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Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model

We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data. Moreover, we prove asymptotic regularization properties of weak solutions.


Introduction
The Penrose-Fife system, proposed by O. Penrose and P. Fife in [26,27], represents a thermodynamically consistent model for the description of the kinetics of phase transition and phase separation processes in binary materials. It couples the singular heat equation (2.3) for the absolute temperature ϑ with a nonlinear relation describing the evolution of the phase variable χ which represents the local proportion of one of the two components. This can be of the fourth order in space (cf. (2.6)-(2.7) below), in case the physical process preserves the total mass of χ (conserved Penrose-Fife model, describing phase separation) or of the second order in space (cf. (2.5) below), in case the total mass of χ is admitted to vary (non-conserved Penrose-Fife model, describing phase transition). In the conserved case, the equation for χ is usually written as a system by introducing an auxiliary variable w called chemical potential. We refer to the next section for a detailed presentation of the equation and to the papers [10,11,33] for further mathematical background.
Due to physical considerations, it is generally accepted to consider no-flux boundary conditions for χ and, in the conserved case, also for w. On the other hand, various types of boundary conditions (for instance, no-flux, non-homogeneous Dirichlet, or Robin conditions) make sense for the heat equation (2.3), which give rise to different mathematical scenarios. We refer the reader to [11,12,13,22,23,33] for the case of Robin (or "third type") conditions, to [14,15] for the Dirichlet case, and to [10,17,18] for the homogeneous Neumann case, which is probably the most difficult one due to lower coercivity properties of the elliptic operator in (2.3).
Our aim in this paper is that of improving existing results on the homogeneous Neumann problem for the Penrose-Fife model both in the non-conserved and in the conserved case. Actually, our results will cover both situations, generally with minor variations in the proofs. As a first property, we will show that the problem admits a unique solution under weak assumptions on the initial data. Actually, noting that the system admits a natural Liapounov functional representing the total energy, we will prove that a (unique) weak solution exists for any initial data (ϑ 0 , χ 0 ) having finite energy. This improves existing results which assume some extra summability condition on ϑ 0 , typically ϑ 0 ∈ L 2 (Ω). As a side effect, we pay the price that the heat equation has to be interpreted in the generalized (H 1 ) ′framework developed by Damlamian and Kenmochi in [13]. Namely, the (thermal part of the) energy has to be intended as a (relaxed) functional operating on the negative order Sobolev space H 1 (Ω) ′ (cf. (2.20) below) and also the relation linking ϑ to its inverse has to be stated properly. On the other hand, if we know in addition that ϑ 0 ∈ L 1 (Ω), then we can prove that ϑ(t) ∈ L 1 (Ω) for all t ≥ 0; moreover, both the energy functional and the relation between ϑ and its inverse can be written in the usual (pointwise) sense (cf. (2.19) below).
In the subsequent part of the paper, we prove our main results, which regard uniform timeregularization properties of weak solutions. In this frame, we will actually present two theorems. Firstly, we will show that, for any T > 0, there exists a constant ϑ > 0 depending only on the "energy" of the initial data and on T such that ϑ(x, t) ≥ ϑ for a.e. (x, t) ∈ (T, +∞) × Ω. Second, we will prove that a similar bound from above (i.e., ϑ(x, t) ≤ ϑ for a.e. (x, t) ∈ (T, +∞) × Ω) and a suitable ϑ > 0) holds provided that the initial temperature ϑ 0 , in addition to the "energy" regularity, satisfies the additional hypothesis ϑ 0 ∈ L 3+ε (Ω) for some ε > 0. This additional condition appears also in other works concerning L ∞ -regularization properties of the solutions to very fast diffusion equations like (2.3) on R 3 (see, e.g., [34], [6] and the references therein). In particular, in three space dimensions, the exponent p = 3 happens to be critical for the boundedness (for strictly positive times) of the solutions to (2.3): starting from initial data in L p (R 3 ), p > 3, implies boundedness of the solutions for strictly positive times (see [6]). For p < 3, the situation is drastically different, as the self similar solution (2.40) shows.
We finally note that the proved uniform bounds permit, by standard methods, to improve furtherly the regularity of solutions for strictly positive times. In particular, our estimates complement a recent paper by Prüss and Wilke [28] who show maximal regularity estimates for the conserved Penrose-Fife model under the conditional (i.e., unproved in their paper) assumption that the temperature ϑ satisfies the uniform bounds ϑ ≤ ϑ(x, t) ≤ ϑ almost everywhere. Thanks to our results, the maximal regularity estimates of Prüss and Wilke hold for all strictly positive times and all weak solutions emanating from initial data satisfying the "energy regularity" plus the condition ϑ 0 ∈ L 3+ε (Ω). It is also worth noting that the uniform bound ϑ ≥ ϑ implies, in the nonconserved case, the so-called "separation from singularities" property for χ in case the configuration potential of the system (i.e., the function b defined in (2.10)) has a bounded domain. For instance, in the physically relevant case of the logarithmic potential

Main results
Let Ω be a smooth bounded domain of R d , d ∈ {2, 3}. For the sake of simplicity, let us assume . For simplicity, we will often write · p in place of · L p (Ω) . Let H := L 2 (Ω), endowed with the standard scalar product (·, ·) and norm · . Let also V := H 1 (Ω). We note by · X the norm in the generic Banach space X. and by ·, · X the duality between X ′ and X.
For any function, or functional z, defined on Ω, we can then set where the integral is substituted with the duality z, 1 in case, e.g., z ∈ V ′ . We also define the elliptic operator Then, for t ∈ (0, +∞), we consider the singular heat equation In the non-conserved case, this is coupled with the equation (2.7) In both cases, we will take the initial conditions The nonlinear function b is assumed to satisfy b ∈ C 0,1 loc (I; R), b(0) = 0, b ′ (r) ≥ 0 a.e. in I, lim r→∂I b(r) sign r = lim r→∂I b ′ (r) = +∞, (2.9) where I, the domain of b, is an open, possibly bounded, interval of R containing 0. We also set b(r) : in such a way that b is a convex function satisfying b(0) = 0. Concerning the initial datum χ 0 , we will always assume To specify the regularity required for the initial temperature ϑ 0 , we first need to introduce some convex analysis machinery. First of all, we set respectively for v < 0 and z > 0. Then, j and j * can be seen as a couple of conjugate functions according to the standard theory (cf., e.g., [7]). This permits to introduce the convex functional where we have implicitly set J(v) = +∞ for those v ∈ V such that j(v) is not summable (this happens, for instance, when v is strictly positive on a set of strictly positive measure). The set {v ∈ V : J(v) < +∞} is called domain of J. Then, the conjugate of J is given by (2.14) We can also introduce the subdifferential of J with respect to the duality between V ′ and V . Namely, In general, the "weak" subdifferential ∂ V,V ′ J is a multivalued maximal-monotone operator. Its structure is analyzed in several papers (see, e.g., [5]; see also [8] and [16] for the slightly different situation where V is substituted by H 1 0 (Ω)). Actually, it in not difficult to prove that On the other hand, for given v ∈ V , the set ∂ V,V ′ J(v) needs not be contained in L 1 (Ω). More precisely, a generic element ζ ∈ ∂ V,V ′ J(v) is a bounded measure which may have a singular part ζ s . A precise characterization is given in [8,Thm. 3] for the H 1 0 -case. However, it is easy to realize that such a characterization extends to the case of V = H 1 (Ω) at least provided Ω is smooth, the only difference being that ζ s can be supported also on the boundary ∂Ω. On the other hand, if we are able to prove that the singular part of some element ζ is 0, then we still have pointwise inclusion. Namely, there holds that That said, the minimal regularity required on ϑ 0 is given by Then, we can define the "strong" energy functional of the system as However, since in general ϑ 0 is just an element of V ′ , it does not make sense to compute E s (ϑ 0 , χ 0 ). Actually, we have to relax E s , by defining Actually, assumptions (2.18) and (2.11) are equivalent to asking that the "relaxed" energy E 0 := E(0) is finite. In the sequel we shall often write E(t) in place of E(ϑ(t), χ (t)). Notice that the last (2.9) ensures the coercivity of E. Notice also that the no-flux conditions entail some conservation properties. Actually, for the conserved system, testing (2.3) and (2.6) by 1, we immediately get while of course for the non-conserved system we only have The following result, stating existence of strong solutions, is well-known (see, for instance, [18] for a proof): Then, in the non-conserved case, there exists a unique triplet (ϑ, u, χ ) satisfying, for all T > 0, Thus, in fact no functional on V ′ appears in the above "strong" formulation.
In the next result we will provide an existence theorem working without the additional regularity (2.23). We will pay the price of the occurrence of relaxed functionals. Moreover, we will also need to intend relation (2.4) in the relaxed sense of [13].  Finally, any V ′ -solutions satisfies, for all t ∈ [0, T ], the energy equality in the non-conserved case. In the conserved case, the same holds provided that the term χ t 2 on the left hand side is replaced by ∇w 2 .
We now turn to discussing regularization properties of solutions. The first result regards the function u.
Theorem 2.4. Let (2.9), (2.18) and (2.11) hold. Then, the V ′ -solution either to the non-conserved or to the conserved problem satisfies Moreover, in the non-conserved case we also have the "separation property" Here and below, Q is a computable nonnegative function, monotone in each of its arguments, whose expression is independent of initial data and of time.
Remark 2.5. It is worth noting that the above regularization properties are in fact instantaneous. Indeed, with minor modification in the proofs one could easily see that (2.32)-(2.33) and (2.34) hold starting from any τ > 0 (and not only from τ = 1 or 2 or 3). Of course, then the functions Q on the right hand sides would also monotonically depend on τ −1 and possibly explode for τ ց 0. The same considerations hold also for what is proved in Theorem 2.7 below.
In case the initial temperature, beyond satisfying (2.18), is an L 1 -function, we can say something more precise on regularity: Then, the triplet (ϑ, u, χ ) given by Theorem 2.3 additionally satisfies, for all T > 0, and the strong identification property (2.4).
It is easy to show that, if ϑ 0 ∈ L p (Ω) for p > 1, then ϑ(t) remains in L p (Ω) for t > 0. What is more interesting is that, if p > 3, then ϑ is asymptotically uniformly bounded: Theorem 2.7. Let (2.9), (2.18) and (2.11) hold and let also Then, the (V ′ ∩ L 1 )-solution either to the non-conserved or to the conserved problem satisfies the additional bound Remark 2.8. Relation (2.37) suggests that 3 should play the role of a critical exponent for equation (2.3) in space dimension 3. Actually, it is easy to check that the related "very-fast diffusion" equation over the whole space (0, +∞) × R 3 admits the similarity solution (see, e.g., [34]) which belongs to L p loc (R 3 ) for all p < 3 and all t ≥ 0 and does not exhibit any instantaneous regularizing effect (of course, it satisfies a delayed regularization property since it extinguishes in a finite time; however, this effect is not expected to hold in the case of a finite domain when we have conservation of mass). However, we do not know what happens in the critical case of an initial datum ϑ 0 belonging to L 3 (Ω).

Proofs
All proofs will be in principle given only for the conserved case which is, actually, more difficult. The properties holding only for the non-conserved case (as well any significant differences in the proofs) will be remarked on occurrence.

Proof of Theorem 2.3
We start by proving existence for weak initial data. Given ϑ 0 satisfying (2.18), we then set, for n ∈ N, ϑ 0,n + n −1 Aϑ 0,n = ϑ 0 . (3.1) The properties of this approximation deserve to be stated in a lemma.
Being ϑ 0 ∈ V ′ , it is clear that ϑ 0,n ∈ V for all n. Moreover, the V ′ -strong convergence in (3.2) can be proved by standard Hilbert techniques. In order to complete the proof of (3.2), we introduce, for any given n ∈ N, the strictly positive sequence ϑ 0,n,k := max{ϑ 0,n , 1 k }, for k ∈ N. Correspondingly, we set u 0,n,k := − 1 ϑ 0,n,k . Note that, by construction, u 0,n,k ∈ V and the map ϑ 0,n → ϑ 0,n,k is monotone. As a consequence, we can write where the subdifferential acts now in the duality between V ′ and V ) for any k. Hence, by definition of subdifferential. On the other hand, since ϑ 0,n,k ≡ ϑ 0,n in Ω ∩ {ϑ 0,n ≥ 1/k}, Thus, collecting the above computations we have Finally, since for any n ∈ N we have that ϑ 0,n,k kր+∞ − −−−− → ϑ 0,n strongly in L p (Ω) for any p ∈ [1, 6) (hence, a fortiori, in V ′ ), we have the following chain of inequalities: i.e., (3.2) holds. Now, we assume that also (2.35) holds, namely we assume that ϑ 0 ∈ V ∩ L 1 (Ω). Note that, thanks to Remark 2.2, in this new regularity framework, (3.4) is nothing else than J * (ϑ 0,n ) ≤ J * (ϑ 0 ), which as been proved above.
Thus, we only need to prove the L 1 -convergence. To this end, we have to be a bit more careful. First, we define the Banach space X := V ′ ∩ L 1 (Ω), endowed with the norm and introduce the unbounded linear operator A on X defined as Av := Av with domain where ∆ is the usual distributional Laplace operator. Then, we have that A is an accretive operator on the space X. Indeed, by [4, Prop. II.3.1], this corresponds to checking that, if λ > 0 and for f i ∈ X, then Actually, the analogue of (3.11) w.r.t. the V ′ -norm can be obtained by testing the difference where I is the identity mapping of H (and, hence, On the other hand, the L 1 -analogue of (3.11) is obtained by testing (3.12) by sign(x 1 − x 2 ) and applying the Brezis-Strauss theorem [9, Lemma 2]. Moreover, we have that D(A) is dense in V ′ ∩ L 1 (Ω). To see this, let us take z ∈ V ′ ∩ L 1 (Ω). Then, setting where ρ n is the standard mollifer, it is clear that z n is smooth (hence, in particular, it belongs to D(A)). Moreover, the convergence z n → z in L 1 (Ω) follows from standard properties of convolutions, while the convergence z n → z in V ′ follows from the density of H in V ′ and from the fact that the mapping z → z n is a contraction w.r.t. the V ′ -norm (this may be verified for z ∈ H by using Fubini's theorem and then extended to V ′ by density). These facts permit to apply [4, Prop. 3.2 (e)], which gives exactly the convergence property (3.3), which concludes the proof.
Thus, taking ϑ 0,n as an initial datum for equation (2.3) (while the initial datum χ 0 is kept fixed), existence of a corresponding solution (ϑ, u, χ ) is guaranteed by Theorem 2.1. Our aim will be now that of removing the approximation of the initial datum letting n ր +∞. With this aim, we start by recalling a couple of basic a-priori estimates. The procedure is detailed only in the conserved case, the differences occurring in the non-conserved case being pointed out at the end. For the meanwhile, we will not emphasize the dependence on n in the notation. Energy estimate. We test (2.3) by 1 + u, (2.6) by w and (2.7) by χ t . This formal procedure will be justified at the end, when we prove (2.31). We obtain d dt E + ∇u 2 + ∇w 2 = 0, (3.14) where E was defined in (2.20). Using also the properties of A, we immediately get Here and below, the letters c and κ will denote generic positive constants, independent of initial data and of time, whose value possibly varies on occurrence, κ being used in estimates from below. In particular, the above estimate is uniform with respect to T . A generalized Poincaré inequality. To estimate the full V -norm of u (and not just the H-norm of its gradient), we need a proper form of Poincaré's inequality (cf., e.g., [19, Lemma 5.1] for a similar tool), which we prove just for the sake of completeness: in Ω. Then, setting K := Ω (log v) + , the following estimate holds:

16)
the constants C 1 and C 2 depending only on Ω.

Proof.
First of all, we recall that for any function z ∈ W 1,1 (Ω) such that |E 0 | > 0 (with E 0 := {x ∈ Ω : z(x) = 0}) the following Poincaré type inequality (see [21, Lemma 5.1, pag. 89]) holds: where the constant C can be explicitely computed and depends only on Ω. Now, let v be a function in the hypothesis of the Lemma. Set K := Ω (log v) + and note that, thanks to the Chebychev inequality, we have, for any fixed N > 0, As a consequence, the inequality (3.17), with z = (v −N ) + and (3.20), entails which is (3.16) with C 1 = 2/|Ω| and C 2 = 2C, C being the constant in (3.17).
Consequences of the energy estimate. Using the above lemma, (3.15) additionally gives Applying standard techniques to system (2.6)-(2.7), we also have Moreover, testing (2.7) by χ − χ Ω and proceeding, e.g., as in the Appendix of [24], it is not difficult whence a comparison of terms in (2.7) and estimate (3.15) also give w L 2 (t,t+1;V ) ≤ Q(E 0 ). (3.25) Passage to the limit. We will now let n ր +∞, still referring to the conserved case. With this aim, we rename as (ϑ n , u n , χ n ) the solution to the n-approximation. By (3.15) and (3.22)-(3.25), we then have, for any T > 0, This is sufficient to take the limit of equation (2.3), but not of (2.4). Actually, to identify ϑ in terms of u, we have to work a little bit more. Namely, we have to integrate (2.3) with respect to time both at the n-level and in the limit and then test, respectively, by u n and by u. Notice that, even at the limit level, the use of u as a test function is guaranteed by the fact that u ∈ L 2 (0, T ; V ) and all terms in (2.3) lie at least in L 2 (0, T ; V ′ ). Then, at the n-level, we obtain where * denotes convolution in time. Taking the supremum limit in the above relation and comparing the result with the limit equation, we obtain lim sup almost everywhere in (0, T ). Then, the standard integration by parts formula [7, p. 73], applied in the duality between V ′ and V , gives Thanks to this formula, in the non-conserved case for any V ′ -solution we are allowed to test (2.3) by u and (2.5) by χ t . Integrating over (0, t) for arbitrary t > 0, we obtain exactly (2.31). In the conserved case, instead, we have to test (2.3) by u, (2.6) by w and (2.7) by χ t . Note that this is still possible for any V ′ -solutions. Indeed, thanks to (2.27) and the properties of A, we have that w ∈ L 2 (0, T ; V ). Thus, w can be used as a test function in (2.6) (which is a relation in L 2 (0, T ; V ′ )) and χ t can be used as a test function in (2.7) (which is a relation in L 2 (0, T ; V ) thanks to the above discussion). However, we have to notice that, while it is not expected to be true that, separately, A χ ∈ L 2 (0, T ; V ) and b( χ ) ∈ L 2 (0, T ; V ). Nevertheless, as shown, e.g., in [29, Lemma 4.1], property (3.38) is sufficient to prove that almost everywhere in (0, T ). Thus, we still have (2.31), of course with ∇w 2 in place as χ t 2 .

Proof of Theorem 2.4
We start by deducing an additional a-priori estimate. As before, we present it just in the conserved case, the variations in the non-conserved case being given at the end. Second estimate -local version. We test (2.3) by tu t = tϑ t /ϑ 2 and add the result to (2.6) multiplied by tw t . Then, we add also the time derivative of (2.7) multiplied by t χ t . We obtain d dt Then, noting that integrating (3.41) between 0 and t ∈ (0, 1] and taking advantage of (3.15), we obtain Second estimate -global version. We test (2.3) by u t = ϑ t /ϑ 2 and add the result to (2.6) multiplied by w t . Then, we add also the time derivative of (2.7) multiplied by χ t . Proceeding as above, we obtain d dt Integrating between 1 and t ≥ 1 and recalling (3.15) and (3.43), we infer Using again the logarithmic Poincaré inequality (Lemma 3.16), we also have i.e., the first (2.32).
In the non-conserved case, the procedure is similar. In place of (3.45) we rather obtain As a further consequence, we can look at equation (2.7) in the conserved case ((2.5) in the nonconserved case, respectively). Thanks to estimates (3.45)-(3.46) for u and w (respectively, to estimate (3.47) for u), applying standard regularity results for elliptic equations with monotone nonlinearities, we then obtain (2.33).
Conclusion of the proof. To obtain the uniform boundedness of u it is now sufficient to notice that, by the first (2.32), u(t) ∈ L 6 (Ω) for all t ≥ 1. Then, (3.50) holds. Moreover, (3.49) are a consequence of (3.45) (or, in the non-conserved case, (3.47)), which gives the required regularity for f = − χ t , and of (3.46), which gives the required regularity for u. The second (2.32) is then a consequence of the lemma (applied with the choice of S = 2). It is also worth noting that, at the level of V ′ -solution, the identification property in (3.48) needs not hold in the strong (pointwise) form (but just in the "weak" sense (2.30)). However, one can apply Lemma 3.3 at the n-regularized level, and then pass to the limit noting that the procedure yields n-uniform estimates.
To complete the proof we have to show (2.34) in the non-conserved case. Of course, such a property is significant only in the case when I, the domain of b, does not coincide with the real line, i.e., we are in presence of a singular potential (like the logarithmic one (1.1)). Otherwise, (2.34) is (also in the conserved case, of course) an immediate consequence of (2.33).
That said, let us prove the upper bound, the lower one working in a similar way. Being, by the second (2.32), |u(t, x)| ≤ U for some U = Q(E 0 ) and a.e. (t, x) ∈ (2, ∞) × Ω, we can then apply the comparison principle to (2.5). This gives that χ is bounded from above by the solution χ + to the forward Cauchy problem

Proof of Theorem 2.6
We start by giving the proof in the non-conserved case, the variations occurring in the conserved case being outlined at the end. Then, we know that any V ′ -solution (ϑ, u, χ ) satisfies the energy equality (2.31). Analogously, if (ϑ n , u n , χ n ) is the approximating solution constructed in the existence proof, the analogue of (2.31) reads t 0 ∇u n 2 + χ n,t 2 = J * (ϑ 0,n ) − J * (ϑ n (t)) + ϑ 0,n , 1 − ϑ n (t), 1 Our task is now to compute the supremum limit of (3.75) and compare it with (2.31). Then, we firstly observe that, by (3.27), the Aubin-Lions lemma, and lower semicontinuity of b, So, it remains to prove that lim sup nր∞ J * (ϑ 0,n ) ≤ J * (ϑ 0 ). (3.78) The proof of this fact is actually a bit more involved. We prepare a Lemma (Ω), j and j * being given by (2.12). Then, Proof.
Let z belong to the domain of J, namely let z ∈ V with J(z) < +∞. Then, for a.e. x ∈ Ω, by definition of subdifferential in R, we have (3.80) Integrating over Ω, we would formally get However, the integral on the left hand side could make no sense since the function zv could not belong to L 1 (Ω Passing to the supremum w.r.t. z varying in the domain of J, we then get the ≤ sign in (3.79).
To prove the converse, we first let, for v as in the statement and N ∈ (1, ∞), Then, for ǫ ∈ (0, 1), we regularize by singular perturbation as in (3.1): Then, where the first inequality follows from definition of conjugate function, the second equality from the fact that z ǫ,N is smooth and bounded, the third equality is trivial, the fourth inequality comes from (3.84), the convergence ε ց 0 from standard properties of elliptic systems, and the convergence N ր ∞ from Lebesgue's theorem. This proves the ≥ in (3.79) and the lemma.
Then, by definition of conjugate function (recall (2.12)), where the last inequality follows from (3.4). In particular, we have that −1 − log ϑ 0 ∈ L 1 (Ω). Thus, applying the above Lemma, we obtain Thus, in particular, we have that, for all t ∈ [0, T ], ϑ(t) ∈ L 1 (Ω) ∩ ∂ V,V ′ J(u(t)), whence the pointwise identification (2.4) follows from (2.17). This concludes the proof in the non-conserved case. Conserved case. To conclude the proof, we outline the differences occurring in the conserved case, which only regard the above L 1 -argument. Actually, the convergence of χ n,t in (3.89) is now replaced by ∇w n → ∇w, strongly in L 2 (0, T ; H). Of course, thanks to the properties of A this also gives χ n,t → χ t strongly in L 2 (0, T ; V ′ ), (3.94) which, however, is not sufficient to proceed as before. On the other hand, we can rely on estimates (3.43) and (3.47) which tell us that t 1/2 χ n,t L 2 (0,T ;V ) + t 1/2 χ n,t L ∞ (0,T ;V ′ ) ≤ c. for all ǫ ∈ (0, 1). Coming back to (3.91), we now have that, for all t ∈ [0, T ], (3.97) Then, integrating over (0, T ), we arrive at where the second inequality is a consequence of (3.96), the thirds follows from Hölder's inequality, and the fourth holds provided that we take ǫ so small that 4 < 3(2 − ǫ). In particular, using (3.94) (note that 8(1 − ǫ)/3(2 − ǫ) is smaller than 2 for ǫ as above), we obtain that the right hand side tends to 0 for large m and n. At this point the proof goes on like in the non-conserved case.

Proof of Theorem 2.7
To start, we need to prove some further a-priori estimates holding under the additional assumption (2.37). In particular, the key step will be that of showing that the L 3+ε regularity of ϑ is conserved uniformly in time. To show this, we will use L p -techniques in equation (2.3) (actually, with p = 3 + ε). However, doing this will require some care since, due to the low regularity of initial data, the "forcing term" χ t needs not belong to L 3+ε (Ω) for small values of the time variable. However, we will see that the L 3+ε -norm of χ t (t) explodes, as t ց 0, in a way which is sufficiently slow for our purpose. As before, the proof is detailed just in the conserved case. That said, we start with the Third estimate -local version. To start, we test (2.3) by ϑ 2+ε , to obtain and that 3 + 3ε where (3.43) has also been exploited. Note that, for ε ∈ (0, 1) the latter exponent lies in (−1, 0). Notice also that in the non-conserved case the exponents are even better since it is sufficient to interpolate between V and H (rather than between V and V ′ ). Thus, integrating (3.103) between 0 and 1, and using once more (3.43), we infer Third estimate -global version. As before, we test (2.3) by ϑ 2+ε . Taking now care also of the gradient term, we get d dt Adding also the inequality (which is true thanks to (3.15)) κ ϑ Thus, setting Y (t) := ϑ(t) 2 3+ε , we obtain from (3.107) where we wrote Q in place of Q(E 0 ). Now, let us set , (3.109) so that it is clear that Z satisfies, i.e., the global analogue of (3.104).
Conclusion of proof. Thanks to estimate (3.111), ϑ satisfies the first (3.113) for any S ≥ 1 (where M is the quantity on the right hand side of (3.111) which is independent of S). Analogously, we have ϑ(S) ∈ L 3+ε for (almost) all S ≥ 1. Moreover, combining (3.15) and (3.45), we have the second of (3.113), still with F independent of S. We then conclude applying the above Lemma over the generic interval (S, S + 2), with S ≥ 1.
Remark 3.6. Of course, with (2.32) and (2.38) at our disposal, equation (2.3) is both nonsingular and nondegenerate. Consequently, we can prove, with standard tools, further regularization properties of solutions. In the non-conserved case, thanks to (2.34), also the (possibly) singular character of b is lost. Thus, the smoothness of the solution is limited only by the differentiability properties of b. For instance, if b ∈ C ∞ , then also the solution is infinitely differentiable for strictly positive times.