STOCHASTIC POROUS MEDIA EQUATIONS WITH DIVERGENCE ITˆO NOISE

. We study the existence and uniqueness of solution to stochastic porous media equations with divergence Itˆo noise in inﬁnite dimensions. The ﬁrst result prove existence of a stochastic strong solution and it is essentially based on the non-local character of the noise. The second result proves exis- tence of at least one martingale solution for the critical case corresponding to the Dirac distribution.


1.
Introduction. We are concerned in the present work with the following stochastic porous media equation div (µ k X (t)) e k dβ k (t) , (0, T ) × O, where O is a bounded open domain in R d , d ≤ 3, with smooth boundary ∂O and the initial datum x is from H −1 (O). We assume that {e k } k∈N is the orthonormal basis in L 2 (O) of eigenfunctions of the homogeneous Dirichlet Laplace operator −∆. We denote by {λ k } k the corresponding eigenvalues −∆e k = λ k e k , k ∈ N.
Through all the paper the sequence {µ k } k∈N is assumed to be such that where λ k are the eigenvalues of the Laplace operator with homogeneous Dirichlet boundary conditions.

IOANA CIOTIR
The sequence {β k } k∈N is formed of mutually independent Brownian motions on a filtered probability space (Ω, F, (F t ) t , P) such that is a cylindrical Wiener process in L 2 (O) .
The constants {µ k } k∈N are assumed to be from R d , i.e. µ k = µ 1 k , µ 2 k , ..., µ d k and the operator Ψ is maximal monotone. We recall that a function Ψ is said to be maximal monotone, i.e. (v 1 − v 2 ) (u 1 − u 2 ) ≥ 0 for all v i ∈ Ψ (u i ) , i = 1, 2, and the range R (I + Ψ) of I + Ψ is all R. A standard example is Ψ (r) = a |r| m−1 r − br where m ≥ 1 and a > 0, b ≥ 0. Through all the paper we shall denote by C a positive constant independent of the approximations, that may change in the chains of estimates.
Note that, for dW (t) = ∞ i=1 e i dβ i (t) ∈ L 2 (O) we have that B (X (t)) dW (t) = ∞ k=1 div (µ k X (t)) e k dβ k (t) . Now we can easily check that B is well defined from L 2 (O) into the Hilbert-Schmidt space L 2 L 2 (O) ; H −1 (O) , i.e., for any u ∈ L 2 (O) we have where C is a constant. Indeed, since d ≤ 3, by the Sobolev embedding, it follows that and we get by elementary computations that (see [8], [9]). This leads to the fact that We compute Going back to (5) we get via assumption (2) that , and we obtain (3).

State of the art
We can easily see that the general existence theory mentioned below is not applicable in the present situation.
First of all, the result from [22] can not be applied in the present case since the equation is not considered in a Gelfand triple and since we don't have the assumption A 2 from [22]. Indeed, the operator B defined above is not Lipschitz Recently, the stochastic porous media equation was studied with different assumptions for the drift, with additive and multiplicative noise. See e.g. [4], [5], [6], [8], [9], [12], [24].
More precisely, the general existence theory is concerned with a stochastic porous media equation, with Itô multiplicative noise in infinite dimensions, as follows is linear and Lipschitz continuous from Recently in [7] the cases of σ : were also studied, but they are not covering the present case.
A case of porous media equation with divergence-type noise is studied in a result from [3], but only for finite dimensions and for a Stratonovich type noise. See also [15], [19] and [25].
With respect to the situations considered above, in the present work we assume an Itô multiplicative noise of divergence type, in infinite dimensions. To the best of our knowledge, this case was never studied before. One can also easily see that it is not covered by the previews situations since the noise is Itô-type in infinite dimensions, but not Lipschitz from H −1 (O) into L 2 L 2 (O) ; H −1 (O) or in the cases covered by [7].

Organization of the paper
The present paper is organized as follows.
After an introduction we have a first section which is concerned with the study of existence and uniqueness of a distributional solution for a stochastic porous media equation with non-local divergence Itô noise of the form where f in an L 1 (O) function. This case can be seen as an intermediary step in the study of equation (1). In fact the function f can be seen as a regular distribution and if we take the Dirac distribution instead of f we have the singular equation (1).
The second section is concerned with the study equation (1). More precisely we shall prove the existence of at least one martingale solution of this equation.

2.
The case with non-local noise. We are concerned in this section with the following stochastic porous media equation where function f is assumed to be from L 1 (O) and x ∈ H −1 (O). We shall assume in this section that, in addition to (2), the following hypotheses are satisfied.

Hypotheses
i) The operator Ψ : R → R is a continuous, differentiable monotonically increasing function on R, which satisfies the following conditions where C i > 0, ∀i ∈ {1, 2, 3, 4} and m ≥ 1. The constant C 4 is assumed to be sufficiently large. ii) The operator Ψ : R → R is strongly monotone, i.e.
where the constant C 5 > 0 is also assumed to be sufficiently large.

Remark 2.
The assumption that C 4 and C 5 are supposed to be sufficiently large is necessary from the technical point of view to compensate the noise. The same result can be obtained if we replace this condition by C 0 sufficiently small.
As in the introduction, we can rewrite equation (6) as where the operator A is defined as previously and is defined by As in the general case, we have that and we can easily check that B f is well defined from L 2 (O) into the Hilbert-Schmidt space L 2 L 2 (O) ; H −1 (O) , i.e., for any u ∈ L 2 (O) we have Indeed, since Keeping in mind that f ∈ L 1 (O) and going back to (8) we get that We can easily see that the general existence theory mentioned before is not applicable in the present case neither. Indeed the operator B f defined above is not Lipschitz from H −1 (O) into L 2 L 2 (O) ; H −1 (O) and the result from [22] can not be applied in the present case, also since the equation is not considered in a Gelfand triple.
We shall prove now existence and uniqueness of the solution for equation (6) in the following sense.
for m as in assumption i) and such that for all j ∈ N, where {e j } j is the orthonormal basis considered above, and for all This type of solution is inspired from [18] and [22] and was already used several times in the study of the stochastic porous media equations. See [8], [9], [14].
Note that this solution is a strong one from the stochastic point of view and a weak one from the point of view of partial differential equations.
We can now formulate the main result of this section.
Theorem 2.2. Assume that (2) and that Hypotheses 1 hold. Then, for each to equation (1) in the sense of Definition 2.1.

Proof. Existence of the solution
The main idea which shall be used in this proof is the approximation of the operator B by using a mollifier, as follows.
We shall first consider a density ρ ∈ We can now define It is well known, by classical theory, that f ε converges strongly in L 1 R d to f for ε → 0 and therefore We shall approximate the operator B as follows: We can now check that the approximating equation has a unique solution in the sense of the definition above.
Since the drift satisfies already the necessary conditions, it is sufficient to check Indeed, we have that B ε f is linear and also that We compute Moreover we have that By replacing (13) in (12) and the result in (11) we get that , by using the assumption (2). Note that the constant C (ε) depends on ε and changes form line to line. We can apply Theorem 2.2 from [8] or the more recent existence result from Chapter 3 of [7], for each ε fixed.
We shall now pass to the limit in for ε → 0. By using an idea similar to the one from Proposition 3.2.1 from [7] we can prove a Itô-type formula for the squared H −1 (O) norm of a solution of equation (10).
More precisely, for any j ∈ N we note first that (X ε (t) , e j ) −1 is an Itô's process and that Then, by applying Itô's formula as detailed in Proposition 3.2.1 from [7] and by taking the expectation, we get directly that From assumption (2) we have that which is used in the previous relation as follows We study the last term of (16) and we get by using (4) and the assumption (2) that where C is a constant independent of ε by (9). By going back to (16) we obtain that where C 4 is assumed to be sufficiently large and in our case this means C 4 − C > 0. Consequently, we can easily see that, via the hypothesis (2), we have

IOANA CIOTIR
We obtain then the existence of and weak* in We shall now study the strong convergence of the approximating solution. To this purpose we shall argue as in (15) for the difference of two approximating solutions X ε and X λ for ε > 0 and λ > 0. We get that Since the operator Ψ is strongly maximal monotone, we get that Now, we only have to study the term from the right-hand side. By using the properties of the operator B f we get that where the constants C and C are independents of ε.
We shall replace now (18) and (19) in (17) and, since the constant C 5 is also assumed to be sufficiently large, we get that is a positive constant independent of ε and λ. Finally, since {f ε } ε is strongly converging to f in L 1 (O) and E t 0 X λ (s) 2 2 ds is bounded uniformly in λ, we get that and strongly in C [0, T ] ; L 2 Ω; H −1 (O) .
In order to pass to the limit in (14) we still have to study what happens in the last term of this relation.
More precisely, keeping in mind that is well defined, we can first see that Since ∞ k=1 t 0 (e j , div (µ k f * X (s)) e k ) −1 dβ k (s) = t 0 (e j , B (X (s)) dW (s)) −1 we get, by using the Itô isometry for stochastic integrals with cylindrical Wiener processes, that See e.g. [17], Proposition 2.3.5 from [23] or Remark 6.3.2 from [7]. On the other hand we compute Going back to (21) and replacing the computation above, we get that which goes to zero for ε → 0.
We finally obtain that, on a subsequence, we have that We can now pass to the limit in (14) and get that In order to finish the proof we only need to show that η = Ψ (X) a.e. on Ω × (0, T ) × O. Since the operator X −→ Ψ (X) is maximal monotone in the duality pair L m+1 (Ω × (0, To prove (23) we shall use the same argument as in [9]. We first note that Computing as in (19) in the last term of the previous relation and using that

POROUS MEDIA EQUATION WITH DIV ITÔ NOISE 387
On the other hand, via Itô's formula applied to (22) and summation over j, we obtain that Combining (24) and (25) we get (23) and this completes the proof of the existence part.

Uniqueness of the solution
Concerning the uniqueness of the solution, it is sufficient to take two solutions X (1) and X (2) with the same starting point, and, by repeating the argument above, we obtain Since Ψ is strongly monotone and the constant C 5 is assumed to be sufficiently large, we obtain that X (1) = X (2) and the proof is complete.
3. The critical case. In this section we shall prove existence of at least one martingale solution for equation (1). Hypothesis i) The operator Ψ : R → R is a C 1 , monotonically increasing function on R, which satisfies the following conditions where C i > 0, ∀i ∈ {1, 2, 3, 4, 6, 7} and m > 1. The constant C 4 is assumed to be sufficiently large. ii) The operator Ψ : R → R is strongly monotone, i.e.
where the constant C 5 > 0 is also assumed to be sufficiently large.
for all j ∈ N, where {e j } j is the orthonormal basis considered above, and for all The martingale solution is a weak solution from the PDE and also from the stochastic point of view. For more details see [17] and see [11] for a similar approach.
The main result of this section is the following. , P, W , X to equation (1). Moreover, we have that Proof. In order to approximate the equation with a mollifier as in the previous case, we shall first rewrite the operator B as where δ is the Dirac function and keeping in mind that δ * u = u. By taking a mollifier sequence {δ ε } ε which is defined as in the previous section, we can approximate the operator B as follows We can easily check now that the approximating equation has a unique solution X ε . Indeed, by arguing as in the previous section we have that B ε is Lipschitz from −1 , and then we have a solution which satisfies P − a.s.
∀j ∈ N and ∀t ∈ [0, T ]. Note that by using Remark 3.1.4 from [7], the previous relation can be equivalently written as By Itô's formula we obtain P − a.s. that where is a continuous local martingale such that and (B ε (X ε (s))) * is the adjoint of We can first easily check that where C is independent of ε. Indeed, by arguing as in the first part of this work, we have that By the Burkholder-Davis-Gundy inequality, we get for all r ∈ [0, T ] that where C is also independent of ε.
Keeping in mind that B ε (X ε ) is a Hilbert-Schmidt operator and therefore By replacing the previous relation in(28) we get that for ε > 0, where C is a positive constant independent of ε. Then, on a subsequence again denoted in the same way, we have the existence of and weak* in Since the weak convergences above are not sufficient to conclude the proof, we shall replace {X ε } by a sequence X ε of processes defined in a probability space Ω, F, P, W such that X ε and X ε have the same law.
To this purpose we consider the sequence of probability measures {ν ε } ε , where ν ε is the law of X ε , and we prove that {ν ε } ε is tight in the space C [0, T ] ; H −1 (O) . We recall that this means that, for each δ > 0 there is a compact subset B of We define for each r > 0 and γ > 0, the set Since the set is uniformly bounded and satisfies a Hölder condition of order 1/2 with a fixed constant γ, we have by Arzelà-Ascoli theorem that B r,γ is compact in We can apply the Itô formula to equation (27) with the Lyapunov function In fact we apply the Itô formula with where J ν = (Id − ν∆) −1 is the resolvent of the Laplace operator, and we get that Keeping in mind that the resolvent of the Laplace operator is strongly convergent in L 2 (Ω × (0, T ) × O) (see e.g. [2]) we can pass to the limit for ν → 0 as in [4], [9], [13] or [14].
We obtain and then, by using the strong monotonicity property of Ψ, we get that From the previous relation we get also that We shall apply Itô's formula to (27) with the Lyapunov function Finally, by applying again the Itô formula to the process we get that We get by (29) that We obtain that . Finally, by using the Tchebychev inequality we can conclude that for each δ there exist γ and r, independent of ε such that and therefore {ν ε } ε>0 is tight.
Then, by the Skorohod theorem, we have a probability space Ω, F, P and the stochastic process X and X ε ε>0 on Ω, F, P such that the law of X ε is the same as the law of X ε and as ε −→ 0. We have also that the law of X is the same as the law of X.
we can use the Egorov theorem to get that Indeed, we have for ∀ δ > 0 a subset A δ of Ω such that P Ω\A δ < δ and We see by using the Hölder inequality and since m > 1 that On the other hand, since we have the uniform convergence on A δ , we get that for each δ there is ε > 0 such that We conclude that Since the law of X ε is the same as the law of X ε and keeping in mind that We shall show now that, for each ε fixed, the process is a square integrable martingale with respect to where B ε X ε (s) * is the adjoint of B ε X ε (s) . All this is true for the process Since X ε and X ε have the same law, we have the previous properties also for M ε . More precisely we get first that E M ε (t) We have also that E M ε (t) , a We shall now check that we can take the limit as ε −→ 0 in the previous relations and then we will get that the process We can easily pass to the limit in T 2 by (30) and get that On the other hand, for T 1 we have that Since ∇ (δ ε * ϕ) = δ ε * ∇ϕ −→ ∇ϕ for ε −→ 0 and keeping in mind that E By the representation theorem (see e.g. [17] Theorem 8.2) we have the existence of a probability space Ω, F, P , a filtration F a Wiener process W and a predictable continuous process X such that The proof of the existence of at least one martingale solution is now complete.