Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.


Introduction
There are many physical phenomena, including fatigue, damage, and fracture of materials, see e.g. [4,8,[20][21][22]26] which are irreversible and grow in one direction following an energy gradient structure; modeling and analysis of such phenomena by nonlinear differential equations is a challenging and interesting problem. The following prototype problem is often considered as a mathematical model of such unidirectional gradient systems More precisely, we consider an energy F : [0, T ] × H 1 0 → R of the form F(t, u) = 1 2 a(u, u) − f (t), u , where a(·, ·) is a coercive, continuous bi-linear form in H 1 0 , f belongs to L 2 (0, T ; L 2 ), and ·, · denotes the L 2 -scalar product. Introducing the operator A : H 1 0 → H −1 defined by a(u, v) = −(Au, v) the Fréchet differential of F(t, ·) clearly reads dF (t, u) where (·, ·) denotes the duality between H 1 0 and H −1 . If Au ∈ L 2 we can write −dF (t, u) In particular, the positive part [Au(t)+f (t)] + turns out to be the L 2 -projection of −∇ L 2 F(t, u) on the cone of positive functions. Therefore, a natural formulation for the unilateral gradient flow is Technically, if u(t) ∈ H 2 the above PDE makes sense whenever Au(t) + f (t) is a locally finite Radon measure and [Au(t) + f (t)] + is its positive part in the sense of Hahn decomposition (see Proposition 2.5).
Now, let us turn to our unilateral gradient flow, introducing the notion of weak solution. For sake of clarity, we start assuming again f ∈ AC(0, T ; L 2 ). In this case, u ∈ H 1 (0, T ; L 2 ) is a unilateral gradient flow if the energy t → F(t, u(t)) is absolutely continuous and for a.e. t ∈ (0, T ) the following power estimate holdṡ (1.5) Here |u| L 2 + and |∂F| L 2 + (t, u) denote respectively a "singular distance", which takes into account the unilateral constraint, and the unilateral slope respectively defined by If |∂F| L 2 + (t, u) is finite then Au + f (t) is a Radon measure with positive part in L 2 and |∂F| L 2 + (t, u) = [Au + f (t)] + L 2 (see Corollary 3.8). Therefore, inequality (1.5) reads, in analogy with (1.4), Note that in the above inequality we cannot write the identity Au(t) + f (t) = −∇ L 2 F(t, u(t)) since in general Au(t) ∈ L 2 and thus the L 2 -gradient is not well defined. When f ∈ L 2 (0, T ; L 2 ) the power balance inequality (1.5) does not make sense since the time derivative of f is not available. However, the same role is played by the power balancė E(u(t)) ≤ − 1 2 |u(t)| 2 (1. 6) where E(u) = 1 2 a(u, u). To be precise, in Definition 2.2 we will employ the time-integral formulation of (1.6) which is more convenient in the proofs, i.e., for every t * ∈ (0, T ).
As we will see, for any f ∈ L 2 (0, T ; L 2 ) and any initial datum other terms, (1.6) provides a criterion to select a unique solution among those of the parabolic equation. Now, let us describe in detail the structure and the content of the article. Sections and results are organized according to the time regularity of the datum f , which plays an important role (both in the analysis and in the applications). First of all we consider the most general case, i.e., f ∈ L 2 (0, T ; L 2 ), which occupies most of the paper. We prove existence and uniqueness of a solution in the sense of (1.6). Existence is obtained by time discretization, employing three different incremental problems of interest in the applications [3,17,26]. Let t n,k = kτ n be a uniform discretization of the interval [0, T ] with τ n = T /n. In the first scheme, given u n,k at time t n,k , the configuration u n,k+1 at time t n,k+1 is simply given by u n,k+1 ∈ argmin F(t n,k+1 , u) + 1 2τn |u − u n,k | 2 In the second scheme, we employ instead an a posteriori truncation, i.e., given u n,k we define u n,k+1 by ũ n,k+1 ∈ argmin F(t n,k+1 , u) The fact that the first minimization is unconstrained makes this scheme very convenient in the numerical implementation [3,26], on the other hand the analysis is slightly more involved. Last, we consider a penalty method, i.e., given u n,k we get u n,k+1 by solving, where α n → +∞ and Each scheme defines a sequence of discrete solutions u n (depending on τ n ) which enjoys suitable compactness properties and which converges (weakly and up to subsequences) to a solution of (1.6); a posteriori we actually prove that the whole sequence converges.
Note that for f ∈ L 2 (0, T ; L 2 ) the time regularity of solutions is rather low, since in general u ∈ H 1 (0, T ; L 2 ). As a consequence, uniqueness does not follow from classical tools, we use instead a contradiction argument [19] based on energy balance and convexity. For the same reason, the energy identity does not follow by the chain rule, which would require at least u ∈ H 1 loc (0, T ; H 1 ), rather, it is proved employing a measure theory argument, employed also in [12,22].
In the second case we consider f ∈ AC(0, T ; L 2 ). Existence and uniqueness is obviously contained in the previous one, however, from a theoretical point of view, it is interesting to know that in this case solutions are of class H 1 loc (0, T ; H 1 0 )∩W 1,∞ loc (0, T ; L 2 ); as a consequence, a better representation holds and few issues, due to the lack of time regularity, are avoided. Lastly, when f is independent of time, besides recovering the classical result of [18], we prove a (rather surprising) property: the unique solution of (1.6) turns out to coincide with the unique solution of the unconstrained L 2gradient flow for the functionalF(u) = F(u) + I + (u − u 0 ), in other terms, the monotonicity constraint can be replaced by a fixed obstacle, given by the initial datum u 0 ; this property, however, does not hold when f depends on time.
To complete our analysis, we prove a comparison principle and a (nonquantitative) continuous dependence property for solutions of (1.6); moreover, for the interested reader, we provide in the appendix further properties, representations and remarks on the unilateral slope.

Setting and statement of the main results
Let us consider an open, bounded domain Ω ⊂ R d . Throughout the paper we employ the shorthand notation L 2 for L 2 (Ω) and similarly for other functional spaces. We use the notation ·, · for the scalar product in L 2 , while (·, ·) denotes the duality between H −1 and H 1 0 . We consider a coercive, continuous and symmetric bi-linear form a(·, ·) in H 1 0 × H 1 0 given by where B = B T ∈ L ∞ (Ω, R d×d ) and b ∈ L ∞ (Ω); in order to guarantee coercivity we assume that B is positive definite (i.e., there exists C B > 0 such that ξ · B(x)ξ ≥ C B |ξ| 2 , for every ξ ∈ R d and a.e. x ∈ Ω) and that C Ω ess-inf{b} + C B > 0, where C Ω is the Poincaré constant of Ω. Let A : H 1 0 → H −1 be the associated operator, given by (Au, v) = −a(u, v). Accordingly, we introduce the stored energy E : H 1 0 → R given by In (2.1) and in the sequel we drop, whenever possible, the dependence on x ∈ Ω. Clearly a 1/2 (u, u) is the energy-norm which is equivalent to the standard norm in H 1 0 . Let [0, T ] be a time interval and let f ∈ L 2 (0, T ; L 2 ). Let us choose a representative of f (defined for every t ∈ [0, T ]) and consider the free energy (in the sequel we will see that the evolution is actually independent of the choice of the representative). Clearly, for u, v ∈ H 1 0 and t ∈ [0, T ] the Fréchet differential of F(t, ·) reads Here, following the theory of curves of maximal slope [5] we provide first a notion of unilateral slope. To this end, for w ∈ L 2 let us introduce the following notation Accordingly, we will say that v → u in L 2 + when |v − u| L 2 + → 0, i.e. when v ≥ u and v → u in L 2 . Finally, let us introduce the unilateral L 2 -slope, defined as follows.
In Sect. 3 we will see that is a locally finite Radon measure with positive part in L 2 . For equivalent ways of writing the slope, in terms of a "singular metric" and in terms of a "unilateral subdifferential", see Appendix A and B.

The case
where the identity holds in L 2 (and thus a.e. in Ω). Ifu(t) ≥ 0 (a.e. in Ω) for a.e. t ∈ (0, T ) then u(t 2 ) ≥ u(t 1 ) (a.e. in Ω) for every 0 ≤ t 1 ≤ t 2 ≤ T , and thus we will say that u is monotone in [0, T ]. The fact that the evolution u, appearing in Definition 2.2, is monotone in [0, T ] is implicitly written in (2.5); indeed for t * = T (2.5) implies that then |u(t)| L 2 + < +∞ for a.e. t ∈ (0, T ), i.e.,u(t) ≥ 0 for a.e. t ∈ (0, T ). Note that (2.5) is independent of the choice of the representative of the datum f . The next theorem contains the main result: existence, uniqueness, and power identity.
Note that the above power identity implies that |u(t)| L 2 + = |∂F| L 2 + (t, u(t)). The proof of the above theorem is split in different sections: existence is proved by means of three different time-discrete schemes, contained in Sects. 4.1, 4.2, and 4.3, while energy identity and uniqueness are proved in Sect. 4.4.
In order to better understand and justify the above definition of solution we state here the following results, dealing respectively with strong and weak solutions of the parabolic problemu(t) = [Au(t) + f (t)] + . Theorem 2.5. Let u be the unilateral gradient flow provided by Theorem 2.3, then u solves the parabolic partial differential equation where Au(t)+f (t) is a (locally finite) Radon measure and [Au(t)+f (t)] + ∈ L 2 is its positive part. Remark 2.6. As far as Theorem 2.4, note that in general Au(t) + f (t) ∈ H −1 \ L 2 for a.e. t ∈ (0, T ); an explicit example is given in Sect. 5.1. In particular, if Ψ : L 2 → [0, +∞] is given by Ψ(v) = 1 2 v 2 L 2 +I + (v) (where I + is the indicator function of the set {v ≥ 0}) it is not always possible to re-write (2.7) in the form since ∂Ψ ⊂ L 2 and thus Au(t) + f (t) should be in L 2 . The latter equation, in the formu Weak solutions for unidirectional gradient flows Page 9 of 46 59 is adopted e.g. in [1,2] under stronger regularity on the data (e.g. if u 0 ∈ H 2 and f is suitably controlled) which ensure Au(t)+f (t) in L 2 for a.e. t ∈ (0, T ), see e.g. [2, Theorem 2.6].
Remark 2.7. In Sect. 5.1 we will see that solutions of (2.8) are not unique. Lack of uniqueness is essentially due to the unilateral constraint, since the solution of the unconstrained gradient flow for F would be unique (see e.g. [10]).
Remark 2.8. Due to the lack of time regularity of the datum f , in general the energy t → F(t, u(t)) is not absolutely continuous, for this reason in (2.6) we employ the derivative of the energy t → E(u(t)). The absolute continuity of the energy t → F(t, u(t)) is recovered under more restrictive assumptions of the datum f , for instance when f ∈ AC(0, T ; L 2 ), see Proposition 2.12. Finally, note that (2.6) can be equivalently written in integral form as: for and P ext (t,u(t)) = f (t),u(t) denote respectively the dissipation and the power of external forces.
Unilateral gradient flows, in the sense of Definition 2.2, enjoy comparison principle and continuous dependence; on the contrary, by lack of uniqueness, solutions of (2.8) do not satisfy them.
Moreover, following [18], the unique solution u of Theorem 2.3 is also the unique solution in and ∂Φ(u) ⊂ H −1 denotes its subdifferential. Finally, the solution u belongs also to the space W 1,∞ loc (0, T ; L 2 ). Inequality (2.12) provides, in the non-autonomous case, a notion of curve of maximal (unilateral) slope corresponding to [5,Definition 1.3.2]. Moreover, as a consequence of Proposition 2.12, for every 0 ≤ t * ≤ T the energy identity reads (2.14)

The case f independent of time
The existence and uniqueness of solutions when f is independent of time has been already treated in the literature, see e.g. [18]; however, in this case we show that the constraint onu can be replaced by a fixed obstacle, as a consequence we provide a further and simple characterization (2.15) of solutions, in the spirit of the recent [1, Remark 5.3] but with a different proof.
Then, u turns out to be the (unconstrained) L 2 -gradient flow for the func-tionalF(u) = F(u) + I + (u − u 0 ). Moreover, u is also the unique solution of the following parabolic obstacle problem: for a.e. t ∈ (0, T ).
(2.15) As we will see in Sect. 7 the above characterization does not hold when the datum f depends on time.

Generalizations and applications
To conclude this section, let us mention that several of the above results can be extended or partially extended to more general functionals, with few modifications in the definitions and in proofs. The choice of quadratic functionals is motivated by sake of simplicity and by the fact that quadratic, or separately quadratic, energies are mostly used in applications.
As far as generalizations, let us mention the following. Let p ∈ (1, +∞) such that W 1,p ⊂ L 2 (by Sobolev embedding), let f ∈ L 2 (0, T ; L 2 ) and con- for some λ < 0 and for every z 0 , z 1 ∈ R and s ∈ (0, 1). The double-well potential w(z) = z 2 (z − 1) 2 , appearing in the Allen-Cahn equation, is a prototype λ-convex functions for λ ≤ min z w (z). Under these assumptions, we can define the stored energy E and the free energy F respectively given by In this case, adapting the arguments of the following sections, it is not difficult to see that for u 0 ∈ W 1,p 0 there exists an evolution u ∈ H 1 (0, T ; L 2 ) ∩ L ∞ (0, T ; W 1,p ) which satisfies the energy identity (2.10). However, in this weak setting, uniqueness is still open since the arguments of Sect. 4.4 do not apply.
As far as applications, let us mention the phase-field approach to damage and fracture, based on the "Ambrosio-Tortorelli" energy [6] In this context the unilateral gradient flow takes the form [3,8,21] Here the monotonicity constraint models irreversibility, with v = 0 corresponding to the sound material and v = 1 to the fully damaged or fully cracked material. We remark that, beside the non-linearity, the regularity of the forcing term t → F (t, v(t)) is usually quite low.

Energy and unilateral slope
To the purpose of this section, fix t ∈ [0, T ] and let Given Taking the supremum on the right hand side we get |∂F| L 2 and then, for n 1, is linear, taking the supremum in (3.2) yields also the following Corollary.
where the supremum is taken pointwise in [0, T ]. Measurability follows.

Corollary 3.4. Let f n (t) f(t) in L 2 and consider the energies
Weak solutions for unidirectional gradient flows Page 13 of 46 59 Proof. The weak lower semicontinuity of the energy is obvious.
3) follows by taking the supremum with respect to z.
To conclude, we provide in Corollary 3.8 an L 2 "representation" of the slope, which is fundamental to connect the unilateral gradient flow and the parabolic problem (2.8); its proof is a direct consequence of the next abstract lemmas on Radon measures.
Proof. For the first part see [22,Lemma A.2] or [11]. We give a short proof of the second part. By density we can write which concludes the proof.
Lemma 3.7. Let ζ be as in the previous lemma and z ∈ L 2 with z ≥ 0 and Proof. Since (z − ζ) is a positive measure, it follows that ζ is a measure as well. Write ζ = ζ + − ζ − , where ζ ± are supported on the disjoint sets Ω ± . In particular we have z ≥ ζ + as measures and functions in Ω + . As z ≥ 0 it follows that z ≥ ζ + also in Ω − .

Solutions for
In the following subsections we prove existence of unilateral gradient flows, in the sense of Definition 2.2, by means of three discrete schemes, which take into account the monotonicity constraint in different ways. We remark that all these ways of introducing monotonicity are currently employed in applications to phase-field fracture, with suitable adaptations. We provide complete proofs, however, those parts which are quite similar are not repeated.

Constrained incremental problem
Let τ n = T /n and t n,k = kτ n for k = 0, ..., n. First of all, for every index Define u n,0 = u 0 at time t n,0 , and then, given u n,k at time t n,k , let the configuration at time t n,k+1 be given by Note that a unique minimizer exists by standard arguments and that u n,k+1 ≥ u n,k . Define u n : [0, T ] → L 2 and u n : [0, T ] → L 2 respectively as the piecewise affine interpolation and the piecewise constant backward (left-continuous) interpolation of the values u n,k in the points t n,k . In this section we will prove the following proposition. We write for simplicity u n,k+1 = u n (t n,k+1 ) anḋ u n,k+1 = (u n,k+1 − u n,k )/τ n instead ofu n (t) for t ∈ (t n,k , t n,k+1 ).
Proof. Using (4.2) and being F n (t n,k+1 , ·) convex we get, for t ∈ (t n,k , t n,k+1 ), Writing Using the interpolation u n , u n , and f n we get (4.6).
Proof of Proposition 4.1. Using (4.6) for 0 ≤ k ≤ n − 1 and (4.2) we get . Being f n bounded in L 2 (0, T ; L 2 ) the sequence u n turns out to be bounded in H 1 (0, T ; L 2 ) and thus, up to subsequences (not relabeled), u n u in H 1 (0, T ; L 2 ). By weak convergenceu n ≥ 0 impliesu ≥ 0. We will identify u with its absolutely continuous representative, i.e.
By coercivity of the stored elastic energy, we deduce that u n is bounded in L ∞ (0, T ; H 1 0 ) and thus u n is bounded in L ∞ (0, T ; H 1 0 ) as well. Given t ∈ (0, T ] let k n such that t n,kn < t ≤ t n,kn+1 . Remembering that u n (t) = u n (t n,kn+1 ) we can write Clearly t n,kn+1 → t * and the sequence u n,kn+1 = u n (t * ) converges weakly to (t n,kn+1 )).

NoDEA
Weak solutions for unidirectional gradient flows Page 17 of 46 59 By weak convergence in H 1 (0, T ; L 2 ) we get All the above · L 2 -norms can be replaced with | · | L 2 + sinceu n andu are nonnegative. Since f n (t) → f (t) in L 2 and u n (t) u(t) in H 1 for a.e. t ∈ (0, T ), by (3.3) we get (4.10) For last term in the left hand side of (4.8) we easily have Taking the liminf in (4.8) gives the thesis.
Next lemma follows closely the corresponding one in the previous subsection.
Note that formally (since the sequences do not coincide) the only difference between (4.6) and (4.19) is the slope: in the former it is evaluated in (t, u n (t)) while in the latter in (t,ũ n (t)).
Proof of Proposition 4.5. Following line by line the first step in the proof of Proposition 4.1 we get that the sequence u n is bounded in L ∞ (0, T ; H 1 0 ) and in H 1 (0, T ; L 2 ). Hence u n u in H 1 (0, T ; L 2 ), upon extracting a (non-relabeled) subsequence.
We claim thatũ n (t) where we used the continuity of the bi-linear form. Since u n is bounded in L ∞ (0, T ; H 1 0 ) the right hand side is bounded uniformly with respect to n and k; thus there exists C > 0 s.t. where in the second line we used coercivity. As f n,k+1 → f (t) in L 2 , a simple algebraic estimate yields ũ n,k+1 H 1 0 ≤ C . Since u n,k = u n (t n,k ) → u(t) in L 2 it follows thatũ n (t) =ũ n,k+1 → u(t) in L 2 . Beingũ n,k+1 bounded uniformly in H 1 0 we getũ n (t) u(t) in H 1 0 . To conclude the proof it is enough to argue as in the proof of Proposition 4.1

Unconstrained incremental problem with penalty
Let α > 1. For v ∈ L 2 let us denote Clearly, when α is large |v| L 2 α penalizes v − . Note that Ψ α can be equivalently seen as a Yosida regularization of the indicator function of the set {v ∈ L 2 : v ≥ 0}. Moreover Before proceeding, let us prove this lemma.
Proof of Proposition 4.9. As |u n | L 2 αn ≥ u n L 2 , arguing as in the proof of Proposition 4.1, it follows that the sequence u n is bounded in H 1 (0, T ; L 2 ) and in L ∞ (0, T ; H 1 0 ). Thus, upon extracting a subsequence (non relabeled) u n u in H 1 (0, T ; L 2 ) and u n * u in L ∞ (0, T ; H 1 0 ). Given t * ∈ (0, T ] let k n such that t n,kn < t * ≤ t n,kn+1 ; by the previous lemma we have E(u n (t n,kn+1 )) + t n,kn+1 0 1 2 |u n (t)| 2 As in the proof of Proposition 4.1 we get that u n (t) u(t) in H 1 0 and thus by convexity of the energy E(u(t * )) ≤ lim inf n→+∞ E(u n (t n,kn+1 )).

Energy identity, uniqueness and strong convergence
This section contains the proof of Theorem 2.3. We give first a short proof of the following Lemma.
. Let w j be the piecewise affine interpolation of (the continuous representative) of w in the points t j,i . Then Hence, by Jensen inequality Taking the sum for i = 0, ..., I j −1 yields the estimate (4.25). For t ∈ (t j,i , t j,i+1 ) we can write where in the last inequality we used (4.27). Hence Taking the sum for i = 0, ..., I j − 1 yields the estimate (4.26).
Given t * let 0 < t * < t * with |∂F| L 2 + (t * , u(t * )) < +∞. We can find a sequence of finite subdivisions strongly in L 2 (t * , t * ), and (It is enough to apply the Riemann sum argument to the couple (|∂F| L 2 + , f) ∈ L 2 (0, T ; R ⊗ L 2 ); for sake of clarity, we also remark that, in general, the points {t j,i } do not coincide with the points t n,k = kτ n appearing in the discrete scheme). By convexity of F(t j,i+1 , ·) we write Denote by u j the piecewise affine interpolation of u(t j,i ). Writing explicitly the energies F(t j,i , u(t j,i+1 )) and F(t j,i , u(t j,i )) we get Using the above estimate for i = 1, ..., I j we get, in terms of the functions S j and F j appearing in (4.28) and (4.29), By (4.28) we known that S j (·) → |∂F| L 2 + (·, u(·)) strongly in L 2 (t * , t * ) and by (4.29) that F j → f strongly in L 2 (t * , t * ; L 2 ). By Lemma 4.12 we getu j u in L 2 (t * , t * ; L 2 ) and u j L 2 u L 2 in L 2 (t * , t * ). In summary, we can pass to the limit in (4.30) and get, by Young's inequality, Taking the liminf of the right hand for t * → 0 + we get which concludes the proof.
Proof of Theorem 2.3. Clearly, using (2.5) and Lemma 4.13 it follows that for every 0 ≤ t * ≤ T we get the energy identity Let t * such that u I (t * ) = u II (t * ). Define u = 1 2 (u I + u II ). Writing the energy identity (4.31) for both u I and u II we get (for i = I, II) Taking the sum for i = I, II and using the strict convexity of the energy E, the convexity of 1 2 · 2 L 2 , the convexity of 1 2 |∂F| 2 L 2 + (t, ·) (see Corollary 3.2) and the linearity of f (t), · we get Hence |∂F| L 2 + (·, u (·)) belongs to L 2 (0, t * ) and clearly u ∈ H 1 (0, T ; L 2 ). The previous inequality is a contradiction with Lemma 4.13.
Since the limit evolution is unique it is not necessary to extract any subsequence in Propositions 4.1, 4.5 and 4.9. Proof. Given t * ∈ [0, T ] let us first prove that u n (t * ) → u(t * ) in H 1 0 . Since u n (t * ) u(t * ) in H 1 0 it is enough to show that E(u n (t * )) → E(u(t * )), which implies that u n (t * ) → u(t * ) in H 1 0 . Let k n such that t n,kn < t * ≤ t n,kn+1 and recall (4.8), i.e., E(u n (t * )) = E(u n (t n,kn+1 )) ≤ E(u 0 ) − t n,kn+1 0 1 2 |u n (t)| 2 Then, where, in the last line, we used (4.9) and (4.11) from Proposition 4.1 together with the energy identity (4.31). As a consequence, all inequalities above turn into equalities and u n (t * ) → u(t * ) in H 1 0 ; hence u n (t n,kn+1 ) = u n (t * ) → u(t * ) in H 1 0 . A similar argument shows that u n (t n,kn ) → u(t * ) in H 1 0 . Being u n (t * ) a convex combination of u n (t n,kn ) and u n (t n,kn+1 ) it converges strongly to u(t * ) as well.

Parabolic equation
Proof of Theorem 2.4.
In a similar way, ∇F(t, u(t)) = −Au(t) − f (t), in particular Under the regularity assumptions of Theorem 2.4 we know, e.g. by [10,Lemma 3.3], that the energy t → E(u(t)) is absolutely continuous and that a.e. in (0, T ) it holdṡ Assume that u is a solution of (2.7), i.e.u(t) = [−∇F(t, u(t))] + , then Taking the integral in time gives Definition 2.2. Conversely, assume that u is the unilateral gradient flow given by Theorem 2.3. Beingu ≥ 0 we geṫ  Proof of Theorem 2.5. Let u be the unilateral gradient flow for F with initial datum u 0 , in the sense of Definition 2.2. By the energy identity we know that u(t) L 2 = |∂F| L 2 + (t, u(t)) is a.e. finite in [0, T ]. Hence, by Corollary 3.8, for a.e. t ∈ [0, T ] we have Now, let us show that −dF(t, u(t)) ≤u(t) in H −1 for a.e. t in [0, T ]. By uniqueness, we can rely on the discrete scheme of Sect. 4.1. By (4.4) for every t ∈ (t n,k , t n,k+1 ) we have −dF n (t, u n (t))[φ] ≤ u n (t), φ for every φ ∈ C ∞ 0 with φ ≥ 0, which reads, by symmetry of a(·, ·), Passing to the limit, by the strong convergence of u n (t) in L 2 (0, T ; L 2 ) together with the strong convergence of f n in L 2 (0, T ; L 2 ) and the weak convergence of Since the above inequality holds for any choice of t 1 < t 2 , for a.e. t ∈ (0, T ) we have which reads (Au(t) + f (t), φ) ≤ u(t), φ . Therefore, applying Lemma 3.7 we get [Au(t) + f (t)] + ≤u(t). By (5.2), we have u(t) L 2 = [Au(t) + f (t)] + L 2 , it follows thatu(t) = [Au(t) + f (t)] + .

Non-uniqueness for the parabolic PDE in L 2
We provide an example in which the parabolic problem (2.8) has many solutions, and thus it is not equivalent to Definition 2.2. Let u 0 ∈ H 1 0 (−1, 1) be defined by u 0 (x) = 1 − |x|. We will denote by u and u the first and second space derivatives, respectively. Clearly u 0 = −2δ 0 , where δ 0 denotes the Dirac delta in the origin. Note that u 0 > 0 in (−1, 1) and [u 0 ] + = 0.
Let f = u 0 and consider the energy F : H 1 0 (−1, 1) → R given by Clearly On the other hand u does not satisfy the energy identity which is equivalent to (2.5). Indeed, E(u(t)) Solution of the unilateral gradient flow. The solution of the unilateral gradient flow, in the sense of Definition 2.2, is computed hereafter. Let us start considering the sub-interval (0, 1) and the following parabolic problem for x ∈ (0, 1).
Consider the auxiliary function u * = u 0 + |u 1 − u 0 | and note that We claim that F r (u * ) ≤ F r (u 1 ). Indeed, since (u 1 − u 0 ) and |u 1 − u 0 | belong to H 1 0 (0, 1) we can write Since u * ≥ u 1 and f = u 0 > 0 we have F r (u * ) ≤ F r (u 1 ). As |u * −u 0 | = |u 1 −u 0 | from the latter inequality it follows that and then, by uniqueness of the minimizer in (5.5), that Next, let us see that u k+1 ≥ u k for k ≥ 1. In this case, we employ the Euler-Lagrange equation for u k , i.e., (0,1) As in the case k = 0, we check that F r (u * ) ≤ F(u k+1 ) for u * = u k +|u k+1 −u k |.
Choosing φ = u k+1 − u k and φ = |u k+1 − u k | in the Euler-Lagrange equation yields, respectively, It is well known that up to subsequences the piecewise affine interpolation u τ converges weakly in H 1 (0, T ; L 2 (0, 1)) to the unique solution u r of (5.4). Therefore, u r is monotone non-decreasing in time.

Proposition 5.2. The function u defined by
is the unilateral gradient flow for the functional F with initial conditions u 0 (x) = 1 − |x|.
Moreover, in terms of u l and u r the derivatives of u reaḋ where L and δ 0 denote respectively the Lebesgue measure and the Dirac delta. We remark thatu(t, 0) = 0. Note that [u r (t, 0)−u l (t, 0)] = 2u r (t, 0) < 0, hence In By the regularity of f we can characterize the unilateral gradient flow for F as in Proposition 2.12, i.e. 1)) the chain rule and the fact thatu(t, 0) = 0 yieldḞ which concludes the proof.

Comparison principle
Proof of Proposition 2.10. Since the unilateral gradient flow is unique it is enough to prove the maximum principle for the discrete solutions provided in Sect. 4.2; indeed, if u n,k ≤ v n,k for every k ≥ 0 then u n ≤ v n in [0, T ] and passing to the limit weakly in H 1 (0, T ; L 2 ) we get u ≤ v in [0, T ]. To this end, fix τ n > 0 and assume that u n,0 = u 0 ≤ v 0 = v n,0 . We will show by induction that u n,k ≤ v n,k for every index k ≥ 1. We recall that u n,k+1 = max{ũ n,k+1 , u n,k } and that In a similar way, there exists ζ n,k+1 ∈ ∂F(t n,k+1 ,ṽ n,k+1 ) such that ζ n,k+1 + (f − g) + 1 τn (ṽ n,k+1 − v n,k ) = 0 in H −1 . Assume by induction that u n,k ≤ v n,k , we claim thatũ n,k+1 ≤ṽ n,k+1 from which we get u n,k+1 ≤ v n,k+1 . Using [ũ n,k+1 −ṽ n,k+1 ] + ∈ H 1 0 as a test function we get , [ũ n,k+1 −ṽ n,k+1 ] + = 0. The first term is non-negative by T -monotonicity, see Remark 3.5, the second is non-negative because f ≤ g, hence for the last term we can write Since u n,k ≤ v n,k the integrand in the left hand side is non-positive; it follows that [ũ n,k+1 −ṽ n,k+1 ] + = 0 and thusũ n,k+1 ≤ṽ n,k+1 .

Continuous dependence
Proof of Proposition 2.11. We adopt the scheme of [23]. Denote F m (t, u) = E(u) − u, f m (t) . By definition we know that for every 0 ≤ t * ≤ T it holds We recall that by Theorem 2.5 and Corollary 3.
. Hence, choosing t * = T above we obtain Since u m 0 → u 0 in H 1 0 we have E(u m 0 ) → E(u 0 ); since f m → f in L 2 (0, T ; L 2 ) the above estimate implies that the sequenceu m is bounded in L 2 (0, T ; L 2 ). It follows that u m is bounded in H 1 (0, T ; L 2 ). Moreover, by (5.6) and by coercivity of the stored energy, for every 0 ≤ t * ≤ T we have Hence, the sequence u m is bounded also in L ∞ (0, T ; H 1 0 ). In conclusion, there exists a subsequence (non relabeled) such that u m u in H 1 (0, T ; L 2 ); clearly u m is monotone non-decreasing. Moreover, arguing as in the proof of Proposition 4.1, we get that u m (t) u(t) in H 1 0 for every t ∈ [0, T ]. It remains to show that u is the unilateral gradient flow for F with initial condition u 0 . By (5.6) for every t * ∈ (0, T ] we can write Sincė u m u in L 2 (0, T ; L 2 ) we have u 2 L 2 (0,t * ;L 2 ) ≤ lim inf m→+∞ u m 2 L 2 (0,t * ;L 2 ) . As u m u in H 1 0 a.e. in (0, T ) we can apply Corollary 3.4 and then by Fatou's lemma we get Finally, ,u(t) dt by strong-weak convergence in L 2 (0, t * ; L 2 ). In conclusion, we get which is (2.5).
In order to prove that u m (t) → u(t) strongly in H 1 0 for every t ∈ [0, T ] it is enough to follow the proof of the strong convergence in Sect. 4.4.

Solutions for f ∈ AC(0, T ; L 2 )
In this section we consider the case in which f ∈ AC(0, T ; L 2 ) and we will prove the assertions contained in Proposition 2.12.

Characterization by differential inclusions in H 1 0
The results of this section are essentially an adaption of those contained in [18]; we provide some short alternative proofs, for sake of completeness. Proof. We will employ the discrete evolutions u n obtained by the implicit Euler scheme of Sect. 4.1. We will show that given 0 < T < T the sequence u n is bounded in W 1,∞ (T , T ; L 2 ) ∩ W 1,2 (T , T ; H 1 0 ). Let us denoteu n,k = (u n,k − u n,k−1 )/τ n . For k ≥ 0 by (4.3) we have u n,k+1 2 L 2 + dF n (t n,k+1 , u n,k+1 )[u n,k+1 ] = 0 . (6.1) For k ≥ 1 by (4.4) we get dF n (t n,k , u n,k )[u n,k+1 ] + u n,k ,u n,k+1 ≥ 0.
A simple algebraic calculation gives − u n,k ,u n,k+1 + u n,k+1 ,u n,k+1 ≥ 1 2 u n,k+1 Hence by coercivity (6.2) reads, for k ≥ 1, − cτ n u n,k+1 Neglecting the H 1 0 -norm and denoting a k = u n,k L 2 we obtain the discrete inequality a 2 k+1 −a 2 k ≤ b k a k+1 where b k = 2 f n,k+1 −f n,k L 2 . By an elementary algebraic calculation we get a k+1 ≤ a k + b k . Then for every k 0 < k we can write Let k n such that τ n (k n −1) < T ≤ τ n k n . Given k > k n the estimate a k ≤ a k0 + C holds for every index k 0 such that 1 ≤ k 0 ≤ k n with C = 2 f AC(0,T ;L 2 ) . Taking the sum of a k ≤ a k0 + C for k 0 = 1, ..., k n (and k fixed) and dividing by k n yields Hence, for every k > k n we have where C(T ) > 0 is independent of n because the sequence u n is bounded in H 1 (0, T ; L 2 ). (Note that C(T ) diverges as T → 0 + ). Taking the supremum with respect to k > k n it follows that the sequenceu n is bounded in L ∞ (T , T ; L 2 ) for every T > T > 0. Let us go back to (6.3). For every k > k n , now we can write Proof. Sinceu(t) ∈ H 1 0 , for a.e. t ∈ (0, T ), by classical results in convex analysis we have

A characterization for f independent of time
In this section we will prove Proposition 2.13.
The aim of this last section is to provide, to the unfamiliar reader, the equivalence between gradient flows and curves of maximal slope (for our family of functionals) in the case of unconstrained evolutions. As we have already observed, for unilateral evolutions these notions are not always equivalent.
If Au ∈ L 2 then the L 2 -gradient of the energy E reads In a similar way, ∇F(t, u) = −Au − f (t).