Phase transitions in porous media

The full quasistatic thermomechanical system of PDEs, describing water diffusion with the possibility of freezing and melting in a visco-elasto-plastic porous solid, is studied in detail under the hypothesis that the pressure-saturation hysteresis relation is given in terms of the Preisach hysteresis operator. The resulting system of balance equations for mass, momentum, and energy coupled with the phase dynamics equation is shown to admit a global solution under general assumptions on the data.


Introduction
A model for fluid flow in partially saturated porous media with thermomechanical interaction was proposed and analyzed in [2,4]. The model was subsequently extended in [15] by including the effects of freezing and melting of the water in the pores. Typical examples in which such situations arise are related to groundwater flows and to the freezing-melting cycles of water sucked into the pores of concrete. Due to the specific volume difference between water and ice, this process produces important pressure changes and represents one of the main reasons for the degradation of construction materials in buildings, bridges, and roads. The model of [15] still neglects the influence of changes of microstructure, as for example the breaking of pores, but the main thermomechanical interactions between the state variables are taken into account.
The model is based on the assumption that slow diffusion of the fluid through the porous solid is a dominant effect, so that the Lagrangian description is considered to be appropriate. It is assumed that volume changes of the solid matrix material are negligible with respect to the pore volume evolution

The model
Consider a bounded domain Ω ⊂ R 3 of class C 1,1 filled with an elastoplastic solid matrix material with pores containing a mixture of H 2 O and gas, where we assume that H 2 O may appear in one of the two phases: water or ice. We state the balance laws in referential (Lagrangian) coordinates. We have in mind construction materials where large deformations are not expected to occur. This hypothesis enables us to reduce the complexity of the problem and assume that the deformations are small in order to avoid higher degree nonlinearities. We denote for x ∈ Ω and time t ∈ [0, T ] p(x, t) ... capillary pressure; u(x, t) ... displacement vector in the solid; ε(x, t) = ∇ s u(x, t) ... linear strain tensor, (∇ s u) ij := 1 2 ∂ui ∂xj + ∂uj ∂xi ; θ(x, t) ... absolute temperature; χ(x, t) ∈ [0, 1] ... relative amount of water in the H 2 O part. The model derived in [15] aims at coupling the effects of capillarity, interaction between a deformable solid matrix material and H 2 O in the pores which may undergo water-ice phase transitions, and energy exchange between the individual components of the system. Hysteresis is included following the modeling section of [4]. The full system consists of equations describing mass balance (1.1), mechanical equilibrium (1.2), energy balance (1.3) and phase evolution (1.4) in the form (χ+ρ * (1−χ))(G[p] + div u) t = div (μ(p)∇p), (1.1) − div (B∇ s u t + P [∇ s u]) + ∇(p(χ+ρ * (1−χ)) + β(θ − θ c )) = g, (1.2) C V (θ) t − div (κ(θ)∇θ) = B∇ s u t : ∇ s u t + μ(p)|∇p| 2 + D P [∇ s u] t *

C. Gavioli and P. Krejčí
NoDEA represents the pressure component due to the phase transition. Finally, the energy balance (1.3) and the inclusion (1.4) for the evolution of the phase parameter are derived from the principles of thermodynamics with the aid of the energy balance for both the plasticity and the pressure-saturation operator (1.5) where U P , U G and D P , D G are the potential and dissipation operators, and · * is a seminorm in the space R 3×3 sym of symmetric 3 × 3 tensors. In (1.3), C V (θ) is the caloric component of the internal energy, κ(θ) is the heat conductivity coefficient, L is the latent heat, I [0,1] is the indicator function of the interval [0, 1] and ∂I [0,1] is its subdifferential, γ(θ, div u) is the phase relaxation time (which we assume to explicitly depend on both θ and div u for technical reasons).
Note that the values of G have to be naturally confined between 0 and 1, so that the system is degenerate in the sense that we do not control a priori the time derivatives of p in (1.1). Another difficulty is related to the lack of spatial regularity of χ. The temperature field is problematic as well: Eq. (1.3) contains high order heat source terms, which are difficult to handle and prevent the temperature from being regular. We complement the system (1.1)-(1.4) with initial conditions (1.6) and boundary conditions where p * is a given outer pressure, θ * is a given outer temperature, α : ∂Ω → [0, ∞) is the permeability of the boundary and ω : ∂Ω → [0, ∞) is the heat conductivity of the boundary. The solution to (1.1)-(1.4) was only constructed in [15] under the assumption that shear stresses in the momentum balance Eq. (1.2) as well as all hysteresis effects are neglected. Then (1.2) turns into an ODE for the relative volume change div u, which considerably simplifies the analysis. Here we prove existence of a global solution for the full problem under suitable hypotheses.

Hysteresis in capillarity phenomena
The operator G is considered as a sum where f is a monotone function satisfying Hypothesis 3.1 (vi) in Sect. 3 below, and G 0 is a Preisach operator that we briefly describe here.

11)
C. Gavioli and P. Krejčí NoDEA provided we define the Preisach potential U 0 and the dissipation operator D 0 by the integrals (2.12) The energy identity in (1.5) then holds with the choice (2.13) With the notation we can also write 15) thus separating the hysteretic from the non-hysteretic part.
For our purposes, we adopt the following hypothesis on the Preisach density.

Hypothesis 2.2. There exists a function
A straightforward computation shows that G 0 (and, consequently, G) are Lipschitz continuous in C[0, T ]. Indeed, by (2.5) and Hypothesis 2.2 we obtain for p 1 , p 2 ∈ C[0, T ] and t ∈ [0, T ] that (2.16) Moreover, the Preisach potential is continuous from From Hypothesis 2.2 and identity (2.7) for the play we also obtain The Preisach operator admits also a family of "nonlinear" energies. As a consequence of (2.7), we have for a. e. t the inequality for every nondecreasing function h : R → R. Hence, for every absolutely continuous input p, a counterpart of (2.11) in the form holds with a modified potential This is related to the fact that for every absolutely continuous nondecreasing functionĥ : R → R, the mapping Gĥ := G 0 •ĥ is also a Preisach operator, see [13].

Statement of the problem
We introduce the spaces 1) for some q * > 2 that will be specified below in Theorem 3.3. Taking into account the boundary conditions (1.7), we consider (1.1)-(1.4) in variational form for a. e. t ∈ (0, T ) and all test functions φ ∈ X, ψ ∈ X 0 and ζ ∈ X q * . Note that we split the capillary hysteresis terms in hysteretic and non-hysteretic part according to (2.1), (2.13)-(2.15). This is done in view of the regularization performed in Sect. 4, where only the non-hysteretic part will be affected by the cut-off. We assume the following hypotheses to hold.
We also assume that there exist constants f > f > 0, ν , γ > γ > 0 such that the nonlinearities satisfy the following conditions: and G 0 is the Preisach operator from Sect. 2 with density function satisfying Hypothesis 2.2 and with potential U 0 and dissipation D 0 as in (2.12 is the constitutive operator of elastoplasticity with dissipation operator D P defined below in (3.6)-(3.9).

Remark 3.2.
In this remark we comment on the more technical hypotheses.
(vi) The growth condition for f is in agreement with the physical requirement that f has to degenerate when p → ±∞. The specific form of the lower bound will play a substantial role in the Moser iteration argument. (viii) The growth condition for c V will be of fundamental importance in Sect.
5.6 where, in order to estimate div u t in L q (0, T ; L 2 (Ω)) with an exponent q > 4, we will need a higher integrability (in space) for the temperature than simply L ∞ (0, T ; L 1 (Ω)). (ix) The tangled boundâ 7 − 2b for the growth exponent of the function κ is required in Sect. 5.9, where we apply an iterative method in order to derive higher order estimates for the temperature.
(x) The dependence of the relaxation coefficient γ on both θ and div u is uncommon but crucial for obtaining estimates (4.34) and (5.45).
We model the elastoplasticity following [17]. We assume that a convex subset 0 ∈ Z ⊂ R 3×3 sym with nonempty interior representing the admissible plastic stress domain is given in the space R 3×3 sym of symmetric tensors, and that the constitutive relation between the strain tensor ε and the stress tensor σ involves two fourth order tensors A h (the kinematic hardening tensor) and A e (the elasticity tensor). We define the constitutive operator P by the formula where σ p is the solution of the variational inequality for a given ε ∈ W 1,1 (0, T ; R 3×3 sym ), where Q Z : R 3×3 sym → Z is the orthogonal projection onto Z. The variational inequality (3.7) has a unique solution σ p ∈ W 1,1 (0, T ; R 3×3 sym ) and the solution mapping is strongly continuous, see [12]. It holds The energy potential U P and the dissipation operator D P associated with P are defined by the formula Let M Z * denote the Minkowski functional of the polar set Z * to Z. The energy identity where · * = M Z * (·) is a seminorm in R 3×3 sym , and the inequalities hold for all inputs ε ∈ W 1,1 (0, T ; R 3×3 sym ). The operator P can be extended to a continuous operator in the space a. e., (3.13) (3.14) with a constant C depending only on A h and A e . For inputs ε ∈ L 2 (Ω; W 1,1 (0, T ; R 3×3 sym )) we obtain from (3.14) similarly as in [4,Formula (6.25)] the inequality The main result of the paper reads as follows.  1] a. e.
The reason why we do not specify the precise value of q * here is that it relies on a certain number of intermediate computations that cannot be detailed at this stage. The proof of Theorem 3.3 will be divided into several steps. In order to eliminate possible degeneracy of the functions f and μ, we start by regularizing the problem by means of a large parameter R. Then we prove that this regularized problem admits a solution by the standard Faedo-Galerkin method: here the parameter R will be of fundamental importance in order to gain some regularity. Once we have derived suitable estimates, we pass to the limit in the Faedo-Galerkin scheme. The second part of the proof will consist in the derivation of a priori estimates independent of R, which will allow us to pass to the limit in the regularized system and infer the existence of a solution with the desired regularity.
In what follows, we denote by C any positive constant depending only on the data, by C R any constant depending on the data and on R and by C R,η any constant depending on the data, on R and on η, all independent of the dimension n of the Galerkin approximation. Furthermore, we denote by |v| r the L r (Ω)-norm of a function v ∈ L r (Ω) or v ∈ L r (Ω; R 3 ) for r ∈ [1, ∞], and the norm of a function v ∈ W 1,r (Ω) will be denoted by |v| 1;r . We systematically use the Korn's inequality (see [19]) 16) for every w ∈ X 0 with a constant c > 0 independent of w. We will also often use the Poincaré inequality (see [9,16]) in the form for functions v ∈ W 1,2 (Ω) provided γ ∈ L ∞ (∂Ω) is such that γ ≥ 0 a. e. and ∂Ω γ(x) ds(x) > 0. Finally, let us recall the Gagliardo-Nirenberg inequality (see [3,9]) for v ∈ W 1,r (Ω) on a bounded Lipschitzian domain Ω ⊂ R N in the form with r < N, s < q < (rN )/(N − r) and with a constant C depending only on q, r, s, where

Cut-off system
We choose a regularizing parameter R > 1, and first solve a cut-off system with the intention to let R → ∞.
For z ∈ R we denote by the projection of R onto [−R, R]. Then we cut-off some nonlinearities by setting for p, θ, div u ∈ R. Note that by Hypothesis 3.1 (vi) we deduce that |f R (p)| ≤ |f (0)| + f |p|, from which C. Gavioli and P. Krejčí NoDEA and also, from Hypothesis 3.1 (x), We replace (3.2)-(3.5) by the cut-off system for all test functions φ, ζ ∈ X and ψ ∈ X 0 . For the system (4.8)-(4.11) the following result holds true.
We split the proof of Proposition 4.1 in two steps. First, in Sect. 4.1, we further regularize the system by means of a small parameter η > 0 in order to obtain some extra-regularity for the gradient of the capillary pressure. Then, in Sect. 4.2, we solve this new problem by Galerkin approximations. Here the extra-regularization will be of fundamental importance in order to pass to the limit in the nonlinearity Q R (|∇p (n) | 2 ), where n is the dimension of the Galerkin scheme. As a last step, we let η → 0.

W 2,2 -regularization of the capillary pressure
We define the functions for p, θ ∈ R, and introduce the new variables v = M R (p), z = K R (θ). We then choose another regularizing parameter η ∈ (0, 1) and consider the following system in the unknowns v, u, z, χ: with test functions φ ∈ W 2,2 (Ω), ψ ∈ X 0 and ζ ∈ X.

Galerkin approximations
For each fixed R > 1 and η ∈ (0, 1), system (4.13)-(4.16) will be solved by Faedo-Galerkin approximations. To this end, let W = {e i : i = 0, 1, 2, . . . } ⊂ L 2 (Ω) be the complete orthonormal systems of eigenfunctions defined by where the coefficients v i , z k : [0, T ] → R and u (n) , χ (n) will be determined as the solution of the system (4.20) for i, k = 0, 1, . . . , n and for all ψ ∈ X 0 , and with p (n) : This is an ODE system coupled with a nonlinear PDE (4.18). It is nontrivial to prove that such a system admits a unique strong solution. We proceed as follows. For a given function w ∈ L r (Ω × (0, T )) consider the equation (4.22) which is to be satisfied for every ψ ∈ X 0 a. e. in (0, T ) together with an initial condition u(x, 0) = u 0 (x), u 0 ∈ X 0 ∩ W 1,r (Ω; R n ) and boundary condition u = 0 on ∂Ω.
Step 1. As an application of the L r -regularity theory for elliptic systems in divergence form (see e. g. [21]) we can conclude that for every

Step 2. Let us define the convex and closed subset
). Note that the trace operator is well defined on this space, so that the initial condition makes sense. Letû ∈ U r , and let u be the solution of the equation the existence of which follows from Step 1. We prove that the mappingû t → u t is a contraction with respect to a suitable norm.
Indeed, letû 1 ,û 2 be given, and let u 1 , u 2 be the corresponding solutions. The differenceū = u 1 − u 2 is the solution of the equation According to Step 1, we have (4.23) By inequality (3.14) we have for a. e. (x, t) with a constant C > 0. Hence, by Hölder's inequality, (4.24) Now, set

NoDEA
It follows from (4.23) and (4.24) that We now multiply both sides of the above inequality by e −Ct r , and after an integration over t ∈ [0, T ] we obtain from the Fubini Theorem This means that the mappingû t → u t is a contraction in L r (0, T ; X 0 ∩ W 1,r (Ω; R 3 )) with respect to the weighted norm hence it has a unique fixed point which is a solution of (4.22).
Step 3. The mapping which with a right-hand side w ∈ L r (Ω × (0, T )) associates the solution u t ∈ L r (0, T ; X 0 ∩ W 1,r (Ω; R 3 )) of (4.22) is Lipschitz continuous. Indeed, consider w 1 , w 2 and the corresponding solutions u 1 , u 2 , and set as beforew = w 1 − w 2 ,ū = u 1 − u 2 . As a counterpart of (4.23) we get and the computations as in (4.24) yield We obtain the Lipschitz continuity result when we test by e − C p t r and integrate over t ∈ [0, T ], similarly as in Step 2. Now, coming back to our Eq. (4.18), we see that it is of the form (4.22 its associated solution operator, we conclude that (4.17)- (4.20) give rise to a system of ODEs with a locally Lipschitz continuous right-hand side containing the operator S. We now derive a series of estimates. Note that we decompose the auxiliary variables v and z instead of p and θ into a Fourier series with respect to the basis W because we are going to test Eqs. (4.17) and (4.19) by nonlinear expressions of p and θ, namely, by their Kirchhoff transforms (4.12). Indeed, the Galerkin method allows only to test by linear functions and their derivatives.
Moreover, we do not discretize the momentum balance equation because considering the full PDE is the only way to deduce compactness of the sequence {∇ s u (n) t }, which is needed in order to pass to the limit in some nonlinear terms.
Indeed, we will not be able to control higher derivatives of u (n) , and this will prevent us from applying the usual embedding theorems.

Estimates independent of n Estimate 1.
We test (4.17) by v i and sum up over i = 0, 1, . . . , n, and (4.18) by ψ = u (n) t . Then we sum up the two equations to obtain (4.25) where we exploited also the energy identity (3.10). We now define and introduce the modified Preisach potential as a counterpart to (2.20) according to (2.19). Note that (4.2) and (4.3) together with Hypothesis 3.1 (vi) and (vii) yield for all p ∈ R, with some positive constants c R , C R depending only on R. Moreover, the estimate

C. Gavioli and P. Krejčí
NoDEA holds as a counterpart of (2.18). By the definition of v (n) and Hypothesis Moreover, thanks again to Hypothesis 3.1 (vii), the boundary term is such that Young's inequality and Hypothesis 3.
Moreover, by Hölder's inequality and Hypothesis 3.1 (ii), where in the last line we used first Korn's inequality (3.16) and then Young's inequality. Neglecting some lower order positive terms on the left-hand side, exploiting estimates (4.6), (4.26) and (4.27), and the fact that 31) where we used also Hypothesis 3.1 (i), Young's inequality and the pointwise inequality t | on the left-hand side together with the term coming from (4.30).
We need to control |χ To this aim note that (4.20) is of standard form, namely, , (4.33) or, equivalently, . This yields, thanks to (2.18), (4.6) and (4.7), . We now come back to (4.31) and integrate in time

C. Gavioli and P. Krejčí
NoDEA from which we deduce a bound also for the term div u (n) since ∇ s u (n) t is dominant. Thus, coming back to (4.47) and using Hypothesis 3.1 (iii), (iv) and estimate (4.45), we finally obtain Applying Grönwall's lemma and Poincaré's inequality (3.17) we finally obtain the estimates sup ess where the strong convergences are obtained by compact embedding, see [3]. We also need strong convergence of the sequences {∇ s u (n) } and {∇ s u (n) t } in order to pass to the limit in some nonlinear terms. Taking the difference of (4.49) for indices n and m, and testing by ψ = u we obtain, arguing as for Step 3 of the existence part, with the notation of (4.33), where we have by virtue of (2.16) for a. e. x ∈ Ω that Note that by (4.39) Π is bounded above independently of n. We further have

and (4.53) is of the form
Thus, from Fubini's theorem and Grönwall's lemma we obtain
Note that as a side product, from (4.58) we get that the estimate holds also in the limit as η → 0. Hence, arguing as for (4.38) we get M R (p) 2 L 2 (0,T ;W 2,2 (Ω)) ≤ C R . This, by Sobolev embedding, yields as well as But then we can argue as in Sect. 4.4 and obtain an inequality similar to (4.55), but with a constant independent of η. This entails the strong convergence of the sequences ∇ s u (η) and ∇ s u

Estimates independent of R
We now come back to our cut-off system (4.8)-(4.11). We are going to derive a series of estimates independent of R. More precisely, after proving that the temperature stays away from zero, we will perform the energy estimate and the Dafermos estimate in order to gain some regularity for the temperature. Subsequently, a key-step will be the derivation of a bound for p in an anisotropic Lebesgue space. Then an analogous estimate based on the particular structure of Eq. (4.9) is obtained for ∇ s u t . We finally show that this is sufficient for starting the Moser iteration and obtain an L ∞ bound for p. After deriving some higher order estimates for the capillary pressure and for the temperature, we will be ready to let R tend to ∞ in (4.8)-(4.11).

Positivity of the temperature
For every nonnegative test function ζ ∈ X we have, by virtue of (4.10), where in the last line we used Hypothesis 3.1 (i) together with inequality (4.32), and also estimate (4.7). Then, by Young's inequality, with a constant C depending on L, θ c , β, B, γ . Let now ϕ(t) be the solution of the ODE d dt C V (ϕ(t)) + Cϕ 2 (t) = 0, ϕ(0) =θ withθ from Hypothesis 3.1. Then ϕ is nondecreasing and positive. Taking into account the fact that C V (ϕ) t = −Cϕ 2 and ∇ϕ = 0, for every nonnegative test function ζ ∈ X we have in particular Consider now the following regularization of the Heaviside function

C. Gavioli and P. Krejčí
NoDEA for ε > 0, and set ζ(x, t) = H ε (ϕ(t) − θ(x, t)) which is an admissible test function. This yields By the Lebesgue Dominated Convergence Theorem we can pass to the limit in the above inequality for ε → 0, getting Owing again to the monotonicity of C V and ϕ, we conclude that, independently of R, We now pass to a series of estimates independent of R.

Energy estimate
Since we proved that the temperature stays positive, from now on we will write We test (4.8) by φ = p, (4.9) by ψ = u t and (4.10) by ζ = 1. Summing up the three resulting equations we obtain Note that some of the terms cancel out. Moreover, recalling the notation introduced in (4.3) and the energy balance (2.11), the identities hold true. Hence we obtain, using also (3.10) and (4.11), (5.4) We now integrate in time τ 0 dt. On the left-hand side Young's inequality, (3.11) and (4.32) entail By the definition of Q R in (4.1), it holds |∇p| 2 ≥ Q R (|∇p| 2 ). The boundary term is such that thanks to Young's inequality and Hypothesis 3.1 (iii) and (iv). Concerning the right-hand side of (5.4), the time integration gives where the term containing the initial conditions is controlled by using Hölder's inequality and observing that Hence by Young's inequality and Hypothesis 3.
where we used also Korn's inequality (3.16). The first term in the last line is absorbed by (5.5). Finally, the initial conditions are kept under control thanks to (2.18), (3.9), (4.6) and Hypothesis 3.1 (v). Hence what we eventually get is C. Gavioli and P. Krejčí and applying Grönwall's lemma we finally obtain the estimates sup ess Estimate (5.7) also gives sup ess where b is from Hypothesis 3.1 (viii).

Mechanical energy estimate
In order to estimate the capillary pressure in a suitable anisotropic Lebesgue space, we first need to find a bound for div u t in L 2 (Ω × (0, T )), independently of R. To this purpose, we test (4.8) by φ = p, (4.9) by ψ = u t and sum up to obtain, with the notation of the previous subsection, Note that some terms cancel out. Owing to (5.2) and exploiting also the energy identity (3.10), what we eventually get is Now, (4.11) yields where in the last line we used Young's inequality and (4.7), and where the constant C is independent of R. Moreover, from the pointwise inequality (4.32) and arguing as for (4.30) we get Hence we obtain, exploiting also Hypothesis 3.1 (i) to absorb the terms coming from the two estimates above, where the boundary term was handled as in (5.6). We now integrate in time τ 0 dt for some τ ∈ [0, T ]. The right-hand side is bounded thanks to estimate (5.15), whereas the initial conditions are kept under control thanks to (2.18), (3.7), (3.9), (4.6) and Hypothesis 3.1 (v). Hence, neglecting some already estimated positive terms, we finally obtain independently of R. This, together with (5.8) and Poincaré's inequality (3.17), yields p 2 L 2 (0,T ;W 1,2 (Ω)) ≤ C. (5.18)

Estimate for the capillary pressure
We choose an even function λ : R → (0, ∞) such that λ (p) ≥ 0 for p > 0 and pλ(p) ∈ X. Then we test (4.8) by φ = pλ(p). We obtain The term under the time derivative has the form and introduce the modified Preisach potential as a counterpart to (2.20) according to (2.19). Note that V λ,R (p) > 0 and U λ [p] ≥ 0 for all p = 0. Then (5.20) can be rewritten as Now, using Young's inequality as in (5.16) we obtain and similarly Note that so that (5.19) and a time integration Now that we got rid of χ t and derived a manageable estimate, we choose λ(p) = |p| 2k with k ≥ ν/2 which will be specified later. Here ν is as in Hypothesis 3.1.

NoDEA
Note that this is an admissible choice, that is, pλ(p) ∈ X. Indeed, by estimates (4.45), (4.59) and by the anisotropic embedding formulas, see [3], we have p ∈ L q (0, T ; C(Ω)) (5.24) for any q ∈ [1,4). The bound also depends on R, but for a fixed R and each k > 0 the function p|p| 2k (·, t) belongs to X for a. e. t ∈ (0, T ). With this choice (5.23) takes the form Note also that, from Hypothesis 3.1 (vi) and an analogous version of (2.3), Moreover Finally, by Young's inequality with conjugate exponents 2+2k 1+2k , 2 + 2k , we see that the boundary term is such that Hence, using also Hypothesis 3.1 (vii), we obtain From Hypothesis 3.1 (iv) and (v) it follows that the above inequality is of the form 1 + k and with a constant C independent of τ , R and k. We now show that for a suitably chosen k, the right-hand side of (5.29) is dominated by the lefthand side, which will imply a bound for the left-hand side. By the Gagliardo-Nirenberg inequality (3.18) with q = q k , s = s k , r = 2 and N = 3 we have (5.30) We now choose k in such a way that δ k q k = 2, that is, 3q k = 6 + 2s k , which yields By Hypothesis 3.1 we have s k ≥ 1. Hence, by (5.30), Since k < 1, we conclude from (5.29) that there exists a constant C independent of R such that, in particular, Invoking (5.28), we obtain for p the estimates sup ess We now distinguish two cases: ν ≤ 1/3 and ν > 1/3. For ν ≤ 1/3 (that is, For ν > 1/3 (that is, 3(1 − ν) < 2) we use again the Gagliardo-Nirenberg inequality (3.18) with q = 2, s = 3(1 − ν), r = 2 and N = 3, obtaining so that for Hence by virtue of (5.18), (5.27), and (5.31) we obtain with a constant C > 0 independent of R, according to the notation (5.27). Note that q ν ≥ 6 thanks to Hypothesis 3.1 with ν ≤ 1/2.

Further estimates
Now that we have obtained a suitable estimate for the capillary pressure in (5.33), we derive an analogous estimate for div u t . To this aim we test (4.9) by ψ = u t , which yields By (3.8), Hypothesis 3.1 (v) and (5.17) we have hence using Hypothesis 3.1 (i) and Young's inequality as in Sect. 5.4 we conclude that the estimate holds for a. e. t ∈ (0, T ) with a constant C > 0 independent of R. We want to find and estimate for ∇ s u t in the norm of L q (0, T ; L 2 (Ω)) for a suitable q.
To this aim we apply the Gagliardo-Nirenberg inequality (3.18) toθ with the choices q = r = 2, s = 1 + b (with b from Hypothesis 3.1) and N = 3. We obtain that, for t ∈ (0, T ), so that for C. Gavioli and P. Krejčí

NoDEA
Hence by virtue of (5.9) and (5.14) we get independently of R. Note that our hypotheses on b imply q b ≥ 6. Thus, coming back to (5.34) we have obtained that there exists q := min{q ν , q b } ≥ 6 such that, thanks to (5.32) or (5.33) and (5.35), independently of R, according to the notation (5.27).
We continue our analysis with the inequality (5.25) again. Unlike in Sect. 5.5, we do not keep the exponent k bounded, but we let k → ∞ in a controlled way. As in (5.28), we define auxiliary functions w k = p|p| k and rewrite (5.25) (5.37) with a constant C ≥ 1 independent of k, withh given by (5.26), and with It follows from (5.32) or (5.33), (5.35), and (5.36) thath ∈ L 6 (0, T ; L 2 (Ω)) and Repeating exactly the argument of the proof of [8, Proposition 6.2] we obtain the following result.
The main consequence of Proposition 5.1 is that, since we aim at taking the limit as R → ∞ in (4.8)-(4.11), we can restrict ourselves to parameter values R > R σ , with R σ from (5.38), so that the cut-off (4.2), (4.3), (4.3) is never active and γ R (p, θ, div u) = γ(θ, div u). Hence we can rewrite (4.8)- (4.11) in the form for all test functions φ, ζ ∈ X, ψ ∈ X 0 , withθ = Q R (θ) and with initial conditions (1.6). In order to pass to the limit as R → ∞, we still need to derive some higher order estimates. for p ∈ R, so that μ(p)∇p = ∇M (p). We would like to test (5.39) by φ = M (p) t = μ(p)p t which, however, is not an admissible test function since p t / ∈ X. Hence we choose a small h > 0 and test by φ