On the equivalence of pathwise mild and weak solutions for quasilinear SPDEs

The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.


Introduction
The aim of this paper is to relate two solution concepts for quasilinear SPDEs, namely weak and pathwise mild, with a particular emphasis on cross-diffusion systems. Such systems arise in numerous applications, for example, they can be used to describe the dynamics of interacting population species. A well-known model is the deterministic Shigesada-Kawasaki-Teramoto population system, which was introduced in [39] in order to analyze population segregation between two species by induced cross-diffusion. This system can be formally derived from a random-walk model on lattices for transition rates which depend linearly on the population densities. Generalized population cross-diffusion models are obtained when the dependence of the transition rates on the densities is nonlinear, see for instance [43].
In order to model population densities for n ≥ 2 species, we consider cross-diffusion systems of the form (1.1) du = div (B(u)∇u) dt + σ(u) dW t , t > 0, on an open, bounded domain O ⊂ R d (d ≥ 1), with smooth boundary ∂O. Here B = (B ij ) is an n × n diffusion matrix, σ is a nonlinear term and W = (W 1 , . . . , W n ) is a cylindrical Wiener process. The precise assumptions on the coefficients will be stated in Section 2 and 3.
The previous system rewrites componentwise as (1.2) du i − div In order to investigate mild solutions for (1.1), we write it as an abstract quasilinear Cauchy problem where the linear operator A u is given by A u v := div(B(u)∇v).
Another solution concept, so-called pathwise mild solution, for semilinear parabolic SPDEs with nonautonomous random generators was introduced by Pronk and Veraar in [36], where they bypassed the issue of non adaptive integrand in the definition of the Itô integral by the use of integration by parts. This solution concept was then extended to the case of quasilinear parabolic SPDEs, including stochastic SKT system, by Kuehn and the last named author [27] where they proved the existence of a unique local-in-time pathwise mild solution for the equations of the form (1.5). A stochastic process u is called a pathwise mild solution of (1.5) if (1.6) u(t) = U u (t, 0)u 0 − t 0 U u (t, s)A u (s) t s σ(u(τ ))dW τ ds + U u (t, 0) t 0 σ(u(s))dW s , where U u (·, ·), is the random evolution family generated by A u , see Section 2 for further details. This formula can be motivated using integration by parts and overcomes the nonadaptedness of the random evolution family U u (·, ·) required in order to define the stochastic convolution as an Itô integral. Pathwise mild solutions for hyperbolic SPDEs with additive noise were analyzed in [30].
The non-adaptedness of the integrand in the definition of a stochastic integral was firstly discussed by Alós, León and Nualart in [3,28] using the Skorokhod-and the Russo-Vallois [38] forward integral. Similar to [36], in [28] such problems arise for semilinear SPDEs with random, nonautonomous generators. Furthermore, in [3,28] it was shown that a Skorokhodmild solution for such SPDEs does not satisfy the weak formulation, whereas the forward mild (based on the Russo-Vallois integral) does. In [36], the authors showed that the pathwise mild solution is equivalent to the forward mild one.
The equivalence results for weak, pathwise-and forward-mild solutions for semilinear SPDEs obtained in [36] and the existence of pathwise mild solutions for quasilinear SPDEs obtained in [27], raise a natural question regarding the equivalence of these solution concepts in the quasilinear case. Such an aspect is also important to study from a numerical point of view, since numerical schemes are aligned to the solution concept at hand. The same holds true for dynamical systems. In fact, many results regarding dynamics and asymptotic behavior of semilinear SPDEs, rely on a semigroup approach. If we take the SKT system as a motivation, then there are deterministic results regarding the existence of attractors using weak [34] as well as mild [41] solution concepts.
We emphasize that for semilinear PDEs and SPDEs numerous results regarding the equivalence of weak and mild solutions are well-known, see e.g. [5,6] and [7] for PDEs and e.g. [11] and [40] for SPDEs. On the other hand, for quasilinear PDEs and SPDEs the literature discussing the equivalence of various solution concepts is very scarce. Therefore, we contribute to this aspect and establish the equivalence between pathwise mild and weak solutions for quasilinear SPDEs of the form (1.2), see Theorem 3.4 and Theorem 3.5. Results regarding the existence of strong solutions for quasilinear PDEs are available in [4], respectively for SPDEs in [24]. For certain elliptic-parabolic PDEs, using accretive operators and nonlinear semigroups, assertions regarding the equivalence of weak and mild solutions have been derived in [22,32] and the references specified therein. For nonlinear degenerate problems, the concept of entropy solution was introduced by Carrillo [10]. In [25], the equivalence between weak and entropy solutions was established for an elliptic-parabolic-hyperbolic degenerate PDE. However, to the best of our knowledge there are no other works in the literature discussing the equivalence of various solution concepts for quasilinear SPDEs such as (1.5).
The outline of our paper is as follows. In Section 2, we introduce basic notations and collect results from the theory of evolution families generated by nonautonomous, random sectorial operators. These are necessary in order to introduce the concept of a pathwise mild solution for (1.5). Section 3 contains our main result, which establishes under suitable assumptions on the coefficients, the equivalence of pathwise mild and weak solutions for (1.5). The main idea is to approach the quasilinear SPDE (1.5) as a semilinear SPDE where the nonlinear map A u is viewed as a linear nonautonomous map for a fixed u, which is the pathwise mild solution of (1.5). Thereafter we employ similar tools as in [36,27] to prove the equivalence of weak and pathwise mild solutions for (1.5). Finally, we provide in Section 4 examples of quasilinear SPDEs, to which the theory developed in this paper applies. These include the stochastic SKT system.

Preliminaries
Let T > 0 be arbitrary but fixed and (Ω, F, (F) t∈[0,T ] , P) be a filtered stochastic basis. Let X ( · X , ·, · X ), Y ( · Y , ·, · Y ) and Z ( · Z , ·, · Z ) be separable Hilbert spaces such that the embeddings Z ֒→ Y ֒→ X where {β n (·)} n∈N are mutually independent real valued standard Brownian motions and {e n } n∈N denotes an orthonormal basis of X and the sequence (2.1) converges in H P-a.s. The space of Hilbert-Schmidt operators from H to X will be denoted by L 2 (H; X) and will be endowed with the norm We recall some auxiliary results related to the regularity of the stochastic integral with respect to a cylindrical Wiener process.
Recalling (1.5), we write Therefore, A v is a linear time-dependent random operator. In order to highlight this dependence we use the notation A v (t, ω) := A v(t,ω) . To simplify the notation, we drop the parameter ω and simply write A v (t). We now collect essential results regarding evolution families. These are extracted from [36]. For further details regarding evolution systems for nonautonomous operators, we refer to the monographs by Pazy [33] and Yagi [42] as well as to [2].
In order to deal with the time and ω-dependence of A v we impose, as in [36], that the Acquistapace-Terreni conditions hold for every ω ∈ Ω. These were introduced in [2] for time-dependent generators and involve a sectoriality condition on A v together with a suitable Hölder-regularity. Since we are dealing with nonlinear generators we additionally impose a certain Lipschitz continuity assumption [42,27].  (A1) A v is a sectorial operator on X, i.e., there exists a ϑ ∈ ( π 2 , π), such that for every (A2) The resolvent operator (λId − A v ) −1 satisfies the Hile-Yosida condition, i.e., there exists a constant M ≥ 1 such that for every (t, ω) ∈ [0, T ] × Ω, (A3) There exist two exponents ν, δ ∈ (0, 1] with ν + δ > 1 such that for every ω ∈ Ω there exists a constant L(ω) ≥ 0 such that for all s, t ∈ [0, T ] Then, for every ω ∈ Ω there exists a constant L(ω) > 0 such that The conditions (A1)-(A3) will be referred to as the (AT) conditions.
Since we aim to relate mild and weak solutions for (1.5), we impose similar assumptions on the adjoint A * v defined on X * of the operator A v with parameters ϑ * , M * , ν * and δ * . Recall that X is a Hilbert space and we identified it with its dual X * . Assumption 2. (Assumptions adjoint operators) (A1 * ) A * v is a sectorial operator, i.e., there exists a ϑ * ∈ ( π 2 , π), such that for all (t, ω) (A3 * ) There exist two exponents ν * , δ * ∈ (0, 1] with ν * + δ * > 1 such that for every ω ∈ Ω there exists a constant L * (ω) ≥ 0 such that for all s, t ∈ [0, T ] (A4 * ) Let 0 < ν * ≤ 1 be fixed. Then, for every ω ∈ Ω there exists a constant L * (ω) > 0 such that We refer the reader to Appendix A for details on the fractional power of the operator A u and its adjoint.
(1) We assume that the random variables L(ω), L * (ω) are uniformly bounded with respect to ω. This assumption can be dropped by a suitable localization argument, see [36,Section 5.3].
(2) The assumption (A1) implies that −A v is a sectorial operator. Alternatively, one can assume as in [42], that the spectrum of A v (t, ω) is contained in an open sectorial domain with angle 0 < ϕ < π 2 for every t ∈ [0, T ] and ω ∈ Ω, i.e. σ(A v (t, ω)) ⊂ Σ ϕ := {λ ∈ C : | arg λ| < ϕ}, which would imply that A v is a sectorial operator.  Definition 2.1. To emphasize the fact that we are working with constant domains, we introduce the following notations for t ∈ [0, T ] and ω ∈ Ω : In general, conditions (A1) and (A2) can be difficult to verify for a given system, but in the Hilbert space framework, it suffices to apply the following criterion. According to [42, Chapter 2.1], we can associate to −A v a bilinear form a(v; ·, ·) on a separable Hilbert space V which is densely and continuously embedded in X. More precisely, we set In this case, in order to verify (A1) and (A2) it suffices to show that [42, Chapter 2.
The assumptions (A1)-(A3) allow us to apply pathwise the deterministic results from [2] for the generation of an evolution family for nonautonomous operators and obtain as in [36, Theorem 2.2] the following statement.
(T5) for every s < t it holds pointwise in Ω that Moreover, there exists a mapping C : Ω → R + such that In [36,Prop. 2.4] the following measurability result for the evolution family U v was established. This fact prevents us from defining the stochastic convolution as an Itô-integral.
Proposition 2.4. The evolution system U v : ∆ × Ω → L(X) is strongly measurable in the uniform operator topology. Moreover, for each t ≥ s, the mapping ω → U v (t, s, ω) ∈ L(X) is strongly F t -measurable in the uniform operator topology.
In the following, we point out spatial-and time-regularity results of the evolution family U v , cf. [36,Lemma 2.6]. For the convenience of the reader, we provide a brief overview on fractional powers of sectorial operators in Appendix A.
Remark 2.4. Lemma 2.5 and Lemma 2.6 remain valid if the roles of A v (t) and (A v (t)) * are interchanged. This is due to the fact that we identified X with its dual X * .
For the sake of completeness, we point out the following statement on the adjoint of an evolution family U v . Regarding Assumption 2 we conclude by Theorem 2.3 that for ev- Based on Lemma 2.6, we motivate a very useful identity resembling the fundamental theorem of calculus. This will be extensively used in Section 3. For further details we refer to [36].
v , the following identity holds true for every ω ∈ Ω and x ∈ X: where ·, · denotes the inner product in X.
Sketch of the proof. Since A v (t, ω) satisfies (A1)-(A3), by Theorem 2.3 there exists an evolution family U v : ∆ × Ω → L(X). In particular, (2.7) holds. Assume x ∈ D v , then integrating (2.7) with respect to time leads to Testing the above identity by x * ∈ X * , we obtain Next we verify that (2.16) holds for x * ∈ D * v and x ∈ X. This is justified by (2.13), which implies that for every s < t, and fixed v, the operator , for λ > δ * and γ < δ * , can be uniquely extended to a bounded linear operator on X. The claim follows regarding that where we used (2.13) in the last step.

The main result
In this section we introduce two solution concepts: pathwise mild and weak, for the quasilinear SPDE We are interested in showing the equivalence of these solution concepts. As already mentioned, Pronk and Veraar introduced in [36] the concept of pathwise mild solution for semilinear SPDEs with random nonautonomous generators, which was later extended in [27] to quasilinear problems. The precise definition of the pathwise mild solution is given below.
where U u is the evolution family generated by the operator A u .
} is a maximal local pathwise mild solution of (3.1), then the stopping time τ is called its lifetime.
In [27], the existence of a maximal local pathwise mild solution was established under additional regularity assumptions on the nonlinear term σ. These were necessary for the fixed point argument. For the convenience of the reader, we recall these assumptions.
Theorem 3.1. Let Assumption 4 hold true. Then, there exists a unique maximal local pathwise mild solution u for (3.1) such that u ∈ L 0 Ω; We now give the definition of a weak solution of (3.1). Definition 3.3. We call an (F t ) t≥0 -adapted Z-valued process u a weak solution of (3.1), if the following identity holds P-a.s.
Remark 3.1. Note that the test functions ϕ are allowed to depend on time and on ω. We do not make any assumption regarding the adaptedness of these test functions, but only on their spatial and temporal regularity.
Remark 3.2. In comparison to the standard weak formulation of (3.1), the weak formulation (3.3) contains two additional terms The term L 1 appears quite naturally when we test The key idea. We describe the intuition behind the approach we use in order to prove the equivalence of pathwise mild and weak solutions for (3.1). Such an idea is standard in the context of quasilinear problems. Similar to the proof of existence of solutions [42,27], we firstly work with nonautonomous equations with random coefficients. More precisely, instead of treating (3.1) we consider the linear equation where A v u = div(B(v)∇u) as specified in (2.2). This is a linear nonautonomous Cauchy problem, which due to [36], has a unique pathwise mild solution Its weak formulation reads as for every test function ϕ ∈ L 0 Ω; C 1 ([0, T ]; D * u )) . According to [36,Section 4.4], the weak solution (3.7) is equivalent with the pathwise mild solution (3.6). Returning to the quasilinear problem (3.1), one can show by means of fixed point arguments as in [27], that under Assumption 4 is the pathwise mild solution of (3.1). In conclusion, we approach the quasilinear SPDE (3.1) as a semilinear SPDE where the nonlinear map A u is viewed as a linear nonautonomous map for a fixed u, which is the pathwise mild solution of the quasilinear SPDE (3.1). In this case, the tools developed in [36,Section 4.4] can be employed in order to establish the equivalence of weak (3.3) and pathwise mild (3.2) solutions in the quasilinear case.
In order to prove our main result, we make the following assumptions.
Similar to [36] we introduce an appropriate space that incorporates the time and space regularity of the test functions.
In the following, we use test functions of the form ϕ(s) = U (t, s) * x * for x * ∈ D * u . Thus, we need to show that such a ϕ ∈ Γ u t,β for some β > 0. This is established in the next lemma. We recall that the parameters δ respectively δ * stand for the Hölder exponents in the (AT) conditions for A u and A * u as specified in (A3) and (A3 * ), respectively.
Proof. In order to show that ϕ(s) ∈ Γ u t,β we verify the three conditions of Definition 3.4. For (1), using the definition of the norm on D where λ ∈ (β, δ * ). Next we verify condition (2). Using [1, Prop. 2.9], we obtain By Lemma 2.5, we immediately see that ϕ is continuously differentiable for s < t. For the case s = t, we refer to [2, Theorem 6.5]. Now from the above identity, we have  Proof. For the first term we obviously have For the second integral, we obtain For the third term, setting ϕ(s) := U u (t, s) * x * for x * ∈ D * u , we have The last term is bounded for λ ∈ (β, δ * ). Collecting all the previous deliberations, we now state the main result of this paper.

Proof. 1.) We start by showing that a pathwise mild solution of (3.1) is also a weak solution.
Assume that (3.2) holds and fix t ∈ [0, T ]. Let λ ∈ (β, δ * ) and u be the mild solution of (3.1) with zero initial condition, i.e.
Applying x * ∈ D λ * u to (3.8) and using that further leads to We use the identity (2.14) for x = t 0 σ(u(τ ))dW τ and rewrite the first term on the righthand-side as Using (3.10) in (3.9), we obtain In order to show the equivalence with the weak solution, we need to use test functions ϕ ∈ Γ u t,β . To this aim we choose x * = (−A u (s)) * ϕ(s) in (3.11) and integrate over time to obtain where we used Fubini's theorem in the last step. Using (2.7), we obtain for all x ∈ X and 0 ≤ r ≤ t ≤ T , Note that the expressions above are well-defined. Indeed, for ϕ ∈ Γ u t,β and x ∈ X, choosing 1 > λ > θ > 0 and applying(2.13), we infer that Setting x = r 0 σ(u(τ ))dW τ in (3.13) further leads to U u (t, r)A u (r) r 0 σ(u(τ ))dW τ , ϕ(t) + r 0 σ(u(τ ))dW τ , (−A u (r)) * ϕ(r) (3.14) Next we use (3.14) to deal with the right-hand-side of (3.12). From (3.12) we get Thus, using (3.14) in (3.15) The above expression on simplification results in Moreover, choosing x * = ϕ(t) in (3.11), we get Using  Further, choosing x * = ϕ ′ (t) in (3.11), we can express the integrand of the last term in the right-hand-side of (3.18) as

(3.19)
Using Fubini's theorem and plugging in the relation (3.19) in (3.18), we infer that From the previous expression we conclude that u satisfies (3.3) and is therefore a weak solution of (3.1).
2.) Let u be a weak solution of (3.1). Then u ∈ Z and satisfies the weak formulation (3.3), namely In particular, by Theorem 2.3 there exists an evolution family U u : ∆ × Ω → L(X) generated by the operator A u . Using the relation between a(u; ·, ·) and A u , recall (2.4), the previous expression rewrites as From Lemma 3.2, for a fixed t ∈ [0, T ] and x * ∈ D * u , ϕ(s) = U u (t, s) * x * belongs to Γ u t,β . Now with this choice of test functions and using that ϕ ′ (s) = (−A u (s)) * ϕ(s), we obtain from (3.20) that which simplifies further to All in all, we obtained that Splitting the stochastic integral in the first term on the right-hand-side, into two stochastic integrals results in Using the identity (2.14) with x = t 0 σ(u(τ ))dW τ , the first term on the right-hand-side of (3.21) can be written as Using the above identity in (3.21) entails In the previous deliberations x * ∈ D * u . Since D * u is dense in X, the result can be extended to every x * ∈ X by the Hahn-Banach theorem. This justifies that the weak solution of (3.1) satisfies the mild formulation (3.2).
For time-independent test functions, we recover the equivalence of the pathwise mild solution with the following standard weak formulation Proof. 1.) Let t ∈ [0, T ]. We consider only the case σ(u(t)) ∈ D u such that the process t → A u (t)σ(u(t)) is adapted and belongs to L 0 (Ω; L p (0, T ; L 2 (H, X))).
For x * ∈ D * u , we simply set ϕ(t) := x * . However, such a test function does not belong to Γ u t,β , since ϕ / ∈ D 1+β u . Though, the additional spatial regularity of σ enables us to perform the same proof as before. From (3.12), we obtain for the test function Testing with (−A u (s)) * x * instead of (−A u (s)) * ϕ(s) entails Analogously to the proof of Theorem 3.4, the result now follows for σ(·) ∈ D u . The general case follows from a suitable approximation argument for σ(·) as in [36,Theorem 4.9].

Examples
In this section, we present two examples of parabolic quasilinear SPDEs, to which the theory developed in this paper applies. The existence theory for pathwise mild-, martingaleand weak solutions for these problems is well-known, see [27,15,23]. After introducing these SPDEs and recalling the corresponding existence results, we show that they satisfy our assumptions. Therefore, the pathwise mild and weak solution concepts are equivalent in these cases. be an open bounded domain with C 2 boundary. We fix parameters α 1 , α 2 , δ 11 , δ 21 > 0. We are interested in studying a cross-diffusion SPDE, which was originally introduced by Shigesada, Kawasaki and Teramato [39] in the deterministic setting, to analyze population segregation by induced cross-diffusion in a two-species model. Note that the nonlinear drift term correspond to those arising in the classical Lokta-Volterra competition model. The stochastic SKT system is given by for t ∈ [0, T ] and x ∈ O and is supplemented with the following boundary and initial conditions: is an H-valued cylindrical Wiener process. The solution u := (u 1 , u 2 ), where u 1 = u 1 (x, t) and u 2 = u 2 (x, t) denote the densities of two competing species S 1 and S 2 at certain location x ∈ O, at time t. The coefficients θ 11 , θ 22 > 0 denote the intraspecies competition rates in S 1 , respectively in S 2 and θ 12 , θ 21 > 0 stand for the interspecies competition rates between S 1 and S 2 . Furthermore, the terms ∆(β 1 u 2 1 ) and ∆(β 2 u 2 2 ) represent the self-diffusions of S 1 and S 2 with rates β 1 , β 2 ≥ 0, and ∆(γ 1 u 1 u 2 ), ∆(γ 2 u 1 u 2 ) represent the cross-diffusions of S 1 and S 2 with rates γ 1 , γ 2 ≥ 0.
The SKT system (4.1) can be rewritten as an abstract quasilinear SPDE: The nonlinear term F corresponds to the Lotka-Volterra type competition model In order to ensure the positive definiteness of the matrix B, the following restriction on the parameters is necessary: This assumption is required in order to show that A u generates a parabolic evolution system U u for u ∈ Z, for a natural choice of Z, see below. The (AT) conditions (A1)-(A3) are satisfied for a standard choice of Hilbert spaces Z ֒→ Y ֒→ X, where X := L 2 (O), Z := H 1+ε (O) and Y := H 1+ε 0 (O) for 0 < ε 0 < ε, see [42,Chapter 15.2.2]. Moreover, according to [42,Chapter 1.8.2], (A1 * )-(A3 * ) are also satisfied by the adjoint operator (A u ) * . We also emphasize that the domains of the fractional powers of A u , for u ∈ Z, can be identified with Sobolev spaces, see [42,Prop. 15.3]. More precisely, we have incorporates the Neumann boundary conditions. Furthermore, the nonlinear drift term is locally Lipschitz continuous on X. Letting u 0 ∈ Z a.s. and assuming a local Lipschitz continuity on σ (recall Assumption 4), [27,Theorem 4.3] provides the existence of a local-in-time pathwise mild solution u of (4.2) such that u ∈ L 0 (Ω; 2 ). The main result establishes the equivalence of this pathwise mild solution with the weak solution. This statement is a direct consequence of Theorem 3.4 and Theorem 3.5. Note that the results in Theorem 3.4 and Theorem 3.5 hold if an additional drift term F is incorporated.
Example 4.2. We let d ≥ 1 and consider a quasilinear parabolic stochastic partial differential equation on a d-dimensional domain O ⊂ R d with smooth boundary ∂O of the form where (W t ) t∈[0,T ] is a cylindrical Wiener process taking values in a Hilbert space H ⊃ X := L 2 (O).
Such equations have been extensively studied in the literature, see [14,13,23], under the following assumptions on the coefficients B and σ: (1) The coefficients B : R → R d×d are nonlinear functions, such that the diffusion matrix B = (B ij ) d i,j=1 is of class C 1 b , symmetric, uniformly positive definite and bounded, i.e. there exist constants κ, C > 0 such that κI ≤ B ≤ CI.
(2) For each u ∈ X we consider a mapping σ(u) : H → X defined by where σ k ∈ C(O × R). We further suppose that σ satisfies usual Lipschitz and linear growth conditions, i.e.
In particular, these assumptions imply that σ maps X to L 2 (H, X). Thus, given a predictable process u that belongs to L 2 Ω, L 2 (0, T ; X) , the stochastic integral is a well-defined X-valued process.
The previous assumptions on σ can be relaxed and an additional regular drift term can be incorporated [14,13,23]. An example of such a drift term is given by div(F (u)) , where Next, we give the definition of a weak solution of (4.6). Under the previous assumptions on B and σ, together with suitable regularity conditions on the initial data, the existence of a weak solution was established in [14,13].
Moreover, assuming higher spatial regularity on σ, the regularity of this weak solution can be improved [14,Theorem 2.6]. For the convenience of the reader we indicate this statement. To this aim we let η > 0, set D T := [0, T ] × O and consider the Hölder space C η/2,η (D T ) with different time and space regularity, endowed with the norm Theorem 4.2. Assume that • u 0 ∈ L m Ω; C ι (Ō) for some ι > 0 and all m ∈ [2, ∞), and u 0 = 0 on ∂O a.s.
. Then the weak solution u of (4.6) belongs to L m Ω; C η/2,η (D T ) , for all m ∈ [2, ∞). Now, we are ready to state the main result for this example, based on Theorem 4.2.
where A u u := div(B(u)∇u) and U u (·, ·) is the evolution family generated by A u . Moreover, Hence, we can infer from Theorem 2.3 that the operator A u u = div(B(u)∇u) generates an evolution family U u . According to Theorem 3.5 part (2), this evolution family along with the weak solution u satisfies (4.8). Consequently, we obtain that u is a pathwise mild solution of (4.6).
Remark 4.4. Note that the regularity on σ assumed in Theorem 4.2 is necessary in order to obtain the Hölder regularity of the solution.

Appendix A. Fractional powers of sectorial operators
Let X be a Banach space with norm · X and A be a linear sectorial operator of X with angle 0 ≤ ϑ A < π. As before an open sectorial domain Σ ϑ for ϑ A < ϑ < π is given by Σ ϑ = {λ ∈ C; | arg λ| < ϑ} , ϑ A < ϑ < π.
We define, for each complex number z with Re z > 0, the bounded linear operator Theorem A.1. For any 0 < φ < π 2 , as z → 0 with z ∈ Σ ϑ \ {0}, A −z converges to Id strongly on X.
It also holds that A −z satisfies the law of exponent, i.e.
Theorem A.2. The L(X)-valued function A −z is an analytic semigroup defined in the halfplane {z ∈ C; Re z > 0}.
The fractional power A α , for every real number −∞ < α < ∞ is defined, see [42, Chapter 2.7.2] for details. The following theorem lists some of the properties of the fractional power A α , cf. [42, Theorem 2.23].
Next, we compare domains of fractional powers of two sectorial operators A and B of X for which D(A) ⊂ D(B) continuously, i.e. there exists a constant C > 0 such that Bv X ≤ C Av X , v ∈ D(A).