1. Introduction
We are interested in the uniqueness and existence of the stochastic entropy solution for the following stochastic scalar balance law:
with a non-random initial condition:
Here ∘ is the Stratonovich convention and the use of the Stratonovich differential stems from the fact that ordinary differential equations with time dependent converging Brownian motion give rise stochastic differential equations of Stratonovich’s.
In (
1),
is a scalar random field.
is an
n-dimensional standard Wiener process on the classical Wiener space (
), i.e.,
is the space of all continuous functions from
to
with locally uniform convergence topology,
is the Borel
-field,
is the Wiener measure,
is the natural filtration generated by the coordinate process
. The flux function
is assumed to be of class
, i.e.,
The force
A is supposed to satisfy that
For every
, we assume
When
, (
1) reduces to a deterministic partial differential equation known as the balance law
The first pioneering result on the well-posedness of weak solutions for (
6) is due to Kružkov [
1]. Under the smoothness hypothesis on
F and
A, he obtained the existence in company with uniqueness of the admissible entropy solutions. For a completely satisfactory well-posedness theory for balance laws, one can consult to [
2].
When
vanish and
, the equation has been discussed by Lions, Perthame and Souganidis [
3,
4]. Under the presumption that
, they developed a path-wise theory with quasi-linear (i.e.,
B is independent of the derivatives of
) multiplicative stochastic perturbations.
Recently there has been an interest in studying the effect of stochastic force on the corresponding deterministic equations, especially for the uniqueness and existence of solutions. Most of works are concentrated on the following form:
where
is a 1-dimensional Wiener process or a cylindrical Wiener process,
is a bounded domain or
. When
, the bounded solution has been founded by Holden and Risebro [
5], and Kim [
6] for the forces
and
, respectively, under assumptions that
and
A has compact support. For general
A, even the initial data is bounded, the solution is not bounded since the maximum principle is not available. Therefore,
(
) is a natural space on which the solutions are posed. When the force
A is time independent, Feng and Nualart [
7] developed a general theory for
-solutions
, but the existence was true only for
. Since then, Feng and Nualart’s result was generalized in different forms. For example, Bauzet, Vallet and Wittbold [
8], Biswas and Majee [
9] established the weak-in-time solutions, Karlsen and Storrøsten [
10] derived the existence and uniqueness of stochastic entropy solutions for general
. At the same time, by using a different philosophy, Chen, Ding and Karlsen [
11], Debussche and Vovelle [
12], Hofmanová [
13] also founded the well-posedness for
-solutions
for any
. Furthermore, there are many other works devoted to discussing the Cauchy problem (
7), (
2), such as existence and uniqueness for solutions on bounded domains [
14,
15,
16], existence of invariant measures [
17,
18] and long time behaviors [
19] for solutions. For more details in this direction for random fluxes, we refer the readers to [
20,
21], and for more details for Lévy noises to see [
22,
23,
24].
If we regard the last term in (
7) as a multiplicative perturbation for the scalar conservation law:
then the spatial average satisfies
So the mass is not preserved in general. But if one considers the noise given in (
1), then the mass is preserved exactly. It is one of our motivations to discuss the balance law
with the noise give by the form
. However, as far as we know the existing results for weak solutions to (
1), (
2) are few and all the results are concentrated on the following special case [
25,
26]:
Further investigations are still needed. By using kinetic theory, we will prove the uniqueness and existence of the stochastic entropy solution to (
1), (
2). Here the stochastic weak solution and stochastic entropy solution are defined as follows:
Definition 1. is a stochastic weak solution of (1), (2), if for every,is an-semi-martingale and with probability one, the below identityholds true, for all. Remark 1. Our motivation to define the weak solution comes from the classical theory of partial differential equations, i.e., ρ is a weak solution if it satisfies the equation in the sense of distributions: for every,holds. Since ρ is continuous in time, the above identity is equivalent to (8). Definition 2. A stochastic weak solution of (1), (2) is a stochastic entropy solution, if for every,in the sense of distributions, i.e., for everyand almost allwhereand Remark 2. We define the stochastic entropy solution by the inequality (9), and the source or motivation for this definition comes from thelimit of the following equation Indeed, if one multiplies the above identity by, it yields that Since η is convex, with the help of the chain rule,Therefore, So the vanishing viscosity limit in the proceeding inequality leads to (9). We state our first main result on the Cauchy problem (
1), (
2).
Theorem 1 (Stochastic kinetic formulation)
. Suppose that (3)–(5) hold. (i) Let ρ be a stochastic entropy solution of (1), (2) and set. Thenand it is a stochastic weak solution of the following linear stochastic transport equation (i.e., it is-adapted and satisfies the equation in the sense of distributions)supplied with Here,, satisfying, for everyand for almost all, m is bounded on, supported in(), and for every, (ii) Suppose that. Ifis a stochastic weak solution of (11)–(14). We set, thenand it is a stochastic entropy solution of (1), (2). Remark 3. (i) If u is a stochastic weak solution of (11)–(14), then (11) admits an equivalent representation: for every, every,is-adapted and with probability one, (ii) To the present case, we only study (1) with. However, if F depends on spatial variables, i.e.,, we can also establish a stochastic kinetic formulation up to a long and tedious calculations. In particular, for,andis replaced by, we refer to [27], and for,and, to [28], and some related work, to [29]. Our second result is on the uniqueness of the stochastic entropy solution.
Theorem 2 (Uniqueness)
. Let, that Then there is at most one stochastic entropy solution ρ of (1), (2). As a corollary, we have
Corollary 1 (Comparison Principle).
Letandbe two stochastic entropy solutions of (1), with initial valuesand, if, then with probability one,. To make Theorem 2 more clear, we exhibit two representative examples here.
Example 1. The first example is concerned with the Buckley-Leverett equation (see [2]), which provides a simple model for the rectilinear flow of immiscible fluids (phases) through a porous medium. To be simple, nevertheless, to capture some of the qualitative features, we consider the case of two-phase flows (oil and water) in 1-dimensional space. In this issue, the Buckley-Leverett equation, with an external force, and a stochastic perturbation readswhereis a constant, W is a 1-dimensional standard Wiener process,and The flux function F is determined using Darcy’s law and incompressibility of the two phases and is given by [30]:denote the mobility of the oil and water phase, respectively, and,
represent the relative permeability of oil and water, respectively.andare non-negative smooth functions and. Applying Theorem 2, we obtain
Corollary 2. Assume that. Then there exists at most one stochastic entropy solution ρ of (19). Moreover, if the initial data is non-negative, then the unique stochastic (if it exists) is non-negative as well. Example 2. The second example is concerned with a generalized Burgers equation (see [31]). This equation with a nonlinear stochastic perturbation of Brownian type, and a nonlinear nonhomogeneous term readsassociated with the initial value, whereis a fixed vector,are constants,,is a d-dimensional standard Wiener process. From Theorem 2, we have
Corollary 3. Let. If the stochastic entropy solutions of (22), (2) exists, then it is unique. In addition,implies the unique stochastic entropy solution (if it exists). Our third result is on the existence of the stochastic entropy solution. And now we should assume the growth rates on the coefficients
, i.e.,
is at most linear growth in
, and regularity property of
A on spatial variables (e.g., Lipschitz continuous). In this case, we will establish the existence for stochastic entropy solutions. Up to a tedious calculation which is not technique, all calculations for
and
are the same as
and
. To make our result present in a concise form, we only discuss the following stochastic balance law:
here
, (
).
Theorem 3 (Existence)
. Let F, σ and A satisfyThen there exists a stochastic entropy solution of the Cauchy problem (23), (2). If one argues Buckley-Leverett Equations (
19)–(
21) again, then by Theorems 2 and 3, we obtain
Corollary 4. Let F, ϑ and A be given in Example 1 and assume. Then there exists a unique stochastic entropy solution ρ of (19). Moreover, if, then. The rest of the paper is structured as follows. In
Section 2, we give some preliminaries. In
Section 3 we present the proof of Theorem 1. The uniqueness and existence of stochastic entropy solutions are proved in
Section 4 and
Section 5.
Section 4 is devoted to the proof of the uniqueness and in
Section 5, we study the existence.
We end up the section by introducing some notations.
Notations.,
,
,
and
stand for the sets of all smooth functions on
,
,
,
and
with compact supports, respectively. Correspondingly,
,
,
and
represent the non-negative elements in
,
,
,
and
, respectively.
denotes the duality between
and
.
is the duality between
and
.
denotes a positive constant depending only on
T, whose value may change in different places. a.s. is the abbreviation of “almost surely”. The stochastic integration with a notation ∘ is interpreted in Stratonovich sense and the others is Itô’s. For a given measurable function
g,
is its positive portion, defined by
, and
.
.
is natural numbers and
. For notational simplicity, we set
3. Proof of Theorem 1
For every
,
so (
10) implies (
15), and vice versa. We need to check the rest of (i) and (ii) in Theorem 1.
Let
be a stochastic entropy solution of (
1), (
2) fulfilling the statement (i) in Theorem 1. For every
, it renders that
for almost all
, where
For every
, then
Observing that
and
. On account of (
4), it follows that
thus
.
Similarly, by using conditions (
3) and (
5), one computes in the sense of distributions that
From (
31), one derives the identity (
11). In order to prove the assertion of Theorem 1 (i), it suffices to show that
m satisfies all the properties described in (i).
Noting that
is bounded local-in-time, from (
28) and (
29), for every fixed
, and almost all
,
m is supported in
, with
. Accordingly, it remains to examine that
m is bounded and continuous in
t. And it is sufficient to show that
is bounded and continuous in
t.
Since
and it is supported in a compact subset for
v in
, we obtain
for every
.
Thanks to (
30),
for every
and
, where
.
On account of Hypotheses (
3)–(
5), by using Lemma 2, it leads to
where
Using the Itô isometry and Lemma 1,
where
Obviously, (
33) holds ad hoc for
, where
,
,
For this fixed
, by an approximation demonstration, one can fetch
By letting
, we gain from (
33) and (
34) (by choosing a subsequence if necessary), that
which suggests that for every given
,
m is bounded on
and
.
Specially, when
, we obtain
The arguments employed above for 0 and
T adapted to every
now, yields that
which hints
m is continuous in
t. So
u is a stochastic weak solution of (
11)–(
13) with
m satisfying (
14).
Let us show the reverse fact. Since
m satisfies (
14) and
solves (
11)–(
13), for every
, then
is
-adapted. It remains to show the inequality (
9).
Given
and
, set
then
is convex,
, and
In a consequence of
solving (
11)–(
13) with
m satisfying (
14), it follows that
for every
,
,
, where
Applying the partial integration, one deduces
when
is large enough, for
m yields the properties stated in Theorem 1 (i).
Upon using (
30) and (
39), from (
37), we derive
by taking
to infinity, here
On the other hand
and
for almost everywhere
.
If one lets
approach to zero in (
40), we attain the inequality (
9), thus
is a stochastic entropy solution.
Remark 4. Our proof for Theorem 1 is inspired by Theorem 1 in [36], but the demonstration here appears to be finer, and for more details, one can see [36] and also see [37] for nonlocal conservation laws. 4. Proofs of Theorem 2 and Corollary 1
We begin our discussion in this section to prove Theorem 2. Let
and
be two stochastic entropy solutions of (
1), with initial values
and
, respectively. Then
and
are stochastic weak solutions of (
11) with nonhomogeneous terms
and
, initial datum
and
, respectively.
Let
and
be two regularization kernels described in Lemmas 3 and 4, respectively. Let
be another regularization kernel in variable
v, i.e.,
For
, set
then
(
) yields that
here
, and
For every
, we set
. For
, if one uses Itô’s formula for
first, and lets
tend to 0 next, it follows that
where
,
and
are given by (
13), (
35) and (
38), respectively.
Analogue calculations also yield that
From (
43) and (
44), one infers
where
Observing that for every
, and almost all
,
and
are bounded on
, supported in
, where
From (
41), with the aid of assumptions (
3)–(
5) and Lemma 2,
is continuous in
v in a neighborhood of zero. Besides, for almost everywhere
,
Hence for large
(
) and every
,
Moreover, due to (
30) and the fact
, if one chooses
large enough, then
On account of (
42), thanks to conditions (
3) and (
16), and Lemma 3, then,
and
On the other hand, for fixed
, we have
where
By Lemma 4 and (
5),
and by virtue of Lemma 3
For
and
(
is big enough) be fixed, if one lets
tend to zero first,
approach to zero next,
incline to zero last, with the aid of (
47)–(
53) and Lemma 1, from (
45), it leads to
Observing that
, and
we have
.
Since
and
are supported in
for
v, if one chooses
, it follows from (
54) that
By taking
to infinity, with the help of (
17), (
18), then
where
K is given by (
46).
From (
56), we complete the proof.
It remains to prove Corollary 1. Indeed, if one mimics the above calculations, then
Observing that
hence
The Grönwall inequality applies, one concludes
which implies
Remark 5. As a special case, one confirms the uniqueness of stochastic entropy solutions forwhen. However, we can not give an affirm answer on the problem whether the weak solution is unique or not, when F is non-regular (such as). 5. Proof of Theorem 3
The conclusion will be reached in three steps, and to make the expression simpler and clearer, we use instead of .
Step 1:. Now (
11), (
12) become to
We begin with building the existence of weak solutions for (
57) by using the Bhatnagar-Gross-Krook approximation, i.e., for
, we regard (
57) as the
limit of the integro-differential equation
where
Assertion 1: (
58) is well-posed in
.
Clearly, (
58)
1 grants an equivalent presentation
here
Due to the assumptions
and
, there is a unique global solution to the ODE
for every
.
Therefore, along the direction (
59),
i.e.,
Define
, thanks to Euler’s formula, then
whence, the inverse of the mapping
exists and it forms a flow of homeomorphic. We thus have
where
, i.e.,
and
.
For every
, we define a mapping
by:
here
We claim that is well-defined in and locally (in time) contractive in .
Initially, we collate that (
62) is well-defined. Indeed,
and for every
,
thus (
62) is meaningful.
For every
, an analogue calculation of (
64) also leads to
where
and
.
In particular, if
, from (
65), for every
Given above
we select
so small that
. Then we apply the Banach fixed point theorem to find a unique
solving the Cauchy problem (
58). By (
63),
, so
. We then repeat the argument above to extend our solution to the time interval
. Continuing, after finitely many steps we construct a solution existing on the interval
for any
. From this, we demonstrate that there exists a unique
solving the Cauchy problem (
58).
Assertion 2: (Comparison principle). For every
, the allied solutions
and
of (
58) satisfy
Furthermore, if
, for almost all
, and almost all
,
Equation (
69) holds mutatis mutandis from (
66) and
, it is sufficient to show (
66)–(
68). Since the calculations for (
67) and (
68) are analogue of (
66), we only show (
66) here. Let
, by an approximation argument, it leads to
in
, with the initial data
Obviously, we have the following facts:
and
Indeed, when
, (
73) is nature and reversely,
By (
72), (
73), from (
70) it follows that
which suggests that for every
,
For every
, we can choose
such that for every
,
, then by letting
k tend to infinity, one deduces
On account of the fact: for every
,
from (
74), by (
71) and a Grönwall type argument, one arrives at (
66).
Assertion 3: With locally uniform convergence topology, is pre-compact in and is pre-compact in .
From (
66) (with a slight change), we have for every
,
,
With the aid of (
75), then for
, it follows that
which implies for every
,
is contained in a compact set of
,
is pre-compact in
. Hence by appealing to the Arzela-Ascoli theorem, with any sequence
,
as
, is associated two subsequences (for ease of notation, we also denote them by themselves)
and
, such that
On the other hand, by (
63) and the lower semi-continuity,
Let
be fixed, assuming without loss of generality that
, define
Hence is non-decreasing on and non-increasing on . On the other hand, , we conclude .
Since
, owing to (
60), (
61) and (
68), and the condition
, then
where
.
For the above fixed
,
Combining (
68), we arrive at
Whence is bounded uniformly in .
By extracting a unlabeled subsequence, one achieves
In order to show that m yields the properties stated in Theorem 1, it suffices to check that it is continuous in t, and by a translation, it remains to demonstrate the continuity at zero. But this fact is obvious, so the required result is complete.
Assertion 5: and
solves (
23), (
2) with
(
). In addition, for every
, the related solutions
u and
of (
57) fulfill
Furthermore, if
, for almost all
, and almost all
,
In particular, if , then , .
Observing that
,
and
, so
and then
. Moreover,
is a weak solution of (
57).
With the help of (
66)–(
69), the rest of the assertion is clear.
Step 2: Existence of stochastic weak solutions to the Cauchy problem:
Before handling the general
, we review some notions. For any
, set
by
and the pullback mapping of
m by
is defined by
for every
Let us consider the Cauchy problem below
The arguments employed in (
57) for
adapted to
in (
81) now, produces that there is a
solving (
81). Note that
is
-adapted with values in
, thus for every
,
is
-adapted. Besides, by Assertion 5,
.
Hence, upon using Itô-Wentzell’s formula (see [
38]) to
, one gains
Let
, then
, which is
-adapted, and
Thanks to (
82) and Remark 3, hence there exists a stochastic weak solution to (
80).
Step 3: Existence of stochastic entropy solutions to (
23), (
2).
Due to Step 2, one claims that
Theorem 1 (ii) applies,
is a stochastic entropy solution of (
23), (
2).
Remark 6. When, then analogue calculations of (77), (78) also yield that Whence for every, If there is a positive real numbersuch that, then with probability one, the unique stochastic entropy solution ρ is exponentially stable. If for some real number, ξ possesses the below formwhere, then from (83),which implies ρ is asymptotically stable. 6. Conclusions
In recent years, people have made broad research about the uniqueness and existence of solutions for the conservation law
with a stochastic perturbation. Most of these works are concentrated on the multiplicative type:
where
is a 1-dimensional Wiener process or a cylindrical Wiener process,
is a bounded domain or
. However, for Equation (
85), if we take the spatial average for
, then it satisfies
It seems difficult to provide any bound on the average for the last term in the above identity. So the mass is not preserved in general. But if one considers the scalar conservation (
84) with the noise given by
,
then
Therefore, with such noise, the mass is preserved exactly. From the point of this view, the noise given here is more reasonable, and compared with the existing research works [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18], this idea is new.
On the other hand, when we discuss the conservation law (
84),
is a natural space on which the solutions are well-posed. But if one perturbs the Equation (
84) by the noise
, even the initial data is bounded, the solution is not bounded since the maximum principle is not available. Therefore,
is not a natural space on which the solutions exist. Even though, if we assume further that
A has compact support, then
solutions will exist [
5,
6]. However, in the present paper, by using the stochastic kinetic formulation, we also found the existence for bounded solutions without the compact support assumptions on coefficients for stochastic balance law (
1). Moreover, we prove the uniqueness for stochastic entropy solutions without any assumptions on the growth rates of the coefficients to (
1). Compared with the known results, the existence and uniqueness for stochastic entropy solutions established in the present paper are new as well.