Stochastic partial differential equations (SPDEs) often result from the efforts to model complex physical phenomena where uncertainty/randomness is inherent. A brief survey of relevant literature reveals the use of SPDEs in a wide variety of studies in areas that include physics, biology, engineering and finance. As examples we mention the studies of neural dynamics, the spread of infectious disease (a tracer or a passive population in a flow subject to possibly random external forces), Navier–Stokes-type flow under random forces, ferromagnetism under random influences (stochastic Landau–Lifschitz–Gilbert equation) and the modeling of forward rate curve in finance. A popular way to account for the randomness has been to add some noise to the deterministic models and, most often, the noise is assumed to be Gaussian/Brownian. However, the data collected through surveys/experiments often exhibit properties such as heavy-tailedness, which can not be adequately explained by a Brownian noise. This inadequacy has lately prompted a lot of interest in models with Lévy-type noise, which necessitates the development of a well-rounded mathematical theory for SPDEs driven by such type of noise. In this article, we embark on the well-posedness study of a class of nonlinear and degenerate parabolic SPDEs with a Lévy noise.
Let be a filtered probability space satisfying the usual hypothesis, i.e., is a right-continuous filtration such that contains all the P-null subsets of . In addition, let be a σ-finite measure space and let be a Poisson random measure on with intensity measure with respect to the same stochastic basis. The existence and construction of such a general notion of Poisson random measure with a given intensity measure are detailed in . We are interested in the Cauchy problem for a nonlinear degenerate parabolic stochastic PDE of the following type:
with the initial condition
where with fixed, is the unknown random scalar-valued function, is a given flux function and is the compensated Poisson random measure. Furthermore, is a real-valued function defined on the domain , and is a given non-decreasing Lipschitz continuous function. The stochastic integral on the right-hand side of (1.1) is defined in the Lévy–Itô sense.
Since ϕ is a real-valued non-decreasing and Lipschitz continuous function, we know that the set is not empty in general and hence the problem is called degenerate. Furthermore, A is not negligible either and the problem is strongly degenerate in the sense of .
The analysis of this paper remains valid if the noise in the right-hand side of (1.1) is of jump-diffusion type. In other words, the same analysis holds if we add the term in the right-hand side of (1.1), where denotes a cylindrical Brownian motion. Moreover, we will carry out our analysis under the structural assumption that , where is a subset of the Euclidean space. The measure m on E is defined as the product measure , where λ is a Radon measure on and μ is a so-called Lévy measure on . In such a case, the noise in the right-hand side would be called an impulsive white noise with the jump position intensity λ and the jump size intensity μ. We refer to  for more on Lévy sheet and related impulsive white noises.
In so far as the techniques are concerned, one anticipates to follow closely the methodology used in analyzing a version of (1.1) with a Brownian noise in . However, there will be additional difficulties specific to the discontinuous character of the Lévy noise. Note that the solution of problem (1.1) is interpreted in the entropy sense as in [9, 3]. The entropy inequalities will have an Itô–Lévy correction term, and they will have non-localities resulting from the jump nature of the noise term in the right-hand side of (1.1). This non-locality tends to derail the stability analysis, and proving the uniqueness becomes a trickier affair. We are able to manage this added difficulty here and establish the uniqueness, but under a slightly restrictive assumption (iv) on the jump vector η (see Section 2.1).
Equation (1.1) becomes a multi-dimensional deterministic degenerate parabolic-hyperbolic equation if . It is well-documented in the literature that the solution has to be interpreted in the weak sense and one needs an entropy formulation to prove the well-posedness of the problem. We refer to [1, 9, 11, 12, 4, 20] and the references therein for more on the entropy solution theory for deterministic degenerate parabolic-hyperbolic equations.
1.1 Studies on degenerate parabolic-hyperbolic equations with a Brownian noise
The study of stochastic degenerate parabolic-hyperbolic equations has so far been limited to mainly equations with a Brownian noise. In particular, hyperbolic conservation laws with a Brownian noise are examples of such problems that have attracted the attention of many. The first documented development in this direction is , where Holden and Risebro established a result of existence of a path-wise weak solution (possibly non-unique) for one-dimensional balance laws via the splitting method. In a separate development, Khanin, Mazel and Sinai  published their celebrated work that described some statistical properties of Burgers equations with a noise. Kim  extended the Kruzhkov entropy formulation and established the well-posedness of the Cauchy problem for one-dimensional balance laws driven by an additive Brownian noise. The multi-dimensional analogue on bounded domains was studied by Vallet and Wittbold . They established a result of well-posedness of the entropy solution with the theory of Young measures.
This approach is not applicable to the multiplicative noise case. This case was studied by many authors [2, 10, 14, 16]. In , Feng and Nualart found a way to recover the necessary information in the form of the strong entropy condition from the parabolic regularization, and they established a result of uniqueness of the strong entropy solution in the -framework for the multi-dimensional case, but the existence of a solution was established only in the one-dimensional case. We also add here that Feng and Nualart  used an entropy formulation which is strong in time but weak in space, which in our view may give rise to problems when the solutions are not shown to have continuous sample paths. We refer to , where a few technical questions are raised on the strong in time formulation and remedial measures have been proposed. In , Debussche and Vovelle obtained the existence of a solution via a kinetic formulation, and Chen, Ding and Karlsen  used the BV solution framework. Bauzet, Vallet and Wittbold  established a result of well-posedness via the Young measure approach. The well-posedness of the problem to a multi-dimensional degenerate parabolic-hyperbolic stochastic problem has been studied by Debussche, Hofmanová and Vovelle  and Bauzet, Vallet and Wittbold . The former adapted the notion of kinetic formulation and developed a well-posedness theory, while the latter revisited [1, 9, 11] and established the well-posedness of the entropy solution via the Young measure theory.
1.2 Relevant studies on problems with a Lévy noise
Over the last decade, there have been many contributions on the larger area of stochastic partial differential equations that are driven by a Lévy noise. A worthy reference on this subject is . However, very little is available on the specific problem of degenerate parabolic problems with a Lévy noise such as (1.1). This article marks an important step in our quest to develop a comprehensive theory of stochastic degenerate parabolic equations that are driven by jump-diffusions. The relevant results in this context are made available recently and they are on conservation laws that are perturbed by a Lévy noise. In recent articles, Biswas, Karlsen and Majee  and Biswas, Koley and Majee  established the well-posedness of the entropy solution for multi-dimensional conservation laws with a Poisson noise via the Young measure approach. In , Biswas et al. developed a continuous dependence theory on nonlinearities within the BV solution setting.
Stochastic degenerate parabolic-hyperbolic equations are one of the most important classes of nonlinear stochastic PDEs. Nonlinearity and degeneracy are two main features on these equations and yield several striking phenomena. Therefore, this requires new mathematical ideas, approaches, and theories. It is well known that due to the presence of nonlinear flux terms, solutions to (1.1) are not smooth even for smooth initial data . Therefore, a solution must be interpreted in the weak sense. Before introducing the concept of weak solutions, we first recall the notion of predictable σ-field. By a predictable σ-field on , denoted by , we mean the σ-field generated by the sets of the form and for any , , . The notion of a stochastic weak solution is defined as follows.
Definition 1.4 (Stochastic weak solution).
We say that an -valued -predictable stochastic process is a weak solution to problem (1.1) provided the following conditions are satisfied:
in the sense of distribution.
For almost every and P-a.s., the following variational formulation holds:
for any .
However, it is well known that weak solutions may be discontinuous and are not uniquely determined by their initial data. Consequently, an admissibility criterion for the so-called entropy solution (see Section 2 for the definition of an entropy solution) must be imposed to single out the physically correct solution.
1.3 Goal of the study and outline of the paper
The case of a strongly degenerate stochastic problem driven by a Brownian noise is studied by Bauzet et al. . In this article, drawing primary motivation from [3, 5, 9], we propose to establish a result of well-posedness of the entropy solution to a degenerate Cauchy problem (1.1) by using the vanishing viscosity method along with few a priori bounds.
The rest of the paper is organized as follows: We state the assumptions, details of the technical framework and the main results in Section 2. Section 3 is devoted to prove the existence of a weak solution for the viscous problem via an implicit time discretization scheme and to derive some a priori estimates for the sequence of viscous solutions. In Section 4, we establish first the uniqueness of the limit of the viscous solutions when the viscosity parameter goes to zero via the Young measure theory, and then we establish the existence of an entropy solution. The uniqueness of the entropy solution is presented in Section 5.
2 Technical framework and statements of the main results
Here and in the sequel, we denote by the space of predictable -valued processes u such that
Moreover, we use the letter C to denote various generic constants. There are situations where the constant may change from line to line, but the notation is kept unchanged so long as it does not impact the primary implication. We denote by and the Lipschitz constants of ϕ and f, respectively. Also, we use to denote the pairing between and .
2.1 Entropy inequalities
We begin this subsection with a formal derivation of the entropy inequalities à la Kruzhkov. Remember that we need to replace the traditional chain rule for deterministic calculus by the Itô–Lévy chain rule.
Definition 2.1 (Entropy flux triple).
A triplet is called an entropy flux triple if is Lipschitz continuous and , is a vector-valued function and is a scalar-valued function such that
An entropy flux triple is called convex if .
For a small positive number , assume that the parabolic perturbation
of (1.1) has a unique weak solution . Note that this weak solution is in . Moreover, for the time being, we assume that it satisfies the initial condition in the sense of (A.2). This enables one to derive a weak version of the Itô–Lévy formula for the solutions to (1.1) as detailed in Theorem A.1 in Appendix A.
Let be an entropy flux triple. Given a non-negative test function , we apply the generalized version of the Itô–Lévy formula (cf. Appendix A) to have, for almost every ,
Let G be the associated Kirchhoff function of ϕ, given by
A simple calculation shows that
Since β and ψ are non-negative functions, we obtain
Clearly, the above inequality is stable under the limit if the family has -type stability. Just as the deterministic equations, the above inequality provides us with the entropy condition. We now formally define the entropy solution.
Definition 2.2 (Stochastic entropy solution).
A stochastic process is called a stochastic entropy solution of (1.1) if the following conditions hold:
For each ,
Given a non-negative test function and a convex entropy flux triple , the following inequality holds:
We point out that, by a classical separability argument, it is possible to choose a subset of Ω of P-full measure such that (2.1) holds on that subset for every admissible entropy triplet and test function.
The primary aim of this paper is to establish a result of existence and uniqueness of an entropy solution for the Cauchy problem (1.1) in accordance with Definition 2.2, and we do so under the following assumptions:
is a non-decreasing Lipschitz continuous function with . Moreover, if η is not a constant function with respect to the space variable x, then has a modulus of continuity such that
is a Lipschitz continuous function with for all .
The space E is of the form , and the Borel measure m on E has the form , where λ is a Radon measure on and μ is a so-called one-dimensional Lévy measure.
There exist positive constants , and with such that
There exist a non-negative function and such that for all
The above definition does not say anything explicitly about the way the entropy solution satisfies the initial condition. However, the initial condition is satisfied in a certain weak sense. Here we state the lemma whose proof follows the same line of argument as the one of [5, Lemma 2.3].
Any entropy solution of (1.1) satisfies the initial condition in the following sense: for every non-negative test function such that ,
Next, we describe a special class of entropy functions that plays an important role in the sequel. Let be a and Lipschitz continuous function satisfying
For any , define by
By simply dropping ϑ, for we define
We conclude this section by stating the main results of this paper.
Theorem 2.5 (Existence).
Theorem 2.6 (Uniqueness).
In addition, if is in for , then it can be concluded that
Furthermore, if and there is such that for and
then for almost every . We sketch a justification of this claim in Section 4.
3 Existence of a weak solution to the viscous problem
Just as in the case of the deterministic problem, here we also study the problem regularized by adding a small diffusion operator and derive some a priori bounds. Due to the nonlinear function ϕ and related degeneracy, one cannot expect a classical solution and instead seeks a weak solution.
3.1 Existence of a weak solution to the viscous problem
For a small parameter , we consider the viscous approximation of (1.1) as
In this subsection, we establish the existence of a weak solution for problem (3.1). To do this, we use an implicit time discretization scheme. Let for some positive integer . Further, set for .
Assume that is small. For any given , there exists a unique with such that P-a.s. for any the following variational formula holds:
Before proving this proposition, we first state a key deterministic lemma, related to the weak solution of parabolic equations. We have the following lemma, a proof of which can be found in [8, p. 19].
Assume that is small and . Then, for a fixed parameter , the following holds:
There exists a unique with such that, for any ,
There exists a constant such that the following a priori estimate holds:
The map is continuous.
Proof of Proposition 3.1.
Let . Take
Then, by assumption (v), we obtain
This shows that almost surely. Therefore, one can use Lemma 3.2 and conclude that for almost surely all there exists a unique satisfying the variational equality (3.2). Moreover, by construction . Thus, due to the continuity of Θ for the measurability and to the a priori estimate (3.3), we conclude that with . We denote this solution u by . Hence the assertion of the proposition follows. ∎
3.1.1 A priori estimate
Note that for any , and hence it holds true for any by a density argument. We choose the test function in (3.2) to have
we see that
Since is arbitrary, one can choose so that
Thanks to the discrete Gronwall lemma, we can deduce from (3.5) that
For fixed , we define
with for . Similarly, we define
A straightforward calculation shows that
Since ϕ is a Lipschitz continuous function with , in view of the above definitions and the a priori estimate (3.6), we have the following proposition.
Assume that is small. Then and are bounded sequences in , and are bounded sequences in , and .
Moreover, in .
Next, we want to find some upper bounds for . Regarding this, we have the following proposition.
is a bounded sequence in and
First, we prove the boundedness of . By using the definition of , assumption (v) and the boundedness of in , we obtain
Thus, is a bounded sequence in .
To prove the second part of the proposition, we see that for any ,
This completes the proof. ∎
3.1.2 Convergence of
Thanks to Proposition 3.3 and the Lipschitz property of f and ϕ, there exist u, and such that (up to a subsequence)
Next, we want to identify the weak limits and . Note that, for any , we can rewrite (3.2) in terms of , and as
is a Cauchy sequence in .
Consider two positive integers N and M and denote and . Then, for any , from (3.8) one gets
Let . Set in (3.9); then one has
Note that .
Similarly, we also have
for some α and . Since are arbitrary, there exist some positive constants , and such that
In view of Proposition 3.3, we notice that
Hence, an application of Gronwall’s lemma yields
This implies that
i.e., is a Cauchy sequence in . ∎
We are now in position to identify the weak limits and . We have shown that and that is a Cauchy sequence in . Thanks to the Lipschitz continuity of ϕ and f, one can easily conclude that and .
In view of the variational formula (3.8), one needs to show the boundedness of
in and then identify its weak limit. To this end, we have the following lemma.
The sequence is bounded in , and
where u is given by (3.7).
To prove the lemma, let , and , where represents the predictable σ-algebra on and represents the Lebesgue σ- algebra on E.
The space represents then the space of the square integrable predictable integrands for the Itô–Lévy integral with respect to the compensated compound Poisson random measure . Moreover, the Itô–Lévy integral defines a linear operator from to and it preserves the norm (see, for example, ).
Again, note that
From (3.2), we see that for any ,
This implies that is a bounded sequence in .
To prove the second part of the lemma, we recall that
in . In view of the first part of this lemma, one can conclude that, up to a subsequence,
This completes the proof. ∎
3.1.3 Existence of a weak solution
As we have emphasized, our aim is to prove the existence of a weak solution to the viscous problem. For this, it is required to pass to the limit as . To this end, let us choose and . Then, in view of the variational formula (3.8), we obtain
Since is a separable Hilbert space, the above formulation (3.16) yields almost surely in Ω,
for almost every .
This proves that u is a weak solution of (3.1). For every , it is easily seen that
Now, by letting , we have
3.2 A priori bounds for the viscous solutions
Note that for a fixed there exists a weak solution, denoted as , which satisfies the following variational formulation: P-almost surely in Ω, and for almost every ,
for any . Let . Applying the Itô–Lévy formula (cf. Theorem A.1) to , one gets that for any ,
Note that as . Taking the expectation, we obtain
An application of Gronwall’s inequality yields
The achieved result can be summarized by the following theorem.
For any , there exists a weak solution to problem (3.1). Moreover, it satisfies the following estimate:
where G is an associated Kirchhoff’s function of ϕ, defined by .
Let us remark that since any solution to (3.1) is an entropy solution, the solution in unique.
4 Existence of an entropy solution
In this section, we will establish the existence of an entropy solution. In view of the a priori estimates given in (3.17), we can apply [5, Lemmas 4.2 and 4.3] (see also ) and show the existence of a Young measure-valued limit process solution , associated with the sequence .
The basic strategy in this case is to apply the Young measure technique and to adapt Kruzhkov’s doubling variable method in the presence of a noise to viscous solutions with two different parameters, and then to send the viscosity parameters to zero. One needs a version of the classical contraction principle (for conservation laws) to get the uniqueness of the Young measure-valued limit and to show that the Young measure-valued limit process is independent of the additional (dummy) variable. Hence, it will imply the point-wise convergence of the sequence of viscous solutions.
4.1 Uniqueness of the Young measure-valued limit process
To do this, we follow the same line of argument as in  for the degenerate parabolic part and  for the Lévy noise. For the convenience of the reader, we have chosen to provide detailed proofs of a few crucial technical lemmas and the rest is referred to appropriate resources. Bauzet et al.  and Biswas et al.  used the fact that . Note that, in this case, . Therefore, we need to regularize by convolution. Let be a sequence of mollifiers in . Since is a solution to problem (3.1), as shown in the proof of Theorem A.1, is a solution to the problem
Note that for fixed .
Let ρ and ϱ be standard non-negative mollifiers on and , respectively, such that and . We define
where δ and are two positive constants. Given a non-negative test function and two positive constants δ and , define
Clearly only if , and hence outside .
Let be the weak solution to the viscous problem (3.1) with parameter and initial condition . Moreover, let J be the standard symmetric non-negative mollifier on with support in and for . We now simply write down the Itô–Lévy formula for against the convex entropy triple , multiply by for and then integrate with respect to s, y and k. Taking the expectation of the resulting expressions, we have
We now apply the Itô–Lévy formula to (4.1) and obtain
We now add (4.2) and (4.3) and look for the passage to the limit with respect to the different approximation parameters. Note that and are both non-negative terms, they are the left-hand sides of inequalities (4.2) and (4.3), respectively. Hence we can omit these terms. Let us consider the expressions and . Recall that
By using the properties of Lebesgue points, convolutions and approximations by mollifications, one is able to pass to the limit in and , and conclude the following lemma.
It holds that
It follows that
In view of Lemma 4.1, we see that
Let and be the Young measure-valued narrow limits associated with the sequences and , respectively. With these at hand, one can use a similar argument as in [3, Lemma 3.4] and arrive at the conclusion that (4.4) holds. ∎
We consider the terms originating from the initial conditions. Note that as . Under a slight modification of the same line of arguments as in , we arrive at the following lemma.
It holds that
We now turn our attention to . Note that and that β and are even functions. A simple calculation gives
One can now pass to the limit in and have the following conclusion.
It holds that
In regard to and , we have the following lemma.
It holds that
Let us consider the passage to the limits in . To do this, we define
Note that, for all ,
By using (4.5), we have
where is a compact set depending on ψ and δ.
almost surely for all . Therefore, by the dominated convergence theorem, we have
Moreover, one can use the property of convolution to conclude
Passage to the limit as : Let
Note that for all ,
Therefore, by (4.6) we have
This proves the first part of the lemma.
To prove the second part, let us recall that
A classical property of Lebesgue points and convolution yields
Recall that . Making use of Green’s type formula along with the Young measure theory and keeping in mind that and are Young measure-valued limit processes associated with the sequences and , respectively, we arrive at the following conclusion:
This completes the proof of the lemma. ∎
It holds that
In view of the above, we now want to pass to the limit as . We follow a line of argument similar to the proof of the second part of [5, Lemmas 5.7 and 5.8], and arrive at the following lemma.
Assume that , and . Then
We now focus on and establish the following lemma.
For fixed and β, it holds that
where the second line follows by the Cauchy–Schwarz inequality. Similarly, we have . This completes the proof. ∎
Next we consider the stochastic terms . To this end, we cite  and assert that for two constants with ,
where J is a predictable integrand and X is an adapted process.
For each with and each non-negative , we define
where , . Furthermore, we extend the process for negative time simply by if . With this convention, it follows from (4.7) that
and hence we have and
Our aim is to pass to the limit into the stochastic terms . This requires the following moment estimate of , a proof of which can be found in .
Let be a function such that and let p be a positive integer of the form for some . If , then there exists a constant such that
and the following identities hold:
It holds that
Note that satisfies for all ,
Now apply the Itô–Lévy formula on to get
Note that for fixed κ and ε. One can use Young’s inequality for convolution and replace by to adapt the same line of argument as in  and conclude
Again, it is routine to pass to the limit in and arrive at the conclusion that
This completes the proof. ∎
It holds that
Assume that , and . Then
By using arguments similar to the ones used in the proof of [5, Lemma 5.11], we arrive at
where the constant depends only on ψ and is in particular independent of ε. We now let , and , yielding
This concludes the proof as
thanks to (4.9). ∎
Our aim is to pass to the limit in as . For this, we need some a priori estimates on . Here we state the required lemma whose proof can be found in .
The following holds:
Moreover, if has a modulus of continuity , then
We now shift our focus back to the expression and prove the following lemma.
Let be a modulus of continuity of . Then, thanks to Lemma 4.13, we obtain
Hence, we have
Put . Then, by our assumption (i), we see that
To conclude the proof, it is now required to show
which follows easily from the fact that is Lipschitz continuous and that
The following proposition combines all of the above results.
Let and be the predictable process with initial data and , respectively, which have been extracted out of Young measure-valued sub-sequential limits of the sequences and , respectively. Then, for any non-negative function with compact support, the following inequality holds:
First we add (4.2) and (4.3) and then pass to the limit . Invoking Lemmas 4.2, 4.3, 4.4, 4.5, 4.6, 4.8 and 4.12, we put in the resulting expression and then let with the second parts of Lemmas 4.3 and 4.4. Keeping in mind Lemmas 4.7, 4.12 and 4.14, we conclude that (4.10) holds for any non-negative test function . It now follows by routine approximation arguments that (4.10) holds for any ψ with compact support such that . This completes the proof. ∎
Note that the same proposition holds without assuming the existence of a modulus of continuity for if η is not a function of x. Indeed, it is possible to pass to the limit first in the parameter ϑ, then δ, in Lemmas 4.3, 4.4, 4.7 and 4.12. Thus, if one assumes that η is not a function of the space variable x, it is also possible to pass to the limit first in the parameter ϑ, then δ, in Lemma 4.12 since one would have
in its proof. Then the result holds following the first situation on [3, p. 523].
where in which could be chosen in such a way that . Also, for each and fixed , we define
A straightforward calculation revels that
Clearly, (4.10) holds with . Thus, for a.e , we obtain
Since on the set , we have from (4.11),
Note that for any . Since ϕ is a Lipschitz continuous function and , inequality (4.12) gives
Now passing to the limit as , and then using a weaker version of Gronwall’s inequality, we obtain for a.e. ,
Thus, if we assume that , then we arrive at the conclusion
which says that for almost all , a.e. and a.e. , we have the equation . On the other hand, we conclude that the whole sequence of viscous approximation converges weakly in . Since the limit process is independent of the additional (dummy) variable, the viscous approximation converges strongly in for any and any bounded open set .
4.2 Existence of an entropy solution
Fix a non-negative test function , and a convex entropy flux triple . Now apply the Itô–Lévy formula (3.1) and conclude
Let the predictable process be the pointwise limit of for a.e. almost surely. One can now pass to the limit in (4.13) (same argument as in ) except the first term. The pointwise limit of is not enough to pass to the limit in the first term of the inequality because is in a gradient term. For this, we proceed as follows: Fix . Define
Note that is uniformly bounded and in . Also, converges to f pointwise (up to a subsequence), where
and the right-hand side is integrable, one can apply the dominated convergence theorem to conclude in . Moreover, we have in , and therefore, by Fatou’s lemma for weak convergences,
Thus, we can pass to the limit in (4.13) as and arrive at following inequality:
We are now in a position to prove the result of existence of an entropy solution for the original problem (1.1).
Proof of the theorem 2.5.
The uniform moment estimate (3.17) together with a general version of Fatou’s lemma give
For any and any given convex entropy flux triple , inequality (4.14) holds for every . Hence, the inequality
We now close this section with a sketch of the justification of our claim in Remark 2.7. To see this, let denote a smooth even convex approximation of defined for positive x by the following: vanishes at 0, and uniquely recovered from its second-order derivative defined as if and if . It holds that , and there exists such that . Furthermore, it is easily seen that
Note that the weak Itô–Lévy formula in Theorem A.1 makes sense for as is bounded. This enables us to write, for almost every ,
We can now use the properties of and the assumptions on η to arrive at
and, by a weak Gronwall inequality,
for almost all t. This implies
by the monotone convergence theorem. The solution u will inherit the same property by Fatou’s lemma.
If is bounded and for , with given M, then consider the non-negative regular convex function , where . Since and h vanishes where η is active, Itô’s formula yields
and is uniformly bounded by K. Again, the solution u will inherit the same property by passing to the limit.
5 Uniqueness of the entropy solution
To prove the uniqueness of the entropy solution, we compare any entropy solution to the viscous solution via Kruzhkov’s doubling variables method and then pass to the limit as the viscosity parameter goes to zero. We have already shown that the limit of the sequence of viscous solutions serves to prove the existence of an entropy solution to the underlying problem. Now, let be any entropy solution and let be a viscous solution for problem (3.1). Then one can use exactly the same arguments as in Section 4, and end up with the following equality:
This implies that, for almost every , one has for almost every and . In other words, this proves the uniqueness of the entropy solution.
A Weak Itô–Lévy formula
Let u be an -valued -predictable process and assume that it is a weak solution to the SPDE
In addition, in view of (3.7), we further assume that . Moreover, u satisfies the initial condition in the following sense: P -almost surely
for every . We have the following weak version of the Itô–Lévy formula for .
Let assumptions (i)–(v) hold and let be an -valued weak solution of equation (A.1), as described in Section 3.1.3, which satisfies equation (A.2). Then for every entropy triplet and , it holds P-almost surely that
for almost every .
Let be a standard sequence of mollifiers on . Then for every we have that
holds P-almost surely. For every , define
It follows by standard approximation arguments that (A.3) is still valid if we replace by . Afterwards, we invoke the right continuity of the stochastic integral and standard facts related to Lebesgue points of Banach space valued functions to pass to the limit and conclude that, for almost all ,
holds P-almost surely. Above, we have used that the weak solution satisfies the initial condition in the sense of (A.2). Let β be the entropy function mentioned in the statement and let ψ be the test function specified. Now we apply the Itô–Lévy chain rule to to have, for almost every ,
P-almost surely. We now apply the Itô–Lévy product rule to and integrate with respect to x to obtain for almost every ,
almost surely. Note that and in as . Therefore, by the Lipschitz continuity of β, we have
in . By a similar reasoning,
Furthermore, note that
and . Therefore,
and in as . Therefore,
in as . By the same reasoning,
in as .
Also, it may be recalled that and in as . Therefore,
as in . To this end, we denote
It follows from straightforward computations that
Therefore, we can invoke the Itô–Lévy isometry and pass to the limit as , in the martingale term in (A.4). This completes the validation of the passage to the limit as in every term of (A.4). The assertion is now concluded by simply letting in (A.4) and rearranging the terms. ∎
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About the article
Published Online: 2017-09-12
The authors are profoundly thankful for the generous support from IFCAM, which allowed them to travel between India and France and made this collaboration possible. In addition, the first author acknowledges the support of INSA. The third author acknowledges the support of ISIFoR.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 809–844, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0113.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0