Abstract
In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers \(A^{2r}\) and \(B^{2\sigma }\) (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space \(L^2(\Omega )\), for some bounded and smooth domain \(\Omega \subset {{\mathbb {R}}}^3\), and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter \(\sigma \) appearing in the operator \(B^{2\sigma }\) decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.
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1 Introduction
A research project that the three of us recently carried out in [15,16,17] deals with the well-posedness, regularity and optimal control for the abstract evolutionary system
where \(A^{2r}\) and \(B^{2\sigma }\), with \(r>0\) and \(\sigma >0\), denote fractional powers of the linear operators A and B, respectively. These operators are supposed to be densely defined in \(H:=L^{2}(\Omega )\), with \(\Omega \subset {{\mathbb {R}}}^3\), selfadjoint and monotone, and to have compact resolvents. The above system is a generalization of the standard or viscous Cahn–Hilliard system (depending on whether \(\tau =0\) or \(\tau >0\)), which models a phase separation process taking place in the container \(\Omega \). The particular sample case \(A^{2r}=B^{2\sigma }=-\Delta \) with homogeneous Neumann boundary conditions is included, indeed. The physical variables \(\,\varphi \,\) and \(\,\mu \,\) stand for the order parameter and the chemical potential, respectively, while f is a given source term. Moreover, \(\,F\,\) denotes a double-well potential. We offer three physically significant examples for F, namely,
where the constants \(c_i\) in (1.5) and (1.6) satisfy \(c_1>1\) and \(c_2>0\) in order that all the three functions \(F_\mathrm{reg}\), \(F_\mathrm{log}\), \(F_{2\mathrm{obs}}\) are nonconvex (they are just semiconvex, indeed). We point out that \(F_\mathrm{reg}\), \(F_\mathrm{log}\), \(F_{2\mathrm{obs}}\) are called the classical regular potential, the logarithmic potential, and the double obstacle potential, respectively. In irregular situations like (1.6), one has to split F into a nondifferentiable convex part \({{\widehat{\beta }}}\) (the indicator function of \([-1,1]\), in the case of (1.6)) and a smooth perturbation \({{\widehat{\pi }}}\). At the same time, one has to replace the derivative of the convex part by the subdifferential and to interpret (1.2) as a differential inclusion or, equivalently, as a variational inequality involving \({{\widehat{\beta }}}\) rather than its subdifferential, as actually done in [15].
Fractional versions of the Cahn–Hilliard system have been considered by different authors and are the subject of several papers. As for references regarding well-posedness and related problems, a rather large list of citations is given in [15]; we recall some concerned and recent literature also here, by mentioning [1, 2, 8, 21, 30, 33]. Moreover, one can find a number of results regarding the asymptotic behavior of solutions, for the standard Cahn–Hilliard equations, for variants thereof, and for systems including the Cahn–Hilliard equations: without any claim of completeness, we can quote, e.g., [3, 6, 9,10,11,12, 14, 18,19,20, 23,24,25, 31, 32, 34, 35]. These works mainly deal with the asymptotics with respect to parameters, or the study of the trajectories and related topics, or the existence of global or exponential attractors and their properties. A special role in our citations is played by the paper [13], where the longtime behavior of the solutions as well as an asymptotic analysis similar to the one we address here are investigated for a fractional system involving the Allen–Cahn equation.
In this paper, we consider the viscous case \(\tau >0\) within the system (1.1)–(1.3) and study the asymptotic behavior of the solution as the parameter \(\sigma \) involved in the operator \(B^{2\sigma }\) tends to zero. In this analysis, a crucial role is played by the orthogonal projection operator \(P:H\rightarrow H\) onto the kernel \(\ker B\) of B. Indeed, if \((\varphi _{\!\sigma },\mu _\sigma )\) denotes the solution to system (1.1)–(1.3) for an arbitrary \(\sigma >0\), we prove that \((\varphi _{\!\sigma },\mu _\sigma )\) converges as \(\sigma \searrow 0\) to a solution \((\varphi ,\mu )\) to the system
In general, the convergence occurs along a subsequence, but in the case when the limit pair \((\varphi ,\mu )\) uniquely solves (1.7)–(1.9), then the whole family \((\varphi _{\!\sigma },\mu _\sigma )\) converges to \((\varphi ,\mu )\) in the sense made precise by the statement of Theorem 2.5. Moreover, let us point out that the component \(\varphi \) of the pair \((\varphi ,\mu )\) is always uniquely determined, as it follows from the continuous dependence result given by Theorem 2.10. In the last part of the paper, we also discuss the limiting problem by proving a class of regularity results, quite interesting in our opinion, for which we have to use some sophisticated tools of interpolation theory. Our approach may be considered as an extension and further investigation with respect to the asymptotic results of [13, Section 7], in which a phase relaxation problem is obtained at the limit. Also in the present paper, Eq. (1.8) can be seen as an ordinary differential equation, but with a nonlocal structure due to the presence of the projection operator P. Our contribution here gives account of a new line of investigation that in our opinion should be further explored.
The rest of the paper is organized as follows: in the next Sect. 2, we list our assumptions and state our results. The corresponding proofs are given in the last two Sects. 3 and 4.
2 Statement of the problem and results
In this section, we state precise assumptions and notations and present our results. Our framework is the same as in [15], and we briefly recall it here, for the reader’s convenience. First of all, the open set \(\Omega \subset {{\mathbb {R}}}^3\) is assumed to be bounded, connected and smooth. We use the notation
and denote by \(\mathopen \Vert \,\cdot \,\mathclose \Vert \) and \((\,\cdot \,,\,\cdot \,)\) the standard norm and inner product of H. As for the operators involved in our system, we postulate that
We denote by \(\{\lambda _j\}\) and \(\{\lambda '_j\}\) the nondecreasing sequences of the eigenvalues of \(\,A\,\) and \(\,B\), and by \(\{e_j\}\) and \(\{e'_j\}\) the corresponding (complete) systems of orthonormal eigenvectors, that is,
where \(\delta _{ij}\) denotes the Kronecker index. The power \(A^r\) of A with an arbitrary positive real exponent r is given by
In principle, we could endow \(V_A^{r}\) with the standard graph norm in order to make \(V_A^{r}\) a Hilbert space. However, we will choose an equivalent Hilbert structure later on. In the same way, for \(\sigma >0\), we define the power \(B^\sigma \) of B. For its domain, we use the notation
At this point, we can start listing our assumptions. First of all,
As for the linear operators, we postulate, besides (2.2), that
In [15], some remarks are given on the above assumptions. Moreover, it is shown that an equivalent Hilbert structure on \(V_A^{r}\) is obtained by taking the norm defined by
and the corresponding inner product, which we term \((\,\cdot \,,\,\cdot \,)_{A,r}\). This equivalence is trivial if \(\lambda _1>0\). In the opposite case \(\lambda _1=0\), with the notation
for the mean value of the generic function v, the equivalence relies on the inequality
which is of Poincaré type, since the term \((v,e_1)\) appearing in (2.11) and involving the constant function \(e_1\) (see (2.9)) is proportional to \({\mathrm{mean}}\,v\). Next, the nonlinear potential F appearing in (1.2) is split as follows:
Notice that these assumptions are fulfilled by all of the important potentials (1.4)–(1.6). We set, for convenience,
Moreover, we term \(D({{\widehat{\beta }}})\) and \(D(\beta )\) the effective domains of \({{\widehat{\beta }}}\) and \(\beta \), respectively, and notice that \(\beta \) is a maximal monotone graph in \({{\mathbb {R}}}\times {{\mathbb {R}}}\). The same symbol \(\beta \) is used for the maximal monotone operators induced in \(L^{2}(\Omega )\) and \(L^{2}(Q)\). Finally, we introduce
As for the data of our problem, we allow the forcing term appearing in (1.2) to depend on \(\sigma \) and assume that:
At this point, we make the notion of solution precise. In the following, we use the notations
A solution to our system is a pair \((\varphi _{\!\sigma },\mu _\sigma )\) fulfilling the regularity requirements
and satisfying the following weak formulation of Eqs. (1.1)–(1.3):
We notice that (2.23) implies that \({{\widehat{\beta }}}({\varphi _{\!\sigma }}(t))\in L^{1}(\Omega )\) \(\hbox {for a.a.}~t\in (0,T)\), so that (2.25) has a precise meaning. In the same inequality, one obviously has to read \(\int _\Omega {{\widehat{\beta }}}(v)=+\infty \) if \(v\in V_B^{\sigma }\) and \({{\widehat{\beta }}}(v)\not \in L^{1}(\Omega )\).
Remark 2.1
The regularity (2.22) of the second component of the solution is expected even though (2.24) just suggests \(\mu _\sigma \in L^{2}(0,T;V_A^{r})\). Indeed, \(\hbox {for a.a.}~t\in (0,T)\) the variational equation has the form
with \(g\in H\). From this, one easily derives that \(u\in V_A^{2r}\) and \(\mathopen \Vert A^{2r}u\mathclose \Vert \le \mathopen \Vert g\mathclose \Vert \) (one can formally test by \(A^{2r}u\), but a regularization procedure makes the argument rigorous). Since \(\partial _t\varphi _{\!\sigma }\in L^{2}(0,T;H)\), we thus have the regularity (2.22) as well as
i.e., the equation holds in its strong form.
Remark 2.2
In the sequel, the symbol \(\mathbf{1}\) denotes the constant function on \(\Omega \) that takes the value 1 at every point. With this notation, we remark that (2.9) implies that \(A^r(\mathbf{1})\) vanishes if \(\lambda _1=0\), so that (2.24) and (2.26) imply that
in this case. On the contrary, if \(\lambda _1>0\), no conservation property is expected.
The well-posedness result (cf. [15, Thm. 2.6]) reads as follows:
Theorem 2.3
Let the assumptions (2.2), (2.8)–(2.10), and (2.14) on the structure of the system and (2.17)–(2.19) on the data be fulfilled. Then, there exists a pair \((\varphi _{\!\sigma },\mu _\sigma )\) satisfying (2.21)–(2.23) and solving problem (2.24)–(2.26). Moreover, the component \(\varphi _{\!\sigma }\) of the solution is unique.
Remark 2.4
No uniqueness for the component \(\mu _\sigma \) of the solution can be expected, in general. However, in particular situations, \(\mu _\sigma \) is unique, too. This is the case if \(\lambda _1>0\). Indeed, this assumption implies that \(A^{2r}\) is invertible so that (2.27) can be uniquely solved for \(\mu _\sigma \). On the contrary, the case \(\lambda _1=0\) is much more delicate. A sufficient condition that ensures uniqueness for \(\mu _\sigma \) is the following (see [15, Rem. 4.1]): \({{\widehat{\beta }}}\) is an everywhere defined \(C^1\) function and \(\varphi _{\!\sigma }\) is bounded. We notice that the same argument used in the quoted remark also applies if \(D(\beta )\) is an open interval and \(\beta \) is a continuous single-valued function on it (like in the case (1.5) of the logarithmic potential) provided that all of the values of \(\varphi _{\!\sigma }\) belong to a compact subset of \(D(\beta )\).
Let us come to the results of this paper. The first deals with the behavior of the solutions to problem (2.24)–(2.26) as \(\sigma \) tends to zero.
Theorem 2.5
Besides the assumptions of Theorem 2.3, assume that
Then, for every \(\sigma >0\) there is a solution \((\varphi _{\!\sigma },\mu _\sigma )\) to problem (2.24)–(2.26) such that
as \(\sigma \searrow 0\), possibly along a subsequence, for some triplet \((\varphi ,\mu ,\zeta )\) satisfying
Moreover, under the additional assumption,
the following holds true: whenever \((\varphi _{\!\sigma },\mu _\sigma )\) is a solution to problem (2.24)–(2.26) for \(\sigma >0\) and (2.30)–(2.32) hold for some triplet \((\varphi ,\mu ,\zeta )\), then \(\zeta =\varphi -P\varphi \) and the pair \((\varphi ,\mu )\) is a solution to the system
Remark 2.6
The above statement looks a little involved. Besides the assumption (2.34) we are going to discuss in a while, we point out that no uniqueness for the solution \((\varphi _{\!\sigma },\mu _\sigma )\) is required. On the contrary, if additional assumptions were made that guarantee uniqueness for \((\varphi _{\!\sigma },\mu _\sigma )\) (see Remark 2.4) and (2.34) were assumed, then the statement would look much simpler, namely: as \(\sigma \) tends to zero, the solution \((\varphi _{\!\sigma },\mu _\sigma )\) converges (in the sense of (2.30)–(2.31), possibly along a subsequence) to a solution \((\varphi ,\mu )\) to problem (2.35)–(2.37). If, in addition, uniqueness holds for the solution \((\varphi ,\mu )\) to the limiting problem, then the whole family \(\{(\varphi _{\!\sigma },\mu _\sigma )\}\) converges to \((\varphi ,\mu )\) as \(\sigma \) tends to zero.
Remark 2.7
As observed in the forthcoming Remark 3.3, if (2.34) is not assumed, a weaker conclusion can anyway be obtained: the variational inequality (2.36) is fulfilled by all the test functions \(v\in V_B^{\sigma _0}\). Indeed, it is stressed in the remark that assumption (2.34) is used in the proof of Theorem 2.5 just to extend to any \(v\in H\) the validity of (2.36) proved for test functions \(v\in V_B^{\sigma _0}\).
Remark 2.8
So, if (2.34) is assumed, then every limiting pair \((\varphi ,\mu )\) satisfies (2.36) with arbitrary test functions \(v\in H\). This has the following important consequence: there exists some \(\xi \) satisfying
Indeed, if we set
then \(\xi \) belongs to \(L^{2}(0,T;H)\), Eq. (2.39) is satisfied, and (2.36) becomes
But this exactly means that \(\xi (t)\in \partial {{\widehat{\beta }}}(\varphi (t))=\beta (\varphi (t))\) \(\hbox {for a.a.}~t\in (0,T)\), i.e., the second condition in (2.38). If instead (2.34) is not assumed, then (2.36) is satisfied only for test functions \(v\in V_B^{\sigma _0}\), as said in Remark 2.7. Nevertheless, the definition (2.40) still yields \(\xi \in L^{2}(0,T;H)\) and implies that (2.39) is satisfied. However, in this case, (2.41) is only true for \(v\in V_B^{\sigma _0}\), and this means that \(\hbox {for a.a.}~t\in (0,T)\) the function \(\xi (t)\) belongs to the subdifferential of the function \(V_B^{\sigma _0}\ni v\mapsto \int _\Omega {{\widehat{\beta }}}(v)\). Notice that this subdifferential is a subset of the dual space \((V_B^{\sigma _0})^*\) and might contain elements that do not belong to H (in the sense of the Hilbert triplet \({(V_B^{\sigma _0},H,(V_B^{\sigma _0})^*)}\,\)). Moreover, if a function \(u\in H\) belongs to such a subdifferential, then it is not clear whether it also belongs to the subdifferential in H (i.e., that of the function \(H\ni v\mapsto \int _\Omega {{\widehat{\beta }}}(v)\)), so that we cannot conclude that \(\xi \in \beta (\varphi )\) \(\hbox {a.e. in}~Q\). About this matter, let us quote the paper [5] for related issues.
Remark 2.9
A sufficient condition for (2.34) to hold true is the following (satisfied in all of the concrete cases, at least if \(\sigma _0\) is small enough):
In order to construct the sequence \(\{v_n\}\) for a given \(v\in H\), we solve the Neumann boundary value problem
Since \(v\in H\), we have that \(v_n\in H^{2}(\Omega )\) and thus \(v_n\in V_B^{\sigma _0}\), by (2.42). Now, if we take \(z=v_n\) in (2.43) and use the Cauchy–Schwarz inequality in the right-hand side, then we easily find that
Hence, there are a subsequence \(\{v_{n_k}\}\) and some \(w\in H\) such that \(v_{n_k}\rightarrow w\) weakly in H. Moreover, since \(\frac{1}{n} \nabla v_n\rightarrow ( 0,0,0)\) strongly in \(H\times H\times H\) by (2.44), we easily infer from (2.43) that \(w=v\). A fortiori, by the uniqueness of the limit point, the entire sequence \(\{v_n\}\) converges weakly in H to v. But then, by the weak sequential lower semicontinuity of norms,
where the latter inequality follows from (2.44). We thus have \(\Vert v\Vert =\lim _{n\rightarrow \infty }\Vert v_n\Vert \), and the uniform convexity of H yields that \(v_n\rightarrow v\) strongly in H.
Now, denoting by \({{\widehat{\beta }}}_\varepsilon \) and \(\beta _\varepsilon \) the Moreau–Yosida \(\varepsilon \)-approximations of \({{\widehat{\beta }}}\) and \(\beta \), respectively (see, e.g., [7, p. 28]), we account for the definition of the subdifferential \(\beta _\varepsilon =\partial {{\widehat{\beta }}}_\varepsilon \) and the identity obtained by testing (2.43) by \(\beta _\varepsilon (v_n)\in H^{1}(\Omega )\). We have that
and we deduce that
by letting \(\varepsilon \) tend to zero. This implies the inequality “\(\le \)” in (2.34). Since the opposite inequality clearly follows from the lower semicontinuity of the function \(z\mapsto \int _\Omega {{\widehat{\beta }}}(z)\) in H, we finally deduce the validity of (2.34).
Notice that Theorem 2.3 ensures the existence of at least one solution to the limiting problem (2.35)–(2.37) with the regularity specified in (2.33). Our next result deals with partial uniqueness and continuous dependence of the solution. This will be proved in the last Sect. 4, which is devoted to the study of the limiting problem.
Theorem 2.10
Let the general assumptions on the structure be fulfilled, and assume that \(\varphi _0\) satisfies (2.18). Moreover, let \(f_i\in L^{2}(0,T;H)\), \(i=1,2\), be two choices of the forcing term f appearing in (2.36), and let \((\varphi _i,\mu _i)\in H^{1}(0,T;H)\times L^{2}(0,T;V_A^{2r})\) be two corresponding solutions to problem (2.35)–(2.37) with \(f=f_i\). Then, we have
with a constant \(C_{cd}\) that depends only on \(\tau \), the Lipschitz constant \(L_\pi \), and T. In particular, if \(f\in L^{2}(0,T;H)\), the first component \(\varphi \) of the solution \((\varphi ,\mu )\) to problem (2.35)–(2.37) is uniquely determined.
In our final result, we require some regularity of the data and further assumptions on the structure that are satisfied in all of the concrete cases of interest, and we prove a regularity result. As a byproduct, we obtain a sufficient condition for the uniqueness of the second component \(\mu \) of the solution. Sufficient conditions for uniqueness in a different direction are given in the forthcoming Remark 4.5.
Theorem 2.11
Let the general assumptions on the structure be fulfilled. In addition, assume that
and let \((\varphi ,\mu )\) with
be a solution to problem (2.35)–(2.37). Then, \(\varphi \) enjoys the further regularity
and there exists some \(\xi \) satisfying (2.38)–(2.39). In particular, even the second component \(\mu \) of the solution is unique if \(\beta \) is single-valued.
Throughout the paper, we widely use the Cauchy–Schwarz and Young inequalities, the latter in the form
Moreover, in performing a priori estimates, we use the same small letter c for (possibly) different constants that depend only on the structure of our system but \(\sigma \), and on the assumptions on the data. In particular, the values of c do not depend on the regularization parameter \(\lambda \) we introduce in the next section. On the contrary, some precise constants are denoted by different symbols (see, e.g., (2.13), where a capital letter with an index is used).
3 Asymptotic analysis
This section is devoted to the proof of Theorem 2.5. The construction of the solutions \((\varphi _{\!\sigma },\mu _\sigma )\) mentioned in the statement relies on a priori estimates on the solutions to a regularized problem, as done in [15] to solve problem (2.24)–(2.26) with a fixed \(\sigma \). Hence, we briefly recall that regularization procedure. For \(\lambda >0\) (small enough if needed), let \(\beta _\lambda \) be the Yosida approximation of \(\beta \) at the level \(\lambda \) (see, e.g., [7, p. 28]). The corresponding Moreau regularization \({{\widehat{\beta }}}_\lambda \) of \({{\widehat{\beta }}}\) is thus given by
since \(\beta _\lambda (0)=0\,\) due to (2.14). Then, the regularized problem is to find a pair \((\varphi _{\!\sigma }^\lambda ,\mu _\sigma ^\lambda )\) satisfying the regularity requirements
and solving the following system:
We notice that the variational inequality (2.25) is replaced by the equality (3.3) in the approximating problem (since \(\beta _\lambda \) is an everywhere defined Lipschitz continuous function). The existence part of Theorem 2.3 is proved by solving the above regularized problem (cf. [15, Thm. 5.1]) and showing that its solution \((\varphi _{\!\sigma }^\lambda ,\mu _\sigma ^\lambda )\) converges as \(\lambda \searrow 0\) (in a suitable topology, possibly just along a subsequence) to a pair \((\varphi _{\!\sigma },\mu _\sigma )\) which turns out to solve problem (2.24)–(2.26). This solution, where now \(\sigma \) is a varying parameter that we intend to approach zero, will be the good candidate for Theorem 2.5.
Before starting to estimate, it is worth observing that Remark 2.1 applies to both equations (3.2) and (3.3). This is obvious for the former. As far as the latter is concerned, one has to replace A and r by B and \(\sigma \), respectively, and notice that \(\beta _\lambda \) is Lipschitz continuous, so that \(\mu _\sigma ^\lambda +f_{\!\sigma }-\beta _\lambda (\varphi _{\!\sigma }^\lambda )-\pi (\varphi _{\!\sigma }^\lambda )\in L^{2}(0,T;H)\). This justifies the last regularity condition for \(\varphi _{\!\sigma }^\lambda \) in (3.1) (in contrast to (2.21)) and implies the strong form of both equations, i.e.,
We also recall the convention on the symbol c for possibly different constants made at the end of Sect. 2. Moreover, since (2.29) implies that \(f_{\!\sigma }\) is bounded in \(L^{2}(0,T;H)\), we allow c to also depend on a bound for the corresponding norm.
First a priori estimate We test (3.2) written at the time s by \(\mu _\sigma (s)\). At the same time, we insert \(+ \varphi _{\!\sigma }^\lambda (s) \) to both sides of (3.6) written at the time s and multiply it by \(\partial _t\varphi _{\!\sigma }^\lambda (s)\), then integrating over \(\Omega \). We sum up both equalities, noting that a cancellation occurs, and integrate over (0, t) with respect to s. We obtain
Even the last integral on the left-hand side is nonnegative. We estimate the terms on the right-hand side by accounting for the assumptions (2.18) and (2.29) on \(\varphi _0\) and \(f_{\!\sigma }\), respectively, and owing to the Lipschitz continuity of \(\pi \). Recalling also (2.7), we have that
Therefore, by rearranging and applying the Gronwall lemma, we conclude that
From this and (3.5), we deduce that
Second a priori estimate Our aim is to improve the estimate concerning \(\mu _\sigma ^\lambda \). Indeed, for the following we need that
We notice at once that this and (3.8) would imply that
The desired estimate trivially follows from (3.7) if \(\lambda _1>0\). So, we now deal with the other case \(\lambda _1=0\) and apply a well-known trick based on the assumption (2.19) and the subsequent inequality
which holds for some \(C_0>0\) and every \(s\in {{\mathbb {R}}}\) and \(\lambda \in (0,1)\), where \(\delta _0\) is such that the interval \([m_0-\delta _0,m_0+\delta _0]\) is included in the interior of \(D(\beta )\) (cf. [28, Appendix, Prop. A.1]; see also [22, p. 908] for a detailed proof). Inequality (3.11) implies that
and this can be used when testing equation (3.3) by \({\varphi _{\!\sigma }^\lambda }-m_0{\mathbf{1}}\). To this concern, we recall that \(\mathbf{1}\in V_B^{\sigma }\) by (2.10) and notice that the conservation property (2.28) also holds for \(\varphi _{\!\sigma }^\lambda \), i.e., \({\mathrm{mean}}\,\varphi _{\!\sigma }^\lambda (t)=m_0\) for every \(t\in [0,T]\). So, \(\hbox {for a.a.}~t\in (0,T)\), we test (3.3) by \(\varphi _{\!\sigma }^\lambda (t)-m_0{\mathbf{1}}\) and make some minor adjustments. However, we omit writing the time t for a while. We also write k instead of \(k\mathbf{1}\) if k is a real number. We have \(\hbox {a.e. in}~(0,T)\) that
The left-hand side of this equality can be estimated from below by virtue of (3.12). The first term on the right-hand side can be dealt with by accounting for the Poincaré-type inequality (2.13) as follows:
the last equality being due to \(A^r\mathbf{1}=0\). Therefore, by recalling (3.7), we have that the whole right-hand side of (3.13) is bounded in \(L^2(0,T)\) and conclude that
At this point, we can test the second equation (3.3) by \(\mathbf{1}\) and deduce a bound for \({\mathrm{mean}}\,\mu _\sigma ^\lambda \) in \(L^2(0,T)\). This and (3.7) imply (3.9). As already noticed, (3.10) is proved as well.
First conclusion As already remarked, in the proof of [15, Thm. 5.1] with a fixed \(\sigma \) it is shown that \((\varphi _{\!\sigma }^\lambda ,\mu _\sigma ^\lambda )\) converges as \(\lambda \) tends to zero (in a proper topology, possibly along a subsequence) to some pair \((\varphi _{\!\sigma },\mu _\sigma )\), and it is proved that such a pair is a solution to problem (2.24)–(2.26). We prove that the family \(\{(\varphi _{\!\sigma },\mu _\sigma )\}_{\sigma >0}\) constructed in this way satisfies all the requirement of the statement. The starting point is the conservation of the bounds just proved in the limit as \(\lambda \searrow 0\). We have that
and we conclude that (2.30)–(2.32) hold true for some triplet \((\varphi ,\mu ,\zeta )\) satisfying (2.33). This ends the proof of the first part of the statement.
Let us come to the second part. So, we assume that \(\{(\varphi _{\!\sigma },\mu _\sigma )\}_{\sigma >0}\) is a family of solutions to problem (2.24)–(2.26) and that (2.30)–(2.32) hold true for some triplet \((\varphi ,\mu ,\zeta )\) satisfying (2.33) as \(\sigma \searrow 0\), possibly for a subsequence (however, we always write \(\sigma \) instead of the elements of some subsequence \(\{\sigma _k\}\), for brevity). We have to prove that \(\zeta =\varphi -P\varphi \) and that \((\varphi ,\mu )\) solves problem (2.35)–(2.37), by also assuming (2.34).
First characterization We are going to show that \(\zeta =\varphi -P\varphi \) by proving that
To this end, we use the eigenvalues \(\lambda '_j\) and the eigenfunctions \(e'_j\) of B and notice that \(e'_j\) is orthogonal to \(\ker B\) if \(\lambda '_j>0\) while \(\lambda '_j=0\) if \(e'_j\in \ker B\). We set, for convenience,
for \(\psi \in L^2(0,T)\) and \(j=1,2,\dots \), and we notice that (3.14) follows if we prove that
for every \(\psi \) and j as before, since the linear combinations of the products \(\psi \,e'_j\) of such real functions and eigenfunctions of B form a dense subspace of \(L^{2}(Q)\). So, we fix \(\psi \) and j. As for j, we distinguish two cases. Assume first that \(\lambda '_j>0\). Then, \((\lambda '_j)^\sigma \) tends to 1 as \(\sigma \) tends to zero. Moreover, (2.30) holds. We thus deduce that
the last equality being due to the orthogonality between \(P\varphi (t)\) and \(e'_j\). Assume now that \(\lambda _j=0\). Then, we trivially have that \(A^\sigma _j(\psi )=0\) for every \(\sigma >0\). On the other hand, we also have that \(A^0_j(\psi )=0\) since \(e'_j\in \ker B\) and \(\varphi (t)-P\varphi (t)\) is orthogonal to \(\ker B\) \(\hbox {for a.a.}~t\in (0,T)\). Therefore, (3.15) is proved in any case.
Remark 3.1
The same argument shows that, for every \(v\in L^{2}(0,T;V_B^{\sigma _0})\), the weak convergence \(B^\sigma v\rightarrow v-Pv\) in \(L^{2}(0,T;H)\) holds true as \(\sigma \) tends to zero. In fact, the convergence is strong:
Indeed, \(\hbox {for a.a.}~t\in (0,T)\), \(B^\sigma v(t) \rightarrow v(t)-Pv(t)\) strongly in H by [13, Lem. 7.5]. Moreover, the Lebesgue dominated convergence theorem can be applied since
\(\hbox {for a.a.}~t\in (0,T)\) and every \(\sigma \in (0,\sigma _0]\), and \(\mathopen \Vert v(\,\cdot \,)\mathclose \Vert _{B,{\sigma _0}}^2\) belongs to \(L^1(0,T)\).
To conclude the proof, we have to show that \((\varphi ,\mu )\) solves problem (2.35)–(2.37) under the further assumption (2.34). The first equation obviously follows from (2.24) due to (2.30)–(2.31), and the initial condition (2.37) is satisfied as well since (2.30) implies weak convergence in \(C^{0}([0,T];H)\). So, it remains to verify the variational inequality (2.36). To this concern, it is convenient to give different formulations of both (2.25) and (2.36). This procedure is based on the lemma stated below, which follows from the classical theory of variational inequalities of elliptic type in the framework of Convex Analysis.
Lemma 3.2
Let V be a Hilbert space, \(V^*\) its dual space, \(\mathopen \langle \,\cdot \,,\,\cdot \,\mathclose \rangle \) the duality pairing between \(V^*\) and V, and \(a:V\times V\rightarrow {{\mathbb {R}}}\) a continuous bilinear form. Moreover, assume that
Then, for every \(u\in V\) and \(g\in V^*\), the variational inequalities
are equivalent to each other.
As announced, we use this lemma to replace both (2.25) and (2.36) by different variational inequalities.
First alternative formulation We first observe that (2.25) for every \(v\in V_B^{\sigma }\) as required implies the same inequality for every \(v\in V_B^{\sigma _0}\) since \(V_B^{\sigma _0}\subset V_B^{\sigma }\). Now, by recalling that \(L_\pi \) is the Lipschitz constant of \(\pi \), we replace the latter variational inequality by an equivalent one by applying the lemma above, with the choices
Notice that \({{\widehat{\gamma }}}_2\) actually is convex (since \(\pi '+L_\pi \ge 0\) a.e. in \({{\mathbb {R}}}\)) and Gâteaux differentiable and that its derivative \(\gamma _2\) is given by \(\mathopen \langle \gamma _2(u),v\mathclose \rangle =(\pi (u)+L_\pi u,v)\). Hence, we deduce that the variational inequality (2.25) required just for every \(v\in V_B^{\sigma _0}\) is equivalent to
where, for brevity, we have set
We fix what we have established:
Second alternative formulation Similarly, we would like to show that (2.36) is equivalent to
Unfortunately, this does not seem to be true, in general, and we prove the following:
To this aim, it suffices to apply the lemma with the same \({{\widehat{\gamma }}}_i\) as before and obvious u and g, but with \(V=V_B^{\sigma _0}\) and a defined by \(a(u,v):=(u-Pu,v)-L_\pi (u,v)\) for \(u,v\in V_B^{\sigma _0}\).
Conclusion of the proof In view of (3.23) and (3.25), our aim is first to verify (3.24) by starting from (3.21) (implied by (2.25)), while (2.36), as it is, will be proved at the end by accounting for (2.34). However, the left-hand side of (3.21) contains the quadratic term associated with the map \(v\mapsto -L_\pi \int _\Omega |v|^2\). This term is unpleasant since the related map is concave. To get rid of it, we adapt the procedure introduced in [13] to the present case. We set, for convenience,
and we notice that \(w\mapsto \smash {\int _Qe^{-2\kappa t}w^2}\) is the square of an equivalent norm in \(L^{2}(Q)\). At this point, we pick an arbitrary \(v\in L^{2}(0,T;V_B^{\sigma _0})\), write (3.21) by taking v(t) as test function, multiply by \(e^{-2\kappa t}\), and integrate over (0, T). We obtain
Well, we want to take the limit as \(\sigma \) tends to zero in this inequality. As for the first term on the left-hand side, we have that
By observing that \(\rho _\sigma \) converges to \(\rho \) weakly in \(H^{1}(0,T;H)\), thus weakly in \(C^{0}([0,T];H)\), so that \(\rho _\sigma (T)\) converges to \(\rho (T)\) weakly in H, we therefore have that
Next, by (3.14) and the lower semicontinuity of the norms, we have that
By also recalling (3.16), we can write
By taking the difference, we deduce that
the last equality being due to the orthogonality between \((\varphi -P\varphi )(t)\in (\ker B)^\perp \) and \((P\varphi -Pv)(t)\in \ker B\). Moreover, by observing that the functional \(w{{}\mapsto {}}\int _Qe^{-2\kappa t}{{\widehat{\alpha }}}(w)\) is lower semicontinuous on \(L^{2}(Q)\), and recalling that \(\varphi _{\!\sigma }\) converges to \(\varphi \) weakly in \(L^{2}(Q)\), we deduce that
This ends the treatment of the terms on the left-hand side of (3.27). Concerning the right-hand side, we have to overcome the difficulty due to the coupling between \(\mu _\sigma \) and \(\varphi _{\!\sigma }\). To this end, we introduce the notation
and deduce from (2.27) that
Hence, we have that
Now, from (2.31) we deduce that \(1*\mu _\sigma \) converges to \(1*\mu \) weakly in \(H^{1}(0,T;V_A^{2r})\). Since the embedding \(H^{1}(0,T;V_A^{2r})\subset L^{2}(0,T;V_A^{r})\) is compact, we infer that \(1*\mu _\sigma \) converges to \(1*\mu \) strongly in \(L^{2}(0,T;V_A^{r})\). In view of (2.35) and (2.37), we deduce that
Finally, by recalling (2.29), we see that the term involving \(f_{\!\sigma }\) and the last one of (3.27) do not give any trouble. Therefore, we conclude that
and this holds for every \(v\in L^{2}(0,T;V_B^{\sigma _0})\). On the other hand, (3.28) is equivalent to
\(\hbox {for a.a.}~t\in (0,T)\) and every \(v\in V_B^{\sigma _0}\). By multiplying by \(e^{2\kappa t}\) and recalling that \(\kappa =L_\pi /\tau \), we obtain (3.24) as claimed. Recalling (3.25), we have proved that the variational inequality (2.36) is satisfied for every test function \(v\in V_B^{\sigma _0}\). At this point, we account for (2.34), not yet used up to now, and show that (2.36) actually holds for every \(v\in H\). To this end, for a given \(v\in H\) with \({{\widehat{\beta }}}(v)\in L^{1}(\Omega )\) without loss of generality, it suffices to take a sequence \(\{v_n\}\) given by (2.34), test (3.24) by \(v_n\) and let n tend to infinity. One obtains (3.24) for v without any trouble. This completes the proof.
Remark 3.3
Going back to the above proof, one justifies what has been announced in Remark 2.7: if (2.34) is not assumed, one anyway arrives at the variational inequality (2.36) required for every \(v\in V_B^{\sigma _0}\) instead of for every \(v\in H\). Indeed, (2.34) has been only used at the end, in order to extend to any \(v\in H\) the validity of (2.36) already proved for test functions \(v\in V_B^{\sigma _0}\).
4 The limiting problem
In this section, we prove Theorems 2.10 and 2.11. As far as the former is concerned, some preliminaries are needed. We refer to [15, Sect. 3] for more details. We set
and we use the symbol \(\mathopen \langle \,\cdot \,,\,\cdot \,\mathclose \rangle _{A,r}\) for the duality pairing between \(V_A^{-r}\) and \(V_A^{r}\). It is understood that H is identified with a subspace of \(V_A^{-r}\) in the usual way, i.e., in order that \(\mathopen \langle v,w\mathclose \rangle _{A,r}=(v,w)\) for every \(v\in H\) and \(w\in V_A^{r}\). Moreover, we introduce the subspaces \(V_0^{\pm r}\) of \(V_A^{\pm r}\) by setting
Next, we define \(A_0^{2r}:V_0^{r}\rightarrow V_A^{-r}\) by the formula
It turns out that the range of \(A_0^{2r}\) is \(V_0^{-r}\) and that \(A_0^{2r}\) is an isomorphism between \(V_0^{r}\) and \(V_0^{-r}\). Thus, we can set \(A_0^{-2r}:=(A_0^{2r})^{-1}\) and obtain an isomorphism between \(V_0^{-r}\) and \(V_0^{r}\). It also turns out that
Finally, the following formula holds true:
In particular,
Proof of Theorem 2.10
We just prove the continuous dependence part, since uniqueness for the first component follows as a consequence. We set, for convenience, \(f:=f_1-f_2\), \(\varphi :=\varphi _1-\varphi _2\), and \(\mu :=\mu _1-\mu _2\). Now, we write equation (2.35) at the time s for these solutions and take the difference. Then, we test the resulting identity by \(\,v=A_0^{-2r}\varphi (s)\), where we observe that \(\varphi (s)\in V_0^{-r}\), since \(\varphi \in C^{0}([0,T];H)\) by (2.33) and \({\mathrm{mean}}\,\varphi (s)=0\) if \(\lambda _1=0\) by the conservation property (2.28), so that v is a well-defined element of \(V_A^{r}\). Moreover, we have that \(A_0^{-2r}\varphi \in L^{\infty }(0,T;V_A^{r})\). Integrating over (0, t) with respect to s, where \(t\in (0,T)\) is arbitrary, we obtain the identity
Now, the first term on the left-hand side is nonnegative by (4.2). Hence, by also noting that \(\mu \in L^{2}(0,T;V_A^{r})\) and applying (4.1), we deduce that
At the same time, we write (2.36) for \(f_i\) and \((\varphi _i,\mu _i)\), \(i=1,2\), test them by \(\varphi _2\) and \(\varphi _1\), respectively, add the resulting inequalities to each other, and integrate over (0, t) as before. Then, the terms involving \({{\widehat{\beta }}}\) cancel out. By denoting by I the identity map of H and rearranging, we have that
We observe that \(I-P\) is the projection operator on the orthogonal subspace \((\ker B)^\perp \). It follows that \(((I-P)v,v)=((I-P)v,(I-P)v)\ge 0\) for every \(v\in H\), so that the second term on the left-hand side of (4.4) is nonnegative. By adding (4.3) and (4.4) to each other, and accounting for this observation, an obvious cancellation, the Lipschitz continuity of \(\pi \) and the Schwarz and Young inequalities, we deduce that
By applying the Gronwall lemma, we conclude that the desired estimate (2.45) holds true with a constant \(C_{cd}\) as in the statement. \(\square \)
Finally, we prove Theorem 2.11. The proof we give is based on the study of the auxiliary problem of finding \(\phi \in H^{1}(0,T;H)\) satisfying
for given
We have subtracted the constant \(\pi (0)\) from \({\pi }(\phi (t))\) in (4.5) in order to use the inequality \(|\pi (s)-\pi (0{)}|\le L_\pi \,|s|\) for \(s\in {{\mathbb {R}}}\) without any additive constant. This is needed in the sequel, indeed. Since \({{\widehat{\beta }}}\) is convex, P is linear and \(\pi \) is Lipschitz continuous, this problem has a unique solution \(\phi \) provided that the initial datum also satisfies
In the forthcoming Lemma 4.2, we prove a regularity result by applying a particular case of [29, Sect. I, Thm. 2] which we present here in the form of a lemma.
Lemma 4.1
Let \({\mathcal {A}}_0\), \({\mathcal {A}}_1\), \({\mathcal {B}}_0\) and \({\mathcal {B}}_1\) be four Banach spaces with the continuous embeddings \({\mathcal {A}}_0\subset {\mathcal {A}}_1\) and \({\mathcal {B}}_0\subset {\mathcal {B}}_1\), and let \({\mathcal {T}}:{\mathcal {A}}_1\rightarrow {\mathcal {B}}_1\) be a nonlinear operator satisfying \({\mathcal {T}}v\in {\mathcal {B}}_0\) for every \(v\in {\mathcal {A}}_0\). Assume that
for some positive constants \(C_1\) and \(C_2\). Then, for every \(\vartheta \in (0,1)\) and \(p\in [1,+\infty ]\), we have that
with a constant C that does not depend on \({\mathcal {T}}\).
In the above lemma, the symbol \(\mathopen \Vert \,\cdot \,\mathclose \Vert _X\) stands for the norm in the generic Banach space X. The same convention is followed in the rest of the section, where \(\mathopen \Vert \,\cdot \,\mathclose \Vert _X\) also denotes the norm in the power \(X^3\) (however, we keep the short notation \(\mathopen \Vert \,\cdot \,\mathclose \Vert \) without indices if \(X=H\)). Moreover, \((X,Y)_{\vartheta ,p}\) is the real interpolation space between the Banach spaces X and Y (for basic definitions and properties see, e.g., [27, Sect. 1.1]).
Lemma 4.2
Let the general assumption on the structure be fulfilled and assume that the data g and \(\phi _0\) satisfy
for some \(\eta \in {{}(0,1]}\), as well as (4.8). Then, the solution \(\phi \) to problem (4.5)–(4.6) enjoys the further regularity
and there exists some \(\xi \) satisfying
Proof
By still denoting by \({{\widehat{\beta }}}_\lambda \) and \(\beta _\lambda \) the Moreau–Yosida approximations of \({{\widehat{\beta }}}\) and \(\beta \), respectively, we introduce the approximating problem of finding \(\phi _\lambda \in H^{1}(0,T;H)\) that satisfies
and the initial condition (4.6). For any data satisfying (4.7) (while (4.8) is not needed here), also this problem has a unique solution \(\phi _\lambda \). We perform some a priori estimates. As usual, the symbol c stands for possibly different constants. In this proof, the values of c can only depend on \(\tau \), \(\pi \), \(\Omega \), T and the eigenfunctions \(e'_j\) associated with the zero eigenvalues of B (if any). In particular, they do not depend on \(\lambda \), nor on the data of problem (4.5)–(4.6). Symbols like C and \(C_i\) denote particular values of c we want to refer to. The first three estimates we perform are in the direction of the inequalities (4.9) and (4.10) which we want to satisfy with a suitable choice of the spaces and the operator. For this reason, they are obtained under different regularity assumptions on the data.
First a priori estimate Let \(g_i\) and \(\phi _{0,i}\), \(i=1,2\), be two choices of the data satisfying (4.7) and let \(\phi _{\lambda ,i}\) be the corresponding solutions to the approximating problem. We set for brevity \(\phi _\lambda :=\phi _{\lambda ,1}-\phi _{\lambda ,2}\), \(g:=g_1-g_2\) and \(\phi _0:=\phi _{0,1}-\phi _{0,2}\). We write (4.16) for both solutions, take the difference and multiply it by \(\phi _\lambda \). Then, we integrate over \(Q_t\). We obtain that
Since \(\beta _\lambda \) is monotone, all of the terms on the left-hand side are nonnegative. By estimating the right-hand side on account of the Lipschitz continuity of \(\pi \) and the Schwarz and Young inequalities, and then applying the Gronwall lemma, we easily conclude that
It trivially follows that
Second a priori estimate We assume (4.7) on the data. By multiplying (4.16) by \(\phi _\lambda \) and integrating over \(Q_t\), we obtain that
All of the terms on the left-hand side are nonnegative since \(\beta _\lambda \) is monotone and \(\beta _\lambda (0)=0\). If we estimate the right-hand side by using the Lipschitz continuity of \(\pi \) and the Schwarz and Young inequality, we immediately deduce that
Third a priori estimate We set \(V:=H^{1}(\Omega )\) for brevity and assume that the data satisfy \(g\in L^{2}(0,T;V)\) and \(\phi _0\in V\). Before going on, we make an observation. Assume first that \(\ker B=\{0\}\). Then, \(P=0\) and (4.5) is an ordinary differential equation where the space variable is just a parameter. In the opposite case, the presence of the nonlocal operator P could be unpleasant. However, we are reduced to the same situations as before by moving the term \(P\phi _\lambda \) to the right-hand side and treating it as a datum. More precisely, in this case, \(\ker B\) has a finite dimension \(m>0\) and is spanned by the first m eigenfunctions (those corresponding to the zero eigenvalues). Since every eigenfunction of B belongs to the domain \(V_B^{n}\) of \(B^n\) for every \(n\in {{\mathbb {N}}}\) and we are assuming (2.46), the eigenfunctions (we are interested in) belong to V, and we have the identities
Namely, we have that \(Pv\in V\) even though v only belongs to H. Therefore, in any case, the solution \(\phi _\lambda \) enjoys some space regularity. Precisely, it belongs to \(L^{2}(0,T;V)\) as well as its time derivative and we have that
By multiplying this equation by \(\nabla \phi _\lambda \) and integrating over \(Q_t\), we obtain that
All of the terms on the left-hand side are nonnegative. The volume integrals on the right-hand side, except the one involving P, can be easily treated thanks to the boundedness of \(\pi '\) and the Schwarz and Young inequalities. If \(P=0\), then we can apply the Gronwall lemma and obtain an estimate of \(\nabla \phi _\lambda \). Recalling (4.19), we conclude that
We claim that the same estimate holds true even though \(\ker B\) is nontrivial. In this case, we recall the representation formula (4.20) and apply it to \(\phi _\lambda \). By also accounting for standard inequalities, we obtain that
So, it suffices to recall (4.19) and apply the Gronwall lemma to obtain (4.21) also in this case. Therefore, (4.21) is established and it trivially implies that
Interpolation Now, let the data satisfy (4.12) with \(\eta \in (0,1)\). We choose
and apply Lemma 4.1 to the operator \({\mathcal {T}}:{\mathcal {A}}_1\rightarrow {\mathcal {B}}_1\) that associates to the pair \((g,\phi _0)\) the solution \(\phi _\lambda \) to problem (4.5)–(4.6). Then, (4.18) and (4.22) yield (4.9) and (4.10), respectively. Moreover, by setting \(\vartheta :=1-\eta \), we have that
so that \((g,\phi _0)\in ({\mathcal {A}}_0,{\mathcal {A}}_1)_{\vartheta ,2}\) by (4.12). It follows that
with a constant C that does not depend on \(\lambda \). Notice that (4.23) with \(\eta =1\) (i.e., \(\vartheta =0\)) is ensured by (4.22).
Fourth a priori estimate We are close to the conclusion, and we thus assume that the data g and \(\phi _0\) are as in the statement. By multiplying (4.16) by \(\partial _t\phi _\lambda \), integrating over \(Q_t\), and rearranging, we have that
Since \({{\widehat{\beta }}}_\lambda \) is nonnegative and \({{\widehat{\beta }}}_\lambda ({\phi _0})\le {{\widehat{\beta }}}({\phi _0})\) \(\hbox {a.e. in}~\Omega \), owing to the Schwarz and Young inequalities and the Lipschitz continuity of \(\pi \), and accounting for (4.8) and (4.19), we infer that
A comparison in (4.16) then yields that
Conclusion At this point, we let \(\lambda \) tend to zero based on (4.23)–(4.25), the compact embedding \(H^{\eta }(\Omega )\subset H\) for \(\eta \in (0,1]\), and the well-known Aubin–Lions lemma (see, e.g., [26, Thm. 5.1, p. 58]). We deduce that there exists a pair \((\phi ,\xi )\) such that
possibly only for a subsequence \(\lambda _k\searrow 0\). Then, \(\phi (0)=\phi _0\), and (4.15) is verified. Moreover, by also applying, e.g., [4, Lemma 2.3, p. 38], we infer that \((\phi ,\xi )\) satisfies the inclusion in (4.14). On the other hand, all this implies (4.5) since \({{\widehat{\beta }}}\) is convex, so that \(\phi \) is the solution to problem (4.5)–(4.6). This completes the proof of the lemma. \(\square \)
Proof of Theorem 2.11
We apply Lemma 4.2 by choosing
Notice that conditions (4.12) are satisfied due to (2.47)–(2.48). We thus obtain the existence of some \(\xi \) satisfying (4.14) and (4.15). The latter reads
But \(\varphi \) satisfies this equation (see Remark 2.8) since \((\varphi ,\mu )\) is a solution to problem (2.35)–(2.37) by assumption, and this implies (4.5) for \(\varphi \) since \({{\widehat{\beta }}}\) is convex. On the other hand, we have that \(\varphi (0)=\varphi _0=\phi _0\). Since the solution \(\phi \) to problem (4.5)–(4.6) is unique, we conclude that \(\phi =\varphi \). Therefore, (2.38)–(2.39) are proved. The last sentence of the statement trivially follows. \(\square \)
Remark 4.3
We observe that in Theorem 2.11 we start from a solution \((\varphi ,\mu )\) to problem (2.35)–(2.37) without using sufficient conditions for the existence of such a solution. In particular, (2.34) is not accounted for. We also notice that the argument followed in the above proof provides the existence of a unique solution \(\varphi \) to both equation (2.39) and the variational inequality (2.36) for a given \(\mu \) without the use of (2.34).
Remark 4.4
It is possible to slightly modify the proof of Lemma 4.2 in the application of Lemma 4.1 and to obtain different regularity results in Theorem 2.11. One can play with the index p in the interpolation argument, indeed. If we want to maximize the time regularity, we change the choice of the spaces \({\mathcal {B}}_i\) by taking
and start from (4.17) and (4.21) in place of (4.18) and (4.22). Then, we apply Lemma 4.1 still with \(\vartheta =1-\eta \), but with \(p=\infty \). Instead of (4.23), we obtain that
still with a constant C that does not depend on \(\lambda \). Then, everything can proceed as before. At the end of the proof of Theorem 2.11, we arrive at the regularity
for the first component \(\varphi \) of the solution \((\varphi ,\mu )\) to problem (2.35)–(2.37). We avoid the troubles that may arise with the exponent \(\infty \) and do not offer a different representation of the space appearing in (4.28). We just remark that the regularity (4.28) is neither better nor worse than (2.38), since it yields some better time regularity at the expense of a lower space regularity. One can prove that \((L^{\infty }(0,T;V),L^{\infty }(0,T;H))_{\vartheta ,\infty }\subset L^{\infty }(0,T;H^{\eta -\varepsilon }(\Omega ))\) for every \(\varepsilon >0\) (in particular, the Aubin–Lions lemma can be applied also in the modified proof of Lemma 4.2) so that the Sobolev-type regularity for \(\varphi \) we can obtain is
Remark 4.5
Concerning uniqueness for the second component \(\mu \) of the solution to problem (2.35)–(2.37), we can give sufficient conditions in a different direction. The situation is similar to the one encountered for problem (2.24)–(2.26) and mentioned in Remark 2.4. Let us give some detail. Assume that \((\varphi ,\mu _i)\), \(i=1,2\), are solutions corresponding to some data \(\varphi _0\) and f (with the same first component, due to Theorem 2.10). By writing (2.35) for both solutions and taking the difference, we immediately obtain that \((A^r(\mu _1-\mu _2),v)=0\) for every \(v\in V_A^{r}\) and \(\hbox {a.e. in}~(0,T)\), that is
This implies that \(\mu _1=\mu _2\) if \(\lambda _1>0\). In the opposite case \(\lambda _1=0\), we can arrive at the same conclusion under additional conditions, as we show at once by following the ideas of [15, Rem. 4.1]. However, in the present case, the condition we assume on the solutions is difficult to verify, unfortunately. Suppose that \(D(\beta )\) is an open interval, the restriction of \({{\widehat{\beta }}}\) to \(D(\beta )\) is a \(C^1\) function, and all of the values attained by \(\varphi \) belong to a compact subinterval \([a,b]\subset D(\beta )\). Now, choose \(\delta _0\) such that the interval \([a-\delta _0,b+\delta _0]\) is contained in \(D(\beta )\). Then, for an arbitrary \(\delta \in (0,\delta _0)\) and \(\hbox {for a.a.}~t\in (0,T)\), we can choose \(v=\varphi (t)-\delta \) (whence \(\varphi (t)-v=\delta \)) and \(v=\varphi (t)+\delta \) (whence \(\varphi (t)-v=-\delta \)) in the variational inequality (2.36) written for \((\varphi ,\mu _1)\) and \((\varphi ,\mu _2)\), respectively. Then, by adding the resulting inequalities, we deduce that
Division by \(\delta \) then yields that
Taking the limit as \(\delta \searrow 0\), we conclude from the Lebesgue dominated convergence theorem that
Interchanging the roles of \(\mu _1\) and \(\mu _2\), we then infer that \({\mathrm{mean}}\,\mu _1 = {\mathrm{mean}}\,\mu _2\) \(\hbox {a.e. in}~(0,T)\). By combining this with (4.29), we conclude that \(\mu _1 = \mu _2\).
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Acknowledgements
This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati,” University of Pavia. In addition, PC gratefully mentions some other support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), and PC and CG point out their affiliations as Research Associates to the IMATI—C.N.R. Pavia, Italy.
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Colli, P., Gilardi, G. & Sprekels, J. An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators. J. Evol. Equ. 21, 2749–2778 (2021). https://doi.org/10.1007/s00028-021-00706-1
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DOI: https://doi.org/10.1007/s00028-021-00706-1