On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
-
Imran H. Biswas
Abstract
In this article, we deal with the stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasis is on analyzing the effects of a multiplicative Lévy noise on such problems and on establishing a well-posedness theory by developing a suitable weak entropy solution framework. The proof of the existence of a solution is based on the vanishing viscosity technique. The uniqueness of the solution is settled by interpreting Kruzhkov’s doubling technique in the presence of a noise.
1 Introduction
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
with the initial condition
where
Remark 1.1.
Since ϕ is a real-valued non-decreasing and Lipschitz continuous function, we know that the set
Remark 1.2.
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
Remark 1.3.
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 [3]. 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
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 [17], 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 [15] published their celebrated work that described some statistical properties of Burgers equations with a noise. Kim [18] 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 [21]. 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 [16], 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
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 [19]. 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 [5] and Biswas, Koley and Majee [6] established the well-posedness of the entropy solution for multi-dimensional conservation laws with a Poisson noise via the Young measure approach. In [6], 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
Definition 1.4 (Stochastic weak solution).
We say that an
It holds
in the sense of distribution.
For almost every
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. [3]. 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
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
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
An entropy flux triple
For a small positive number
of (1.1) has a unique weak solution
Let
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
Definition 2.2 (Stochastic entropy solution).
A stochastic process
For each
Given a non-negative test function
(2.1)
Remark 2.3.
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:
The space E is of the form
There exist positive constants
There exist a non-negative function
one has
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].
Lemma 2.4.
Any entropy solution
Next, we describe a special class of entropy functions that plays an important role in the sequel.
Let
and
For any
Then
where
By simply dropping ϑ, for
We conclude this section by stating the main results of this paper.
Theorem 2.5 (Existence).
Let assumptions (i)–(v) be true and that the
Theorem 2.6 (Uniqueness).
Let assumptions (i)–(v) be true and that the
Remark 2.7.
In addition, if
Furthermore, if
then
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
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
Define
Proposition 3.1.
Assume that
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].
Lemma 3.2.
Assume that
There exists a unique
There exists a constant
(3.3)The map
Proof of Proposition 3.1.
Let
Then, by assumption (v), we obtain
This shows that
3.1.1 A priori estimate
Note that
Since
we see that
In view of assumption (v), inequality (3.4), the Itô–Lévy isometry and the fact that
for any
Since
Thanks to the discrete Gronwall lemma, we can deduce from (3.5) that
For fixed
with
where
A straightforward calculation shows that
Since ϕ is a Lipschitz continuous function with
Proposition 3.3.
Assume that
Moreover,
Next, we want to find some upper bounds for
Proposition 3.4.
Proof.
First, we prove the boundedness of
Thus,
To prove the second part of the proposition, we see that for any
Therefore, in view of (3.6) and assumption (v), we have
This completes the proof. ∎
3.1.2 Convergence of
u
Δ
t
(
t
,
x
)
Thanks to Proposition 3.3 and the Lipschitz property of f and ϕ, there exist u,
Next, we want to identify the weak limits
Lemma 3.5.
Proof.
Consider two positive integers N and M and denote
Let
Note that
Therefore,
Similarly, we also have
Combining (3.10), (3.11) and (3.12), we obtain
for some α and
In view of Proposition 3.3, we notice that
In addition,
where we have used Proposition 3.4 and assumption (v). Thus, we get
We combine (3.14) and (3.15) in (3.13) and have
Hence, an application of Gronwall’s lemma yields
This implies that
i.e.,
We are now in position to identify the weak limits
In view of the variational formula (3.8), one needs to show the boundedness of
in
Lemma 3.6.
The sequence
where u is given by (3.7).
Proof.
To prove the lemma, let
The space
Thanks to Propositions 3.3 and 3.5,
Again, note that
From (3.2), we see that for any
and hence
This implies that
To prove the second part of the lemma, we recall that
in
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
We make use of (3.7) and Lemmas 3.5 and 3.6 to pass
to the limit as
Since
for almost every
This proves that u is a weak solution of (3.1).
For every
Therefore,
Now, by letting
3.2 A priori bounds for the viscous solutions
Note that for a fixed
for any
Note that
An application of Gronwall’s inequality yields
The achieved result can be summarized by the following theorem.
Theorem 3.7.
For any
where G is an associated Kirchhoff’s function of ϕ, defined by
Remark 3.8.
Let us remark that since any solution to (3.1) is an entropy solution, the solution
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 [2]) and show the existence of a Young measure-valued limit process solution
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
4.1 Uniqueness of the Young measure-valued limit process
To do this, we follow the same line of argument as in [3] for the degenerate parabolic part and [5] 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. [2] and Biswas et al. [5] used the fact that
Note that
Let ρ and ϱ be standard non-negative mollifiers on
where δ and
Clearly
Let
i.e.,
We now apply the Itô–Lévy formula to (4.1) and obtain
i.e.,
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
By using the properties of Lebesgue points, convolutions and approximations by mollifications,
one is able to pass to the limit in
Lemma 4.1.
It holds that
and
Lemma 4.2.
It follows that
Proof.
In view of Lemma 4.1, we see that
Let
We consider the terms
Lemma 4.3.
It holds that
and
We now turn our attention to
One can now pass to the limit in
Lemma 4.4.
It holds that
and
In regard to
Lemma 4.5.
It holds that
and
Proof.
Let us consider the passage to the limits in
Note that, for all
By using (4.5), we have
Then
where
Note that
almost surely for all
Moreover, one can use the property of convolution to conclude
Passage to the limit as
Note that for all
Therefore, by (4.6) we have
One can justify the passage to the limit as
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
This completes the proof of the lemma. ∎
Next, we want to pass to the limits in
Lemma 4.6.
It holds that
and
In view of the above, we now want to pass to the limit as
Lemma 4.7.
Assume that
We now focus on
Lemma 4.8.
For fixed
Proof.
Note that
where the second line follows by the Cauchy–Schwarz inequality.
Similarly, we have
Next we consider the stochastic terms
where J is a predictable integrand and X is an adapted process.
For each
where
and hence we have
Our aim is to pass to the limit into the stochastic terms
Lemma 4.9.
Let
and the following identities hold:
Lemma 4.10.
It holds that
Proof.
Note that
Now apply the Itô–Lévy formula on
Therefore, from (4.8) and Lemma 4.9 we have
Note that
Again, it is routine to pass to the limit in
This completes the proof. ∎
Let us consider the additional terms
Lemma 4.11.
It holds that
and
Now we add these terms (cf. Lemma 4.11) with the terms resulting from Lemma 4.10 and have the following lemma.
Lemma 4.12.
Assume that
Proof.
In view of Lemmas 4.10 and 4.11, we see that
where
By using arguments similar to the ones used in the proof of [5, Lemma 5.11], we arrive at
where the constant
This concludes the proof as
thanks to (4.9). ∎
We now turn our attention back to the terms which are originating from Lemmas 4.2 and 4.5. To this end, define
where
Our aim is to pass to the limit in
Lemma 4.13.
The following holds:
Moreover, if
and
We now shift our focus back to the expression
Lemma 4.14.
It holds
Proof.
Let
and
Hence, we have
Put
To conclude the proof, it is now required to show
which follows easily from the fact that
The following proposition combines all of the above results.
Proposition 4.15.
Let
Proof.
First we add (4.2) and (4.3) and then pass to the limit
Remark 4.16.
Note that the same proposition holds without assuming the existence of a modulus of continuity for
in its proof. Then the result holds following the first situation on [3, p. 523].
Our aim is to show the uniqueness of
where
A straightforward calculation revels that
Clearly, (4.10) holds with
Since
Note that
Now passing to the limit as
Thus, if we assume that
which says that for almost all
4.2 Existence of an entropy solution
In this subsection, using the strong convergence of the sequence of viscous solutions and the a priori bounds (3.17), we establish the existence of an entropy solution to problem (1.1).
Fix a non-negative test function
Let the predictable process
Note that
Since
and the right-hand side is
Thus, we can pass to the limit in (4.13) as
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
holds P-almost surely. This shows that
We now close this section with a sketch of the justification of our claim in Remark 2.7. To see this,
let
Note that the weak Itô–Lévy formula in Theorem A.1 makes sense for
We can now use the properties of
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
and
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
This implies that, for almost every
Funding statement: 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.
A Weak Itô–Lévy formula
Let u be an
In addition, in view of (3.7), we further assume that
for every
Theorem A.1.
Let assumptions (i)–(v) hold and let
for almost every
Proof.
Let
holds P-almost surely. For every
It follows by standard approximation arguments that (A.3) is still valid if we replace
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
P-almost surely. We now apply the Itô–Lévy product rule to
almost surely.
Note that
and
in
Furthermore, note that
and
and
in
in
Also, it may be recalled that
as
It follows from straightforward computations that
Therefore, we can invoke the Itô–Lévy isometry and pass to the limit as
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- Higher-order anisotropic models in phase separation
- Well-posedness and maximum principles for lattice reaction-diffusion equations
- Existence of a bound state solution for quasilinear Schrödinger equations
- Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth
- Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems
- A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications
- Diffusive logistic equations with harvesting and heterogeneity under strong growth rate
- On viscosity and weak solutions for non-homogeneous p-Laplace equations
- Periodic impulsive fractional differential equations
- A result of uniqueness of solutions of the Shigesada–Kawasaki–Teramoto equations
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Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in
- Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term
- The elliptic sinh-Gordon equation in a semi-strip
- The Gelfand problem for the 1-homogeneous p-Laplacian
- Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary
- Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth
- Multiple solutions for an elliptic system with indefinite Robin boundary conditions
- New solutions for critical Neumann problems in ℝ2
- A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
- Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system
- Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
- Nonlocal perturbations of the fractional Choquard equation
- A pathological example in nonlinear spectral theory
- Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four
- On Cauchy–Liouville-type theorems
- Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions
- Besov regularity for solutions of p-harmonic equations
- The classical theory of calculus of variations for generalized functions
- On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
- Hölder gradient estimates for a class of singular or degenerate parabolic equations
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Critical and subcritical fractional Trudinger–Moser-type inequalities on
- Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
- Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations
- An elliptic system with logarithmic nonlinearity
- The Caccioppoli ultrafunctions
- Equilibrium of a production economy with non-compact attainable allocations set
- Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
- The higher integrability of weak solutions of porous medium systems
- Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature
- Regularity results for p-Laplacians in pre-fractal domains
- Carleman estimates and null controllability of a class of singular parabolic equations
- Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
- On a measure of noncompactness in the space of regulated functions and its applications
- p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities
- On the well-posedness of a multiscale mathematical model for Lithium-ion batteries
- Global existence of a radiative Euler system coupled to an electromagnetic field
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On the existence of a weak solution for some singular
- Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth
- Clustered solutions for supercritical elliptic equations on Riemannian manifolds
- Ground state solutions for the Hénon prescribed mean curvature equation
- Quasilinear equations with indefinite nonlinearity
- Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
- Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
