On kinetic formulation of first-order hyperbolic quasilinear systems
We give kinetic formulation of measure valued and strong measure valued solutions to the Cauchy problem for a first-order quasilinear equation. For the corresponding kinetic equation the class of existence and uniqueness to the Cauchy problem is extracted. This class consists of so-called entropy so...
Gespeichert in:
| Datum: | 2004 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2004
|
| Schriftenreihe: | Український математичний вісник |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/124631 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On kinetic formulation of first-order hyperbolic quasilinear systems / E.Yu. Panov // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 548-563. — Бібліогр.: 14 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-124631 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1246312017-10-01T03:03:15Z On kinetic formulation of first-order hyperbolic quasilinear systems Panov, E.Yu. We give kinetic formulation of measure valued and strong measure valued solutions to the Cauchy problem for a first-order quasilinear equation. For the corresponding kinetic equation the class of existence and uniqueness to the Cauchy problem is extracted. This class consists of so-called entropy solutions, which correspond to strong measure valued solutions of the original problem. In the last section we generalized these results to the case of symmetric generally nonconservative multidimensional systems and introduce the notion of a strong measure valued solution, based only on the kinetic approach under consideration. 2004 Article On kinetic formulation of first-order hyperbolic quasilinear systems / E.Yu. Panov // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 548-563. — Бібліогр.: 14 назв. — англ. 1810-3200 2000 MSC. 35L60, 35L45. http://dspace.nbuv.gov.ua/handle/123456789/124631 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We give kinetic formulation of measure valued and strong measure valued solutions to the Cauchy problem for a first-order quasilinear equation. For the corresponding kinetic equation the class of existence and uniqueness to the Cauchy problem is extracted. This class consists of so-called entropy solutions, which correspond to strong measure valued solutions of the original problem. In the last section we generalized these results to the case of symmetric generally nonconservative multidimensional systems and introduce the notion of a strong measure valued solution, based only on the kinetic approach under consideration. |
| format |
Article |
| author |
Panov, E.Yu. |
| spellingShingle |
Panov, E.Yu. On kinetic formulation of first-order hyperbolic quasilinear systems Український математичний вісник |
| author_facet |
Panov, E.Yu. |
| author_sort |
Panov, E.Yu. |
| title |
On kinetic formulation of first-order hyperbolic quasilinear systems |
| title_short |
On kinetic formulation of first-order hyperbolic quasilinear systems |
| title_full |
On kinetic formulation of first-order hyperbolic quasilinear systems |
| title_fullStr |
On kinetic formulation of first-order hyperbolic quasilinear systems |
| title_full_unstemmed |
On kinetic formulation of first-order hyperbolic quasilinear systems |
| title_sort |
on kinetic formulation of first-order hyperbolic quasilinear systems |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/124631 |
| citation_txt |
On kinetic formulation of first-order hyperbolic quasilinear systems / E.Yu. Panov // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 548-563. — Бібліогр.: 14 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT panoveyu onkineticformulationoffirstorderhyperbolicquasilinearsystems |
| first_indexed |
2025-07-09T01:45:26Z |
| last_indexed |
2025-07-09T01:45:26Z |
| _version_ |
1837131920426663936 |
| fulltext |
Український математичний вiсник
Том 1 (2004), N 4, 548 – 563
On Kinetic Formulation of First-order
Hyperbolic Quasilinear Systems
Evgeniy Yu. Panov
(Presented by E. Ya. Khruslov)
Abstract. We give kinetic formulation of measure valued and strong
measure valued solutions to the Cauchy problem for a first-order quasi-
linear equation. For the corresponding kinetic equation the class of ex-
istence and uniqueness to the Cauchy problem is extracted. This class
consists of so-called entropy solutions, which correspond to strong mea-
sure valued solutions of the original problem. In the last section we gen-
eralized these results to the case of symmetric generally nonconservative
multidimensional systems and introduce the notion of a strong measure
valued solution, based only on the kinetic approach under consideration.
2000 MSC. 35L60, 35L45.
Key words and phrases. symmetric first-order quasilinear systems,
Cauchy problem, kinetic formulation, entropy solutions, measure valued
solutions.
1. Introduction
In the half-space (t, x) ∈ Π = (0,+∞) × Rm consider firstly a scalar
conservation law
ut + divxϕ(u) = 0, (1.1)
u = u(t, x), ϕ(u) = (ϕ1(u), . . . , ϕm(u)) ∈ (C1(R))m.
Nonlocal theory of generalized entropy solutions (briefly — g.e.s.) to
the Cauchy problem for the equation (1.1) with initial condition
u(0, x) = u0(x) ∈ L∞(Rm) (1.2)
was constructed by S. N. Kruzhkov in paper [1]. Remind the definition
of g.e.s.
Received 2.12.2003
Supported by the Russian Foundation for Basic Research (grants N 03-01-00444, N 02-
01-00483), the Ministry of Education of Russian Federation (grant N E02-1.0-216) and
the Program “Universities of Russia”(project N YP.04.01.044)
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
E. Yu. Panov 549
Definition 1.1. A bounded measurable function u = u(t, x) is called a
g.e.s. to the Cauchy problem (1.1), (1.2) if
a) ∀ k ∈ R
|u− k|t + divx[(ϕ(u) − ϕ(k)) sign(u− k)] ≤ 0 (1.3)
in the sense of distributions on Π (in D′(Π));
b) ess lim
t→0
u(t, ·) = u0 in L1
loc(R
m),
i. e. there exists a set of null Lebesgue measure E ⊂ (0,+∞) such that
for t /∈ E u(t, ·) ∈ L1
loc(R
m) and u(t, ·) →
t→0,t/∈E
u0 in L1
loc(R
m).
Condition (1.3) means that for any nonnegative test function g =
g(t, x) ∈ C∞
0 (Π)
∫
Π
[
|u−k|gt+
m∑
i=1
(ϕi(u)−ϕi(k)) sign(u−k)gxi
]
dt dx ≥ 0.
Taking in (1.3) k = ±‖u‖∞ we derive that ut+divxϕ(u) = 0 and a g.e.s.
u(t, x) satisfies the equation (1.1) in the distributional sense. As is known
(see [1]), there always exists a unique g.e.s. to the problem (1.1), (1.2).
Kinetic formulation of conservation laws was proposed in papers [2, 3]
and was further developed in [4]. A function u = u(t, x) was shown to be
a g.e.s. to the problem (1.1), (1.2) if and only if the function
f = f(t, x, v) = χu(t,x)(v), where for u, v ∈ R
χu(v) = θ(u−v)−θ(−v), θ(λ) =
{
1, λ > 0
0, λ ≤ 0
is the Heaviside function,
is a generalized solution (g.s.) to the corresponding problem for a kinetic
equation
∂
∂t
f + (ϕ′(v),∇xf) =
∂
∂v
µ, (1.4)
(here (·, ·) denotes scalar product in Rm) with some nonnegative locally
finite measure µ = µ(t, x, v) on Π × R that has compact support with
respect to the variable v, and with initial condition
f(0, x, v) = f0(x, v), (1.5)
where f0(x, v) = χu0(x)(v). One should understand the initial condition
(1.5) in a similar way to the condition b) in Definition 1.1, that is,
ess lim
t→0
f(t, ·, ·) = f0 in L1
loc(R
m+1). (1.6)
In papers [5, 6] the results above were extended to the more general case
of measure valued solutions.
We recall (see [7, 8]) that a measure valued function on a measurable
subset Ω of some Euclid space is a weakly measurable map x 7→ νx of
550 On Kinetic Formulation...
Ω into the space Prob0(R) of probability Borel measures with compact
support in R. The weak measurability of νx means that ∀ p(λ) ∈ C(R)
the function x 7→
∫
p(λ)dνx(λ) on Ω is measurable. We say that a mea-
sure valued function νx is bounded if there exists M > 0 such that
supp νx ⊂ [−M,M ] for almost all x ∈ Ω. We shall denote by ‖νx‖∞ the
smallest of such M . Finally, we say that the measure valued functions of
the kind νx(λ) = δ(λ− u(x)), where u(x) ∈ L∞(Ω) and δ(λ − u) is the
Dirac measure at u ∈ R, are regular. We identify these measure valued
functions and the corresponding functions u(x), so that there is a natural
embedding L∞(Ω) ⊂MV (Ω), where MV (Ω) is the set of bounded mea-
sure valued functions on Ω. In similar way the vectorial measure valued
functions x→ νx ∈ Prob0(R
n) are defined.
We now consider the measure valued solutions (briefly — m.s.) νt,x ∈
MV (Π) of the Cauchy problem for (1.1) with initial condition
ν0,x = ν0
x ∈MV (Rm). (1.7)
Definition 1.2 (see [8]). A bounded measure valued function νt,x is
called a m.s. of the problem (1.1), (1.7) if
a) ∀ k ∈ R
∂
∂t
∫
|λ−k| dνt,x(λ)+divx
∫
(ϕ(λ)−ϕ(k)) sign(λ−k) dνt,x(λ) ≤ 0 (1.8)
in D′(Π);
b) ∀ p(λ) ∈ C(R) ess lim
t→0
∫
p(λ) dνt,x(λ) =
∫
p(λ) dν0
x(λ)
in L1
loc(R
m).
Remark that Definition 1.2 is consistent with Definition 1.1 in that
u(t, x) is a g.e.s. of the problem (1.1), (1.2) if and only if this function, un-
derstood as the regular measure valued function νt,x(λ) = δ(λ− u(t, x)),
is a m.s. of the problem (1.1), (1.7) with regular initial function ν0
x(λ) =
δ(λ− u0(x)).
We can treat a measure valued function νt,x as the random field u(t, x)
where the value of u(t, x) for fixed (t, x) ∈ Π is a random variable with
distribution νt,x. Then condition (1.8) shows that u(t, x) satisfies condi-
tion (1.3) “on the average”, that is,
E(|u− k|)t + divxE((ϕ(u) − ϕ(k)) sign(u− k)) ≤ 0 in D′(Π)
(here E denotes expectation). Setting k = ±M , where M = ‖νt,x‖∞,
we conclude that u(t, x) satisfies (1.1) “on the average”. Some results on
relations between measure valued and statistic solutions of the problem
(1.1), (1.7) can be found in [9].
E. Yu. Panov 551
As is known (see [10, 11]), there always exists a m.s. of the Cauchy
problem (1.1), (1.7), but it is nether unique if the initial function ν0
x is
not regular. In [10] (see [11] for more details) the notion of a strong
measure valued solution (briefly — s.m.s.) was introduced and existence
and uniqueness of s.m.s. to the problem (1.1), (1.7) were also established.
Definition 1.3 (see [10, 11]). A bounded measure valued function νt,x
is called a s.m.s. of the problem (1.1), (1.7) if for all λ ∈ (0, 1) the
function u(t, x, λ) = inf{ v | νt,x((v,+∞)) ≤ λ } is a g.e.s. of the problem
(1.1), (1.2) with initial function u0(x, λ) = inf{ v | ν0
x((v,+∞)) ≤ λ }.
For a regular m.s. νt,x(λ) = δ(λ− u(t, x)) we have u(t, x, λ) = u(t, x)
and νt,x is a s.m.s. Note (see [11]) that for fixed (t, x) ∈ Π the measure
νt,x is the image of the Lebesgue measure dλ on (0, 1) under the map
λ 7→ u(t, x, λ): νt,x = u(t, x, ·)∗dλ ( similarly, ν0
x = u0(x, ·)∗dλ ). There-
fore,
∀ k ∈ R
∫
|λ− k| dνt,x(λ) =
1∫
0
|u(t, x, λ) − k| dλ,
∫
(ϕ(λ) − ϕ(k)) sign(λ− k) dνt,x(λ)
=
1∫
0
(ϕ(u(t, x, λ)) − ϕ(k)) sign(u(t, x, λ) − k) dλ
and inequality (1.8) can be written in the form
1∫
0
[
|u(t, x, λ)− k|t+divx((ϕ(u(t, x, λ))−ϕ(k)) sign(u(t, x, λ)− k))
]
dλ≤0.
(1.9)
Thus, Definition 1.3 contains strengthening of the condition (1.8), namely
the integrand in (1.9) is required to be nonpositive (in D′(Π)) for all
λ ∈ (0, 1). In particular any s.m.s. is a m.s. in the sense of Definition 2
as well. The unique solvability of the problem (1.1), (1.2) implies easily
existence and uniqueness of s.m.s. of the problem (1.1), (1.7) (see details
in [11]).
2. Kinetic formulation of m.s.
Denote by FM the space of distribution functions f(v) = F (ν)(v) =
ν((v,+∞)) of measures ν with support in the segment [−M,M ]. Func-
tions f(v) ∈ FM don’t increase and are continuous from the right with
552 On Kinetic Formulation...
respect to v ∈ R; f(v) = 0 for v ≥M , f(v) = 1 for v < −M . Denote also
by Fc =
⋃
M>0
FM the space of distribution functions of measures having
compact supports on R.
If νx ∈MV (Ω), Ω is a measurable domain in some Euclid space, then
f(x, v) = F (νx)(v) ∈ FM for a.e. x ∈ Ω, M = ‖νx‖∞ and by weak
measurability of the map x 7→ νx the function f(x, v) = F (νx)(v) is mea-
surable on the set of variables (x, v). Let FM (Ω) be the class of functions
with above properties. Inclusion f(x, v) ∈ FM (Ω) is equivalent to exis-
tence of a measure valued function νx ∈MV (Ω) such that ‖νx‖∞ ≤M
and f(x, v) = νx((v,+∞)).
As was shown in [5], a distribution function f(t, x, v) of a m.s. νt,x ∈
MV (Π) can be described as a solution of the corresponding problem (1.4),
(1.5) for the kinetic equation. In [5] we also presented kinetic formulation
of s.m.s.
Theorem 2.1 ([5]). A bounded measure valued function νt,x is a m.s.
of the problem (1.1), (1.7) if and only if the corresponding distribution
function f(t, x, v) = νt,x((v,+∞)) is a g.s. of the problem (1.4), (1.5)
with initial function f0(x, v) = ν0
x((v,+∞)). Besides, νt,x is a s.m.s. of
the problem (1.1), (1.7) if and only if for each nondecreasing function
s(u) ∈ C([0, 1]) the function s(f(t, x, v)) is a g.s. to the problem (1.4),
(1.5) with initial data s(f0(x, v)). (It is understood that the measure µ
in the right-hand side of (1.4) depends on the function s).
It turns out that the class of strong m.s. correspond to the following
important class of solutions of the kinetic problem (1.4), (1.5).
Definition 2.1. A function f(t, x, v) is called an entropy solution (e.s.
for short) of the problem (1.4), (1.5) if f(t, x, v) ∈ FM (Π) for some
M > 0, for any function g(v) ∈ Fc
∂
∂t
∫
(f(t, x, v)− g(v))2dv+divx
∫
ϕ′(v)(f(t, x, v)− g(v))2dv ≤ 0 (2.1)
in D′(Π), and the initial condition (1.6) is satisfied.
Remark that Definition 2.1 does not include the measure µ, which
plays the role of a free parameter in the equation (1.4). In this sense, the
notion of e.s. solution of the kinetic problem seems to be more natural
in comparison with the notion of g.s. We prove the following result
Theorem 2.2. There exists an unique e.s. f(t, x, v) of the problem (1.4),
(1.5). Besides, f(t, x, v) = νt,x((v,+∞)), where νt,x is the corresponding
strong m.s. of the problem (1.1), (1.7).
E. Yu. Panov 553
Proof. Let f0(x, v) = ν0
x((v,+∞)) and νt,x ∈ MV (Π) be the unique
s.m.s. of the problem (1.1), (1.7), f(t, x, v) = νt,x((v,+∞)) be the
corresponding distribution function. Suppose that M > ‖νt,x‖∞ and
g(v) ∈ FM ∩ C1(R). It is clear that the integrands in (2.1) vanish for
|v| > M . Then, by Theorem 2.1 for s(u) = u, u2
∂
∂t
f + (ϕ′(v),∇xf) =
∂
∂v
µ1, (2.2)
∂
∂t
f2 + (ϕ′(v),∇xf
2) =
∂
∂v
µ2 in D′(Π), (2.3)
where µ1, µ2 are nonnegative locally finite measures on Π × (−M,M).
Let h(t, x) ∈ C∞
0 (Π), h ≥ 0. Then using the equalities (2.2), (2.3), we
obtain
∫
Π×(−M,M)
(f(t, x, v) − g(v))2{ht(t, x) + (ϕ′(v),∇xh)} dt dx dv
=
∫
Π×(−M,M)
(f(t, x, v))2{ht(t, x) + (ϕ′(v),∇xh)} dt dx dv
− 2
∫
Π×(−M,M)
f(t, x, v)g(v){ht(t, x) + (ϕ′(v),∇xh)} dt dx dv
=
∫
Π×(−M,M)
hv(t, x) dµ2(t, x, v) − 2
∫
Π×(−M,M)
h(t, x)g′(v) dµ1(t, x, v)
= −2
∫
Π×(−M,M)
h(t, x)g′(v) dµ1(t, x, v) ≥ 0.
Using approximation of an arbitrary function g(v) ∈ FM by a sequence
of smooth functions, we derive that ∀g(v) ∈ FM ∀h ∈ C∞
0 (Π), h ≥ 0
∫
Π×R
(f(t, x, v) − g(v))2{ht(t, x) + (ϕ′(v),∇xh)} dt dx dv ≥ 0.
Here M takes an arbitrary great enough value and we obtain that the
relation (2.1) holds. The initial condition (1.6) is also fulfilled by The-
orem 2.1. Thus, f(t, x, v) is a e.s. of the problem (1.4), (1.5). By
construction f(t, x, v) satisfies the last assertion of the theorem.
Uniqueness of e.s. is proved by using of the Kruzhkov’s method of
doubling variables, see the more general case of systems below (Theo-
rem 3.1 and Remark 3.2).
554 On Kinetic Formulation...
Remark 2.1. As it follows from the proofs of Theorem 2.2 and The-
orem 2.1, f(t, x, v) is an e.s. of the problem (1.4), (1.5) if and only if
f(t, x, v) and f2(t, x, v) are g.s. of this problem with the corresponding
initial functions. Therefore, in addition to Theorem 2.1, we can con-
clude that the sufficient condition for a bounded measure valued function
νt,x to be a strong m.s. is that the only two functions s(f(t, x, v)), with
s(u) = u, u2, are g.s. of the problem (1.4), (1.5).
Remark 2.2. In the paper [6] one approximation scheme of relaxation
type was proposed. Namely, the following problem was considered
ft + (ϕ′(v),∇xf) = r(g − f), f(0, x, v) = f0(x, v) ∈ FM (Rm), (2.4)
f = f(t, x, v) ∈ L∞(Π × R), r = const > 0, the nonlinear operator f 7→
Ff = g is defined as follows
g = g(t, x, v) =
1, v ≤ −M,
0, v ≥M,
l(f∗(t, x; v)), v ∈ (−M,M),
where f∗(t, x; v) is a nonincreasing rearrangement (see [12]), of the func-
tion v 7→ f(t, x, v) with respect to v ∈ (−M,M); l(f)=max(min(f, 1), 0)
is a cut off function.
As was proved in [6], there exists an unique g.s. fr(t, x, v) of the prob-
lem (2.4), and some subsequence of the sequence fr converges strongly
to a distribution function of some m.s. of (1.1), (1.7). Now we can revise
this result:
Theorem 2.3. The sequence fr(t, x, v) converges in L1
loc(Π × R) to the
e.s. f(t, x, v) ∈ FM (Π) of the problem (1.4), (1.5).
3. The case of hyperbolic systems
We consider the following first-order symmetric quasilinear system
B(u)ut +
m∑
i=1
Ci(u)uxi = 0, (3.1)
u = u(t, x) ∈ Rn, (t, x) ∈ Π = R+ × Rm,
with symmetric matrices B(u) > 0 ( i.e. B(u) is positively definite ) and
Ci(u), i = 1, . . . ,m. Remark that a general (nonstrictly) hyperbolic n×n
system
ut +
m∑
i=1
Ai(u)uxi = 0 (3.2)
E. Yu. Panov 555
could be reduced to the symmetric form (3.1) in the cases when m = 1
and when n ≤ 2 (and only in these cases, see [13]). The system (3.1)
is equivalent to the hyperbolic system (3.2) with the matrices Ai(u) =
B−1(u)Ci(u).
Suppose that the matrices B(u) and Ci(u) depend continuously on
u ∈ Rn and the matrices Ai(u) have bounded eigenvalues λij , j =
1, . . . , n. This and symmetry of matrices Ai(u) with respect to a scalar
multiplication (f, g) → (B(u)f, g) imply that for some constant C > 0
∀u, f ∈ Rn
(B(u)Ai(u)f,Ai(u)f) ≤ C2(B(u)f, f). (3.3)
LetX be a Hilbert space of vector fields f(u) = (f1(u), . . . , fn(u)), u ∈ Rn
with finite value of ‖f‖2 =
∫
Rn
(B(u)f(u), f(u)) du (here (·, ·) is a scalar
product in Rn), and the scalar product (f, g) =
∫
Rn
(B(u)f(u), g(u)) du.
Remark that the correspondence f 7→ Ai(u)f(u) yields a bounded sym-
metric linear operator Ãi on X for each i = 1, . . . ,m, and ‖Ãi‖ ≤ C.
Fix some convex closed set H ⊂ X, the bounded closed subsets of
which are compact. By analogy with the scalar case we study the Cauchy
problem for the kinetic system corresponded to (3.1), (3.2)
ft +
m∑
i=1
Ai(u)fxi = 0, (3.4)
where the unknown vector f = f(t, x, u) ∈ Rn belongs to the set
L2
loc(Π, H) (which is a convex closed subset of the space L2
loc(Π, X) de-
termined by condition: f(t, x, ·) ∈ H for a.e. (t, x) ∈ Π), with initial
condition
f(0, x, u) = f0(x, u) ∈ L2
loc(R
m, H). (3.5)
Definition 3.1. A function f = f(t, x, u) ∈ L2
loc(Π, H) is called an
entropy solution (e.s.) of the problem (3.4), (3.5) if ∀ g = g(u) ∈ H
∂
∂t
‖f(t, x, ·)−g‖2+
m∑
i=1
∂
∂xi
(
Ãi(f(t, x, ·)−g), f(t, x, ·)−g
)
≤ 0 in D′(Π);
(3.6)
ess lim
t→0
‖f(t, x, ·) − f0(x, ·)‖ = 0 in L2
loc(R
m). (3.7)
Theorem 3.1. The e.s. of the problem (3.4), (3.5) is unique.
Sketch of the proof. We make use of the Kruzhkov’s method of doubling
variables developed in [1]. Let f = f(t, x, u), f̄ = f̄(t, x, u) be two e.s. of
the problem (3.4), (3.5). Then, by (3.6) with g(u) = f̄(τ, y, u), (τ, y) ∈ Π
556 On Kinetic Formulation...
∂
∂t
‖f(t, x, ·) − f̄(τ, y, ·)‖2
+
m∑
i=1
∂
∂xi
(
Ãi(f(t, x, ·) − f̄(τ, y, ·)), f(t, x, ·) − f̄(τ, y, ·)
)
≤ 0
in D′(Π × Π).
In the same way, changing places of variables (t, x) and (τ, y), and also
solutions f and f̄ , we obtain that
∂
∂τ
‖f(t, x, ·) − f̄(τ, y, ·)‖2
+
m∑
i=1
∂
∂yi
(
Ãi(f(t, x, ·) − f̄(τ, y, ·)), f(t, x, ·) − f̄(τ, y, ·)
)
≤ 0
in D′(Π × Π).
Putting these relations together we derive that in D′(Π × Π)
(
∂
∂t
+
∂
∂τ
)
‖f(t, x, ·) − f̄(τ, y, ·)‖2 +
m∑
i=1
(
∂
∂xi
+
∂
∂yi
)
×
(
Ãi(f(t, x, ·) − f̄(τ, y, ·)), f(t, x, ·) − f̄(τ, y, ·)
)
≤ 0. (3.8)
Applying the inequality (3.8) to the test function p = p(t, x; τ, y) =
g(t, x)ρν(t − τ, x − y) where g(t, x) ∈ C∞
0 (Π), g ≥ 0, and the sequence
ρν ∈ C∞
0 (Rm+1), ν ∈ N approximates the δ-function, we obtain, after
passing to the limit as ν → ∞, that
∂
∂t
‖f(t, x, ·) − f̄(t, x, ·)‖2
+
m∑
i=1
∂
∂xi
(
Ãi(f(t, x, ·) − f̄(t, x, ·)), f(t, x, ·) − f̄(t, x, ·)
)
≤ 0
in D′(Π).
Applying the obtained relation to test functions approximated the indi-
cator function of the subcharacteristical cone KR = { (t, x) ∈ Π | |x| <
R−Nt }, N =
√
mC, C is the constant from (3.3), we derive that for a.e.
t > 0 ∀ r > 0
∫
|x|<r
‖f(t, x, ·)−f̄(t, x, ·)‖2dx = 0, i.e. f(t, x, u) = f̄(t, x, u)
a.e. on Π×R, and the e.s. f(t, x, u) is unique. The proof is complete.
To prove existence of e.s. we make use of the following approximation
scheme of relaxation type. Namely, let PH : X 7→ H be the projection
E. Yu. Panov 557
map so that ‖f −PHf‖ = min
h∈H
‖f −h‖. Because X is a Hilbert space and
H is its convex closed subset the projection PHf is well-defined, and the
contraction property holds:
‖PHf − PHg‖ ≤ ‖f − g‖ ∀ f, g ∈ X. (3.9)
Fix r > 0 and consider the problem
ft +
m∑
i=1
Ai(u)fxi = −r(f − PHf), (3.10)
f(0, x, u) = f0(x, u) ∈ L2
loc(R
m, X). (3.11)
We shall consider g.s. f(t, x, u) ∈ L2
loc(Π, X) of the problem (3.10),
(3.11), which satisfy the equation (3.10) in D′(Π × R) and the initial
condition (3.11) in the sense of relation (3.7). Now we are ready to es-
tablish the following
Theorem 3.2. There exists a unique g.s. of the problem (3.10), (3.11).
Proof. For the proof fix some R > 0 and define the cone KR = { (t, x) ∈
Π
∣∣ |x| < R − Nt }, and the Banach space L ⊂ L2
loc(KR, X) consisting
of vectors f = f(t, x, u) with finite norm
‖f‖ = ess sup
t>0
( ∫
|x|≤R−Nt
‖f(t, x, ·)‖2dx
)1/2
.
Let g = g(t, x, u) ∈ L and f = f(t, x, u) = Φ(g) ∈ L2
loc(KR, X) be the
unique g.s. to the Cauchy problem for the linear system ft+
m∑
i=1
Ai(u)fxi
= −r(f − PHg), with the fixed initial data (3.11). If g1, g2 ∈ L, f1 =
Φ(g1), f2 = Φ(g2) then, as it is easily verified, for a.e. t > 0
∫
|x|≤R−Nt
‖f1(t, x, ·) − f2(t, x, ·)‖2 dx
≤ r
t∫
0
e−r(t−τ)
∫
|x|≤R−Nτ
‖PHg1(τ, x, ·) − PHg2(τ, x, ·)‖2 dx dτ,
which implies that the map Φ is a contraction with coefficient 1−e−rR/N .
By Banach theorem there exist the unique fixed point f ∈ L of the map
Φ, i.e. problem (3.10), (3.11) has the unique g.s. f = fR in the cone
558 On Kinetic Formulation...
KR. By uniqueness solutions fR1 and fR2 coincide in their common
region. Therefore, we can define the function f(t, x, u) ∈ L2
loc(Π, X)
setting f(t, x, u) = fR(t, x, u) where R > |x|+Nt is arbitrary. It is clear
that f(t, x, u) is a g.s. of the problem (3.10), (3.11). Uniqueness of g.s.
f(t, x, u) follows from its uniqueness in any cone KR.
Our next task is to prove convergence of solutions f = fr(t, x, u) of
(3.10), (3.11) as the parameter r → ∞. To this end we need some a
priory estimates for g.s. of the problem (3.10), (3.11), which are uniform
with respect to the parameter r > 0. Firstly, we have the following
Theorem 3.3. Let f1(t, x, u), f2(t, x, u) be g.s. of the problem (3.10),
(3.11) with initial data f01(x, u), f02(x, u) respectively. Then
1) in D′(Π)
(
‖f1(t, x, ·) − f2(t, x, ·)‖2
)
t
+
m∑
i=1
(
Ãi(f1(t, x, ·) − f2(t, x, ·)), f1(t, x, ·) − f2(t, x, ·)
)
xi
≤ 0; (3.12)
2) for almost each t > 0 ∀R > 0
∫
|x|≤R
‖f1(t, x, ·) − f2(t, x, ·)‖2 dx ≤
∫
|x|≤R+Nt
‖f01(x, ·) − f02(x, ·)‖2 dx.
(3.13)
Proof. To prove (3.12) we multiply the relation
(
f1(t, x, ·) − f2(t, x, ·)
)
t
+
m∑
i=1
[
Ãi
(
f1(t, x, ·) − f2(t, x, ·)
)]
xi
= −r
[(
f1(t, x, ·) − f2(t, x, ·)
)
−
(
PHf1(t, x, ·) − PHf2(t, x, ·)
)]
by f1(t, x, ·) − f2(t, x, ·) scalarly in X and use that
(
f1(t, x, ·) − f2(t, x, ·), PHf1(t, x, ·) − PHf2(t, x, ·)
)
≤ ‖f1(t, x, ·) − f2(t, x, ·)‖2
in view of (3.9).
At last, (3.13) follows from (3.12) after application to test functions,
which approximate indicator functions of subcharacteristical cones.
Applying relation (3.13) to pairs f1 = f(t, x, ·), and f2 = f(t+∆t, x+
∆x, ·) we obtain estimates of continuity modulus of solutions f = fr of the
E. Yu. Panov 559
problem (3.10), (3.11) in L2
loc(Π, X), which do not depend on parameter
r. This, together with compactness of bounded subsets in H, allows us
to establish strong convergence of the sequence gr = PHfr(t, x, ·) to some
function f = f(t, x, ·) ∈ L2
loc(Π, H). Besides, ∀h ∈ H
− ∂
∂t
‖fr(t, x, ·) − h‖2 −
m∑
i=1
∂
∂xi
(
Ãi(fr(t, x, ·) − h), fr(t, x, ·) − h
)
= 2r
(
fr(t, x, ·)− gr(t, x, ·), fr(t, x, ·)−h
)
≥ 2r‖fr(t, x, ·)− gr(t, x, ·)‖2.
(the last inequality easily follows from properties of the projection PH).
The relation above implies that for any nonnegative function ρ(t, x) ∈
C∞
0 (Π)
∫
Π
‖fr(t, x, ·) − gr(t, x, ·)‖2ρ(t, x)dtdx ≤ c(ρ)/(2r) →
r→∞
0,
i.e. fr(t, x, u) − gr(t, x, u) → 0 in L2
loc(Π, X) as r → ∞. Therefore,
fr → f as well.
Passing to the limit as r → ∞ in the inequality (3.12) with f1 = fr,
f2 ≡ g ∈ H we conclude that the limit function f = f(t, x, u) satisfies
condition (3.6). The initial condition (3.5) is also easily verified. Thus,
f(t, x, u) is an e.s. of (3.4), (3.5). In view of uniqueness of e.s., the
limit function f(t, x, u) doesn’t depend on the choice of a convergent
subsequence. Therefore, the original sequence fr(t, x, u) converges to
f(t, x, u) in L2
loc(Π, X). We proved the following
Theorem 3.4. Let fr = fr(t, x, u) be a g.s. of the problem (3.10), (3.11),
r ∈ N. Then the sequence fr converges in L2
loc(Π, X) to the unique e.s.
f(t, x, u) of the problem (3.4), (3.5).
In the important particular case when H is a cone the condition (3.6)
could be rewritten in different form.
Theorem 3.5. Let H be a closed convex cone in X. Then (3.6) is
equivalent to conditions
∀h ∈ H
∂
∂t
(
f(t, x, ·), h
)
+
m∑
i=1
∂
∂xi
(
Ãif(t, x, ·), h
)
≥ 0 in D′(Π) and
(3.14)
∂
∂t
‖f(t, x, ·)‖2 +
m∑
i=1
∂
∂xi
(
Ãif(t, x, ·), f(t, x, ·)
)
= 0 in D′(Π).
(3.15)
560 On Kinetic Formulation...
Corollary 3.1. Let H be a closed linear subspace of X. Then condition
(3.6) is equivalent to the identity: ∀h ∈ H
∂
∂t
(
f(t, x, ·), h
)
+
∂
∂x
(
Ãf(t, x, ·), h
)
= 0 in D′(Π). (3.16)
Remark 3.1. In the case when H is a closed linear subspace existence
and uniqueness of e.s. to the problem (3.4), (3.5) can be proved without
the condition of strong compactness of bounded closed subsets of H
(which holds only for finite-dimensional subspaces). Indeed, we do not
use this condition in the proof of the uniqueness, and the existence can
be established on the base of weak convergence of the sequence fr =
fr(t, x, u) of g.s. to (3.10), (3.11). Obviously, the limit function f(t, x, u)
must satisfy condition (3.16), which is equivalent to (3.6) by Corollary
above.
Remark 3.2. In the case of a scalar equation (1.1) ut+
m∑
i=1
ai(u)uxi = 0,
ai(u) = ϕ′
i(u) we set H = { f(u)−θ(−u) | f ∈ FM }, where M > 0, θ(λ)
is the Heaviside function. We see, with using of the Helly theorem, that
H ⊂ L2(R) is a compact convex set. Therefore, there exists an unique e.s.
f(t, x, u)−θ(−u) of the corresponding kinetic equation ft+
m∑
i=1
ai(u)fxi =
0 with initial data f(0, x, u) = f0(x, u)− θ(−u), f0(x, ·) ∈ FM . It is clear
that f(t, x, u) coincides with e.s. in the sense of Definition 2.1. As follows
from Theorem 2.2 this solution is also e.s. with respect to the wider class
H̃ = { λf | f ∈ H,λ ≥ 0 } that is the conic envelope of H.
Consider some closed convex set H ⊂ X bounded closed subset of
which are compact, and suppose in addition that divf(u) = l+ν ∀f(u) ∈
H where ν ∈ Prob0(R
n) is a finite Borel probability measure on Rn,
l ∈ D′(Rn) is some fixed distribution. Denote
M = {divf(u) − l | f(u) ∈ H} ⊂ Prob0(R
n).
Let ν0
x = divf0(x, u)− l and νt,x = divf(t, x, u)− l where f(t, x, u) ∈
L2
loc(Π, H) is a unique e.s. of the problem (3.4), (3.5) with initial data
f0(x, u). By analogy with the scalar case n = 1 we call the measure valued
function νt,x a strong measure valued solution to the original problem
(3.1) with the measure valued initial data ν0
x. Of course, this notion
depends on choice of “kinetic class” H, which is not uniquely determined
by the family of measures M . We can restrict ourselves to the class of
potential vectors f(u) = ∇ϕ(u). Then (for l = 0) the vector f(u) is
E. Yu. Panov 561
uniquely defined by its divergence since the function ϕ(u) must be an
unique g.s. to the Poisson equation ∆ϕ = ν and therefore
f(u) = ∇ϕ(u) =
∫
∇E(u− v)dν(v),
where E(u) is the fundamental solution of the Laplace equation. The
natural restriction to the class of measures ν ∈M is the condition f(u)∈X.
To demonstrate connection strong measure valued solutions with mea-
sure valued solutions in DiPerna sense (see [8]) we consider the case
when the matrices Ai(u) are symmetric and the matrix B(u) is the
identity matrix. By analogy with a scalar equation we consider wider
conical kinetic class H, which consists of potential vectors f(u) ∈ X,
f(u) = ∇p(u) in D′(Rn) such that the distribution divf = ∆p is non-
negative finite Borel measure in Rn (not necessary — probability). As-
sume as usual that H is a convex closed cone, bounded closed subsets
of which are compact. Then for any initial data f0(x, u) ∈ L2
loc(R, H)
there exists the unique e.s. f = f(t, x, u) ∈ L2
loc(Π, H) of the problem
(3.4), (3.5). We consider measure valued (in wider sense) functions
ν0
x = divuf0(x, u) and νt,x = divuf(t, x, u). Let continuous functions
p(u), q(u) = (q1(u), . . . , qm(u)) be such that ∇p(u) ∈ H,
div(∇qi(u) −Ai(u)∇p(u)) = 0 in D′(Rn) (3.17)
and p, q → 0 as u → ∞. We call the function p(u) an entropy of
the system (3.1) and the function q(u) a corresponding entropy flux .
Remark that p(u) is a subharmonic function, which generalizes the usual
convexity requirement. The condition (3.17) is a weaker form of the
relation ∇qi(u) = Ai(u)∇p(u) postulated for classical entropy pairs (p, q)
in the Lax sense. In contrast to the Lax relation ( which is overdetermined
for n > 2 ), the flux vector q is uniquely defined by (3.17) for any general
enough entropy p(u).
By Theorem 3.5 the relation (3.14) is fulfilled with h = ∇p. Integrat-
ing by parts we conclude that
(f(t, x, ·),∇p) = −
∫
p(u) dνt,x(u);
(Ãif(t, x, ·),∇p) = (f(t, x, ·),∇qi) = −
∫
qi(u) dνt,x(u)
and by (3.14) we derive the DiPerna entropy condition
∂
∂t
∫
p(u) dνt,x(u) +
m∑
i=1
∂
∂xi
∫
qi(u) dνt,x(u) ≤ 0 in D′(Π).
562 On Kinetic Formulation...
Remark in conclusion that some results of this paper including the case
of systems with one space variable were published with detailed proofs in
the preprint [14].
References
[1] S. N. Kruzhkov, First order quasilinear equations in several independent vari-
ables // Mat. Sbornik. 81 (1970), No 2, 228–255; English transl. in Math. USSR
Sb. 10 (1970), No 2, 217–243.
[2] Y. Giga, T. Miyakawa, A kinetic construction of global solutions of first order
quasilinear equations // Duke Math. J. 50 (1983), No 2, 505–515.
[3] Y. Giga, T. Miyakawa, S. Oharu, A kinetic approach to general first order quasi-
linear equations // Trans. Amer. Math. Soc. 287 (1985), No 2, 723–743.
[4] P. L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional
scalar conservation laws and related equations // J. Amer. Math. Soc. 7 (1994),
No 1, 169–191.
[5] E. Yu. Panov, On kinetic formulation of measure valued solution of a first-order
quasilinear equation // Fundamental’naya i prikladnaya matematika. 4 (1998),
N 1, 317–332 (in russian).
[6] E. Yu. Panov, An approximation scheme for measure-valued solutions of a first-
order quasilinear equation // Mat. Sbornik. 188 (1997), No. 1, 83–108; English
transl. in Sbornik: Mathematics. 188 (1997), No. 1, 87–113.
[7] L. Tartar, Compensated compactness and applications to partial differential equa-
tions // Research notes in mathematics, nonlinear analysis, and mechanics:
Heriot-Watt Symposium. 4 (1979), 136–212.
[8] R. J. DiPerna, Measure-valued solutions to conservation laws // Arch. Rational
Mech. Anal. 88 (1985), 223–270.
[9] E. Yu. Panov, On statistical solutions of the Cauchy problem for a first-order
quasilinear equation // Matemat. Modelirovanie. 14 (2002), No. 3, 17–26 (in
russian).
[10] E. Yu. Panov, Strong measure-valued solutions of the Cauchy problem for a first-
order quasilinear equation with a bounded measure-valued initial function // Vest-
nik Mosk. Univ. Ser.1 Mat. Mekh. (1993), No. 1, 20–23; English transl. in Moscow
Univ. Math. Bull. 48 (1993), No. 1.
[11] E. Yu. Panov, On measure valued solutions of the Cauchy problem for a first order
quasilinear equation // Izvest. Ross. Akad. Nauk. (1996), No 2, 107–148; English
transl. in Izvestiya: Mathematics. 60 (1996), No 2, 335–377.
[12] G. G. Hardy, G. E. Littlewood, and G. Pólya, Inequalities, Cambridge University
Press. Cambridge 1952.
[13] E. Yu. Panov, Symmetrizability of first-order hyperbolic systems // Dokl. Akad.
Nauk. 396 (2004), No 1, 28–31; English transl. in Doklady Mathematics. 69
(2004), No. 3, 341–343.
[14] E. Yu. Panov, On kinetic formulation of measure valued and strong
measure valued solutions to the Cauchy problem for hyperbolic first-
order quasilinear equations // Preprint. 2002. Published electronically in
http://www.math.ntnu.no/conservation/2002/049.html
E. Yu. Panov 563
Contact information
E. Yu. Panov Mathematical Analysis Department,
Novgorod State University,
B. St.-Peterburgskaya 41,
173003 Velikiy Novgorod,
Russia
E-Mail: pey@novsu.ac.ru
|