Abstract
In this paper, we prove the existence of global-in-time weak solutions for an evolutionary PDE system modelling nonisothermal Landau–de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier–Stokes system for the macroscopic velocity \(\mathbf{u}\) is coupled to a nonlinear convective parabolic equation describing the evolution of the \(Q\)-tensor \(\mathbb {Q}\), namely a tensor-valued variable representing the normalized second-order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature \(\vartheta \) are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential \(f\) introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the \(Q\)-tensor equation, a term that is at the same time singular in \(\mathbb {Q}\) and degenerate in \(\vartheta \). To treat it, a careful analysis of the properties of \(f\), particularly of its blow-up rate, is carried out.
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1 Introduction
The main aim of this paper is to continue the analysis of nonisothermal Landau–de Gennes nematic liquid crystal flows with singular potential started in our recent contribution [16]. Our purpose is to consider the evolution of the flow assuming that the mixing term in the bulk potential is singular and, in particular, takes the form introduced by Ball and Majumdar in [4]:
where \(\vartheta \) denotes the absolute temperature and the \(Q\)-tensor \(\mathbb {Q}\) is a symmetric traceless tensor representing in a suitable way the normalized second-order moments of the probability distribution function of molecules. Here, \(G\) is a smooth function of \(\mathbb {Q}\) while \(f: R^{3 \times 3}_\mathrm{sym,0} \rightarrow [-K,+\infty ], K\ge 0\) is a convex function smooth on its domain, which consists of \(\mathbb {Q}\)-tensors whose eigenvalues are inside the physical interval \((-1/3,2/3)\).
The use of \(Q\)-tensors permits to capture fine properties of the crystal configuration (like biaxiality), which cannot be described in the framework of vector-based models. In order to introduce \(Q\)-tensors mathematically, we start from a probability measure \(\mu _x\) on the unit sphere \(\mathbb {S}^2\), representing the orientation of the molecules at a point \(x\) in space. Then, we can associate to \(\mu _x\) a symmetric and traceless \(3\times 3\) matrix defined as
Namely, \(\mathbb {Q}\) is intended to measure how much the probability measure \(\mu _x\) deviates from the isotropic measure \(\bar{\mu }\) where \(d\bar{\mu }=\frac{1}{4\pi }\mathrm{d}A\), see [12]. In the Onsager model (cf. [12, 25]), \(\mu _x\) is assumed to be absolutely continuous with density \(\rho =\rho (\mathbf {p})\). In this case, we have
The fact that \(\mu _x\) is a probability measure imposes a constraint on the eigenvalues of \(\mathbb {Q}\), namely that they are bounded between the values \(-1/3\) and \(2/3\), see [4]. Thus, not any traceless \(3\times 3\) matrix is a physical \(\mathbb {Q}\)-tensor but only those whose eigenvalues are in \((-1/3,2/3)\).
The Ball–Majumdar potential is then defined as follows:
Then, it turns out (see, e.g., [4]) that the effective domain \({\mathcal D}[f]\) (i.e., the set where \(f\) assumes finite values) coincides precisely with the set of symmetric traceless tensors whose eigenvalues belong to the physical interval. Moreover, as we prove in detail in Sect. 4 below, \(f\) blows up logarithmically fast as \(\mathbb {Q}\) approaches the boundary of \({\mathcal D}[f]\) (or, equivalently, one of its eigenvalues tends to \(-1/3\)). The main mathematical features of \(f\), which are required in the existence proof for the purpose of obtaining the necessary a priori estimates, are thoroughly discussed in Sect. 4. Here, it is just worth noting that, for large values of the temperature, the convex part \(f\) of the energy \(\psi _B\) is prevailing, whereas when \(\vartheta \) is close to \(0, \psi _B\) may exhibit a multiple well structure.
Dealing with the bulk potential (1.1) generates severe mathematical difficulties, since the term \(\vartheta f(\mathbb {Q})\) is both singular in \(\mathbb {Q}\) (since the eigenvalues of \(\mathbb {Q}\) may approach, at least in some small region, the limiting values \(-1/3,2/3\), and degenerate in \(\vartheta \) (since we can only prove that \(\vartheta >0\) almost everywhere, not excluding, however, that the (essential) infimum of \(\vartheta \) might reach 0). Because of several technical problems associated with the above-mentioned difficulties, our previous contribution [16] focused on the case when the bulk potential in the free-energy functional is given by (cf. also [27, 37])
where \(U\) is assumed to be a convex and decreasing function of \(\vartheta \) with restricted growth at \(\infty \). This form of the bulk potential allowed us to get rid of the degenerate character in \(\vartheta \) and, in fact, to prove strict positivity of \(\vartheta \) in that case. Although formula (1.1) leads to the same stationary problem as (1.4), they give rise to different solutions if the time evolution of the system is studied. In accordance with the underlying physical arguments (see [25]), it is (1.1) rather than (1.4) that should be considered. Indeed, this choice, combined with the standard principles of thermodynamics, yields the entropy of the system in the form
where \(\mathcal {F}\) denotes the free-energy functional and \(c\) represents its purely caloric part [cf. also (1.4)]. On the other hand, in the case (1.4), one has
and the singular potential \(f\) appears only in the internal energy of the system. In particular, this makes the entropy relation easier to treat mathematically with the choice (1.4).
In our model, the evolution of the \(Q\)-tensor is derived as a balance of the following free-energy functional:
where \(\lambda ,a\ge 0\) and the exponent \(m\ge 1\) will be fixed below. The term depending on \(\nabla _x\mathbb {Q}\) describes the interfacial energy, whereas \(a \vartheta ^m, a>0\) prescribes a power-like heat conductivity.
The macroscopic velocity of the crystal flow is described by a Navier–Stokes type system [cf. (2.3) below] for the velocity field \(\mathbf{u}\), in which the stress tensor contains a nonstandard part depending on the tensor \(\mathbb {Q}\). The evolution of \(\mathbb {Q}\) [cf. (2.5)] is ruled by the variational derivative of the free-energy functional, displaying a singular character due to the presence of the singular Ball–Majumdar energy (1.1). In the physical formulation of the model, the evolution of temperature is described by the heat Eq. (2.8). However, due to the presence of quadratic terms in the right hand side (especially those depending on \(\mathbf{u}\)), dealing with (2.8) appears out of reach from a mathematical perspective. For this reason, following an idea originally developed by Bulíček et al. [9] for the nonisothermal Navier–Stokes system, we replace it by an equality describing conservation of total energy conservation, combined with a weak form of the entropy inequality [cf. (2.17) and (2.23) below]. As a drawback of this choice, we have to note the explicit appearance of the pressure \(p\) in the energy balance equation. Actually, \(p\) is no longer just a Lagrange multiplier, but it enters the system as an additional unknown and needs to be controlled carefully. In particular, for this purpose, a suitable choice of the boundary condition is crucial, and indeed, in order to reduce complications to the minimum, we consider periodic b.c.’s [cf. (2.1) below].
Due to the quoted mathematical difficulties, the results we obtain in the present contribution are weaker than those we got in [16]. Actually, here we consider a weaker notion of solution, where the entropy inequality holds only in its integrated (both in space ad in time) form; in other words, we can only prove a global balance of entropy. In addition, in order to deal with the Ball–Majumdar energy (1.1), we need a power-like growth assumptions imposed on the heat conductivity, the specific heat, and the collective rotational viscosity coefficient in the system. Although, mathematically speaking, these assumptions are enforced by the need of sufficiently strong a priori bounds for the temperature \(\vartheta \), they also have a physical justification, at least for some classes of materials (see, e.g., [13]).
The derivation of suitable a priori estimates as well as a rigorous justification of compactness of the family of solutions represents the main novelty of the present paper with respect to [16], while the specific construction of suitable approximate solutions, carried over in detail in [16], is a routine matter. This is the reason why we restrict our discussion to the passage from a priori bounds to compactness (sequential stability), leaving the necessary modifications of the construction of suitable approximate solutions in [16] to the interested reader.
As an additional result, in the last part of the paper, we see that if we additionally assume that the heat flux is singular as the absolute temperature \(\vartheta \) approaches 0 (see Sect. 5 below), then we can recover the entropy inequality in the usual distributional sense. Namely, we can estimate the entropy growth on any region of the space domain and not only globally. Moreover, we can deal with an even more general class of singular potentials. It is worth noting that taking this type of singular flux law is not only a mathematically ad hoc assumption, but is a common choice in several types of phase-transition and phase-separation models both in liquids and in crystalline solids (cf., e.g., [11, 31, 33] where similar growth conditions are assumed). Regarding the choice of power-growing specific heat [cf. the last summand in (1.5)], let us mention the papers [1, 26] for examples in phase transitions and [8, 32, 35] for examples in thermoelastic systems, where this behavior is observed. Note that, however, in our contribution, we can allow the viscosity coefficient to depend on the absolute temperature in a quite general way: it has to be only bounded from below and above without any specific growth condition.
We conclude this introduction by a brief review of other LC models based either on the tensorial variable \(\mathbb {Q}\) or other parameters. To begin, we can quote the standard hydrodynamic theories, where the case of regular bulk potential
is considered (cf. [29, 30] for the analysis of the corresponding isothermal evolution); it is not clear whether this physical constraint on the eigenvectors is preserved or not, while the choice of the Ball–Majumdar free energy (1.1) naturally enforces this constraint.
Moreover, a vectorial quantity \(\mathbf{d}\) could also be used instead of \(\mathbb {Q}\) in order to describe the preferred orientation of the molecules at any point (cf., e.g., [10, 14, 24] and references therein). Let us notice that in the literature, there are few papers dealing with liquid crystal models with singular potential. In the isothermal case, we can quote the very recent contribution [40], where existence and regularity of solution in the 3D and 2D cases are obtained for a tensorial model with Ball–Majumdar potential. Regarding the nonisothermal tensorial case, up to our knowledge, the only contribution, in the case of singular potential (1.4), was given in the paper [16], which we already mentioned in this “Introduction”. Two attempts of considering the nonisothermal case were made in [19] and [15] and refer to vectorial models. In particular, in [15], the stretching and rotation effects of the director field induced by the straining of the fluid were considered, and the existence of global-in-time weak solutions was obtained for the corresponding initial-boundary value problem.
Plan of the paper The remainder of the paper is organized as follows: in Sect. 2, we first introduce the equations of our mathematical model in the strong (physical) form (Sect. 2.1); then, we specify our basic assumptions on coefficients and data (Sect. 2.2). This permits us to introduce (Sect. 2.3) a weak formulation of the initial-boundary value problem for our model and to present a rigorous existence theorem (Theorem 2.2) of weak solutions. The proof is carried out in Sect. 3 and is subdivided into several steps: we first prove (Sect. 3.1) that any weak solution complies, at least formally, with a number of a priori estimates; subsequently, these bounds are shown to be sufficient to provide weak sequential stability (Sect. 3.2). Namely, the sequences of solutions starting from (suitably) bounded families of initial data admit weak limits, which still solve the problem. A key ingredient in the proof is represented by some fine properties of the Ball–Majumdar potential, which are obtained separately in Sect. 4. Finally, in Sect. 5, we show that, assuming a singular heat flux law, even more general classes of potentials can be treated and, moreover, the entropy inequality holds in a stronger sense.
2 The mathematical problem
2.1 The equations of the model
In this part, we write the equations of our mathematical model describing the evolution of the unknown fields \(\mathbf{u}\) (macroscopic velocity), \(\mathbb {Q}\) (\(Q\)-tensor), and \(\vartheta \) (absolute temperature). The model can be physically derived by following closely the argument reported in [16, Section 1]. Indeed, the main difference is represented here by the choice of the Ball–Majumdar potential (1.1). However, this does not affect the way the model is obtained, but only the outcoming form of the equations. These are stated here in their strong form. In Sect. 2.3, we will see that, in order to have a mathematically tractable system, passing to a weak formulation is actually necessary.
In order to avoid complications related to interactions with boundaries, we take periodic boundary conditions for all unknowns. Namely, we assume the system be settled in the 3-dimensional torus
As already noted in the Sect. 1, the above choice is crucial for our existence theorem.
The first two relations in our model describe the evolution of the macroscopic velocity \(\mathbf{u}\):
Here, \(\sigma \) denotes the stress tensor, given by
\(\xi \) being a fixed scalar parameter, measuring the ratio between the rotation and the aligning effect that a shear flow exerts over the directors, see Beris and Edwards [5]. We will denote with \(\omega (\mathbf{u})\) the antisymmetric part of the viscous stress, i.e.,
and with \(\varepsilon (\mathbf{u})\) its symmetric part, i.e.,
The behavior of the \(Q\)-tensor is ruled by the following relation:
The nonnegative function \(\Gamma (\vartheta )\) represents a collective rotational viscosity coefficient, whereas
denotes stretching and rotation effects on the \(Q\)-tensor driven by the macroscopic flow.
Finally, \(-\mathbb {H}\) indicates the first variation of the free energy \({\mathcal F}\) with respect to \(\mathbb {Q}\), namely
where we have chosen \(G(\mathbb {Q})=-\lambda \mathbb {Q}\) in (1.1). The operator \(\mathcal{L}[ \cdot ]\) denotes projection on the space of traceless tensor, that is, \(\mathcal{L}[ \mathbb {M}] = \mathbb {M}- \frac{1}{3} {{\mathrm{tr}}}(\mathbb {M})\) for \(\mathbb {M}\in R^{3\times 3}\). In other words, the expression \(\mathcal {L} \left[ \frac{\partial f(\mathbb {Q})}{\partial \mathbb {Q}} \right] \) represents the subdifferential of \(f\), with respect to the structure of the space of traceless tensors, evaluated at the point \(\mathbb {Q}\). Finally, the evolution of temperature is represented in the strong formulation by the
The model can be derived from the free-energy functional (1.5) by stating the balances of total energy and of entropy. The procedure is completely analogous to that performed in [16, Section 1.3] (see also [20]) to which we refer the reader for details. We just note here that (2.8) corresponds to assuming the internal energy flux being given by (cf. [16, formula (1.30)])
2.2 Assumptions on coefficients and data
We state here our basic assumptions on coefficients and nonlinear terms in system (2.2)–(2.9), as well as on the initial data. First of all, we ask the viscosity \(\mu \) to be bounded with respect to \(\vartheta \), while we prescribe a power-like law for the heat flux and a linear growth of the diffusion coefficient \(\Gamma \) at infinity:
In addition, we take \(a>0\) and assume that the exponents \(k\) in (2.11) and \(m\) in (2.8) are strictly positive and satisfy the relation
see, e.g., [41]. Next, we let the initial data \(\mathbf{u}_0, \mathbb {Q}_0\), and \(\vartheta _0\) satisfy
Finally, we assume that \(f\) satisfies at least the following basic properties:
-
(i)
\(f : R^{3 \times 3}_{\mathrm{sym},0} \rightarrow [-K, +\infty ]\) is convex and lower semi-continuous, with \(K \ge 0\).
-
(ii)
The domain of \(f\), i.e.,
$$\begin{aligned} \mathcal {D}[f] = \left\{ \mathbb {Q}\in R^{3 \times 3}_\mathrm{sym,0} \ | \ f(\mathbb {Q}) < + \infty \right\} , \end{aligned}$$coincides with the (open, convex, and bounded) subset of \(R^{3 \times 3}_{\mathrm{sym},0} \) given by
$$\begin{aligned} \left\{ \mathbb {Q}\in R^{3 \times 3}_\mathrm{sym,0} \ | \ \lambda _i[\mathbb {Q}] \in (- 1/3,2/3) \right\} . \end{aligned}$$ -
(iii)
\(f\) is smooth in \(\mathcal {D}[f],\lambda \ge 0\).
We will see in Sect. 4 below (cf. also [4, Section 3]) that (i)–(iii) hold in the case when \(f\) is given by the Ball–Majumdar potential introduced in (1.3). This is a nontrivial issue since relation (1.3) describes \(f\) implicitly in terms of the orientation distribution(s) \(\rho (\mathbf{p})\) generating \(\mathbb {Q}\). Hence, the properties of \(f\) as a function of \(\mathbb {Q}\) need to be properly demonstrated. On the other hand, (i)–(iii) do not characterize the Ball–Majumdar potential completely and are satisfied by a much wider class of convex singular functions. Indeed, in the proof, we will need that \(f\) satisfies some finer properties, especially related to its blow-up rate as \(\mathbb {Q}\) approaches the boundary of \({\mathcal D}[f]\). The discussion of these properties and the proof of the fact that they are satisfied by the Ball–Majumdar singular potential are postponed to Sect. 4.
2.3 Weak formulation and main results
A proof of existence of global-in-time strong solutions of the model introduced in Sect. 2.1 appears to be out of reach for the available mathematical tools, one of the principal difficulties being related to the presence of quadratic terms on the right hand side of the heat Eq. (2.8). Accordingly, we introduce here a weak formulation of the system along the lines of [16]. The main point stands in the replacement of (2.8) by means of a couple of relations, namely the total energy balance [cf. (2.17) below] and a weak form of the entropy production inequality [cf. (2.23)].
Such relations account for most of the information coming from the heat equation, but are more treatable mathematically. Actually, (2.17) contains no quadratic terms, while inequality (2.23) just states that the global entropy of the system needs to grow (at least) as prescribed by the right hand side. As noted in the “Introduction”, differently from what was done in [16] and due to technical difficulties related to the energy (1.1), we are not able to get here any “pointwise” control of the entropy (actually, this can be reached under different assumptions, cf. Sect. 5 below), but we just get a “global” (i.e., integrated) version of the entropy inequality.
Moreover, as a further drawback of this procedure, we note the explicit appearance of the pressure \(p\) in (2.17), which explicitly enters the system as an unknown. To get a control of it from the momentum equation, specific choices of the boundary conditions are needed (cf. [17] for further comments on this point). In particular, this can be reached in our case of periodic conditions. These being said, we can state the
where
In (2.17), we have noted as \(e\) the internal energy
where the entropy \(s\) is given by
Recall also that the stress \(\sigma \) is given by (2.4).
If one starts from the strong formulation, then (2.17) is (formally) obtained by multiplying (2.3) by \(\mathbf{u}\), (2.5) by \(\mathbb {H}\), and summing the results to (2.8). This procedure involves a number of lengthy, but otherwise straightforward, computations, which are provided in detail in [16, Section 1]. In particular, one uses the identity
which holds for any symmetric matrix \(\mathbb {H}\).
Finally, multiplying the heat Eq. (2.8) by \(1/\vartheta \), one readily deduces the “pointwise version” of the entropy production inequality:
As noted above, in our existence result, we are not able to deal with relation (2.22), even interpreted as a distributional inequality. Indeed, an \(L^p\)-estimate for the term \(s\mathbf{u}\) on the left hand side seems out of reach due to the occurrence of the singular function \(f(\mathbb {Q})\) as a summand in \(s\) [cf. (2.20)]. Hence, we have to replace (2.22), by the following “global” entropy balance [simply obtained integrating (2.22) both in space and in time]:
We are now ready to detail the notion of weak solution to our problem. Recall that the exponent \(A\) was introduced in (2.13).
Definition 2.1
A weak solution to the nonisothermal liquid crystal system with the Ball–Majumdar potential is a quadruplet \((\mathbf{u},p,\mathbb {Q},\vartheta )\) enjoying the regularity properties
where \(X\) is a Sobolev space of negative order depending on \(k\) and \(m\), satisfying the incompressibility condition (2.2), the momentum Eq. (2.3), the \(Q\)-tensor evolution system (2.5), and the total energy balance (2.17) at least in the sense of distributions, and complying with the initial conditions in the following sense:
where
and \(e\) is defined in (2.19).
Then, our existence theorem, which can be considered as the main result of this paper, reads
Theorem 2.2
Under the assumptions stated in Sect. 2.2, the nonisothermal liquid crystal model with Ball–Majumdar potential admits at least one weak solution, which additionally satisfies the integrated entropy production inequality (2.23).
The main steps of the proof of the theorem are carried out in the next section.
Remark 2.3
In the statement, we claimed that the system equations hold at least in the sense of distributions. To be more precise, we can say that by virtue of the regularity properties (2.24)–(2.28), the momentum equation holds in the form
satisfied for any test function \(\varphi \in C^\infty _c([0,T) \times \Omega ; R^3)\). On the other hand, by virtue of the regularity properties (2.24)–(2.28), the order parameter Eq. (2.5) turns out to hold pointwise (i.e., in some \(L^p\)-space) and the periodic b.c. for \(\mathbb {Q}\) holds in the sense of traces. Finally, the total energy balance (2.17) is satisfied as the integral identity
for any \(\varphi \in C^\infty _c([0,T) \times \Omega )\), where \(e\) was defined in (2.19) and \(\widehat{\kappa }\) was introduced in (2.18). Actually, we needed to substitute the expression (2.9) for \(\mathbf{q}\) and manage it by a further integration by parts.
Finally, let us note here that in Sect. 5, we will state another existence result (Thm. 5.1) holding in case of singular heat flux law (5.1) and in that case, the entropy inequality holds true in the usual distributional form [cf. (5.5)].
3 Proof of Theorem 2.2
The remainder of the paper is devoted to the proof of Theorem 2.2. The argument we are going to perform follows closely the lines of the proof given in our previous work [16], the main difference being represented of course by the presence of the Ball–Majumdar potential. In order to reduce complications, we will avoid describing in detail the (rather long and technical) approximation procedure needed to get an existence theorem for a regularized version of the system. Indeed, at this level, we do not see relevant differences with respect to the argument given in [16], to which we refer the reader for more details.
Instead, the subsequent part of the proof is more significantly different from that in [16] and, for this reason, will be presented in an extended way. As a first step, we will prove in Sect. 3.1 that any weak solution satisfies (at least formally) a number of a priori estimates, mainly coming as direct consequences of the energy and entropy balances. In particular, we will get a control of certain norms of the solution components only in terms of the initial data and of the fixed parameters of the system. These basic estimates, however, do not provide any bound on the (subdifferential of the) singular term \(f\). Its control is actually based on a rather delicate argument detailed in Sect. 4 and concerning estimates of its Hessian. This part is the key point of our argument and represents the main novelty of the present paper.
In addition to that, we have to stress that, compared to [16], we have here different (power-like) growth conditions on the viscosity coefficient \(\Gamma (\vartheta )\), on the heat conductivity [cf. (2.11)], and on the specific heat [cf. (1.5)]. Mathematically speaking, these are needed in order to control some \(L^p\)-norm of \(\vartheta \). However, in this case, we can allow for a general dependence of the viscosity coefficient from the absolute temperature [cf. (2.10)].
With the a priori estimates at our disposal, we will prove in Sect. 3.2 that any sequence \((\mathbf{u}_n,p_n,\mathbb {Q}_n,\vartheta _n)\), which complies with the bounds uniformly with respect to \(n\), admits a limit point that is still a weak solution of the system. This part, which we call “weak sequential stability” of weak solutions, can be viewed as a simplified version of the compactness argument needed to remove some regularization of approximation of the system (like that provided in [16]).
3.1 A priori estimates
In this section, we perform formal a priori estimates. As noted above, these estimates can be rigorously justified for certain approximate solutions in the way detailed in [16]. Most of the bounds follow directly from the total energy conservation (2.17) and the entropy inequality (2.22) (indeed, the integrated version (2.23), holding for weak solutions, suffices for our purposes).
Energy-entropy estimate Integrating (2.17) in space and using (2.19) with the periodic boundary conditions, we readily obtain the energy estimate
Due to the lack of coercivity of the internal energy w.r.t. \(\mathbb {Q}\), the above information has to be coupled with the bound coming from the entropy inequality (2.23). Actually, recalling (2.20) and (2.22), we infer
Summing (3.1) to (some positive constant times) (3.2), we immediately obtain the estimates
In particular, we used here the singular character of \(f\), which also gives
Moreover, we have the information coming from the latter integral in (3.2), which we now specify.
First of all, using (2.11), we readily infer
Combining (3.4), the second part of (3.9) and using some generalized version of Poincaré’s inequality (see, e.g., [36, Lemma 3.2]), we get
Next, by (3.5), (3.9) and interpolation, recalling also (2.13), it is easy to infer
Next, by virtue of (2.10) and (2.12), we obtain
As a consequence, by (3.11), (3.12) and interpolation, we have
whence, also,
Interpolating between (3.3) and (3.15) and using that \(A>9\) [cf. (2.13)], we arrive at
Here, \(1+\) denotes a generic exponent strictly greater than \(1\). This kind of notation will be used also in the sequel.
Remark 3.1
It is worth stressing that in all the procedure [cf. for instance estimates (3.5) and (3.9)], we are implicitly assuming that \(\vartheta \) is positive at least almost everywhere. Actually, in [16], we were able to prove this property (and, actually, also a stronger one) by means of maximum principle arguments, even at the approximated level. In the current setting, due to the presence of singular terms on the right hand side of (2.8), ensuring the same property for approximate solutions is more delicate, requiring additional singular terms to be introduced in the process of approximation, see, e.g., [18, Chapter 3].
Estimate of the singular potential The estimate of the (subdifferential of the) Ball–Majumdar potential constitutes the key point of our argument. Recalling estimate (3.13), we compute the \(L^2\)-scalar product of \(\mathbb {H}\) with \(- \vartheta \mathcal{L}\left[ \frac{\partial f(\mathbb {Q})}{\partial \mathbb {Q}} \right] \). The necessary key properties of \(f\) will be proved in the subsequent Sect. 4. Hence, we have
Moreover, by Hölder’s and Young’s inequalities and uniform boundedness of \(\mathbb {Q}\) [cf. (3.8)],
and
Notice that, integrated in time, the last term can be estimated by virtue of the first one in (3.13).
The key point is to control the term \(I_2\) on the left hand side of (3.17). Using Green’s formula, we have
Then, thanks to the key estimate (4.7) below and (3.8),
On the other hand, by Young’s inequality,
Then, integrating (3.17) over \((0,T)\), and taking \(\kappa =\varepsilon /2\), we get
whence, comparing terms in \(\mathbb {H}\) and using (3.8) and the first (3.13), we arrive at
Summarizing, we have obtained an \(L^2\)-control on both the Laplacian of \(\mathbb {Q}\) and the singular term in (2.5).
3.2 Weak sequential stability
In this part, we assume to have a sequence \((\mathbf{u}_n,p_n,\mathbb {Q}_n,\vartheta _n)\) of weak solutions satisfying the estimates proved in the previous subsection uniformly with respect to \(n\). Under these conditions, we show that there exists subsequence that converges in a suitable way to another weak solution to the problem. In order to simplify the notation, we assume all convergence relations appearing in the sequel to hold up to the extraction of (nonrelabelled) subsequences.
First of all, notice that, thanks to the bounds (3.3)–(3.16), we have
the latter relation coming from (3.9) and interpolation. As a consequence of (3.13), we also obtain
for some tensor-valued function \(\overline{\mathbb {H}}\). Here and below we use the upper bar to indicate weak limits that are not yet identified. In particular, we cannot express \(\overline{\mathbb {H}}\) in terms of \(\vartheta \) and \(\mathbb {Q}\) at this level.
Next, by (3.26)–(3.27) and Gagliardo–Nirenberg inequalities (cf., e.g., [28]),
Then, by (2.6), (3.14), \(A>9\) [cf. (2.13)], and uniform boundedness of \(\mathbb {Q}_n\),
By (3.59), (3.31), (3.32) and a comparison of terms in (2.5), we then infer
Hence, by the Aubin–Lions lemma, we get the following strong convergence relations:
To proceed, we now compute the limit \(n\nearrow \infty \) in the momentum equation. To start with, we first deal with the pressure \(p\). This enters the total energy balance (2.17) as an additional unknown; hence, it needs to be estimated directly. To this purpose, one (formally) applies the operator \(\mathrm{div}_x\) to the momentum Eq. (2.3). This gives rise to an elliptic problem for \(p\), which is well posed thanks to the choice of periodic boundary conditions for all unknowns [cf. [9] for more details on this point]. Recalling (2.4) and noting that
thanks to (3.16), (3.25), (3.29) and (3.30), using also the uniform boundedness of \(\mathbb {Q}_n\). It is then not difficult to deduce (cf. [15, Section 4] for more details)
Hence, a comparison of terms in (2.3) gives that
where \(X\) is a suitable Sobolev space of negative order. By the Aubin–Lions lemma and (3.16), we then get
Hence, we have deduced strong (and consequently pointwise) convergence of \(\mathbb {Q}_n\) and \(\mathbf{u}_n\). The next step consists in obtaining an analogous property for the temperature. To this aim, we consider the total energy balance at the level \(n\) and notice that
Using (3.30), (2.6), (3.25), and uniform boundedness of \(\mathbb {Q}_n\), we also get
Moreover, using (3.28) and (2.13), we have
the latter condition being a consequence of (2.13). Recall that \(\widehat{\kappa }\) was defined in (2.18).
Now, it is clear that all summands in the internal energy \(e_n\) [cf. (2.19)] are uniformly \(L^2\)-bounded, with the exception of the last one for which we have
which also follows from (2.13). By (3.46) and (3.16), we then find
Collecting (3.41)–(3.47) and comparing terms in (2.17), we then infer
where \(X\) is, again, some Sobolev space of negative order. On the other hand, a direct computation based on (3.9), (3.11), and the other estimates permits to check that
and also this condition is satisfied due to (2.13). Actually, we used here that
With (3.48) and (3.49) at disposal, we can use once more the Aubin–Lions lemma to conclude that
and, consequently, almost everywhere in \((0,T)\times \Omega \). Note in particular that the limit of \(|\mathbf{u}_n|^2\) is identified as \(|\mathbf{u}|^2\) thanks to (3.40). Moreover, by (3.25), (3.28), and (3.27), we also have
Hence, (3.48), (3.52), and a generalized form of the Aubin–Lions lemma give
where \(X\) is, again, some Sobolev space of negative order. Hence, the Cauchy condition for the total energy balance holds also in the limit, in the sense specified by (2.31).
By (3.51) and (3.40), we also obtain strong convergence of \(e_n\), say, in \(L^{1+}\). Consequently, computing
for a couple of indexes \(n, m\), and using the monotonicity of the function \(\vartheta \mapsto \vartheta ^m\) (recall that the temperature is assumed to be nonnegative) and strong \(L^2\)-convergence of \(\mathbb {Q}_n\) and \(\nabla _x\mathbb {Q}_n\), we readily obtain that \(\{\vartheta _n\}\) is a Cauchy sequence in \(L^1((0,T)\times \Omega )\). Hence, recalling (3.28) and using a proper generalized version of Lebesgue’s theorem, we arrive at
As a consequence, thanks to assumptions (2.10)–(2.12), we infer
Combining (3.28) and (3.29), and recalling assumption (2.12), we also get
Moreover, relation (3.58) also implies \(\overline{\Gamma (\vartheta )\overline{\mathbb {H}}} = \Gamma (\vartheta )\overline{\mathbb {H}}\) [cf. also (3.59)]. This permits us to pass to the limit in the momentum Eq. (2.3) (in particular, the stress \(\sigma \) is identified in terms of the limit functions \(\vartheta , \mathbb {Q}, \overline{\mathbb {H}}\) and \(\mathbf{u}\)). Analogously, we can take the limit in the \(Q\)-tensor Eq. (2.5) (where, however, the function \(\overline{\mathbb {H}}\) is still to be identified) and in the total energy balance (2.17) [we use here also properties (3.44)–(3.48)].
In addition, from (3.55) and (3.10), we also obtain
In particular, if positivity holds (almost everywhere) for \(\vartheta _n\), then it is preserved in the limit \(\vartheta \).
To conclude the proof, we need to take \(n\nearrow +\infty \) in the relations involving the singular potential \(f(\mathbb {Q})\). To be precise, what remains to do is identifying \(\overline{\mathbb {H}}\) in terms of \(\vartheta \) and \(\mathbb {Q}\) and letting \(n\nearrow \infty \) in the weak form (2.23) of the entropy production inequality.
We start with the first task. Thanks to estimate (3.23), we get that
Moreover, by strong convergence of \(\vartheta _n\) and \(\mathbb {Q}_n\) [cf. (3.55) and (3.34)], and using that both \(\mathbb {Q}_n\) and \(\mathbb {Q}\) take their values almost everywhere into \({\mathcal D}[f]\) (otherwise \(f(\mathbb {Q}_n)\) could not be uniformly \(L^1\)-bounded), we obtain
By a generalized form of Lebesgue’s theorem, the combination of these facts entails that
As a consequence of (3.63), \(\overline{\mathbb {H}}\) is identified as \(\mathbb {H}\), in terms of \(\vartheta \) and \(\mathbb {Q}\), namely (2.7) holds. In particular, for \(n\nearrow \infty \), Eq. (2.5) goes to the expected limit.
To conclude the proof, we have to take \(n\nearrow \infty \) in the “weak” entropy production inequality (2.23). To this aim, we notice that, for any nonnegative-valued \(\phi \in C_c^\infty ([0,T)\times \Omega )\), there holds
This is, indeed, a consequence of relations (3.25), (3.28), (3.29) (where, now, \(\overline{\mathbb {H}}=\mathbb {H}\)), of convexity of the above integrand with respect to \(\nabla _x\mathbf{u}+ \nabla _x^t \mathbf{u}, \mathbb {H}\), and \(\nabla _x\vartheta \), and of a standard semicontinuity argument (see, e.g., [22]). Choosing \(\phi \equiv 1\), we can manage the first integral in (2.23). Next, in order to deal with the second integral of (2.23), we simply observe that \(-s\) is the sum of convex function both of \(\vartheta \) and of \(\mathbb {Q}\) and of the function \(ma\vartheta ^{m-1}\) [cf. (2.20)]. Hence, we can take the (infimum) limit of that integral for a.e. \(t\in (0,T)\) by virtue of relations (3.26)–(3.28) and (3.55), and of a further semicontinuity argument. This concludes the proof of Theorem 2.2.
4 Analytical properties of the Ball–Majumdar potential
In this part, we prove a number of fine properties of the Ball–Majumdar potential that were used in the proof. We recall first the definition and the basic features of the Ball–Majumdar potential. As observed in [3, 4], the Ball–Majumdar singular potential \(f\), defined as a function of \(Q\), can be expressed as:
where \((\lambda _1,\lambda _2,\lambda _3)\) are the eigenvalues of \(Q\) and \(F^{BM}\) is defined as
with
and \(\mu _i,i=1,2,3\), are given implicitly as solutions of the system:
The solutions of (4.4) are determined up to an additive constant added to all the \(\mu _i\) (i.e., if \((\mu _1,\mu _2,\mu _3)\) is a solution for a given triplet \((\lambda _1,\lambda _2,\lambda _3)\) then so is \((\mu _1+C,\mu _2+C,\mu _3+C)\) for any \(C\in \mathbb {R}\)).
Nevertheless, let us note that the \(\mu _i,i=1,2,3\), can be taken, without the loss of generality, as real analytic functions of \((\lambda _1,\lambda _2,\lambda _3)\). More precisely, we have
Lemma 4.1
There exists \(\mu _i,i=1,2,3\), real analytic functions of \((\lambda _1,\lambda _2,\lambda _3)\), such that
and solving the system (4.4).
Remark 4.2
The condition (4.5) is just a convenient way of eliminating the 1-dimensional degeneracy in the solutions of the system (4.4).
Proof
We consider the linear space
We aim to apply the analytic implicit function theorem to \(\mathcal {F}:X\times X\rightarrow X\) defined by
in order to express \((\mu _i)_{i=1,2,3}\) as functions of \((\lambda _i)_{i=1,2,3}\).
We note that \(X\) is a \(2\)-dimensional space, with an orthonormal basis \(f_1=\frac{1}{\sqrt{2}}(-1,1,0), f_2=\frac{1}{\sqrt{6}}(1,1,-2)\). As such, in terms of this orthonormal basis, the nondegeneracy condition needed for applying the implicit function theorem is \(\det M\not =0\) for
One can check that \(\det M=36\frac{\partial ^2 \ln Z}{\partial \mu _1^2}\frac{\partial ^2 \ln Z}{\partial \mu _2^2}-36\left( \frac{\partial ^2 \ln Z}{\partial \mu _1\mu _2}\right) ^2\). It is convenient to consider the matrix
and observe that \(\det M=36\det N\). Then, in order to check that \(\det M\not =0\), it suffices to check that for \(e=(e_1,e_2)\in \mathbb {R}^2\), we have \((Ne,e)\ge 0\) with equality if and only if \(e=(0,0)\). Indeed, taking into account the definition (4.3) of \(Z\), we have
where for the last inequality, we used Cauchy–Schwarz, and \(k=1,2,3\), \(\alpha =1,2\). It is clear that equality cannot hold for \(e\not =(0,0)\) hence our claim.\(\square \)
We can now prove the fundamental estimate on the Hessian of \(f\):
Proposition 4.3
There exists \(\varepsilon >0\) such that
Proof
We proceed in several steps:
Step 1: Reduction to proving the concavity of an auxiliary function
We set \(h(\mathbb {Q}):=e^{-\varepsilon f(\mathbb {Q})}\) for all \(\mathbb {Q}\in {\mathcal D}[f]\), where \(\varepsilon >0\) is a positive constant to be chosen later on. It is more convenient to work with \(h\) because the required property (4.7) is implied by \(h\) being concave and the concavity of an isotropic function is in general simpler to check, as it will be seen in the next step.
We claim thus that \(h\) being concave, smooth on \({\mathcal D}[f]\) and positive implies (4.7). We start by noting that for \(\mathbb {Q}\in {\mathcal D}[f]\) and arbitrary \(V\in R^{3 \times 3}_{\mathrm{sym},0} \):
where we denoted \(g(\mathbb {Q},V):=\frac{\mathrm{d}}{\mathrm{d}t}h(\mathbb {Q}+tV)|_{t=0}.\)
Similarly, again for \(\mathbb {Q}\in {\mathcal D}[f]\),
where we denoted \(k(\mathbb {Q},V):=\frac{\mathrm{d^2}}{\mathrm{dt^2}}h(\mathbb {Q}+tV)|_{t=0}\).
Then, the desired estimate (4.7) is equivalent to
which clearly holds as \(h>0\) and \(k:=\frac{\mathrm{d^2}}{\mathrm{dt^2}}h(\mathbb {Q}+tV)|_{t=0}\le 0\) by our assumptions that \(h\) is positive, smooth, and concave.
The assumed positivity of \(h\) is obvious (by the definition of \(h\)), and the smoothness is a consequence of the smoothness of \(f\). We are left with checking the concavity.
Step 2: Checking concavity for a function of eigenvalues
We define now \(H:(-\frac{1}{3},\frac{2}{3})\rightarrow [0,\infty )\) by \(H(\lambda _1,\lambda _2,\lambda _3):=h(\mathbb {Q})\) with \(\mathbb {Q}\) having eigenvalues \(\lambda _i,\,i=1,2,3\) (it is known that such a function is well defined and symmetric, see for instance [2] and the references therein). Then, \(h\) is concave if and only if \(H\) is concave (see for instance Prop \(18.2.4\) in [38]). We note that \(h\) and \(H\) have the same regularity (see for instance [2], Thm \(5.5\) for up to \(C^2\) regularity and the extension up to \(C^\infty \) in [39]). Thus, \(H\) is concave if and only if its Hessian is nonpositive.
Step 3: The concavity of \(H\) and the asymptotics of certain integrals
We have:
Using the definition of \(F^{BM}\) in (4.2), Lemma 4.1 and assuming \(\mu _i,i=1,2,3\), to be differentiable functions of \(\lambda _i,i=1,2,3\), we have
where for the last equality we used (4.4).
On the other hand, taking the partial derivative with respect to \(\lambda _j, j=1,2,3\) in (4.4), we have
Let us denote by \(N=(N_{ij})_{i,j=1,2,3}\) the matrix with components \(N_{ij}:=\frac{\partial ^2 \ln Z}{\partial \mu _i\partial \mu _j}, i,j=1,2,3\). Then, relations (4.9–4.10) show that
Hence, taking into account (4.8)–(4.10), in order to show that \(H\) is concave, we need to prove that for a suitable \(\varepsilon >0\), the matrix:
is positive definite (where we denoted by \(\mu \otimes \mu \) the matrix of components \(\mu _i\mu _j, j=1,2,3\)).
Let us recall now (see for instance [6], Proposition \(1.2.6\)) that if \(P,Q\) are real-valued symmetric matrices with \(Q\) and \(PQ+QP\) positive definite, then \(P\) is positive definite as well. We aim to apply this criterion to \(Q=N\) and \(P=N^{-1}-\varepsilon \mu \otimes \mu \). To this end, we need to show that \(N\) is positive definite and
for suitable \(\varepsilon >0\). The proof of these two claims is quite technical and is postponed to the “Appendix”.\(\square \)
5 The case of a singular heat flux law
In this last section, we consider the case when the heat flux law exhibits a singular behavior for \(\vartheta \sim 0\). Namely, in place of (2.11), we ask that
Indeed, other types of singular behavior may be considered as well. We chose the above expression since, as noted in the Introduction, it is in agreement with the behavior observed in various types of phase-transition and phase-separation models both related to liquids and to solids.
Mathematically speaking, the above choice permits to consider a more general class of singular potentials (still including the Ball–Majumdar case). Moreover, it permits us to get a stronger version of the entropy inequality [namely, it holds as a distributional inequality, and not only in the integrated form (2.23)]. As a drawback, we have a marginal regularity loss for \(\mathbb {Q}\). To state our related result, we first have to introduce the auxiliary function [cf. (2.11)]
needed in the statement of the entropy inequality.
Theorem 5.1
Let the coefficients \(\mu ,\kappa ,\Gamma \) satisfy (2.10), (2.12), and (5.1), with relation (2.13) on exponents. Moreover, let the initial data comply with (2.14)–(2.16). Finally, let \(f\) be any potential fulfilling conditions (i)–(iii) of Sect. 2.2. Then, there exists a quadruplet \((\mathbf{u},p,\mathbb {Q},\vartheta )\) with the regularity (2.24), (2.25), and (2.27), together with
satisfying the incompressibility condition (2.2), the momentum equation (2.3), the \(Q\)-tensor evolution system (2.5) and the total energy balance (2.17) in the sense of distributions and complying with the initial conditions as specified in (2.30)–(2.31). Moreover, the entropy inequality holds in the distributional sense: for any nonnegative function \(\phi \in {\mathcal D}([0,T]\times \overline{\Omega })\),
Remark 5.2
Let us notice that under the current hypotheses, if we have any sufficiently smooth weak solution satisfying (5.5) with the equal sign, then that solution also satisfies the heat Eq. (2.8), or, in other words, it solves the problem in the “physical” sense. Hence, the current formulation and the physical one are in some sense equivalent. Instead, the same does not hold, even for smooth solutions, when we only know that the integrated entropy equality (i.e., (2.22) with the equal sign) holds.
Remark 5.3
Regarding the admissible class of potentials, we may relax even more our assumptions. Indeed, what we need for \(f\) is simply that to be a convex function, with open bounded domain \({\mathcal D}[f]\), smooth inside \({\mathcal D}[f]\). Namely, we do not need \({\mathcal D}[f]\) to be related to the eigenvalues of \(\mathbb {Q}\) [cf. (ii) of Sect. 2.2].
Proof of Theorem 5.1
The argument follows the lines of the proof given for Theorem 2.2. Hence, we only sketch the differences occurring in the a priori estimates and in the weak stability argument.
First of all, we can observe that all the a priori bounds directly deriving from the energy and entropy balances still hold. Additionally, thanks to (5.1), (3.9) is improved as
Combining the last of (5.6) with (3.4) and using a generalized version of Poincaré’s inequality (cf. [36, Lemma 3.2] or [23, Lemma 5.1]), we deduce more precisely
The key point, as before, is constituted by the estimate of the singular potential.
Estimate of the singular potential for singular heat flux Differently from before, we do not need to rely on the explicit properties of the Ball–Majumdar potential. Indeed, we may directly compute
Now, let us observe that
Then, integrating (5.8) in time over \((0,T)\) and using the latter of (3.13), we arrive at
the last inequality following from estimates (3.8) and (5.7). Hence, in place of (3.23)–(3.24), we get now
Using (3.11) and interpolation, we then obtain
where \(\frac{2A}{A+1}>\frac{9}{5}\) thanks to (2.13).
Weak sequential stability for singular heat flux Also, this part of the procedure goes through as before, the only difference being represented by some loss of regularity on \(\mathbb {Q}\). Indeed, in place of (3.26), we now have, due to (5.14),
Combining (5.13) with the second (3.27) and using the Gagliardo–Nirenberg inequalities, it is not difficult to obtain
This leads to some obvious modifications in the exponents in the subsequent relations. For instance, in (3.32)–(3.33), \(L^{12/7}\) turns to \(L^{18/11}\); in the latter of (3.37), \(L^2\) has to be replaced, say, by \(L^{9/5}\); in (3.41), \(L^{4/3}\) becomes \(L^{9/7}\). Finally, \(L^{36/29}\) is substituted by \(L^{6/5}\) in (3.44). A further check shows that the argument used to get the strong convergence (3.53) still works even though the summability properties of the term \(|\nabla _x\mathbb {Q}|^2/2\) in the expression of \(e\) (2.19) are now a little bit lower.
The subsequent convergence relations regarding \(\vartheta \) do not change. Concerning, instead, the identification of the singular potential term, it suffices to notice that, in place of (3.61), we now have only
due to (5.14).
To conclude the proof, we have to see that we can take the (supremum) limit in the distributional entropy inequality (5.5) (of course, we are now assuming it holds in that form at the level \(n\) and want to show that it holds in the same form also in the limit). To this aim, we first notice that the right hand side is treated as in (3.64), taking now a generic (nonnegative) test function \(\phi \). To proceed, we have to deal with the left hand side. We notice first that, by (5.7),
whence, thanks to pointwise convergence [see (3.55)] and Sobolev’s embeddings,
In particular, we identify \(\overline{\vartheta ^{-1}} = \vartheta ^{-1}\).
Next, let us note that, by (5.13), (5.18) and interpolation,
Let us now recall the standard identity
where \(f^*\) is the convex conjugate (see, e.g., [34] [Thm. 23.5, p. 218]) of \(f\). Then, by (ii) of Sect. 2.2, \(f^*\) grows at most linearly at infinity. Combining (5.20), (5.21) and (3.27), we then obtain
Now, interpolating exponents in (3.25) and accounting for the strong convergence (3.40), we get
Moreover, we have
Then, recalling the expression (2.20) for \(s\), combining (5.22) with (5.23), and (5.24) and (3.60) with (3.40), we finally infer
To conclude the proof, we need to take the limit in the third integral in the first row of (2.22), where we recall that \(H\) was defined in (5.2). To this aim, we need a proper compactness tool:
Lemma 5.4
Let \(h:\mathbb {R}\rightarrow [0,+\infty )\) be a continuous function such that \(h(z)\rightarrow \infty \) as \(z\rightarrow \infty \). Suppose that \(\{u_n\}\) is a sequence of real-valued measurable functions defined over \((0,T)\times \Omega \) such that
for some positive constants \(H\) and \(C\). Then,
Proof
Clearly, it is enough to show equi-integrability of the \(L^2\)-norm of \(u_n\), specifically
To see this, we first observe that
Now, applying the Hölder inequality, we get
whence (5.31) follows.
Relations (3.4) and (5.18) allow us to apply the lemma with the choice \(u_n=1/\vartheta _n\) and \(h(u)=\log u\). Hence, relation (5.19) is improved up to
Hence, using (3.55), we have that
and consequently all terms on the left-hand side of (5.5) pass to the limit \(n\nearrow \infty \), which concludes the proof of the theorem.
Notes
Note that the derivatives are computed at \((\theta ,\varphi ,\tilde{\theta },\tilde{\varphi })=(0,\frac{\pi }{2},0,\frac{\pi }{2})\) and we take \((p_1,p_2,p_3)=(\cos \varphi \sin \theta ,\sin \varphi \sin \theta ,\cos \theta )\) with \((\varphi ,\theta )\in [0,2\pi ]\times (0,\pi )\) and \((q_1,q_2,q_3)=(\cos \tilde{\varphi }\sin \tilde{\theta },\sin \tilde{\varphi }\sin \tilde{\theta },\cos \tilde{\theta })\) with \((\tilde{\varphi },\tilde{\theta })\in [0,2\pi ]\times (0,\pi )\).
That \(f(\alpha )\) is bounded when \(\alpha \in [0,1]\) can be more easily seen by looking at the expression of \(f(\alpha )\) in trigonometric form.
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The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 320078.
The work of E.F. was partially supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase).
The work of G.S. was supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).
The work of E.R. was supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).
The work of A.Z. was partially supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase).
Appendix: A technical estimate
Appendix: A technical estimate
We conclude the paper by providing the proof of the key relation (4.11), along with the necessary technical preparations. Our approach is based on the so-called Laplace’s method to evaluate the integrals containing exponentials, see [21]. Let us first note that the positive definiteness of \(N=\left( \frac{\partial ^2\ln Z}{\partial \mu _i\partial \mu _j}\right) _{i,j=1,2,3}\) is proved in [3]. For convenience, we recall the argument, namely
In estimating the positivity of \(\frac{2}{\varepsilon }Id- \left( N\mu \otimes \mu +\mu \otimes \mu N\right) \), we consider the components of the matrix \(N\mu \otimes \mu +\mu \otimes \mu N\), all functions of \((\mu _1,\mu _2,\mu _3)\in X\) [cf. (4.6)]:
It will suffice to show that \({\mathcal {I}}_{ij}(\mu _1,\mu _2,\mu _3), i,j=1,2,3\) is bounded independently of \((\mu _1\mu _2,\mu _3)\in X\). To this end, we consider two cases: when \(\mu _i\not =\mu _j\) for \(i\not =j\) and when two out the three \(\mu _i,i=1,2,3\) are equal.
1.1 Case I: \(\mu _i\not =\mu _j\) for \(i\not =j\)
We let \(\mu _i=\rho \gamma _i\) with \(\rho \ge 0\) and \((\gamma _1,\gamma _2,\gamma _3)\in \mathbb {S}^2\cap X\). It will suffice to prove
uniformly in \((\gamma _1,\gamma _2,\gamma _3)\in \mathbb {S}^2\cap X\).
We first recall the Laplace’s asymptotic expansion method for evaluating integrals (see for instance [21] and also [7]), namely let us consider the integral:
with \(\Omega \subset \mathbb {R}^d\) with \(\Omega \) an open bounded set and \(f,g\in C^\infty (\overline{\Omega },\mathbb {R})\).
If \(f\) has a single strict global minimum point in \(\bar{x}\in \Omega \) such that \(f(\bar{x})=0\), then for any \(n\in \mathbb {N}\), one has
as \(h\rightarrow \infty \) with \(c_k\) coefficients that are explicitly computable in terms of \(f\) and \(g\). More precisely, we will only need the first three, \(c_0, c_1, c_2\) given by
where we denoted, for \(\sigma =\frac{x}{|x|}\):
Let us observe that in [21], the formulae were determined by allowing \(g\) to be potentially singular at the maximum point. However, if \(g\) is smooth (as is our case), then one can see that the odd-index coefficients \(c_{2k+1}\) are zero (see also [7] or directly compute \(c_1\) by the formula above).
In order to apply the previous argument, let us observe that the function \(\gamma _m (p_m^2+q_m^2)\) of variable \((p,q):=((p_1,p_2,p_3),(q_1,q_2,q_3))\in \mathbb {S}^2\times \mathbb {S}^2\) attains its maximum value at two points that depend on the maximum element in the set \(\{\gamma _1,\gamma _2,\gamma _3\}\). Let us assume without the loss of generality that \(\gamma _2,\gamma _3< \gamma _1\). Then, the maximum of the function \(\gamma _m (p_m^2+q_m^2)\) is attained at two points \((p,p)\) with \(p\in \{(1,0,0),(-1,0,0)\}\) so in order to apply Laplace’s method, we need to split \(\mathbb {S}^2\times \mathbb {S}^2\) into two subdomains. Let us denote \(\mathbb {S}_E:=\{p\in \mathbb {S}^2, p\cdot (1,0,0)>0\}\). We then apply the previously mentioned Laplace’s method on each of the sets \(\mathcal {E}:=\mathbb {S}_E\times \mathbb {S}_E\) and \(\mathcal {V}:=\mathbb {S}^2\times \mathbb {S}^2\setminus \mathcal {E}\) chosing \(h=\rho \) and \(f(p_1,p_2,p_3,q_1,q_2,q_3):=\gamma _1-\gamma _i (p_i^2+q_i^2)\) (note that we can multiply both denominator and numerator in \({\mathcal {I}}_{ij}\) by \(e^{-\rho \gamma _1}\)).
The function \(g\) will be chosen to be either \(g\equiv 1\) (for treating denominators) or
(for dealing with the numerators). Then, we take \(d=4\) and for treating the denominator, we note that each of the two integrals (over \(\mathcal {E}\), respectively, \(\mathcal {V}\)) admits an asymptotic expansion of the form: \(\bar{c}_0\rho ^{-2}+o(\rho ^{-2})\). Similarly, for the numerator, the integrals have an expansion of the form \(\rho ^2(\tilde{c}_0\rho ^{-2}+\tilde{c}_1\rho ^{-\frac{5}{2}}+\tilde{c}_2\rho ^{-3}+\tilde{c}_3\rho ^{-\frac{7}{2}}+\tilde{c}_4\rho ^{-4}+o(\rho ^{-4}))\). As previously mentioned, the coefficients of odd index, namely \(\tilde{c}_1,\tilde{c}_3\) are zero. Taking into account the specific form of \(g_{ij}\) (6.7) and the expressions of the coefficients \(\tilde{c}_0\), namely (6.4), respectively, \(\tilde{c}_2\), namely (6.6), we have that both \(g_{ij}\) and its first and second derivatives are zeroFootnote 1 so \(\tilde{c}_0=\tilde{c}_2=0\). Thus, the numerator will have an asymptotic expansion of the form \(\tilde{c}_4\rho ^{-2}+o(\rho ^{-2})\) and overall \({\mathcal {I}}_{ij}\) will be bounded as \(\rho \rightarrow \infty \).
1.2 Case II: Two out the three \(\mu _i,i=1,2,3\) are equal
We assume without the loss of generality that \(\mu _1=\mu _2\) and we denote the common value by \(\mu \). Then, \(\mu _3=-\mu _1-\mu _2=-2\mu \). After a couple of manipulations, using that \((p_1,p_2,p_3)=(\cos \varphi \sin \theta ,\sin \varphi \sin \theta ,\cos \theta )\) with \((\varphi ,\theta )\in [0,2\pi ]\times (0,\pi )\), we have (by denoting \({\bar{Z}}:=2\pi \int _0^\pi e^{3\mu \sin ^2\theta }\sin \theta \,d\varphi \)):
Thus, it suffices to understand what happens with ratios of the type \(\frac{\int _0^{\pi } e^{3\mu \sin ^2\theta }(\mu \sin ^2\theta )^k \sin \theta \,d\theta }{\int _0^{\pi } e^{3\mu \sin ^2\theta }\sin \theta \,d\theta }\) for \(k=1,2\). The case when \(\mu \rightarrow \infty \) can be dealt with using Laplace’s method previously described (but in \(1D\) now, as opposed to \(2D\) domains before), as in this case the maximum of \(3\sin ^2\theta \) occurs in the interior of the interval \((0,\pi )\). We obtain as desired that the limit as \(\mu \rightarrow \infty \) of all the \({\mathcal {I}}_{ij}\) is finite.
In the case \(\mu \rightarrow -\infty \), we note that \(-3\sin ^2\theta \) attains it maximum at the end points of the interval \((0,\pi )\) so Laplace’s method cannot be applied directly. In this case, we let \(-\mu =\alpha ^2\) and consider the change of variables: \(\alpha \sin \theta =y\) which leads to
We let \(f(\alpha ):=\frac{\int _0^\alpha e^{-3y^2}y^{2k+1} \frac{\mathrm{d}y}{\sqrt{1-\frac{y^2}{\alpha ^2}}}}{\int _0^\alpha e^{-3y^2}{y}\frac{\mathrm{d}y}{\sqrt{1-\frac{y^2}{\alpha ^2}}}}\) and then we have
On the other hand,
We assume without the loss of generality that \(\alpha >1\) Footnote 2 and using this assumption as well as the last estimate in (6.8), we obtain
so \(f(\alpha )\) is bounded independently of \(\alpha \).
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Feireisl, E., Schimperna, G., Rocca, E. et al. Nonisothermal nematic liquid crystal flows with the Ball–Majumdar free energy. Annali di Matematica 194, 1269–1299 (2015). https://doi.org/10.1007/s10231-014-0419-1
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DOI: https://doi.org/10.1007/s10231-014-0419-1