Renormalized solutions for stochastic p-Laplace equations with L^1-initial data: The multiplicative case
aa r X i v : . [ m a t h . A P ] M a r Renormalized solutions for stochastic p -Laplace equations with L -initial data:The multiplicative case Niklas Sapountzoglou ∗ Aleksandra Zimmermann † We consider a p -Laplace evolution problem with multiplicative noise on abounded domain D ⊂ R d with homogeneous Dirichlet boundary conditionsfor < p < ∞ . The random initial data is merely integrable. Consequently,the key estimates are available with respect to truncations of the solution. Weintroduce the notion of renormalized solutions for multiplicative stochastic p -Laplace equations with L -initial data and study existence and uniqueness ofsolutions in this framework. In the theory of stochastic partial differential equations (SPDEs), square integrabilityof the initial data is a rather technical assumption. For square integrable initial data u , the stochastic p -Laplace evolution problems can be solved with classical methods fornonlinear, monotone SPDEs (see, e.g. [20], [11], [19] and [8] for systems). In applications,one often has flawed or irregular data and therefore it may be reasonable to study moregeneral, merely integrable random initial conditions. From the results of [6], [3] and [7]it is well known that the deterministic p -Laplace evolution equation with L -data is notwell-posed in the variational setting for < p < d , where d ∈ N is the space dimension. Inthis case the equation can be addressed within the framework of renormalized solutions.The notion of renormalization summarizes different strategies to get rid of infinities (see[10]) that may appear in physical models. It has been introduced to partial differentialequations by Di Perna and Lions in the study of Boltzmann equation (see [12]) and then ∗ Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str.9, 45127 Essen, Germany [email protected] † Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str.9, 45127 Essen, Germany [email protected] v = S ( u ) ,where the nonlinear function S is chosen in order to remove infinite quantities of thesolution u . This strategy has been applied for stochastic transport equations in [1], [9]and for the Boltzmann equation with stochastic kinetic transport in [22].In the study of singular SPDEs such as the Kardar-Parisi-Zhang (KPZ) equation,the idea of renormalization has recently been revisited (see [16], [17] and the referencestherein). For this type of equations, renormalized solutions are obtained as limits of so-lutions to regularized problems with addition of diverging correction terms. These termsarise from a renormalization group which is defined in terms of an associated regularitystructure.Thanks to the new techniques developed in the theories of regularity structures and roughpaths a huge progess in the study of singular SPDEs has been achieved in the last decade.However, the classical L -theory for nonlinear SPDEs of p -Laplace type is an indepen-dent topic which has been recently addressed in [23], where the notion of renormalizedsolutions in the sense of [3] has extended to stochastic p -Laplace evolution equations. Ina subsequent contribution (see, e.g., [24]) existence and uniqueness of renormalized solu-tions to the stochastic p -Laplace evolution problem with random initial data in L (Ω × D ) has been shown in the case of an additive stochastic perturbation, i.e., with an Itô inte-gral R t Φ dβ on the right-hand side of the equation, where Φ is a progressively measurableand square integrable stochastic process. In this contribution, we are interested in well-posedness of the stochastic p -Laplace equation in the multiplicative case, i.e., for an Itôintegral R t Φ( u ) dβ on the right-hand side of the equation, where, roughly speaking, Φ( u ) is a Lipschitz function of the solution u .In the classical L -theory, with the solution of the additive problem and an appropriatecontraction principle at hand, existence of solutions to the corresponding multiplicativeproblem can be written in a few lines applying a fixed point argument.In the multiplicative case with L initial data, the situation changes dramatically dueto new phenomena. The main difficulty is the combination of L -spatial regularity of u with additional terms entering the renormalized formulation from the Itô correction andthe non-cancellation of stochastic integrals in differences of solutions. In our study, weput the spotlight on the new techniques developed for the multiplicative case and referto [24] for known results in order to avoid doubling of arguments. Let (Ω , F , P, ( F t ) t ∈ [0 ,T ] , ( β t ) t ∈ [0 ,T ] ) be a stochastic basis with a countably generatedprobability space (Ω , F , P ) , a filtration ( F t ) t ∈ [0 ,T ] ⊂ F satisfying the usual assumptionsand a real valued, F t -Brownian motion ( β t ) t ∈ [0 ,T ] . Let D ⊂ R d be a bounded Lipschitzdomain, T > , Q T = (0 , T ) × D and p > . Furthermore, let u : Ω → L ( D ) be F -measurable and Φ : Ω × [0 , T ] × R → R a function satisfying the following properties: ( A
1) Φ( ω, t,
0) = 0 for almost every ( ω, t ) ∈ Ω × [0 , T ] ,2 A λ Φ( ω, t, λ ) is Lipschitz-continuous for almost every ( ω, t ) ∈ Ω × [0 , T ] , i.e. thereexists L > such that for all λ , λ ∈ R | Φ( ω, t, λ ) − Φ( ω, t, λ ) | ≤ L | λ − λ | for almost every ( ω, t ) ∈ Ω × [0 , T ] . ( A
3) ( ω, t ) Φ( ω, t, λ ) is progressively measurable for all λ ∈ R .We are interested in well-posedness to the following stochastic p -Laplace evolution prob-lem du − div ( |∇ u | p − ∇ u ) dt = Φ( u ) dβ in Ω × Q T ,u = 0 on Ω × (0 , T ) × ∂D, (1) u (0 , · ) = u ∈ L (Ω × D ) . In Section 2, we will briefly discuss the well-posedness of (1) for square integrable initialdata. In Section 3 we will formulate and prove a contraction principle for strong solutionsthat will serve as a basis for the proof of the existence theorem (see Theorem 6.1) inSection 6. After some technical preliminaries in Section 4, we formulate the notion ofrenormalized solutions for (1) in Section 5. The uniqueness of renormalized solutions informulated in Theorem 7.1 and contained in Section 7 together with its proof.
Theorem 2.1.
Let u ∈ L (Ω × D ) be F -measurable. Then there exists a unique strongsolution to (1), i.e., an F t -adapted stochastic process u : Ω × [0 , T ] → L ( D ) such that u ∈ L p (Ω; L p (0 , T ; W ,p ( D ))) ∩ L (Ω; C ([0 , T ]; L ( D ))) , u (0 , · ) = u in L (Ω × D ) and u ( t ) − u − Z t div ( |∇ u | p − ∇ u ) ds = Z t Φ( u ) dβ in W − ,p ′ ( D ) + L ( D ) for all t ∈ [0 , T ] and a.s. in Ω . Remark 2.2.
A-priori, we do not know that the term R t div ( |∇ u | p − ∇ u ) ds is an ele-ment of L ( D ) for all t ∈ [0 , T ] and a.s. in Ω , but since we know that all other termsin the equation of Theorem 2.1 are elements of L ( D ) for all t ∈ [0 , T ] and a.s. in Ω itfollows that R t div ( |∇ u | p − ∇ u ) ds ∈ L ( D ) for all t ∈ [0 , T ] and a.s. in Ω . Thereforethis equation is an equation in L ( D ) .Proof. Similar as in [24], the existence result is a consequence of [18], Chapter II, Theorem2.1 and Corollary 2.1. The uniqueness result is a consequence of [18], Chapter II, Theorem3.1. 3
Contraction principle
Theorem 3.1.
Let u , v ∈ L (Ω × D ) and u and v strong solutions to the problem (1) with initial value u and v , respectively. Then sup t ∈ [0 ,T ] E Z D | u ( t ) − v ( t ) | dx ≤ E Z D | u − v | dx a.s. in Ω .Proof. We subtract the equations for u and v and we get u ( t ) − v ( t ) = u − v + Z t div ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) ds + Z t Φ( u ) − Φ( v ) dβ (2)for all t ∈ [0 , T ] and a.s. in Ω . Now, for every δ > , we apply the Itô formula pointwisea.s. with respect to x ∈ D with a coercive approximation of the absolute value N δ in (2)which is defined as in Proposition 5 in [26] (see also [24]) and we get Z D N δ ( u ( t ) − v ( t )) dx = Z D N δ ( u − v ) dx + Z t h div( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) , N ′ δ ( u − v ) i ds + Z t Z D (Φ( u ) − Φ( v )) N ′ δ ( u − v ) dx dβ + 12 Z t Z D N ′′ δ ( u − v )(Φ( u ) − Φ( v )) dx ds. Applying the expectation and discarding the nonpositive term coming from the p -Laplaceyields E Z D N δ ( u ( t ) − v ( t )) dx ≤ E Z D N δ ( u − v ) dx + 12 E Z t Z D N ′′ δ ( u − v )(Φ( u ) − Φ( v )) dx ds ≤ E Z D N δ ( u − v ) dx + 1 δ E Z t Z D χ {| u − v |≤ δ } (Φ( u ) − Φ( v )) dx ds ≤ E Z D N δ ( u − v ) dx + 1 δ E Z t Z D χ {| u − v |≤ δ } L δ dx ds ≤ E Z D N δ ( u − v ) dx + δL T | D | . Now, passing to the limit with δ → + yields E Z D | u ( t ) − v ( t ) | dx ≤ E Z D | u − v | dx for all t ∈ [0 , T ] . 4 emark 3.2. In [24], the integrand of the stochastic integral in the equation for u − v isequal to since we consider the additive case there. Therefore, in this case, the integrandof the stochastic integral in the equation for R D N δ ( u ( t ) − v ( t )) dx is equal to as well.But in the multiplicative case the stochastic integral does not vanish and does not tendto uniformly in t ∈ [0 , T ] , a.s. in Ω for δ → + . Hence, in contrast to [24], we haveto apply the expectation before applying the supremum over t ∈ [0 , T ] which leads to aweaker comparison principle as in [24]. For < p < ∞ let p ′ bet the conjugate exponent to p . Let us recall that the spaces L ( D ) and W − ,p ′ ( D ) are continuously embedded into the space of distributions D ′ ( D ) .Moreover, by density of the test functions in L ( D ) and in L p ′ ( D ) , it follows that L ( D ) ∩ W − ,p ′ ( D ) is dense in L ( D ) and W − ,p ′ ( D ) . Therefore, the sum space L ( D ) + W − ,p ′ ( D ) = { w = u + v | u ∈ L ( D ) , v ∈ W − ,p ′ ( D ) } is well defined and a Banach space with respect to the norm |k w k| := inf {k u k L + k v k W − ,p ′ | u ∈ L ( D ) , v ∈ W − ,p ′ ( D ) , u + v = w } , see, e.g., [21], p.23. Moreover, the dual space is given by ( L ( D ) + W − ,p ′ ( D )) ′ = W ,p ( D ) ∩ L ∞ ( D ) . For a Banach space V with dual space V ′ , a function u : [0 , T ] → V is called weaklycontinuous, iff the function [0 , T ] ∋ t
7→ h u ( t ) , v i V,V ′ is continuous for all v ∈ V ′ . The locally convex space of weakly continuous functionswith values in V will be denoted by C w ([0 , T ]; V ) in the sequel. For further details on theproperties these spaces we refer to [15], pp.120-126. In particular, the following resultholds true Lemma 4.1 (see [25], Lemma 1.4, p.263) . Let X and Y be Banach spaces such that X ⊂ Y with a continuous injection. If a function φ belongs to L ∞ (0 , T ; X ) and is weaklycontinuous with values in Y , then φ is weakly continuous with values in X . Proposition 4.1.
Let G ∈ L p ′ (Ω × Q T ) d , g ∈ L (Ω × Q T ) , f ∈ L (Ω × Q T ) beprogressively measurable, u ∈ L (Ω × D ) be F -measurable. Define the continuous, ( D ) + W − ,p ′ ( D ) -valued process u by the equality u ( t ) − u + Z t ( − div G + f ) ds = Z t g dβ (3) for all t ∈ [0 , T ] and a.s. in Ω . If for its dP ⊗ dt -equivalence class we have u ∈ L p (Ω; L p (0 , T ; W ,p ( D ))) and ess sup t ∈ [0 ,T ] E k u ( t ) k L < ∞ , then, for all ψ ∈ C ∞ ([0 , T ] × D ) and all S ∈ W , ∞ ( R ) with S ′′ piecewise continuous such that S ′ (0) = 0 or ψ ( t, x ) = 0 for all ( t, x ) ∈ [0 , T ] × ∂D , we have ( S ( u ( t )) , ψ ( t )) − ( S ( u ) , ψ (0)) + Z t h− div G + f, S ′ ( u ) ψ i ds = Z t ( S ′ ( u ) g, ψ ) dβ + Z t ( S ( u ) , ψ t ) ds + 12 Z t Z D S ′′ ( u ) g ψ dx ds (4) for all t ∈ [0 , T ] and a.s. in Ω , where h− div G + f, S ′ ( u ) ψ i = h− div G + f, S ′ ( u ) ψ i W − ,p ′ ( D )+ L ( D ) ,W ,p ( D ) ∩ L ∞ ( D ) = Z D ( G · ∇ [ S ′ ( u ) ψ ] + f S ′ ( u ) ψ ) dx a.e. in Ω × (0 , T ) . Remark 4.2.
From Lemma 1.4 in [25] we know that L ∞ (0 , T ; L ( D )) ∩ C w ([0 , T ]; L ( D ) + W − ,p ′ ( D )) ⊂ C w ([0 , T ]; L ( D )) . Hence, in Proposition 4.1 we have u ∈ C w ([0 , T ]; L ( D )) a.s. in Ω . Therefore, u ( t ) ∈ L ( D ) makes sense for all t ∈ [0 , T ] , a.s. in Ω .Moreover, since the right-hand side of (3) is an element of L ( D ) for all t ∈ [0 , T ] , evenif the members on the left-hand are not in L ( D ) , (3) holds also in L ( D ) .Proof. See [24], Lemma 4.1. In the statement of this lemma u requires to be an elementof L (Ω; C ([0 , T ]; L ( D ))) . But in the proof of this lemma it is only necessary that u ( t ) ∈ L ( D ) makes sense for all t ∈ [0 , T ] , a.s. in Ω and this is the case because ofRemark 4.2. Let us assume that there exists a strong solution u to (1) in the sense of Theorem 2.1.We observe that for initial data u merely in L , the Itô formula for the square of thenorm (see, e.g., [20]) can not be applied and consequently the natural a priori estimatefor ∇ u in L p (Ω × Q T ) d is not available. Choosing g = Φ( u ) , f ≡ , ψ ≡ and S ( u ) = Z u T k ( r ) dr
6n (4), where T k : R → R is the truncation function at level k > defined by T k ( r ) = ( r , | r | ≤ k,k sign( r ) , | r | > k, we find that there exists a constant C ( k ) ≥ depending on the truncation level k > ,such that E Z T Z D |∇ T k ( u ) | p dx ds ≤ C ( k ) . As in the deterministic case, the notion of renormalized solutions takes this informationinto account :
Definition 5.1.
Let u ∈ L (Ω × D ) be F -measurable. A progressively measurableprocess u : Ω × (0 , T ) → L ( D ) such that u ∈ L (Ω × Q T ) is a renormalized solution to(1) with initial value u , iff(i) ess sup t ∈ (0 ,T ) E k u ( t ) k L < + ∞ and T k ( u ) ∈ L p (Ω; L p (0 , T ; W ,p ( D ))) for all k > ,(ii) For all ψ ∈ C ∞ ([0 , T ] × ¯ D ) and all S ∈ C ( R ) such that S ′ has compact supportwith S ′ (0) = 0 or ψ ( t, x ) = 0 for all ( t, x ) ∈ [0 , T ] × ∂D the equality Z D S ( u ( t )) ψ ( t ) − S ( u ) ψ (0) dx + Z t Z D S ′′ ( u ) |∇ u | p ψ dx ds + Z t Z D S ′ ( u ) |∇ u | p − ∇ u · ∇ ψ dx ds = Z t Z D S ′ ( u ) ψ Φ( u ) dx dβ + Z t Z D S ( u ) ψ t dx ds + 12 Z t Z D S ′′ ( u ) ψ Φ( u ) dx ds (5) holds true a.s. in Ω × (0 , T ) .(iii) The following energy dissipation condition holds true: lim k →∞ E Z { k< | u | For u as in Definition 5.1 such that ( i ) is satisfied, ( ii ) implies that forany S ∈ C ( R ) such that S ′ has compact support with S (0) = 0 there exists a version of S ( u ) with paths in C ([0 , T ]; L ( D ) + W − ,p ′ ( D )) and this version satisfies S ( u ( t )) − S ( u (0)) − Z t div ( S ′ ( u ) |∇ u | p − ∇ u ) ds + Z t S ′′ ( u )[ |∇ u | p − 12 Φ( u ) ] ds = Z t Φ( u ) S ′ ( u ) dβ, (6)7 r equivalently, in differential form, dS ( u ) − div ( S ′ ( u ) |∇ u | p − ∇ u ) dt + S ′′ ( u )[ |∇ u | p − 12 Φ( u ) ] dt = Φ( u ) S ′ ( u ) dβ (7) in W − ,p ′ ( D ) + L ( D ) a.s. in Ω for any t ∈ [0 , T ] . Since the right-hand side of (7) isin L ( D ) , the equation also holds in L ( D ) . From Definition 5.1 it follows that S ( u ) isbounded and therefore by Remark 4.2 it follows that S ( u ) ∈ C w ([0 , T ]; L ( D )) . Remark 5.3. If u is a renormalized solution to (1) , thanks to (7) , the Itô formula fromProposition 4.1 still holds true for S ( u ) for any S ∈ W , ∞ ( R ) with supp( S ′ ) compactsuch that S ( u ) ∈ W ,p ( D ) a.s. in Ω × (0 , T ) . Indeed, in this case (3) is satisfied for theprogressively measurable functions ˜ u = S ( u ) ∈ L p (Ω; L p (0 , T ; W ,p ( D ))) ∩ L (Ω; C ([0 , T ]; L ( D ) + W − ,p ′ ( D ))) ,G = S ′ ( u ) |∇ u | p − ∇ u ∈ L p ′ (Ω × Q T ) d ,f = S ′′ ( u )[ |∇ u | p − 12 Φ( u ) ∈ L (Ω × Q T ) ,g = Φ( u ) S ′ ( u ) ∈ L (Ω × Q T ) . Remark 5.4. Let u be a renormalized solution to (1) with ∇ u ∈ L p (Ω × Q T ) d . For fixed l > , let h l : R → R be defined by h l ( r ) = , | r | ≥ l + 1 l + 1 − | r | , l < | r | < l + 11 , | r | ≤ l. Taking S ( u ) = R u h l ( r ) dr as a test function in (5) , we may pass to the limit with l → ∞ and we find that u is a strong solution to (1) (see also [24]). In this Section, we fix u ∈ L (Ω × D ) F -measurable. Let ( u n ) n ⊂ L (Ω × D ) be an F -measurable sequence such that | u n | ≤ | u | for all n ∈ N and lim n →∞ u n = u in L (Ω × D ) and in L ( D ) for a.e. ω ∈ Ω . A possible choice is u n = T n ( u ) , n ∈ N . Theorem 6.1. Let Φ be bounded. Then, there exists a renormalized solution to (1) withinitial datum u . Theorem 6.1 will be proved succesively in the following Lemmas. Lemma 6.2. There exist constants C ( k ) , C ( k, k ′ ) > only depending on k > or k, k ′ > , respectively, such that(i) E R T R D |∇ T k ( u n ) | dx dt ≤ C ( k ) for all k > and all n ∈ N , ii) E R T R D |∇ θ k ′ k ( u n ) | dx dt ≤ C ( k, k ′ ) for all k, k ′ > and all n ∈ N , where θ k ′ k ( r ) := T k + k ′ ( r ) − T k ( r ) for all r ∈ R .Proof. Since u n is a strong solution to (1) with initial value u n , u n satisfies the equality u n ( t ) − u n − Z t div ( |∇ u n | p − ∇ u n ) ds = Z t Φ( u n ) dβ (8)in L ( D ) for all t ∈ [0 , T ] and a.s. in Ω . Applying Proposition 4.1 with S = R · T k ( r ) dr and ψ ≡ and taking the expectation yields E Z D Z u n ( t )0 T k ( r ) drdx + E Z t Z D |∇ T k ( u n ) | p dx ds = 12 E Z t Z D T ′ k ( u n )Φ( u n ) dx ds + E Z D Z u n T k ( r ) dr dx (9)for all k > , all t ∈ [0 , T ] and a.s. in Ω . The first term on the left hand side of (9) isnonnegative. Now, the Lipschitz continuity of Φ and the estimate | u n | ≤ | u | yield E Z T Z D |∇ T k ( u n ) | p dx ds ≤ T L k | D | k k u k L (Ω × D ) =: C ( k ) , where L is the Lipschitz constant of Φ . This proves ( i ) and assertion ( ii ) is a directconsequence of ( i ) . Lemma 6.3. Passing to not relabelded subsequences if necessary, we have the followingconvergence results:(i) There exists a progressively measurable process u : Ω × [0 , T ] → L ( D ) such that u ∈ L (Ω × Q T ) , u n → u in L (Ω × Q T ) and u n ( t ) → u ( t ) in L (Ω × D ) for a.e. t ∈ (0 , T ) and in L ( D ) a.e. in Ω × (0 , T ) .Moreover, ess sup t ∈ (0 ,T ) E k u ( t ) k L < + ∞ .(ii) ∇ T k ( u n ) ⇀ ∇ T k ( u ) in L p (Ω × Q T ) d for all k > ,(iii) ∇ θ k ′ k ( u n ) ⇀ ∇ θ k ′ k ( u ) in L p (Ω × Q T ) d for all k, k ′ > ,(iv) There exists σ k ∈ L p ′ (Ω × Q T ) d such that |∇ T k ( u n ) | p − ∇ T k ( u n ) ⇀ σ k in L p ′ (Ω × Q T ) d satisfying σ k = σ k +1 χ {| u | The function u from Lemma 6.3 satisfies condition ( ii ) from Definition5.1.Proof. Let u n be a strong solution to (1) with initial value u n , i.e., u n ( t ) − u n − Z t div ( |∇ u n | p − ∇ u n ) ds = Z t Φ( u n ) dβ (12)for all t ∈ [0 , T ] and a.s. in Ω . We apply the Itô formula introduced in Proposition 4.1 toequality (12). Therefore we know that for all ψ ∈ C ∞ ([0 , T ] × D ) and all S ∈ W , ∞ ( R ) such that S ′′ is piecewise continuous and S ′ (0) = 0 or ψ ( t, x ) = 0 for all ( t, x ) ∈ [0 , T ] × ∂D the equality Z D S ( u n ( t )) ψ ( t ) − S ( u n ) ψ (0) dx + Z t Z D S ′′ ( u n ) |∇ u n | p ψ dx ds + Z t Z D S ′ ( u n ) |∇ u n | p − ∇ u n · ∇ ψ dx ds (13) = Z t Z D S ′ ( u n ) ψ Φ( u n ) dx dβ + Z t Z D S ( u n ) ψ t dx ds + 12 Z t Z D S ′′ ( u n ) ψ Φ( u n ) dx ds holds true for all t ∈ [0 , T ] and a.s. in Ω . In the following, passing to a suitable, notrelabeled subsequence if necessary, and taking the limit for n → ∞ , we will show that1113) is also satisfied by u and u respectively and therefore ( ii ) from Definition 5.1 holds.To this end, it is left to show that T k ( u n ) → T k ( u ) in L p (Ω; L p (0 , T ; W ,p ( D ))) for all k > . Similar as in [24], the upcoming technical lemmas which are inspired byTheorem 2 and Lemma 2 in [6] will show that this convergence holds. Lemma 6.6. For n ∈ N , let u n be a strong solution to (1) with respect to the initial value u n . Let H and Z be two real valued functions belonging to W , ∞ ( R ) such that H ′′ and Z ′′ are piecewise continuous, H ′ and Z ′ have compact supports and Z (0) = Z ′ (0) = 0 issatisfied. Then lim n,m →∞ E Z T Z D H ′′ ( u n ) Z ( u n − u m ) |∇ u n | p dx dt = 0 . (14) Proof. Using the Itô product rule (see Proposition 8.1) yields Z D Z ( u n ( t ) − u m ( t )) H ( u n ( t )) dx = Z D Z ( u n − u m ) H ( u n ) dx − Z t Z D |∇ u n | p − ∇ u n ∇ (cid:18) Z ( u n − u m ) H ′ ( u n ) (cid:19) dx ds + 12 Z t Z D H ′′ ( u n ) Z ( u n − u m )Φ( u n ) dx ds + Z t Z D H ′ ( u n ) Z ( u n − u m )Φ( u n ) dx dβ − Z t Z D ( |∇ u n | p − ∇ u n − |∇ u m | p − ∇ u m ) ∇ (cid:18) Z ′ ( u n − u m ) H ( u n ) (cid:19) dx ds (15) + 12 Z t Z D H ( u n ) Z ′′ ( u n − u m )(Φ( u n ) − Φ( u m )) dx ds + Z t Z D H ( u n ) Z ′ ( u n − u m )(Φ( u n ) − Φ( u m )) dx dβ + Z t Z D Z ′ ( u n − u m ) H ′ ( u n )(Φ( u n ) − Φ( u m ))Φ( u n ) dx ds for all t ∈ [0 , T ] and a.s. in Ω . Now we set t = T and take the expectation in equality(15). Since Z ′ , Z ′′ , H ′ and H ′′ have compact support and Φ is Lipschitz continuous it iseasy to see that lim n,m →∞ E Z T Z D H ( u n ) Z ′′ ( u n − u m )(Φ( u n ) − Φ( u m )) dx dt = 0 , lim n,m →∞ E Z T Z D Z ′ ( u n − u m ) H ′ ( u n )(Φ( u n ) − Φ( u m ))Φ( u n ) dx dt = 0 lim n,m →∞ E Z T Z D H ′′ ( u n ) Z ( u n − u m )Φ( u n ) dx ds = 0 . From now on the proof is the same as in [24] or in [6], Theorem 2. Lemma 6.7. For n ∈ N , let u n be a strong solution to (1) with respect to the initialvalue u n . Let u be defined as in Lemma 6.3. Then, lim n,m →∞ E Z T Z D (cid:18) |∇ T k ( u n ) | p − ∇ T k ( u n ) − |∇ T k ( u m ) | p − ∇ T k ( u m ) (cid:19) ·· ( ∇ T k ( u n ) − ∇ T k ( u m )) dx ds = 0 . (16) Especially, we have ∇ T k ( u n ) → ∇ T k ( u ) in L p (Ω × Q T ) d and T k ( u n ) → T k ( u ) in L p (Ω; L p (0 , T ; W ,p ( D ))) for n → ∞ and for all k > .Proof. For k > , we set Z T Z D (cid:18) |∇ T k ( u n ) | p − ∇ T k ( u n ) − |∇ T k ( u m ) | p − ∇ T k ( u m ) (cid:19) ·· ( ∇ T k ( u n ) − ∇ T k ( u m )) dx dt = I n,mk + J n,mk + J m,nk , a.s. in Ω , where I n,mk = Z {| u n |≤ k }∩{| u m |≤ k } ( |∇ u n | p − ∇ u n − |∇ u m | p − ∇ u m ) · ∇ ( u n − u m ) dx dt,J n,mk = Z {| u n |≤ k }∩{| u m | >k } |∇ u n | p − ∇ u n · ∇ u n dx dt a.s. in Ω . J m,nk is the same as J n,mk where the roles of n and m are reversed. Thereforethese two terms can be treated simultaneously.Moreover, we set ≤ J n,mk = J n,m ,k,k ′ + J n,m ,k,k ′ , where J n,m ,k,k ′ = Z {| u n |≤ k }∩{| u m | >k }∩{| u n − u m |≤ k ′ } |∇ u n | p − ∇ u n · ∇ u n dx dt,J n,m ,k,k ′ = Z {| u n |≤ k }∩{| u m | >k }∩{| u n − u m | >k ′ } |∇ u n | p − ∇ u n · ∇ u n dx dt k ′ > k > , a.s. in Ω . Since Lemma 6.3 ( iii ) , ( iv ) , ( v ) and ( vi ) hold true, the samearguments as in [24] yield n,m →∞ E I n,mk = lim n,m →∞ E J n,m ,k,k ′ . The proof that lim n,m →∞ E J n,m ,k,k ′ = 0 is slightly different from [24] and therefore we will give the proof of this equality. We useLemma 6.6, (14) with H = H δk for δ, k > such that ( H δk ) ′′ ( r ) = , | r | < k, − kδ, k ≤ | r | ≤ k + δ , , | r | > k + δ and get lim sup n →∞ lim sup m →∞ E Z {| u n |≤ k } Z ( u n − u m ) |∇ u n | p dx dt ≤ δ · k lim sup n →∞ lim sup m →∞ E Z { k ≤| u n |≤ k + δ } Z ( u n − u m ) |∇ u n | p dx dt ≤ δ · k k Z k ∞ lim sup n →∞ E Z { k ≤| u n |≤ k + δ } |∇ u n | p dx dt . Now applying Proposition 4.1 to the equation of u n with S = R · θ δ k =: ˜ θ δ k , ψ ≡ , g = Φ( u n ) and f ≡ and taking the expectation yields E Z D ˜ θ δ k ( u n ( T )) dx + E Z T Z D χ { k ≤| u n |≤ k + δ } |∇ u n | p dx dt = E Z D ˜ θ δ k ( u n ) dx + 12 E Z T Z D χ { k ≤| u n |≤ k + δ } Φ( u n ) dx dt. The first term on the left hand side is nonnegative and the integrand of the second termon the right hand side is bounded since Φ is bounded.Multiplying by δ and passing to the limit with n → ∞ yields δ · lim sup n →∞ E Z T Z D χ { k ≤| u n |≤ k + δ } |∇ u n | p dx dt ≤ E Z D δ ˜ θ δ k ( u ) dx + 12 δ · C for a constant C > . We can estimate that δ ˜ θ δ k ( u ) → a.e. in Ω × D as δ → and | δ ˜ θ δ k ( u ) | ≤ u + ˜ C for a constant ˜ C > . Therefore Lebesgue’s Theorem yields lim δ → lim sup n →∞ lim sup m →∞ δ · E Z { k ≤| u n |≤ k + δ } Z ( u n − u m ) |∇ u n | p dx dt = 0 . lim n,m →∞ E Z {| u n |≤ k } Z ( u n − u m ) |∇ u n | p dx dt = lim sup n →∞ lim sup m →∞ E Z {| u n |≤ k } Z ( u n − u m ) |∇ u n | p dx dt = 0 . Choosing Z such that Z ( r ) = 1 for | r | ≥ k ′ and Z ≥ on R such that Z (0) = Z ′ (0) = 0 ,it follows ≤ lim n,m →∞ E J n,m ,k,k ′ = lim n,m →∞ E Z {| u n |≤ k }∩{| u m | >k }∩{| u n − u m | >k ′ } |∇ u n | p − ∇ u n · ∇ u n dx dt ≤ lim n,m →∞ E Z {| u n |≤ k } Z ( u n − u m ) |∇ u n | p dx dt = 0 , which finally shows the validity of equality (16). Since equality (16) holds true, it followsthat lim n →∞ E Z T Z D |∇ T k ( u n ) | p − ∇ T k ( u n ) · ∇ T k ( u n ) dx dt = E Z T Z D σ k · ∇ T k ( u ) dx dt. (17)Minty’s trick yields σ k = |∇ T k ( u ) | p − ∇ T k ( u ) . We may conclude by using equality (17)that lim n →∞ k∇ T k ( u n ) k pL p (Ω × Q T ) d = k∇ T k ( u ) k pL p (Ω × Q T ) d . Since L p (Ω × Q T ) d is uniformly convex and ∇ T k ( u n ) ⇀ ∇ T k ( u ) in L p (Ω × Q T ) d it yields ∇ T k ( u n ) → ∇ T k ( u ) in L p (Ω × Q T ) d which ends the proof of Lemma 6.7.For the proof of Theorem 6.1 is left to show that the energy dissipation condition ( iii ) from Definition 5.1 holds true. To this end we have to show the following lemma at first. Lemma 6.8. For n ∈ N , let u n be a strong solution to (1) with respect to the initialvalue u n . Let u be defined as in Lemma 6.3. Then, lim k →∞ lim sup n →∞ E Z { k< | u n | For fixed l > , let h l : R → R be defined as in Remark 5.3. We plug S ( r ) = R r h l ( r )( T k +1 ( r ) − T k ( r )) dr and ψ ≡ in (13) and take the expectation to obtain I + I + I = I + I , (19)15here I = E Z D Z u n ( t ) u n h l ( r )( T k +1 ( r ) − T k ( r )) dr dx,I = E Z { l< | u n | Theorem 7.1. Let u, v be renormalized solutions to (1) with initial data u ∈ L (Ω × D ) and v ∈ L (Ω × D ) , respectively. Then, Z D E | u ( t ) − v ( t ) | dx ≤ Z D E | u − v | dx (24) for all t ∈ [0 , T ] .Proof. This proof is inspired by the uniqueness proof in [7]. We know that S ( u ) satisfiesthe SPDE dS ( u ) − div ( S ′ ( u ) |∇ u | p − ∇ u ) dt + S ′′ ( u ) |∇ u | p dt = Φ S ′ ( u ) dβ + 12 S ′′ ( u )Φ ( u ) dt (25)17or all S ∈ C ( R ) such that supp S ′ compact and S (0) = 0 . Moreover, S ( v ) satisfies ananalogous SPDE. Subtracting both equalities yields S ( u ( t )) − S ( v ( t )) = S ( u ) − S ( v ) + Z t div [ S ′ ( u ) |∇ u | p − ∇ u − S ′ ( v ) |∇ v | p − ∇ v ] ds − Z t (cid:0) S ′′ ( u ) |∇ u | p − S ′′ ( v ) |∇ v | p (cid:1) ds + Z t (Φ( u ) S ′ ( u ) − Φ( v ) S ′ ( v )) dβ (26) + 12 Z t (Φ ( u ) S ′′ ( u ) − Φ ( v ) S ′′ ( v )) ds in W − ,p ′ ( D ) + L ( D ) for all t ∈ [0 , T ] , a.s. in Ω .Now we set S ( r ) := T σs ( r ) for r ∈ R and s, σ > and define T σs as follows: Firstly, wedefine for all r ∈ R ( T σs ) ′ ( r ) = , if | r | ≤ s, σ ( s + σ − | r | ) , if s < | r | < s + σ, , if | r | ≥ s + σ. Then we set T σs ( r ) := R r ( T σs ) ′ ( τ ) dτ . Furthermore we have the weak derivative ( T σs ) ′′ ( r ) = ( − σ sign( r ) , if s < | r | < s + σ, , otherwise . Applying the Itô formula (see 4.1) to equality (26) with S ( r ) = k ˜ T k ( r ) = k R r T k ( r ) dr and ψ ≡ yields Z D (cid:18) k ˜ T k ( T σs ( u ( t )) − T σs ( v ( t ))) − k ˜ T k ( T σs ( u ) − T σs ( v )) (cid:19) dx − Z t h div (( T σs ) ′ ( u ) |∇ u | p − ∇ u − ( T σs ) ′ ( v ) |∇ v | p − ∇ v ) , k T k ( T σs ( u ) − T σs ( v )) i dr = Z D Z t ( − (( T σs ) ′′ ( u ) |∇ u | p − ( T σs ) ′′ ( v ) |∇ v | p ) · k T k ( T σs ( u ) − T σs ( v )) dr dx + Z D Z t (Φ( u )( T σs ) ′ ( u ) − Φ( v )( T σs ) ′ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dβ dx + 12 Z D Z t (Φ ( u )( T σs ) ′′ ( u ) − Φ ( v )( T σs ) ′′ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dr dx + 12 Z D Z t (Φ( u )( T σs ) ′ ( u ) − Φ( v )( T σs ) ′ ( v )) · k χ {| T σs ( u ) − T σs ( v ) | 18e want to pass to the limit with σ → firstly, then we pass to the limit k → andfinally we let s → ∞ . We may repeat the arguments used in [24], proof of Theorem 7.1for the expressions I , I and I to pass to the limit in I σ,k,s , I σ,k,s and I σ,k,s For ω ∈ Ω and t ∈ [0 , T ] fixed. More precisely, from [24], p.21 it follows that lim s →∞ lim k → lim σ → I = Z D | u ( t ) − v ( t ) | − | u − v | dx (28)a.s. in Ω , for all t ∈ [0 , T ] . Then, repeating the arguments from [24], p.21 we get lim inf σ → I σ,k,s ≥ . (29)With the same arguments as on p.22-p.25 in [24] it follows that lim j →∞ lim sup k → lim sup σ → | I σ,k,s j | = 0 , (30)passing to a suitable subsequence ( s j ) j ∈ N with lim j →∞ s j = + ∞ if necessary.In the next steps, we will address I σ,k,s , I σ,k,s and I σ,k,s . Therefore we recall that forany fixed s > , T σs ( r ) → T s ( r ) and ( T σs ) ′ ( r ) → χ {| r |≤ s } pointwise for all r ∈ R as σ → .Since | ( T σs ) ′ | ≤ and | T σs )( r ) | ≤ | r | on R we have ( T σs ) ′ ( u ) → χ {| u |≤ s } in L (Ω × Q T ) and a.e. in Ω × Q T as σ → . An analogous result holds true for v instead of u . Moreover, k T k ( r ) → sign( r ) for k → in R , where sign is the classical(single-valued) sign function. In addition, | k T k ( r ) | ≤ for all k > and all r ∈ R .Now, we write I σ,k,s = I σ,k,s , + I σ,k,s , where I σ,k,s , = Z D Z t Φ( u )(( T σs ) ′ ( u ) − ( T σs ) ′ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dβ dx,I σ,k,s , = Z D Z t ( T σs ) ′ ( v )(Φ( u ) − Φ( v )) · k T k ( T σs ( u ) − T σs ( v )) dβ dx Again, proceeding as in [24], p.25 for the term I and using the boundedness of Φ , itfollows that lim s →∞ lim k → lim σ → I σ,k,s , = 0 a.s. in Ω for all t ∈ [0 , T ] . From the Itô isometry and Lebesgue’s dominated convergencetheorem it follows that lim σ → E (cid:12)(cid:12)(cid:12)(cid:12) I σ,k,s , − Z D Z t ( T s ) ′ ( v )(Φ( u ) − Φ( v )) · k T k ( T s ( u ) − T s ( v )) dβ dx (cid:12)(cid:12)(cid:12)(cid:12) =lim σ → k E Z D Z t (cid:12)(cid:12) [Φ( u ) − Φ( v )]( T σs ) ′ ( v ) T k ( T σs ( u ) − T σs ( v )) − ( T s ) ′ ( v ) T k ( T s ( u ) − T s ( v )) (cid:12)(cid:12) dxdt = 0 , lim σ → I σ,k,s , = 1 k Z D Z t [Φ( u ) − Φ( v )]( T s ) ′ ( v ) 1 k T k ( T s ( u ) − T s ( v )) dβ dx in L (Ω) and, passing to a not relabeled subsequence if necessary, also a.s. in Ω . andsimilarly we obtain lim k → Z D Z t [Φ( u ) − Φ( v )]( T s ) ′ ( v ) 1 k T k ( T s ( u ) − T s ( v )) dβ dx = Z D Z t [Φ( u ) − Φ( v )]( T s ) ′ ( v ) sign( T s ( u ) − T s ( v )) dβ dx (31)in L (Ω) and, passing to a not relabeled subsequence if necessary, also a.s. in Ω . Passingto the limit with s → ∞ in (31), we get lim s →∞ lim k → lim σ → I σ,k,s = Z D Z t [Φ( u ) − Φ( v )] sign( u − v ) dβ dx (32)for all t ∈ [0 , T ] , a.s. in Ω . Now, we write I σ,k,s = 12 Z D Z t (Φ ( u )( T σs ) ′′ ( u ) − Φ ( v )( T σs ) ′′ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dr dx + 12 Z D Z t Φ ( v )(( T σs ) ′′ ( u ) − ( T σs ) ′′ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dr dx Z D Z t ( T σs ) ′′ ( u )(Φ ( u ) − Φ ( v )) · k T k ( T σs ( u ) − T σs ( v )) dr dx := I σ,k,s , + I σ,k,s , Using exactly the same arguments as in [24], p.25 for the expression I , we get that lim sup σ → I σ,k,s , ≤ . (33)In the following we show that there exists a subsequence ( s j ) j ∈ N ⊂ N with lim j →∞ s j =+ ∞ such that lim j →∞ lim sup k → lim sup σ → I σ,k,s j , = 0 (34)We have, for any k ∈ N , | I σ,k,s , | ≤ E Z D Z t | ( T σs ) ′′ ( u ) || Φ ( u ) − Φ ( v ) | · k | T k ( T σs ( u ) − T σs ( v )) | dr dx ≤ δ Z { s< | u | Proposition 8.1. For < p < ∞ , u , v ∈ L (Ω × D ) F -measurable let u, v ∈ L p (Ω × (0 , T ); W ,p ( D )) ∩ L (Ω; C ([0 , T ]; L ( D ))) satisfy u ( t ) = u + Z t ∆ p ( u ) ds + Z t Φ( u ) dβ, (38) v ( t ) = v + Z t ∆ p ( v ) ds + Z t Φ( v ) dβ. (39) Then, for any H, Z ∈ C b ( R ) such that Z (0) = Z ′ (0) = 0( Z (( u − v )( t )) , H ( u ( t ))) = ( Z ( u − v ) , H ( u )) + Z t h ∆ p ( u ) − ∆ p ( v ) , H ( u ) Z ′ ( u − v ) i W − ,p ′ ( D ) ,W ,p ( D ) ds + Z t h ∆ p ( u ) , H ′ ( u ) Z ( u − v ) i W − ,p ′ ( D ) ,W ,p ( D ) ds + Z t (Φ( u ) H ′ ( u ) , Z ( u − v )) dβ + 12 Z t Z D Φ ( u ) H ′′ ( u ) Z ( u − v ) dx ds + 12 Z t Z D (Φ( u ) − Φ( v )) Z ′′ ( u − v ) H ( u ) ds + Z t h Φ( u ) − Φ( v ) , Z ′ ( u − v ) H ( u ) i dβ + Z t (Φ( u ) − Φ( v )) Z ′ ( u − v )Φ( u ) H ′ ( u ) ds (40) for all t ∈ [0 , T ] , a.s. in Ω .Proof. We fix t ∈ [0 , T ] . Since u , v satisfy (39) and (38), it follows that ( u − v )( t ) = u − v + Z t ∆ p ( u ) − ∆ p ( v ) ds + Z t (Φ( u ) − Φ( v )) dβ (41)holds in L ( D ) , a.s. in Ω . For n ∈ N , we use the following classical regularizationprocedure (see, e.g., [14]):We choose a sequence of operators (Π n ) , Π n : W − ,p ′ ( D ) + L ( D ) → W ,p ( D ) ∩ L ∞ ( D ) , n ∈ N such that i. ) Π n ( v ) ∈ W ,p ( D ) ∩ C ∞ ( D ) for all v ∈ W − ,p ′ ( D ) + L ( D ) and all n ∈ N i. ) For any n ∈ N and any Banach space F ∈ { W ,p ( D ) , L ( D ) , L ( D ) , W − ,p ′ ( D ) , W − ,p ′ ( D ) + L ( D ) } . Π n : F → F is a bounded linear operator such that lim n →∞ Π n | F = I F pointwisein F , where I F is the identity on F .Now, we set Φ u,n := Π n (Φ( u )) , Φ v,n := Π n (Φ( v )) , u n := Π n ( u ) , v n := Π n ( v ) , u n :=Π n ( u ) , v n := Π n ( v ) , U n := Π n (∆ p ( u )) , V n := Π n (∆ p ( v )) . Applying Π n on both sides of(41) yields ( u n − v n )( t ) = u n − v n + Z t U n − V n ds + Z t (Φ u,n − Φ v,n ) dβ (42)and applying Π n on both sides of (38) yields u n ( t ) = u n + Z t U n ds + Z t Φ u,n dβ (43)in W ,p ( D ) ∩ L ( D ) ∩ C ∞ ( D ) a.s. in Ω . The pointwise Itô formula in (42) and (43) leadsto Z ( u n − v n )( t ) = Z ( u n − v n )+ Z t ( U n − V n ) Z ′ ( u n − v n ) ds Z t (Φ u,n − Φ v,n ) Z ′ ( u n − v n ) dβ + 12 Z t (Φ u,n − Φ v,n ) Z ′′ ( u n − v n ) ds (44)and H ( u n )( t ) = H ( u n ) + Z t U n H ′ ( u n ) ds + Z t Φ n H ′ ( u n ) dβ + 12 Z t Φ u,n H ′′ ( u n ) ds (45)in D , a.s. in Ω . From (44), (45) and the product rule for Itô processes, which is just aneasy application of the classic two-dimensional Itô formula (see, e.g., [2], Proposition 8.1,p. 218), applied pointwise in t for fixed x ∈ D it follows that Z ( u n − v n )( t ) H ( u n )( t ) = Z ( u n − v n ) H ( u n )+ Z t ( U n − V n ) Z ′ ( u n − v n ) H ( u n ) ds + Z t U n H ′ ( u n ) Z ( u n − v n ) ds + Z t Φ u,n H ′ ( u n ) Z ( u n − v n ) dβ + 12 Z t Φ u,n H ′′ ( u n ) Z ( u n − v n ) ds + Z t (Φ u,n − Φ v,n ) Z ′ ( u n − v n ) H ( u n ) dβ + 12 Z t (Φ u,n − Φ v,n ) Z ′′ ( u n − v n ) H ( u n ) ds + Z t (Φ u,n − Φ v,n ) Z ′ ( u n − v n )Φ u,n H ′ ( u n ) ds (46)23n D , a.s. in Ω . Integration over D in (46) yields I = I + I + I + I + I + I + I + I , where I = ( Z (( u n − v n )( t )) , H (( u n )( t )) ,I = ( Z ( u n − v n ) , H ( u n )) ,I = Z t Z D ( U n − V n ) Z ′ ( u n − v n ) H ( u n ) dx ds,I = Z t Z D U n H ′ ( u n ) Z ( u n − v n ) dx ds,I = Z t (Φ u,n H ′ ( u n ) , Z ( u n − v n )) dβ,I = 12 Z t Z D Φ u,n H ′′ ( u n ) Z ( u n − v n ) dx dsI = Z t ((Φ u,n − Φ v,n ) Z ′ ( u n − v n ) , H ( u n )) dβI = 12 Z t Z D (Φ u,n − Φ v,n ) Z ′′ ( u n − v n ) H ( u n ) dx dsI = Z t ((Φ u,n − Φ v,n ) Z ′ ( u n − v n ) , Φ u,n H ′ ( u n )) ds a.s. in Ω . Repeating the arguments from [24], proof of Proposition 9.1, we show that,passing to a not relabeled subsequence if necessary, lim n →∞ I = ( Z (( u − v )( t )) , H ′ ( u ( t )) , (47) lim n →∞ I = ( Z ( u − v ) , H ′ ( u )) , (48) lim n →∞ I = Z t h ∆ p ( u ) − ∆ p ( v ) , Z ′ ( u − v ) H ( u ) i W − ,p ′ ( D ) ,W ,p ( D ) ds, (49) lim n →∞ I = Z t h ∆ p ( u ) , H ′ ( u ) Z ( u − v ) i W − ,p ′ ( D ) ,W ,p ( D ) ds, (50) lim n →∞ I = Z t Z D Φ( u ) H ′ ( u ) Z ( u − v ) dx dβ, (51) lim n →∞ I = 12 Z t Z D Φ( u ) H ′′ ( u ) Z ( u − v ) dx ds (52)24.s. in Ω . Since (Π n ) n is a sequence of linear operators on L ( D ) converging pointwiseto the identity for n → ∞ , from the Uniform Boundedness Principle and Lebesguesdominated convergence theorem it follows that Φ u,n → Φ( u ) and Φ v,n → Φ( v ) in L (0 , T ; L ( D )) and in L (Ω; L (0 , T ; L ( D )) for n → ∞ . Using the Itô isometry andpassing to a not relabeled subsequence if necessary, it follows that lim n →∞ I = Z t (Φ( u ) − Φ( v )) Z ′ ( u − v ) , H ( u )) dβ, (53) lim n →∞ I = 12 Z t Z D (Φ( u ) − Φ( v )) Z ′′ ( u − v ) H ( u ) dx ds, (54) lim n →∞ I = Z t (Φ( u ) − Φ( v ) Z ′ ( u − v ) , Φ( u ) H ′ ( u )) ds. (55)Now, the assertion follows from (47)-(55). Corollary 8.2. Proposition 8.1 still holds true for H, Z ∈ W , ∞ ( R ) such that H ′′ and Z ′′ are piecewise continuous.Proof. There exists sequence ( H δ ) δ> , ( Z δ ) δ> ⊂ C b ( R ) such that k H δ k ∞ ≤ k H k ∞ , k H ′ δ k ∞ ≤ k H ′ k ∞ , k H ′′ δ k ∞ ≤ k H ′′ k ∞ for all δ > and H δ → H , H ′ δ → H ′ uniformly oncompact subsets, H ′′ δ → H ′′ pointwise in R for δ → and the same results hold true for ( Z δ ) δ> . With these convergence results we are able to pass to the limit with δ → in(40). References [1] Attanasio, S., Flandoli, F.: Renormalized solutions for stochastic transport equationsand the regularization by bilinear multiplication noise. Comm. Partial DifferentialEquations 36 (2011), no. 8, 1455-1474.[2] Baldi, P.: Stochastic Calculus. An Introduction Through Theory and Exercises. Uni-versitext, Springer, 2017.[3] Blanchard, D., Murat, F.: Renormalised solutions of nonlinear parabolic problemswith L data: Existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect. A 127 (6)(1997), 1137-1152[4] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.: An L -theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annalidella Scuola Normale Superiore di Pisa. Classe di scienze 22, 2 (1995), 241-273.[5] Blanchard, D., Redwane, H.: Renormalized solutions for a class of nonlinear evolutionproblems. J. Math. Pures Appl. (9) 77 (1998), no. 2, 117-151.256] Blanchard, D.: Truncations and monotonicity methods for parabolic equations. Non-linear Analysis, Theory, Methods & Applications. 21 (1993), no. 10, 725-743.[7] Blanchard, D., Murat, F., Redwane, H.: Existence and Uniqueness of a Renormal-ized Solution of a Fairly General Class of Nonlinear Parabolic Problems. Journal ofDifferential Equations. 177 (2001), 331-374.[8] Breit, D.: Regularity theory for nonlinear systems of SPDEs. Manuscripta Math. 146(2015), no. 3-4, 329-349.[9] Catuogno P., Olivera C.: L p -solutions of the stochastic transport equation. RandomOper. Stoch. Equ. 21 (2013), 125-134.[10] Delamotte, B.: A hint of renormalization. Am. J. Phys. 72(2), 170-184 (2004).[11] Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. 2. Edition,Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cam-bridge, 1992.[12] DiPerna R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: Globalexistence and weak stability. Ann. Math. 130 (1989), 321-366.[13] Feireisl, E.: Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford LectureSeries in Mathematics and its Applications. Oxford University Press, Oxford 2004.[14] Fellah, D., Pardoux, E.: Une formule d’Itô dans des espaces de Banach, et applica-tion. In: Körezlioˇglu H., Üstünel A.S. (eds) Stochastic Analysis and Related Topics.Progress in Probability, vol. 31. Birkhäuser, Boston, MA, 1992.[15] Gajewski, H., Gröger K., Zacharias K.: Nichtlinear Operatorgleichungen und Oper-atordifferentialgleichungen. Akademie-Verlag, Berlin, 1974.[16] Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singularPDEs. Forum Math. Pi 3 (2015), e6, 75 pp.[17] Hairer, M.: A theory of regularity structures. Invent. Math. 198 (2014), no. 2, 269-504.[18] Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. J. Soviet Math. 16(1981), no. 4, 1233-1277.[19] Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction.Universitext, Springer, 2015.[20] Pardoux, E.: Equations aux dérivées partielles stochastiques non linéaires mono-tones. University of Paris, 1975. PhD-thesis.[21] Farwig, R., Kozono, H., Sohr, H.: An L q -approach to Stokes and Navier-Stokesequations in general domains. Acta Math. 195 (2005), 21-53.2622] Punshon-Smith S., Smith, S.: On the Boltzmann equation with stochastic kinetictransport: global existence of renormalized martingale solutions. Arch. Rational Mech.Anal. 229 (2018), 627-708.[23] Sapountzoglou, N., Zimmermann, A.: Renormalized solutions for a stochastic p -Laplace equation with L initial data. Proceedings of the Fifteenth InternationalConference Zaragoza-Pau on Mathematics and its Applications, Monogr. Mat. GarcíaGaldeano 42, (2020).[24] Sapountzoglou, N., Zimmermann, A.: Well-posedness of renormalized solutions for astochastic p -Laplace equation with L1s } χ {| v |≤ s } dr dx. It is not very hard to see that the first term on the right-hand side of above equationvanishes for k → a.s. in Ω . Concerning the second term, we remark that for ω ∈ Ω fixed, we have lim sup k → k Z D Z t (( T s ) ′ ( v )) (Φ( u ) − Φ( v )) χ {| T s ( u ) − T s ( v ) |≤ k } χ {| u | >s } χ {| v |≤ s } dr dx ≤ lim sup k → k Z { s − k ≤| v |≤ s } k Φ k ∞ dr dx. Consequently, from from Lemma 6 in [7] it follows that there exists a subsequence ( s j ) j ∈ N ⊂ N with lim j →∞ s j = + ∞ such that lim sup j → lim sup k → lim sup σ → I σ,k,s , ≤ . (36)From (28) - (36) it follows that Z D | u ( t ) − v ( t ) | dx ≤ Z D | u − v | dx + Z D Z t [Φ( u ) − Φ( v )] sign( u − v ) dβ dx (37)a.s. in Ω , for all t ∈ [0 , T ] . Taking the expectation in (37), the assertion follows.21 Appendix: The Itô product rule