Existence and extinction in finite time for Stratonovich gradient noise porous media equations

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( \nu \Delta X + \Delta \psi (X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ d\beta_i$ in $]0,T[ \times \mathcal{O}$, with $X(0) = x(\xi)$ in $\mathcal{O}$ and $X = 0$ on $]0,T[ \times \partial \mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

(Communicated by Viorel Barbu) Abstract. We study existence and uniqueness of distributional solutions to the stochastic partial differential equation dX − ν∆X + ∆ψ(X) dt =

1.
Introduction. In this work we consider stochastic porous media equations with Stratonovich gradient noise. In particular, we deal with existence and uniqueness of a solution to such kind of equations, providing also some results concerning its asymptotic behaviour. To be precise, let O ⊂ R d be an open, bounded set with regular boundary and T > 0, then we consider the following stochastic partial differential equation (SPDE) in ]0, T [ × O, dX(t, ξ) − ν∆X(t, ξ) + ∆ψ(X(t, ξ)) dt = N i=1 b i (ξ), ∇X(t, ξ) • dβ i (t), (1) where • denotes that the integration is intended in the Stratonovich sense, ν > 0, ψ : R → 2 R is a maximal monotone function with polynomial growth, b i : O ⊂ R d → R d are C 2 functions and β = (β i ) i=1,...,N is an N -dimensional Brownian motion on a given probability space.
We provide existence of a distributional solution to eq. (1), essentially studying problem (1) with ψ substituted by its Yosida approximation, ψ λ , λ > 0, which, as we will see, admits a solution for all λ > 0, and then showing that the associated sequence of solutions, namely the sequence of solutions of the Yosida approximation scheme related to eq. (1), converges to the one we are looking for, as λ goes to zero. This approach has been employed also in, e.g., [11] to study porous media equations with multiplicative noise.
We then treat two particular examples of eq. (1), namely fast diffusion and self-organized criticality, and prove an asymptotic result concerning the solution to the equation in those frameworks. As we shall see, the solution will be zero from a certain time on with positive probability. It is worth to mention that the method 868 MATTIA TURRA we have used to obtain the aforementioned results is based on the one developed in [8], where the multiplicative noise case is studied.
Perturbing a problem by a Stratonovich noise of gradient type is useful in a wide range of applications, as in the case of image processing, see [24,28], where it has been proved that considering those kind of perturbations improves the solution obtained by the total variation regularization.
Some recent results regarding equations with Stratonovich gradient noise have been studied in the case of p-Laplacian and total variation flow drift. For instance the reader may refer to [6] which deals with p-Laplace equations in the case p > 1, to [15] for the case with Neumann boundary conditions, to [25] for p-Laplace equation and total variation flow on a d-dimensional torus, to [7] for a distributional solution when b i is divergence-free and to [22] for the case of total variation flow with Dirichlet boundary conditions. See also [16,17] for other results related to eq. (1), and more generally pathwise well-posedness and entropy solutions for nonlinear diffusion equations with nonlinear conservative noise.
1.1. Structure of the work. The work is organized as follows. We begin introducing the problem and discussing the assumptions, the definition of solution to eq. (1) and some preliminaries in Section 2. Section 3 is devoted to the construction of the approximating problem and its properties. We prove the existence and uniqueness of a solution to our problem in Section 4. Extinction in finite time for the fast diffusion model is discussed in Section 5. Self-organized criticality model is treated in Section 6. We conclude with some final remarks and considerations in Section 7.
1.2. Notation. Let O ⊂ R d be an open and bounded set with regular boundary ∂O. Then we denote by L p (O), p ∈ [0, ∞], the Banach space of all p-summable (equivalence classes of) functions from O to R and by |·| p its corresponding norm, while we indicate by H k (O), k ∈ N, the Sobolev space of functions in L 2 (O) whose distributional derivatives of order less than k belong to L 2 . H 1 0 (O) is the set of H 1 function vanishing on ∂O, its corresponding norm is given by and its norm is denoted by · −1 , ·, · −1 being its inner product. We indicate by ·, · , or sometimes simply by ·, the scalar product on R N .
be a open and bounded set with smooth boundary ∂O. We aim at providing existence and uniqueness of a solution to the following nonlinear SPDE with Stratonovich gradient noise for X : in O, where ν > 0, b = (b 1 , . . . , b N ) : O → R N ×d , ψ : R → 2 R is a (possibly multivalued) map and β = (β i ) i=1,...,N is an N -dimensional Brownian motion on a filtered probability space (Ω, F, (F t ) t≥0 , P). For the sake of simplicity, we will often omit to write explicitly the dependence of X on (ω, t, ξ) ∈ Ω × [0, T ] × O as well as the dependence of b on ξ ∈ O.
Equation (2) can be equivalently written as the following SPDE in the Itô sense Notice that we can write Before proceeding further we provide the following result, which will be useful in the proof of the existence of a solution to eq. (3). Lemma 2.1. Let b be defined as above and assume that b ij ∈ C 2 (O). Then there existC > 0, depending on O, and γ > 0, depending on b, such that Multiplying the first equation in (6) by v and then integrating we have, by Green's formula and (7), therefore, by (4), we get By eq.
We assume that the following hypotheses on ψ, b and ν hold.
(ii) There exist C > 0 and m ≥ 0 such that Moreover, we assume that (iii) The functions A kj defined in (5) are bounded for any k, j = 1, . . . , d.
(iv) b i ∈ C 2 O; R d for every i = 1, . . . , N and whereC and γ are as in Lemma 2.1.
Notice that condition (8) tells us that the nearer ν is to 0, the stricter the condition on the norm of b is. In particular, the case ν = 0 implies b ≡ 0, reducing eq. (3) to the (deterministic) PDE ∂ t X = ∆ψ(X).
The solution we are looking for is to be intended in the following sense.
where (f j ) j∈N is an orthonormal basis for −∆ in H −1 .
A solution of this type is also referred to as distributional solution since we can equivalently write eq. (9) as where ∆ : 3. The approximating problem. Under Assumption 2.2 we define, for every λ > 0, the resolvent and the Yosida approximation of ψ, respectively, which are known to be Lipschitz-continuous, see, e.g., [4]. We shall consider thus the following approximating problem, for λ > 0, We will need the following result.
Lemma 3.1. We have, for every r ∈ R and λ > 0, Proof. It holds, for every r ∈ R, Moreover, one can also see that r → ψ λ (r)+νr is strictly monotonically increasing, bounded by C(1 + |r| m ) and (ψ λ (r) + νr)r ≥ ν |r| 2 for all r ∈ R. We have, thus, the following existence result, which is a consequence of Krylov and Rozovskii Theorem, see [19] or the more recent book [21]. We conclude this section with the following result.
Lemma 3.2. Let X λ be a solution to eq. (10). Then there exist C 1 , C 2 > 0, depending on ν and on the L 2 -norm of the initial condition x but not on the parameter λ, such that Proof. Let X λ be the solution to eq. (10), then by Itô's formula in L 2 we have Hence, Then, taking the expectations, The maximal monotonicity of r → ψ(r) yields the result.  Proof. As concerns existence, we now prove that the sequence of solutions to eq. (10), (X λ ) λ>0 , is a Cauchy sequence in L 2 (Ω; L ∞ ([0, T ]; H −1 (O))) as λ → 0. Consider X λ and X µ , with λ, µ > 0, then by Itô's formula in H −1 we get which can be written as We shall now provide some estimates of the right-hand side terms in (11). Therefore we divide the remaining part of the proof in different steps.
Consider two solution X and Y of eq. (3), then, by Itô's formula in H −1 and exploiting the estimates in the existence proof, we have Now, taking expectation and recalling the monotonicity of ψ, we obtain Hence, by (8) we have were first used to describe the dynamics of the flow in a porous medium, see, e.g., [20,23]. Indeed, the standard model of diffusion of a gas through a porous media is that where ψ(r) = ρ |r| m−1 r, for every r ∈ R, with ρ > 0 and m > 1, which is the so-called slow diffusion model. More generally, one can consider the case of a continuous monotone function satisfying ρ |r| m+1 ≤ rψ(r) ≤ a 1 |r| q+1 + a 2 r, for every r ∈ R, for q > m > 1, and ρ, a 1 > 0. The case m ∈ ]0, 1[, which we are concerned here, is that of the fast diffusion model. This model is relevant in the description of plasma physics, the kinetic theory of gas or fluid transportation in porous media, as suggested in, e.g., [13,14].
The reader is referred to [26,27] for a complete treatment of porous media equations.
A general feature of the fast diffusion case is that it models diffusion processes with a fast speed of mass transportation and this is one reason why the process terminates within finite time with positive probability. This is, in fact, what we are going to show in this section for the Stratonovich gradient noise case. The result has been proved for the case of linear multiplicative noise, see [8,10]. The approach used in the following is the same as the ones used in those works and in [11,Ch. 3.7].
So, from now on, we will focus on the fast diffusion model and we will work under the following conditions. The meaning of assumption (17) will be clear after the following lemma, which gives us an estimate on X λ (t) 1−m −1 , where X λ is the solution to the approximating problem (10) for λ > 0.
Lemma 5.2. Suppose Assumption 5.1 holds. Then there exists K m > 0 such that, for every λ > 0 and 0 ≤ r ≤ t, Proof. In order to get the estimate on X λ (t) 1−m −1 we start estimating φ ε (X λ (t)), where, for any ε > 0, Notice that By Itô's formula we have which can be rewritten as 1+m and estimating the terms on the right-hand side as we did in the proof of Theorem 4.1, we get ∞ , then integrating with respect to time from r to t, we have Taking ε → 0 yields Set now K m = C 2 (1 − m)/2. By the stochastic Gronwall's lemma we have Recalling that C 1 > 0 because of (8), we get (18).
For the next step, we need an additional assumption concerning d and m. In particular, we would like to have a constant C m ≥ 0 such that which is equivalent to have , with continuous embedding.
By duality, this is equivalent to , with continuous embedding, hence, by Sobolev embedding Theorem, we have that (19) holds provided m m + 1 which explains the meaning of hypothesis (17).
We are now ready to prove our extinction in finite time result.
Proof. By (18), taking into account (19), we have for every t ≥ r ≥ 0 Let λ → 0 to find which can be equivalently written as Recalling that K m = C 2 (1 − m)/2 and defining Y (t) . = e −C2t/2 X(t), this proves that the process Y (t) This implies, for any couple of stopping times τ 1 and τ 2 , that and, in particular, for any t > τ x = inf{t > 0 : X(t) = 0}, we have Now set r = 0 in (21) and take the expectation to get which gives and (20) follows. 6. Self-organized criticality model. In this section we will introduce the selforganized criticality (SOC) model, which is a special case of porous media equation with ψ(r) = ρ sign r + φ(r), for all r ∈ R, where ρ > 0, φ a maximal monotone graph in R × R and {r ∈ R; |r| ≤ 1}, for r = 0.
Such a choice of ψ represents the so-called sand-pile model or Bak-Tang-Wiesenfeld model. The deterministic version of the sand-pile model was first introduced in [1, 2], while its stochastic counterpart has been studied, e.g., in [5,9,12]. In the following we formalize the deterministic model referring to the method treated in [3]. Let O be an N × N discrete region of points, we label each of those points with an integer index i ∈ {1, . . . , N 2 }. We associate a height, X i (t), to every index i at a certain time t. Now, we can select, randomly, a site i and increase X i (t) by 1, leaving the other sites unchanged. A toppling event occurs if the height at a site exceed of a given critical value X c . A site whose height is greater than X c is called activated site.
If X i (t) > X c , then where Consider X(t) = (X i (t)) i , then, the dynamic of the system can be written, starting from equation (22), as where f (t) = (f i (t)) i = H(X i (t) − X c ) i and H is the Heaviside function defined as Noticing that the matrix Z is a discretized version of the Laplace operator ∆, we can claim that equation (23) is the discrete version of the following partial differential equation for X : ]0, +∞[ × O → R, where O ⊂ R 2 is a continuous spatial domain, ∆ is the 2-dimensional Laplace operator and H is the Heaviside function. More generally, one can consider O ⊂ R d , d = 1, 2, 3, and replace H by a continuous function with jump at 0. One has to associate to equation (24) an initial value condition X(0, ξ) = X 0 (ξ), ξ ∈ O, with X 0 : O → R representing the initial configuration of the system, and boundary conditions on ∂O, a common one being the Dirichlet condition X(t, ξ) = 0, (t, ξ) ∈ ]0, +∞[ × ∂O.
We want now to treat eq. (3) in the particular case of self-organized criticality. If we consider ψ(r) = ρ sign(r), r ∈ R, where ρ > 0, then Theorem 4.1 still applies, since ψ satisfies the hypotheses therein. Theorem 5.3 holds with the same proof in the case m = 0, which is exactly the case of self-organized criticality, since ψ(r) = ρ |r| m−1 r = ρ |r| m sign(r).
7. Concluding remarks. Theorem 4.1 provides a nice existence result for eq. (2), however, as we pointed out in Section 2, hypothesis (8) is quite restrictive, since it imposes in particular that ν > 0 to avoid the loss of the noise term. It could be interesting to see if it is possible to gain existence even with ν = 0, but keeping the noise.
As regards the asymptotic behaviour of solutions, Theorem 5.3 ensures extinction in finite time of the solution to the fast diffusion model, while for the SOC model, in Section 6, we only have the result for d = 1. One may wonder what happens in the case d ≥ 2, does extinction in finite time phenomenon still take place? Some asymptotic results for the case of SOC in stochastic porous media equations of the type dX − ∆ψ(X) dt = σ(X) dW, have been provided by V. Barbu, G. Da Prato, and M. Röckner in [11,Ch. 3.8] as well as B. Gess in [18], the latter guaranteeing, under some suitable assumptions, the extinction in finite time of solutions also for d > 1. However, in the case of Stratonovich gradient noise, what happens for d > 1 is still to be proved, up to our knowledge, and it could be the next step to be tackled in future works.