
d
K
k=1
2
∂xi ζik (Tl (uρ ))
φ dx dt.
i=1
Again because of (4.8), it can be easily checked that as δ ↓ 0 (recall q = η f and
r = η a)
(4.12)
for any u ∈ R.
Sη,hl,δ (u) → η(Tl (u)),
Sη,h
(u) → η (Tl (u)),
l,δ
q Sη,hl,δ (u) → q(Tl (u)),
rSη,hl,δ (u) → r(Tl (u))
QUASI-LINEAR DEGENERATE PARABOLIC EQUATIONS
421
Inserting h = hl,δ into (4.7) and using |η | ≤ K, (4.9), (4.11), (4.12) when sending
δ ↓ 0, we get
∂t η(Tl (uρ )) +
d
∂xi qi (Tl (uρ )) −
i=1
(4.13)
d
∂x2i xj rij (Tl (uρ ))
i,j=1
− ρ∆η(Tl (uρ )) − η (Tl (uρ ))F
d
2
K
≤ −η (Tl (uρ ))
∂xi ζik (uρ ) + µK
l,ρ
k=1
in D ((0, T ) × Rd ).
i=1
Equipped with (4.9), passing to the limit ρ ↓ 0 in (4.13) yields that u satisfies the
entropy condition (2.5).
It remains to prove that the chain rule (D.3) in Definition 2.3 holds. For any
ψ ∈ C(R), the classical chain rule gives for k = 1, . . . , K
d
ψ
(Tl (uρ )) = ψ(Tl (uρ ))
∂xi ζik
d
∂xi ζik (Tl (uρ ))
∀l > 0.
i=1
i=1
As in [9], the proof is to observe that this equality continues to hold in the limit as
d
ρ ↓ 0 since uρ converges strongly and i=1 ∂xi ζik (Tl (uρ )) weakly.
Step 2 (unbounded data). Suppose the data u0 and F satisfy (1.2). For n > 1,
introduce the truncated data u0,n = Tn (u0 ) and Fn = Tn (F ). We have u0,n → u0 ,
Fn → F in L1 as n ↑ ∞. Thanks to the L1 contraction property of the solution
operator to (4.1), {un }n>1 is a Cauchy sequence in C(0, T ; L1 (Rd )) and has a limit
point u. From Step 1 we know that each un is a renormalized entropy solution of
(1.1) with u0 and F replaced by u0,n and Fn , respectively. Denote by µK
l,n the corresponding renormalization measure. Lemma 4.1 and (4.10) imply that the following
n-independent a priori estimates hold for each l > 0:
un L∞ (0,T ;L1 (Rd )) ≤ u0 L1 (Rd ) + F L1 ((0,T )×Rd ) ,
K d
2
K
2 divζk (Tl (un )) =
∂xi ζik (Tl (un ))
≤ Cl ,
k=1
k=1
i=1
d
µK
l,n ((0, T ) × R ) ≤
{|u0 |>l}
|u0 | dx +
{|un |>l}
|F | dx dt
for some constant Cl depending on l but not n. Equipped with these estimates and
the strong convergence un → u, we can repeat the steps in the above limiting process
for the viscous approximations {uρ }ρ>0 and prove that the limit point u of {un }n>1
is a renormalized entropy solution of (1.1), with the renormalization measure µu,K
l
being a limit point of {µK
l,n }n>1 . This completes the proof of Theorem 4.1.
Acknowledgment. This work was done while MB visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway, and he is grateful
for the hospitality.
422
MOSTAFA BENDAHMANE AND KENNETH H. KARLSEN
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