Martingale Solution to Stochastic Extended Korteweg-de Vries Equation ()
1. Introduction
The celebrated Korteweg-de Vries equation (KdV for short) [1] , derived from the set of Eulerian shallow water and long wavelength equations, becomes a paradigm in the field of nonlinear partial differential equations. KdV appears as the lowest approximations of wave motion in several fields of physics, see, e.g., monographs [2] [3] [4] [5] [6] and references therein.
KdV is, however, the result of an approximation of the set of the Euler equations within perturbation approach limited to the first order in expansion with respect to parameters assumed to be small. Several authors have extended KdV to the second order (the extended KdV or KdV2), e.g. [7] - [17] , which is a more exact approximation of the Euler equations but far more difficult since it contains higher nonlinear term and higher derivatives. Despite its non-integrability, KdV2 has three forms of exact analytic solutions. There exist solitonic solutions [10] , periodic cnoidal solutions [14] and periodic “superposition” solutions [15] [16] . These solutions have the same form as corresponding solutions to KdV but with slightly different coefficients.
A natural continuation of the study of the extended KdV equation seems to be considering stochastic versions of such equation. KdV2 equation driven by random noise can be a model of several kinds of waves (e.g., surface water waves, waves in plasma) influenced by random factors. Two cases of the stochastic KdV2 equation are possible: the case with additive noise and the case with the multiplicative noise. The additive case was studied by us in [18] , where a mild solution to KdV2 has been established.
In this paper, we consider the stochastic extended Korteweg-de Vries equation with multiplicative random noise. We prove the existence of martingale solution to stochastic KdV2 equation driven by cylindrical Wiener process. In the proof, we generalize the methods used in papers [19] and [20] . We have to emphasize that the method used in [19] for estimations was not suitable in our case. We adapted for our purposes (proof of Lemma 2.4) the approach used by Flandoli and Gątarek in [20] .
The paper is organized as follows. In Section 2, we present the initial value problem for the extended KdV equation. Then, we formulate the definition of the martingale solution to the problem considered. Next, we give the main result formulated in Theorem 2.1 and auxiliary results (Lemmas 2.1 - 2.3). Section 3 contains detail proofs of Lemmas 2.2 and 2.3. In Section 4, Lemma 2.1 is proved in detail. Conclusions are contained in Section 5.
2. Existence of Martingale Solution
We consider initial value problem for Korteweg-de Vries type equation
(2.1)
In (2.1),
,
, is a cylindrical Wiener process,
is a deterministic function,
. Moreover, we assume that
,
, for all
and
, what reflects finitness of solutions to deterministic version of the Equation (2.1) (see, e.g., [10] [15] [16] ). The operator
is a continuous mapping from
to
, the space of Hilbert-Schmidt operators from
to itself. The operator
is such that for any
the following conditions hold:
(2.2)
there exist such functions
with compact support, that the mapping
is continuous in topology
. (2.3)
Condition (2.2) will be used in some auxiliary estimates in proofs of Lemmas 2.1 - 2.3. Condition (2.3) is used in proofs of Lemma 2.1 and Theorem 2.1 of the existence of martingale solution to (2.4) and (2.1), respectively.
From now on, we use the notation
instead of
.
Definition 2.1 We say that the problem (2.1) has a martingale solution on the interval
,
, if there exists a stochastic basis
, where
is a cylindrical Wiener process, and there exists the process
adapted to the filtration
with trajectories belonging to the space
such that
for any
and
.
Now, we can formulate the main result of the paper.
Theorem 2.1 For all real valued functions
and
there exists a martingale solution to (2.1) with conditions (2.2) and (2.3).
Proof. Let
. Consider
(2.4)
Lemma 2.1 For any
there exists a martingale solution to the problem (2.4) with conditions (2.2) and (2.3).
Lemma 2.2 Let
be the martingale solution to (2.4) and let condition (2.2) hold. There exists
, such that
(2.5)
(2.6)
where
.
Lemma 2.3 Let condition (2.2) hold. The family of distributions
, where
is the martingale solution to (2.4), is tight in
.
(Proofs of Lemmas 2.1, 2.2 and 2.3 are given in Sections 3 and Sections 4.)
Substitute in Prohorov’s theorem (e.g., see Theorem 5.1 in [21] )
and
. Since
is tight in S (see, Lemma 2.3), then it is sequentially compact, so there exists a subsequence of
converging to some measure
in
.
Because
is convergent, then it is also weakly convergent. Therefore in Skorohod’s theorem (e.g., see Theorem 6.7 in [21] ) one can substitute
and
. Then there exists a space
and random variables
,
with values in
such that
in
and in
. Moreover
.
Then due to Lemma 2.2, for any
there exist constants
,
such that
Additionally,
Then one can conclude that
weakly in
.
Let
be fixed and denote
Note, that
so,
,
, is a square integrable martingale with values in
, adapted to filtration
with quadratic variation
Substitute in the Doob inequality (e.g., see Theorem 2.2 in [22] )
and
. Then
(2.7)
Assume
and let
be a bounded continuous function on
or
. Let
,
, be arbitrary and fixed. Since
is a martingale and
, then (see [20] , pp. 377-378)
and
Denote
If
, then
and
,
in
. Moreover, since
is continuous, then
,
. Therfeore, if
, then
Additionaly, because (by (2.3))
is a continuous operator in topology
and (2.7) holds, therefore if
, then
and
Then
is also a square integrable martingale adapted to the filtration
with quadratic variation equal
.
Substitute in the representation theorem (e.g., see Theorem 8.2 in [23] ),
,
and
.
Then there exists a process
, such that
,
, and
This implies
so
is a solution to (2.1), what finishes the proof of Theorem 2.1. □
3. Proofs of Lemmas 2.2 and 2.3
Proof of Lemma 2.2 Let
, be a smooth function fulfilling conditions
1) p is increasing in
;
2)
;
3)
;
4)
.
Let
. Applying the Itô formula for
, we obtain
(3.1)
where
We use the following estimates from ( [19] , p. 242). There exist
, such that
Similarly as above, one has
(3.2)
In consequence, we have
(3.3)
Let
be an orthonormal basis in
. Then there exists a constant
, such that
(3.4)
(by (2.2))
Due to (3.3) and (3.4) we have
so,
Let
be fixed. Then for all
one has
what proves (2.5). Moreover one has
what proves inequality (2.6). □
Proof of Lemma 2.3. Let
be arbitrary fixed and let
. Then
(3.5)
Denote
There exists a constant
, that
.
There exists a constant
, such that
Therefore, due to Lemma 2.2, we can write
So, there exists a constant
, such that
Now, we use the result from ( [19] , p. 243). There exists a constant
, that the following inequality holds
(3.6)
This estimate implies the existence of a constant
, such that
Due to Lemma 2.2 there exists a constant
, that we can write
Then, there exists a constant
, such that
We have
Lemma 2.2 implies the existence of a constant
, such that
So, there exists a constant
, such that
.
There exist constants
, that
Due to Lemma 2.2 there exists a constant
, such that
So, there exists a constant
, that
.
Substitute in ( [20] , Lemma 2.1)
,
. Then
and for all
and
there exists a constant
, such that
Then, due to condition (2.2), there exists a constant
, that
Substitution in the above inequality
yields
(3.7)
Let
and
be arbitrary fixed. Note, that the following inclusions hold
and
Then, there exists a constant
, such that
Moreover
and
So, there exist constants
, that
(3.8)
Let
be arbitrary fixed. Due to Lemma 2.2 there exists a constant
, that
(3.9)
Substituting in ( [19] Lemma 2.1)
and using Markov inequality ( [24] p. 114) for
and
, we obtain
Let K be the following mapping for
:
, where
is an increasing sequence of positive numbers, which can, but does not have to, depend on
. Note, that due to ( [19] Lemma 2.1), the set
is compact for all
. Moreover,
, then the family
is tight.
4. Proof of Lemma 2.1
Proof. Let
be an orthonormal basis in space
. Denote by
, for all
, the orthogonal projection on
. Consider finite dimensional approximation of the problem (2.4) in the space
of the form
(4.1)
where
fulfils conditions
(4.2)
Let
be arbitrary fixed and
Then
Note, that
where
, therefore
Analogously,
Moreover
where
, so
Finally,
Additionally, due to the condition (2.2), there exist constants
, such that
(4.3)
then
Therefore, from ( [25] Prop. 3.6 and 4.6), when
and
are as above, for all
, there exists a martingale solution to (4.1). Moreover, applying the same methods as in Section 3, using (2.3), one can show that for all m the following inequalities hold
(4.4)
(4.5)
and the family of distributions
is tight in
. Then application of the same methods, as used already on pages 3-5, leads to the proof of the existence of martingale solution to (2.4) □
5. Conclusions
The existence of martingale solution to highly nonlinear extended KdV equations is proved for all physically relevant initial conditions. The presence of several nonlinear terms in the considered equation required the development of much more involved methods for proving the main theorem than for the case of stochastic KdV equation.
The methods developed in the current paper can be applied to study stochastic hybrid Korteweg-de Vries-Burgers equations, particularly important for nonlinear ion-acoustic waves in plasma physics.