Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals
aa r X i v : . [ m a t h . A P ] O c t GLOBAL WELL-POSEDNESSAND TWIST-WAVE SOLUTIONSFOR THE INERTIAL QIAN-SHENG MODELOF LIQUID CRYSTALS
Francesco De Anna and Arghir Zarnescu , , October 7, 2018
Abstract
We consider the inertial Qian-Sheng model of liquid crystals which couples a hyperbolic-type equation involving a second-order material derivative with a forced incom-pressible Navier-Stokes system. We study the energy law and prove a global well-posedness result. We further provide an example of twist-wave solutions, that issolutions of the coupled system for which the flow vanishes for all times.
1. Introduction
The main aim of this article is to study a system describing the hydrodynamics of nematicliquid crystals in the Q-tensor framework (see for an introduction to the Q -tensor framework[9],[17]) There exists several such models and we will consider the one proposed by T. Qianand P. Sheng in [13]. As most tensorial models, this one provides an extension of the classicalEricksen-Leslie model [6], in particular capturing the biaxial alignment of the molecules, afeature not available in the classical Ericksen-Leslie model.Our main interest in this model is due to the fact that it incorporates systematically acertain term that models inertial effects. Details about the physical relevance of this will beprovided in the Subsection 1.1, below.The inertial term is usually neglected on physical grounds, a fact that is also convenientmathematically since keeping it generates considerable analytical and numerical challenges.From a mathematical point of view the system couples a forced incompressible Navier-Stokessystem, modelling the flow, with a hyperbolic convection-diffusion system for matrix-valuedfunctions that model the evolution of the orientations of the nematic molecules. The inertialterm is responsible for the hyperbolic character of the equation describing the orientation (1) Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA Email: [email protected] (2) IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain(3) BCAM, Basque Center for Applied Mathematics, Mazarredo 14, 48009 Bilbao, Spain
Email: [email protected] (4)“Simion Stoilow” Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei, 010702 Bucharest, Romania of the molecules. This feature is also present in the Ericksen-Leslie model but it is usuallyneglected in the mathematical studies due to the formidable difficulties in treating it in thepresence of the specific unit-length constraint. One can regard our study as a step towardsthe analytical understanding of the inertial Ericksen-Leslie model where one discards theunit-length constraint.In order to clearly describe the system it is convenient to introduce some terminology.The local orientation of the molecules is described through a function Q taking values fromΩ ⊂ R d , into the set of the so-called d -dimensional Q -tensors, that is symmetric and traceless d × d matrices: S ( d )0 := (cid:8) Q ∈ R d × d ; Q ij = Q ji , tr( Q ) = 0 , i, j = 1 , . . . , d (cid:9) The evolution of the Q ′ s is driven by the free energy of the molecules, as well as the transport,distortion and alignment effects caused by the flow.The velocity of the centres of masses of molecules obeys a forced incompressible Navier-Stokes system, with an additional stress tensor, a forcing term modelling the effect that theinteraction of the molecules has on the dynamics of their centres of masses. Explicitly theequations, in non-dimensional form, are:˙ v + ∇ p − β v = ∇ · (cid:18) − L ∇ Q ⊙ ∇ Q + β Q tr { QA } + β AQ + β QA (cid:19) + ∇ · (cid:18) µ Q − [Ω , Q ]) + µ (cid:2) Q, ( ˙ Q − [Ω , Q ]) (cid:3)(cid:19) (1.1) ∇ · v = 0 (1.2) J ¨ Q + µ ˙ Q = L ∆ Q − aQ + b ( Q − d | Q | I d ) − cQ | Q | + ˜ µ A + µ [Ω , Q ] (1.3)where ˙ f = ( ∂ t + v · ∇ ) f denotes a material derivative and for any two d × d matrices M , N , we denote their commutator as [ M, N ] :=
M N − N M . Furthermore, we denote A ij := ( v i,j + v j,i ), Ω ij := ( v i,j − v j,i ), for i, j = 1 , . . . , d , ( ∇ Q ⊙ ∇ Q ) ij := P dk,l =1 Q kl,i Q kl,j (where for a scalar function f , we write f ,j for ∂f∂x j ) and | Q | = p tr( Q ). The I d denotes the d × d identity matrix.The physical relevance of the equations and their meaning is provided in the next sub-section, which can be skipped without impeding on the understanding of the remainingmathematical aspects of the paper. In the following we consider just the d = 3 case ( out of which onecan reduce everything in a standard manner to the d = 2 case) and take the domain Ω tobe R . The velocity v of the centres of masses the molecules satisfies a convection-diffusionfluid-type equation, with forcing provided by the pressure p , the distortion stress σ and the iquid crystal model with inertia 3 viscous stress σ ′ (here and in the following we use the Einstein summation convention, ofsummation over repeated indices):˙ v i = ( − pδ ij + σ ij + σ ′ ij ) ,j , (1.4)where p is the pressure.The fluid is taken to be incompressible so we have the divergence-free constraint: v k,k = 0 . (1.5)The distortion stress σ is given by σ ij := − ∂ F ∂ ( Q αβ,i ) Q αβ,j where we use the simplest form of the Landau-de Gennes free energy density F [ Q ] := L |∇ Q | + ψ B ( Q )modelling the spatial variations through the L |∇ Q | term with positive diffusion coefficient L >
0, and the nematic ordering enforced through the “bulk term” taken to be of thestandard form [9] ψ B ( Q ) = a Q ) − b Q ) + c Q )) . (1.6)The viscous stress σ ′ is given by : σ ′ ij := β Q ij Q lk A lk + β A ij + β Q jl A li + β Q il A lj + 12 µ N ij + µ Q il N lj − µ N il Q lj , where β , β , β , β , µ and µ are viscosity coefficients, A is the rate-of-strain tensor definedby means of A ij = v i,j + v j,i , i.e. the symmetric part of the velocity gradient, and N stands for the co-rotational timeflux of Q , whose ( i, j )-th component is defined as follows N ij := (cid:0) ˙ Q − ω ∧ Q + Q ∧ ω (cid:1) ij = ∂ t Q ij + v k Q ij,k − ε ikl ω k Q lj − ε jkl ω k Q il . N represents the time rate of change of Q ij with respect to the background fluid angularvelocity ω = ∇ × v . Moreover, one can reformulate N making use of the vorticity tensorΩ Ω ij := v i,j − v j,i . Note that in [13] the divergence of a matrix is taken along columns rather than rows as we do in here.However we changed everything consistently to fit our definition of matrix divergence
Francesco De Anna Arghir Zarnescu
Indeed, one can check that N ij = (cid:0) ˙ Q − [Ω , Q ] (cid:1) ij = ˙ Q ij − Ω il Q lj + Q il Ω lj , since we have ω × u = Ω u , for any d -dimensional vector u .We will assume that the viscosity coefficients satisfy the following two constraints β − β = µ ,β + β = 0 . (1.7)For a better understanding of the relation between these conditions and other ones availablein the literature, it is worth making a comparison between the stress tensor σ given in (1.7)and the better-known Leslie stress tensor. Indeed whenever Q is uniaxial (with unitary scalarorder parameter, for simplicity) i.e Q ( t, x ) = s ∗ ( n ( t, x ) ⊗ n ( t, x ) − ) , with n ( t, x ) ∈ S thedirector field, s ∗ = 0, the tensor σ becomes the better-known Leslie stress tensor of theEricksen-Leslie theory, up to some relations involving the viscosity coefficients (see [13]).The well-known Parodi’s relation for the Leslie stress tensor corresponds in our setting to β − β = µ , (1.8)namely the first identity of (1.7).The second condition in (1.7) is not always satisfied by physical materials (though forsome it is nearly satisfied such as for MBBA, see below) however it is sometimes assumed inthe physics literature in the more specialised form β = β = 0 (see for instance [12]).Moreover, we will need to assume that the Newtonian viscosity β is large enough comparedto the other remaining viscosities, in order to obtain the necessary energy dissipation.The evolution of the order tensor Q is driven by J ¨ Q ij = h ij + h ′ ij − λδ ij − ε ijk λ k . (1.9)where ε ijk , the Levi-Civita symbol. The λ, λ k are Lagrange multiplier enforcing the trace-lessness and symmetry of the tensor and in our case they can be easily determined as λ k = 0and λ = − b | Q | I (with I the 3 × h is h ij := − ∂ F ∂Q ij + (cid:18) ∂ F ∂ ( ∂ k Q ij,k ) (cid:19) ,k and the viscous molecular field h ′ is given by: h ′ ij := 12 ˜ µ A ij − µ N ij , (1.10)The definition of ˜ µ requires some clarifications. We note that in the paper [13] of Qianand Sheng, the viscosity coefficient ˜ µ corresponds exactly to µ while other authors take itto be ˜ µ = − µ , see [11, 12]. The two different choices of the sign for ˜ µ provide intrinsical iquid crystal model with inertia 5 differences at the energy level, as it will be seen in Section 2. We will see there that it wouldbe more natural to assume ˜ µ = − µ , otherwise a new continuum variable˙ Q + [Ω , Q ] (1.11)would effect the time-evolution of the flow. However, if we would take alternatively ˜ µ = − µ we would obtain the classical co-rotational time flux N = ˙ Q − [Ω , Q ] (instead of the abovevariable in (1.11)).We assume all the coefficients to be non-dimensional. For a common physical example,the MBBA material, we have the following relations between the coefficients [14]: µ µ ∼ − . , β µ ∼ . , β µ ∼ . , β µ ∼ . , β µ ∼ − .
79 (1.12)Furthermore, because the coefficient β corresponds to the standard Newtonian stress tensorwe can assume β > J in (1.9) stands for the inertial density and it is taken to be greater than 0. This isconsistent with the fact that J has the same sign as the inertia in the Leslie-Ericksen typeof model (see Appendix B in [13]) where it is assumed to be positive (see for instance theassumption that J.L. Ericksen makes in [1]).The inertial term could conceivably play a role when the anisotropic axis is subjected tolarge accelerations, as motivated by F. Leslie (in the context of the director model) in [8].Another interesting feature of the inertia is that it captures the wave-like phenomena,and one of the most mysterious and yet simple manifestation of these is related to the so-called twist-waves, introduced by J.L. Ericksen in [1]. These are very special solutions ofthe coupled system, for which the flow vanishes for all time. The effect of the flow stillremains on the Q -tensor part, by imposing an additional constraint, so these are very specialsolutions. We note first that the system admits a Lyapunov-type functional, upto some relations on the viscosity coefficients. This functional includes the free energy due tothe director field, the kinetic energy of the fluid and most importantly the rotational kineticenergy of the director field.
Theorem 1.1. [Energy law and apriori control of low-regularity norms]
We consider the system (1.1) , (1.2) , (1.3) in R d , d = 2 or d = 3 . Let us assume that theviscosity coefficients fulfill β , β , µ > , (1.14) and the inertia coefficient J as well as the diffusion coefficient L are positive. Furthermore,we assume: Francesco De Anna Arghir Zarnescu β − β = µ ,β + β = 0 . (1.15) Concerning ˜ µ we assume that:if ˜ µ = µ then both of them are set to zero, i.e. ˜ µ = µ = 0 . (1.16) Moreover, in order to have the free energy of the molecules well defined we assume that thematerial coefficient c satisfies c > . (1.17) Then there exists a constant C d depending on ˜ µ , β , β , µ such that if the Newtonianviscosity is large enough, i.e. β > C d then for classical solutions that decay fast enough atinfinity the total energy decays, i.e. ddt Z R d (cid:0) | v | + J | ˙ Q | + L |∇ Q | (cid:1) + ψ B ( Q ) d x ≤ Furthermore: • If d = 2 and a ≥ then ψ B ( Q ) ≥ and for any T > we have the apriori bounds: v ∈ L ∞ (0 , T ; L ( R d )) ∩ L (0 , T ; H ( R d )) , (1.19) Q ∈ L ∞ (0 , T ; H ( R d )) with ˙ Q ∈ L ∞ (0 , T ; L ( R d )) , • If d = 2 and a < , or d = 3 (and a arbitrary) then ψ B ( Q ) can be negative . Thenthere exists ¯ µ = ¯ µ ( a, b, c ) , J := J (¯ µ , a, b, c ) , and ˜ C d = ˜ C d (˜ µ , β , β , µ ) > suchthat if J < J , µ > ¯ µ , β > ˜ C d then the apriori bounds (1.19) hold. The proof of Theorem 1.1 exhibits the main characteristic feature of the system, that isthe mixing of terms that provide the most suitable cancellation of “extraneous” maximalderivatives, i.e. the highest derivatives in v that appear in the Q equation and the highestderivatives in Q that appear in the v equation.It is important to observe that despite these apriori estimates, one cannot expect toconstruct weak solutions just by making use of this energy relation. Indeed, the most commonapproach in order to construct weak solutions is the compactness method, i.e. constructapproximate solutions (satisfying similar apriori bounds) and pass to the limit. In theclassical Navier-Stokes equation one needs to take care in dealing with the nonlinear terms,but the apriori estimates available do provide enough control. However, things are much sufficiently fast to be able to integrate by parts in the proof of the theorem, which happens for instanceif they are in the function spaces (1.19) and thus the energy decay (1.18) does not suffice for providing the apriori bounds (1.19) iquid crystal model with inertia 7 worse in our system (1.1)-(1.3). The main difficulty is inside the stress tensor σ ij , moreprecisely in the nonlinear term( ∇ Q ⊙ ∇ Q ) ij := d X α,β =1 Q αβ,i Q αβ,j (1.20)Let us note that the estimates provided by the apriori bounds (1.19) do not suffice forpassing to the limit in the divergence of (1.20). This is to be contrasted with the case J = 0.One should keep in mind that a positive inertial density J leads the order tensor equationto be hyperbolic-like, in contrast to the parabolic structure that occurs when J is neglected.In the parabolic setting one can make use of regularizing effects, achieving a control on twospatial derivatives of Q (i.e. ∆ Q ), which certainly allows to control the limit of a productas in the divergence of (1.20). This feature is lost when J is positive, so that constructingweak solutions would require a different approach than a rather common compactness onebased on estimates (1.19).Thus one can attempt to construct strong solutions and it will turn out that this canbe done. The most interesting aim is then to construct global in time solutions. We havebeen able to obtain them, using the one of main features of the system namely the dampingprovided by the µ term. Indeed, if one formally takes the flow v to be zero in (1.3) thenthe material derivatives in (1.3) reduce to just time derivatives and the equation becomes anonlinear damped wave equation. Morally speaking it will be this damping that is responsiblefor the global existence even in the case when the flow is present. Thus we have: Theorem 1.2. [Global existence and uniqueness for small initial data]
Consider the system (1.1) , (1.2) , (1.3) .We assume that J < J , µ > ¯ µ , β > ˜ C d (where J := J (¯ µ , a, b, c ) , ¯ µ = ¯ µ ( a, b, c ) , and ˜ C d = ˜ C d (˜ µ , β , β , µ ) > are explicitly computablecoefficients). Furthermore we assume the positivity of a number of coefficients: β , µ > , a > and J, L > .Let ( v , Q ) : R d → R d × S ( d )0 be in H s ( R d ) × H s +1 ( R d ) with s > d and d = 2 or d = 3 .Then there exists ε > , depending on s and d such that if η := k v k H s + k Q k H s +1 + k ˙ Q k H s < ε then there exists a unique strong solution ( v, Q ) of (1.4) - (1.9) , which is global in time.Moreover there exists a positive constant C (independent of the solution) such that k v k L ∞ ( R + ; H s ( R d )) + k∇ v k L ( R + ; H s ( R d )) + k Q k L ∞ ( R + ; H s +1 ( R d )) + k Q k L ( R + ; H s +1 ( R d )) ++ k ˙ Q k L ∞ ( R + ; H s ( R d )) + k ˙ Q k L ( R + ; H s ( R d )) Cη . One important assumption in the above theorem is that the coefficient a >
0. Thiscaptures a regime of physical interest but unfortunately not the most interesting physicalregime (which would be for a ≤ Francesco De Anna Arghir Zarnescu assumption that a > “damping in time” that was usedpreviously in related settings in [16], respectively [5].The difficulties associated with treating the system (1.1),(1.2),(1.3) are generated, as usu-ally (in this type of non-Newtonian fluid) by the forcing term in the Navier-Stokes part. Onecan essentially think of the system as a highly non-trivial perturbation of a Navier-Stokessystem.However the specific difficulty in our system is that the forcing term involves Q whoseequation is now of hyperbolic nature. This is due to the inertial term that contains a secondorder material derivative . We are not aware of any systematic treatment of such a term inother contexts, but it turns out that the most delicate part of the whole proof is related toits treatment. Indeed, one should start by noticing that this is very far from the case when v = 0 (when it is just ∂ t Q ) because it is a highly nonlinear operator, for instance for v and Q smooth we explicitly have:¨ Q = ∂ t Q + 2( v · ∇ ) ∂ t Q + ( v t · ∇ ) Q + (( v · ∇ ) v · ∇ ) Q + v ∇ Qv (1.21)involving an “expensive derivative of v” i.e. ∂ t v and the term v ∇ Qv that competes in asense with the regularizing Laplacian L ∆ Q on the right hand side of the Q -equation.Our main trick in dealing with the double material derivative has been to stay as close aspossible to the standard cancellation appearing in the context of convective derivatives, whichcan be formally written as R R d ( f · ∇ ) vf dx = 0 for f decaying sufficiently fast at infinity.However, in order to implement this we have to use a higher-order commutator estimate thatappears in [2], an estimate which is at the level of homogeneous Sobolev spaces, ˙ H s . This isvery convenient for our purposes because the H s ( R d ) norm does not allow the cancellationof the worst terms, as in obtaining (1.18) in the L ( R d )-setting. This difficulty is partiallydealt with by reformulating the inner product of H s ( R d ) into h ω , ω i L ( R ) + h ω , ω i ˙ H s ( R ) = Z R dξ (cid:0) | ξ | s (cid:1) ˆ ω ( ξ )ˆ ω ( ξ )d ξ, where ˙ H s ( R d ) stands for the homogeneous Sobolev space with index s . It is straightforwardthat this inner product generates the same topology in H s ( R d ) with respect to the commonone.Our main work on proving the existence of classical solutions is to obtain an uniformestimate for our approximate solutions, that is to close an estimate of the type:Φ ′ ( t ) + Ψ( t ) C Φ( t )Ψ( t ) , (1.22)where C is a suitable positive constant, Φ is controlling the H s -norms in space for oursolution and Ψ is an integrable-in-time quantity involving H s -norms. Then, a rather standardargument (see for instance in the Appendix, Lemma 5.1) allows to propagate the smallnesscondition on the initial data (i.e. on Φ(0)). This leads the right-hand side of the aboveequation to be absorbed by the left-hand side, achieving the cited uniform estimates. Finallywe construct our classical solution, through a compactness method. iquid crystal model with inertia 9 The uniqueness of our solutions is proven by evaluating the difference between two solu-tions at a regularity level s = 0, i.e. in L ( R ). Our work is mainly to obtain an estimatethat leads to the Gronwall lemma. Here the main difficulties are handled taking into ac-count a specific feature of the coupling system related to the difference of the two solutions.This feature allows the cancellation of the worst term when considering certain physicallymeaningful combination of terms.It is perhaps interesting to remark that in Theorem 1.2 we do not need consider a positiveconstant c in the bulk free energy density ψ B ( Q ). Usually, this is a necessary condition inorder to have ψ B ( Q ) bounded from below the space of Q -tensors . However we do not needthis restriction on c mainly because we are assuming a smallness condition on the initialdata, smallness which will be propagated by the equation.Finally, one last issue of interest to us are the so-called “twist waves” . These are solutionsof the coupled system, for which the flow v is identically zero. The existence of such solutionsfor the Ericksen-Leslie system was first postulated by J.L. Ericksen in [1], who named them“twist waves” and provided one explicit example.We note that if v = 0, the Q -tensor evolution (1.3) reduces to a nonlinear wave system: J Q tt + µ Q t = L ∆ Q − aQ + b ( Q − | Q | I ) − cQ | Q | (1.23)Nevertheless, there is a part of the momentum equation (1.1) that survives as an additionalconstraint on Q , namely: ∇ p = ∇ · (cid:18) − ∇ Q ⊗ ∇ Q + µ Q t + µ (cid:2) Q, Q t (cid:3)(cid:19) (1.24)Clearly, because of this additional constraint, only very special types of initial data for (1.23)will generate solutions that respect the constraint (1.24). One example is obtained by takingin R d with d = 2 or d = 3 the ansatz: T ( t, x ) := f ( t, | x | ) ¯ H ( x )with f : R + × R → R a function to be determined and ¯ H the “hedgehog” function (see [3]for details about its physical significance) :¯ H ij ( x ) := x i x j | x | − δ ij d , i, j = 1 , . . . , d Then the (1.23) reduces to:
J f tt + µ f t = L (cid:18) f rr + f r d − r − dr f (cid:19) − af + b ( d − d f − c ( d − d f (1.25) also c > Q equationwhere we take u = 0 and J = 0 We can then show:
Proposition 1.3.
Let f : R → R be a smooth function such that f (0) = f r (0) = 0 . Let d = 2 and assume that in (1.25) we have J, L, c, µ > . Then there exists a function f : R + × R → R , that is C on (0 , ∞ ) × (0 , ∞ ) , and such that both f and f t are smoothfunctions of r on [0 , ∞ ) , solution of the equation (1.25) witht f (0 , r ) = f ( r ) so that thefunction T ( t, x ) = f ( t, | x | ) ¯ H ( x ) is a smooth twist wave in R d , i.e. a classical solution of (1.23) satisfying the constraints (1.24) .Remark . A similar statement can be made for d = 3 but for technical reasons we areunable to show the global existence of the twist-wave in this case, but only the local existencein time. Organization of the paper
This paper is organised as follows: in the next section we prove Theorem 1.1 concerningthe energy law, in Section 3 we present apriori estimates for higher norms and prove theglobal existence result, namely Theorem 1.2. Finally in Section 4 we construct a specifictwist-wave solution, providing the proof of Proposition 1.3 . A number of natural openproblems are proposed and discussed in Section 5.
Notations and conventions
We denote by ˙ f the material derivative ˙ f = ∂ t h + v · f , where the fluid velocity v isunderstood from the context. We also use the Einstein summation convention, that issummation over repeated indices. We denote weighted spaces, by specifing the weightedmeasure, for instance L ( R , r dr ) = { f : R → R , R R f r dr < ∞} . We also use the notation k ( f, g ) k X = k f k X + k g k X for any elements f, g ∈ X with X a suitable normed space withnorm k · k X . We denote by F and F − respectively the Fourier transform and its inverse.We use ε ijk , the Levi-Civita symbol, with indices i, j, k from 1 to d , and the comma inthe subscript denotes derivative with respect to particular spatial coordinate. If M ( x ) isa d × d -matrix, then ∇ · M stands for the vector field ( M ij,j ) i =1 ...,d . The | M | denotes theFrobenius norm of the matrix, i.e. | M | = √ M M t .The [ · , · ] stands for the usual commutator bracket [ A, B ] := AB − BA , for any d × d -matrices A and B . We denote = tr( AB ) with A : B . The product ∇ A ⊗ ∇ B is a matrixwith ij component ( ∂ i A : ∂ j B ). iquid crystal model with inertia 11
2. Energy law and apriori boundsProof of Theorem 1.1:
We multiply the equation (1.1) by v i integrate over the spaceand by parts and use (1.2) to cancel some terms. To the result obtained we add equation(1.3) multiplied by ˙ Q ij , integrated over the space and by parts to obtain : ddt Z R d (cid:0) | v | + J | ˙ Q | (cid:1) dx = Z R d LQ kl,j Q kl,i v i,j + (cid:18) L ∆ Q ij − L ∂ψ B ∂Q ij (cid:19) ( ∂ t Q ij + v · ∇ Q ij ) dx − Z R d σ ′ ij v i,j dx + Z R d h ′ ij ˙ Q ij dx = L Z R d Q kl,j Q kl,i v i,j + ∆ Q ij v k Q ij,k dx | {z } := I − L Z R d ∂ t Q ij,k Q ij,k | {z } := I − Z R d ∂ t Q ij ∂ψ B ∂Q ij dx | {z } := I − Z R d v · ∇ Q ij ∂ψ B ∂Q ij | {z } := I − β Z R d A ij v i,j dx | {z } := I − β Z R d Q ij Q lk A lk v i,j dx − β Z R d A il Q lj v i,j dx − β Z R d Q il A lj v i,j dx − µ Z R d (cid:16) ˙ Q ij − Ω ik Q kj + Q ik Ω kj (cid:17) v i,j dx − µ Z R d (cid:16) Q il ˙ Q lj − ˙ Q il Q lj (cid:17) v i,j dx + µ Z R d ( Q il [Ω , Q ] lj − [Ω , Q ] il Q lj ) v i,j dx + ˜ µ Z R d A ij ˙ Q ij − µ Z R d (cid:16) ˙ Q ij − Ω ik Q kj + Q ik Ω kj (cid:17) ˙ Q ij dx (2.1)Noting that thanks to (1.2) we have I = I = 0 and moving I , I , I on the left handsize, the last relation becomes without boundary terms, thanks to our assumptions again without boundary terms thanks to our assumptions where σ ′ was defined in (1.7), h ′ in (1.10) and ψ B in (1.6). The operator L denotes the projection ontotrace-free matrices. ddt Z R d (cid:0) | v | + J | ˙ Q | + L |∇ Q | (cid:1) + ψ B ( Q ) dx + β Z R d |∇ v | dx + µ Z R d | ˙ Q | dx = − β Z R d Q ij Q lk A lk v i,j dx − β Z R d A il Q lj v i,j dx − β Z R d Q il A lj v i,j dx − µ Z R d (cid:16) ˙ Q ij − Ω ik Q kj + Q ik Ω kj (cid:17) v i,j + ˜ µ Z R d A ij ˙ Q ij + µ Z R d (cid:16) ˙ Q il Q lj − Q il ˙ Q lj (cid:17) v i,j dx + µ Z R d (Ω ik Q kj − Q ik Ω kj ) ˙ Q ij dx − µ Z R d ([Ω , Q ] il Q lj − Q il [Ω , Q ] lj ) v i,j dx dx (2.2)Now we analyse each term on the right-hand side of the equality, and we will repeatedly usethat v i,j = A ij + Ω ij and moreover that tr { BC } = B : C is null for any B symmetric and C skew-adjoint. We begin with − β Z R d Q ij Q lk A lk v i,j = − β Z R d Q ij Q lk A lk A ij − β Z R d Q ij Q lk A lk Ω ij = − β Z R d ( Q : A ) − β Z R d ( Q : Ω)( Q : A )= − β Z R d ( Q : A ) , observing that Q : Ω = 0. Furthermore we have: − µ Z R d (cid:16) Q il ˙ Q lj − ˙ Q il Q lj (cid:17) v i,j dx + µ Z R d (Ω ik Q kj − Q ik Ω kj ) ˙ Q ij dx == µ Z R d tr { ( Q ˙ Q − ˙ QQ ) A } | {z } =0 + µ Z R d tr { ( Q ˙ Q − ˙ QQ )Ω } + µ Z R d tr { (Ω Q − Q Ω) ˙ Q } == 2 µ Z R d tr { [Ω , Q ] ˙ Q } . Finally − µ Z R d ([Ω , Q ] il Q lj − Q il [Ω , Q ] lj ) v i,j = µ Z R d tr (cid:26)(cid:0) [Ω , Q ] Q − Q [Ω , Q ] (cid:1) Ω (cid:27) = − µ Z R d tr (cid:8) (Ω Q − Q Ω)[Ω , Q ] (cid:9) = − µ Z R d | [Ω , Q ] | . iquid crystal model with inertia 13 Now we deal with − β Z R d A il Q lj v i,j dx − β Z R d Q il A lj v i,j dx + µ Z R d (cid:0) Ω ik Q kj − Q ik Ω kj (cid:1) v i,j dx == − β Z R d tr { QA ∇ v } − β Z R d tr { AQ ∇ v } + µ Z R d tr { [Ω , Q ] ∇ v } = − β Z R d tr { ( QA + AQ ) ∇ v } − ( β − β ) Z R d tr { AQ ∇ v } + µ Z R d tr { A [Ω , Q ] } = − β Z R d tr { ( QA + AQ ) A } − ( β − β ) Z R d tr { AQ ( A + Ω) } + µ Z R d tr { A [Ω , Q ] } = − β Z R d tr { AQA } − ( β − β ) Z R d tr { AQA } + (cid:18) β − β µ (cid:19) Z R d tr { A [Ω , Q ] } = − ( β + β ) Z R d tr { AQA } + µ Z R d tr { A [Ω , Q ] } . We arrange the remaining terms related to µ , ˜ µ as − µ Z R d ˙ Q ij v i,j + ˜ µ Z R d A ij ˙ Q ij = − µ Z R d ˙ Q : ∇ v + ˜ µ Z R d A : ˙ Q = − µ Z R d ˙ Q : A − µ Z R d ˙ Q : Ω | {z } =0 + ˜ µ Z R d A : ˙ Q = (cid:26) µ = µ − µ R R d ˙ Q : A if ˜ µ = − µ Summarizing all the previous estimates, we get: ddt Z R d (cid:0) | v | + J | ˙ Q | + L |∇ Q | (cid:1) + ψ B ( Q ) d x + β Z R d |∇ v | d x ++ β Z R d ( Q : A ) d x + µ Z R d | ˙ Q − [Ω , Q ] | d x = ˜ µ Z R d ˙ Q : A + µ Z R d tr { A [Ω , Q ] } , (2.3)where we have used the assumption β + β = 0, in (1.15). We recall now the condition weimpose on µ namely (1.16), so that if ˜ µ = µ = 0 then the right hand side of the aboveequation is null. Otherwise, if ˜ µ = − µ , recalling that N = ˙ Q − [Ω , Q ] we obtain˜ µ Z R d ˙ Q : A + µ Z R d tr { A [Ω , Q ] } = ˜ µ Z R d A : N . In both cases, we note out of the above that there exists a constant C d depending on˜ µ , β , β , µ such that if β > C d then the total energy decays, i.e. ddt Z R d (cid:0) | v | + J | ˙ Q | + L |∇ Q | (cid:1) + ψ B ( Q ) d x ≤ d = 2 then the term tr( Q ) vanishes and if furthermore a ≥ ψ B ( Q ) ≥ ψ B ( Q ) can be negative.In order to deal with this we need to obtain apriori control on suitable L p norms of Q .We multiply (1.3) by Q , take the trace and integrate over space, obtaining: J Z R d ¨ Q αβ Q αβ d x + µ Z R d ˙ Q αβ Q αβ d x − µ Z R d (cid:0) Ω αγ Q γβ − Q αγ Ω γβ (cid:1) Q αβ d x | {z } =0 −− Z R d L ∆ Q αβ Q αβ d x = Z R d (cid:0) − aQ αβ Q αβ + bQ αγ Q γβ Q βα − c (cid:0) Q αβ Q αβ (cid:1) (cid:1) d x | {z } := P ( Q ) ++ ˜ µ Z R d A αβ Q αβ d x. Now, let us remark that J Z R d ¨ Q αβ Q αβ d x = J Z R d ∂ t ˙ Q αβ Q αβ + v γ ˙ Q αβ,γ Q αβ d x = J Z R d ∂ t (cid:0) ˙ Q αβ Q αβ (cid:1) + v γ (cid:0) ˙ Q αβ Q αβ (cid:1) ,γ d x − J Z R d ˙ Q αβ ˙ Q αβ d x = J dd t Z R d | ˙ Q + Q | − | ˙ Q | − | Q | d x − J Z R d | ˙ Q | d x and moreover µ Z R d ˙ Q αβ Q αβ d x − L Z R d ∆ Q αβ Q αβ d x = µ t Z R d | Q | d x ++ µ Z R d v γ Q αβ Q αβ,γ d x | {z } =0 + L Z R d |∇ Q | d x. Thus, summarizing, it turns out that12 dd t Z R d J | ˙ Q + Q | − J | ˙ Q | + ( µ − J ) | Q | d x − J Z R d | ˙ Q | d x + L Z R d |∇ Q | d x == P ( Q ) + ˜ µ Z R d tr { AQ } d x. (2.5) iquid crystal model with inertia 15 We will use this estimate together with (2.3) to obtain an estimate of the L norm of Q which then will allow to obtain out of (2.3) the desired apriori estimates on Q, ˙ Q and v .We note that if Q has eigenvalues λ, µ, − λ − µ (as it is traceless) we have | Q | = 2( λ + µ + λµ ) and tr( Q ) = − λµ ( λ + µ ) thus for any δ > | tr( Q ) | ≤ δ | Q | + δ | Q | .Furthermore tr( Q ) = 0 if d = 2 since Q is a two-by-two traceless symmetric matrix. If d = 3 we claim that there exists ¯ µ > a, b and c > µ | Q | + 4 ψ B ( Q ) > ε | Q | (2.6)for some ε > µ | Q | + 4 ψ B ( Q ) − ε | Q | = (¯ µ + 2 a − ε ) | Q | − b tr( Q ) + c | Q | ≥ (¯ µ + 2 a − ε ) | Q | − | b | ( δ | Q | + δ | Q | ) + c | Q | . Thus taking δ | b | = c and letting ¯ µ belarge enough we obtain the claimed relation (2.6). Then, assuming that J < µ and adding(2.5) to twice times (2.3) we obtain ddt Z R d | v | + J (cid:16) | ˙ Q + Q | + | ˙ Q | (cid:17) + L |∇ Q | + 12 ( µ − J ) | Q | + 2 ψ B ( Q ) d x ++ L Z R d |∇ Q | + 2 β Z R d tr { QA } + 2 β Z R d |∇ v | d x + 2 µ Z R d | ˙ Q − [Ω , Q ] | d x = P ( Q ) + J Z R d | ˙ Q | d x + ˜ µ Z R d tr { AQ } d x + 2 ˜ µ Z R d A : N . (2.7)Thus for µ > ¯ µ , J small enough and β large enough we have a Gronwall-type inequalityfor the L norm of Q which then can be combined with (2.3) to obtain the apriori bounds(1.19). (cid:3)
3. Global strong solutions3.1. A priori high-norm estimates.
In this subsection we provide the apriori estimatesthat exhibit in a relatively simple setting the higher-order cancellations and estimates thatwill allow us to prove afterwards the existence of strong solutions through a suitable approx-imation scheme, in the next subsection.We consider the inhomogeneous Sobolev space H s with s > d , equipped with inner product h u, v i H s = h u, v i L + h u, v i ˙ H s . where h u, v i ˙ H s := h ( √− ∆) s u, ( √− ∆) s v i L with ( √− ∆) s u ( ξ ) := F − ( | ξ | s F u ( ξ )).We recall that for s > d we have H s ( R d ) ֒ → L ∞ ( R d ) and, most importantly for ourpurposes, it is an algebra, with k uv k H s ≤ k u k H s k v k H s . We assume that the solutions are suitably smooth and decaying sufficiently fast at infinityto be able to integrate by parts without boundary terms whenever necessary. Taking the H s product between (1.1) and v , we get12 dd t k v k H s + β k∇ v k H s = −h v · ∇ v, v i H s + L h∇ Q ⊙ ∇ Q, ∇ v i H s −− β h tr { AQ } Q, ∇ v i H s − β h AQ, ∇ v i H s − β h QA, ∇ v i H s + − µ h ˙ Q − [Ω , Q ] , ∇ v i H s − µ h [ Q, ˙ Q ] , ∇ v i H s + µ h [ Q, [Ω , Q ]] , ∇ v i H s (3.1)Now, let us observe that h v · ∇ v, v i H s = h v · ∇ v, v i L x | {z } =0 + h v · ∇ v, v i ˙ H s . Since s > d/
2, then H m ( R d ) is continuously embedded in L ∞ ( R d ), for m a natural numberin [ s, s ). Then, by the classical Gagliardo-Niremberg inequality we have k v k L ∞ ( R d ) . k v k θL ( R d ) k v k − θ ˙ H m ( R d ) = k v k θL ( R d ) k∇ v k − θ ˙ H m − ( R d ) . k v k θL ( R d ) k∇ v k − θH m − ( R d ) . k v k θL ( R d ) k∇ v k − θH s ( R d ) , with θ := m − d m . Hence the second term on the right-hand side of the above equality can beestimated as follows: |h v · ∇ v, v i ˙ H s | = |h v ⊗ v, ∇ v i ˙ H s | k v k L ∞ x k v k ˙ H s k∇ v k ˙ H s . k v k θL x k∇ v k − θH s k v k ˙ H s k∇ v k H s , which yields |h v · ∇ v, v i ˙ H s | . k v k θL x k v k ˙ H s k∇ v k − θH s . k v k θL x k v k − θ ˙ H s k v k θ ˙ H s k∇ v k − θH s . k v k H s k v k θ ˙ H s k∇ v k − θH s . (3.2)Since s > d/ > k v k θ ˙ H s = k∇ v k θ ˙ H s − k∇ v k θH s , thus |h v · ∇ v, v i ˙ H s | k v k H s k∇ v k H s . k∇ v k H s k v k H s + c β k∇ v k H s . (3.3)Now, the second term on the right-hand side of (3.1) is h∇ Q ⊙ ∇ Q, ∇ v i H s = h∇ Q ⊙ ∇ Q, ∇ v i L x + h∇ Q ⊙ ∇ Q, ∇ v i ˙ H s . We will see that h∇ Q ⊙ ∇ Q, ∇ v i L x is going to be simplified, while h∇ Q ⊙ ∇ Q, ∇ v i ˙ H s . k∇ Q k L ∞ x k∇ Q k ˙ H s k∇ v k ˙ H s . k∇ Q k H s k∇ Q k H s + c β k∇ v k H s , Finally, the remaining terms on the right-hand side of (3.1) are controlled as follows: β h tr { AQ } Q, ∇ v i H s . k A k H s k Q k H s k∇ v k H s . k∇ v k H s k Q k H s , iquid crystal model with inertia 17 β h AQ, ∇ v i H s + β h QA, ∇ v i H s . k A k H s k Q k H s k∇ v k H s . k∇ v k H s k Q k H s + c β k∇ v k H s ,µ h [Ω , Q ] , ∇ v i H s . k Q k H s k∇ v k H s . k∇ v k H s k Q k H s + c β k∇ v k H s ,µ h [ Q, ˙ Q ] , ∇ v i H s . k Q k H s k ˙ Q k H s k∇ v k H s . k∇ v k H s k Q k H s + c µ k ˙ Q k H s ,µ h [ Q, [Ω , Q ]] , ∇ v i H s . k Q k H s k∇ v k H s . Thus, summarizing the previous estimates we get12 dd t k v k H s + β k∇ v k H s + µ h ˙ Q, ∇ v i H s − L h∇ Q ⊙ ∇ Q, ∇ v i L x .. k∇ v k H s (cid:0) k v k H s + k∇ Q k H s + k Q k H s (cid:1) + c µ k ˙ Q k H s + c β k∇ v k H s + k∇ Q k H s k∇ Q k H s . (3.4)Now, let us take the H s -inner product between the order equation and ˙ Q : J h ¨ Q, ˙ Q i H s + µ k ˙ Q k H s − µ h [Ω , Q ] , ˙ Q i H s + L t k∇ Q k H s − L h ∆ Q, v · ∇ Q i H s == − a h Q ˙ Q i H s + b h Q , ˙ Q i H s − c h Q | Q | } , ˙ Q i H s + ˜ µ h A, ˙ Q i H s . We begin observing that J h ¨ Q, ˙ Q i H s = J t k ˙ Q k H s + J h v · ∇ ˙ Q, ˙ Q i H s and that h v · ∇ ˙ Q, ˙ Q i H s = h v · ∇ ˙ Q, ˙ Q i L | {z } =0 + h v · ∇ ˙ Q, ˙ Q i ˙ H s = h [( √− ∆) s , v · ∇ ] ˙ Q, ( √− ∆) s ˙ Q i L + h v · ∇ ( √− ∆) s ˙ Q, ( √− ∆) s ˙ Q i L | {z } =0 . k∇ v k H s k ˙ Q k H s . k∇ v k H s k ˙ Q k H s + c µ k ˙ Q k H s . (3.5)where for the last inequality we used the commutator estimate from [2]: k [( √− ∆) s , v · ∇ ] B k L = k ( √− ∆) s [( v · ∇ ) B ] − ( v · ∇ )( √− ∆) s B k L x ≤ c k∇ v k H s k B k H s . Moreover µ h [Ω , Q ] , ˙ Q i H s . k∇ v k H s k Q k H s k ˙ Q k H s . k∇ v k H s k Q k H s + c µ k ˙ Q k H s . (3.6) Now, we have h ∆ Q, v · ∇ Q i ˙ H s = h ( √− ∆) s Q αβ,ii , ( √− ∆) s ( v j Q αβ,j ) i L x = − h ( √− ∆) s Q αβ,i , ( √− ∆) s ( v j,i Q αβ,j ) i L x − h ( √− ∆) s Q αβ,i , ( √− ∆) s ( v j Q αβ,ij ) i L x and |h ( √− ∆) s Q αβ,i , ( √− ∆) s ( v j,i Q αβ,j ) i L x | k∇ Q k H s k∇ v k H s . k∇ Q k H s k∇ Q k H s + c β k∇ v k H s , −h ( √− ∆) s Q αβ,i , ( √− ∆) s ( v j Q αβ,ij ) i L x = −h ( √− ∆) s Q ,i , [( √− ∆) s , v · ∇ ] Q ,i i L x −− h ( √− ∆) s Q ,i , v · ∇ ( √− ∆) s Q ,i i L x | {z } =0 . k∇ Q k ˙ H s k [( √− ∆) s , v · ∇ ] Q ,i k L x .. k∇ Q k H s k∇ v k H s . k∇ Q k H s k∇ Q k H s + c β k∇ v k H s , which yields h ∆ Q, v · ∇ Q i ˙ H s . k∇ Q k H s k∇ Q k H s + c β k∇ v k H s . (3.7)Finally − a h ˙ Q, Q i H s = − a t k Q k H s − a h v · ∇ Q, Q i H s , with | a h v · ∇ Q, Q i H s | . k v k H s k∇ Q k H s k Q k H s . k Q k H s k v k H s + c k∇ Q k H s (cid:12)(cid:12)(cid:12) b h ˙ Q, Q i H s (cid:12)(cid:12)(cid:12) . k ˙ Q k H s k Q k H s . k Q k H s k Q k H s + c µ k ˙ Q k H s and (cid:12)(cid:12)(cid:12) c h ˙ Q, Q | Q | i H s (cid:12)(cid:12)(cid:12) . k ˙ Q k H s k Q k H s k Q k H s (cid:0) k ˙ Q k H s + k Q k H s (cid:1) . Thus, summarizing the previous estimates we getdd t (cid:2) J k ˙ Q k H s + L k∇ Q k H s + a k Q k H s (cid:3) + µ k ˙ Q k H s − L h v · ∇ Q, ∆ Q i L x − ˜ µ h A, ˙ Q i H s .. (cid:0) k Q k H s + k∇ Q k H s + k∇ v k H s + k v k H s (cid:1)(cid:0) k ˙ Q k H s + k Q k H s + k∇ Q k H s (cid:1) ++ c β k∇ v k H s + c µ k ˙ Q k H s + c k∇ Q k H s . (3.8)Now, let us consider the H s -inner product between the order tensor equation and Q/ J h ¨ Q, Q i H s + µ h ˙ Q, Q i H s − µ h [Ω , Q ] , Q i H s + L k∇ Q k H s + a k Q k H s == b h Q , Q i H s − c h Q | Q | , Q i H s + ˜ µ h A, Q i H s . First, let us observe that h ¨ Q, Q i H s = h ¨ Q, Q i L x + h ¨ Q, Q i ˙ H s . iquid crystal model with inertia 19 We have h ¨ Q, Q i L x = h ∂ t ˙ Q, Q i L x + h v · ∇ ˙ Q, Q i L x = ddt h ˙ Q, Q i L x − h ˙ Q, ∂ t Q i L x − h ˙ Q, v · ∇ Q i L x = ddt h ˙ Q, Q i L x − k ˙ Q k L x . Moreover h ∂ t ˙ Q, Q i ˙ H s = dd t h ˙ Q, Q i ˙ H s − h ˙ Q, ∂ t Q i ˙ H s and h v · ∇ ˙ Q, Q i ˙ H s = h ( √− ∆) s (cid:16) v · ∇ ˙ Q (cid:17) , ( √− ∆) s Q i L x = h [( √− ∆) s , v · ∇ ] ˙ Q, ( √− ∆) s Q i L x − h ( √− ∆) s ˙ Q, v · ∇ ( √− ∆) s Q i L x = h [( √− ∆) s , v · ∇ ] ˙ Q, ( √− ∆) s Q i L x ++ h ( √− ∆) s ˙ Q, [( √− ∆) s , v · ∇ ] Q i L x − h ˙ Q, v · ∇ Q i ˙ H s , (3.9)Thus, summarizing, we get J h ¨ Q, Q i H sx = J t h ˙ Q, Q i H s − J k ˙ Q k H s ++ J h [( √− ∆) s , v · ∇ ] ˙ Q, ( √− ∆) s Q i L x + J h ( √− ∆) s ˙ Q, [( √− ∆) s , v · ∇ ] Q i L x (3.10)with the estimate J h [( √− ∆) s , v · ∇ ] ˙ Q, ( √− ∆) s Q i L x + J h ( √− ∆) s ˙ Q, [( √− ∆) s , v · ∇ ] Q i L x . k∇ v k H s k ˙ Q k H s k Q k H s . k∇ v k H s k Q k H s + c µ k ˙ Q k H s . (3.11)Furthermore J t h ˙ Q, Q i ˙ H s = J t (cid:2) k ˙ Q + Q k H s − k ˙ Q k H s − k Q k H s (cid:3) . (3.12)On the other hand µ h ˙ Q, Q i H s = µ h ∂ t Q, Q i H s + µ h v · ∇ Q, Q i H s = µ t k Q k H s + µ h v · ∇ Q, Q i H s , with µ h v · ∇ Q, Q i H s . k v k H s k∇ Q k H s k Q k H s . k Q k H s k v k H s + c k∇ Q k H s . Then µ h [Ω , Q ] , Q i H s . k∇ v k H s k Q k H s . k Q k H s k Q k H s + c β k∇ v k H s , (cid:12)(cid:12)(cid:12)(cid:12) b h Q , Q i H s (cid:12)(cid:12)(cid:12)(cid:12) . k Q k H s . k Q k H s k Q k H s + c a k Q k H s , (cid:12)(cid:12)(cid:12) c h Q | Q | , Q i H s (cid:12)(cid:12)(cid:12) . k Q k H s k Q k H s and finally (cid:12)(cid:12)(cid:12)(cid:12) ˜ µ h A, Q i H s (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ˜ µ (cid:12)(cid:12)(cid:12)(cid:12) k∇ v k H s k Q k H s | ˜ µ | a (1 − ε ) k∇ v k H s + a − ε ) k Q k H s Then, summarizing the previous estimates, we getdd t h J k ˙ Q + Q k H s − J k ˙ Q k H s + µ − J k Q k H s i − J k ˙ Q k H s + a ε k Q k H s −− | ˜ µ | a (1 − ε ) k∇ v k H s + L k∇ Q k H s . (cid:16) k∇ v k H s + k Q k H s (cid:17)(cid:16) k Q k H s + k v k H s (cid:17) ++ c β k∇ v k H s + c µ k ˙ Q k H s + c a k Q k H s + c k∇ Q k H s . (3.13)Finally, taking the sum between (3.4), (3.8) and (3.13) and assuming c β , c µ , c and c a small enough, we get for a suitable δ > t h k v k H s + J k Q + ˙ Q k H s + J k ˙ Q k H s + (cid:0) a µ − J (cid:1) k Q k H s + L k∇ Q k H s i ++ (cid:0) β − | ˜ µ | − ε ) a − δ (cid:1) k∇ v k H s + (cid:0) µ − J − δ (cid:1) k ˙ Q k H s + ( a ε − δ ) k Q k H s + ( L − δ ) k∇ Q k H s . (cid:16) k∇ v k H s + k ˙ Q k H s + k Q k H s + k∇ Q k H s (cid:17)(cid:16) k v k H s + k Q k H s + k ˙ Q k H s + k∇ Q k H s (cid:17) , where we have used h∇ Q ⊙ ∇ Q, ∇ v i L x + h v · ∇ Q, ∆ Q i L x = 0 . The last estimate allows, under suitable relations on the coefficients, to obtain a differentialinequality of the type (1.22) with Φ( t ) := k v k H s + k Q k H s + k ˙ Q k H s + k∇ Q k H s and Ψ( t ) := k∇ v k H s + k ˙ Q k H s + k Q k H s + k∇ Q k H s which allows to control apriori these norms globally intime, for small data (see also the Lemma 5.1 in the Appendix). We divide the proof into the existenceand uniqueness parts. The existence is based on a Friedrichs-type scheme that preserves thestructure exhibited in the higher-order energy laws, which allows to construct approximatesolutions. The uniqueness is achieved afterwards through to an H s -type energy estimate. Existence part:
In order to construct global strong solutions, we use the classicalFriedrichs scheme and obtain estimates similar to the ones in the previous section. Wedefine the mollifying operator J n f ( ξ ) := F − (cid:0) { − n | ξ | n } F f (cid:1) , iquid crystal model with inertia 21 The approximate momentum equation then reads as follows: J n ∂ t v ( n ) + P J n (cid:0) J n v ( n ) · ∇ J n v ( n ) (cid:1) + β J n v ( n ) = − L ∇ · P n J n (cid:0) ∇ J n Q ( n ) ⊗ ∇ J n Q ( n ) (cid:1)o ++ ∇ · P n β J n (cid:0) J n Q ( n ) tr { ( J n Q ( n ) J n A ( n ) ) (cid:1) + β J n (cid:0) J n A ( n ) J n Q ( n ) (cid:1) + β J n (cid:0) J n Q ( n ) J n A ( n ) (cid:1)o + ∇ · P n µ (cid:0) J n ˙ Q ( n ) − J n [ J n Ω ( n ) , J n Q ( n ) ] (cid:1) + µ J n (cid:2) J n Q ( n ) , ( J n ˙ Q ( n ) − [ J n Ω ( n ) , J n Q ( n ) ] (cid:1) ] o , (3.14)where P denotes the Leray projector onto divergence-free vector fields, and we denoteΩ ( n ) ij := v ( n ) i,j − v ( n ) j,i , A ( n ) ij := v ( n ) i,j + v ( n ) j,i , i, j = 1 , . . . , d Similarly, the approximate order tensor equation is
J J n ¨ Q ( n ) + µ J n ˙ Q ( n ) = L ∆ J n Q ( n ) − aJ n Q ( n ) + bJ n ( J n Q ( n ) J n Q ( n ) ) − b tr (cid:8) J n ( J n Q ( n ) J n Q ( n ) ) (cid:9) Id d + cJ n (cid:16) J n Q ( n ) tr { ( J n Q ( n ) J n Q ( n ) ) } (cid:17) + ˜ µ J n A ( n ) + µ J n [ J n Ω ( n ) , J n Q ( n ) ]where we have used the abuse of notation˙ f ( n ) := ∂ t f ( n ) + J n ( J n v ( n ) · ∇ J n f ( n ) ) . The system above can be regarded as an ordinary differential equation in L verifyingthe conditions of the Cauchy-Lipschitz theorem. Thus it admits a unique maximal so-lution ( v ( n ) , Q ( n ) ) in C ([0 , T n ) , L ). As we have ( P J n ) = P J n and J n = J n , the pair( J n v ( n ) , J n Q ( n ) ) is also a solution of the previous system. Hence, by uniqueness we get that( J n v ( n ) , J n Q ( n ) ) = ( v ( n ) , Q ( n ) ), and moreover the pair ( v ( n ) , Q ( n ) ) belongs to C ([0 , T n ) , H ∞ )and solves the system ∂ t v ( n ) + P J n ( v ( n ) · ∇ v ( n ) ) + β v ( n ) = − L ∇ · P n J n (cid:0) ∇ Q ( n ) ⊗ ∇ Q ( n ) (cid:1)o ++ ∇ · P n β J n (cid:0) Q ( n ) tr { Q ( n ) A ( n ) } (cid:1) + β J n (cid:0) A ( n ) Q ( n ) (cid:1) + β J n (cid:0) Q ( n ) A ( n ) (cid:1)o ++ ∇ · P n µ (cid:0) ˙ Q ( n ) − J n [Ω ( n ) , Q ( n ) ] (cid:1) + µ J n (cid:2) Q ( n ) , ( ˙ Q ( n ) − [Ω ( n ) , Q ( n ) ] (cid:1)(cid:3) o ,J ¨ Q ( n ) + µ ˙ Q ( n ) = L ∆ Q ( n ) − aQ ( n ) + bJ n ( Q ( n ) Q ( n ) ) − b tr (cid:8) J n ( Q ( n ) Q ( n ) ) (cid:9) Id d + cJ n (cid:16) Q ( n ) tr { ( Q ( n ) Q ( n ) ) } (cid:17) + ˜ µ A ( n ) + µ J n [Ω ( n ) , Q ( n ) ] (3.15)Arguing similarly as in the proof of the apriori estimates and taking advantage of the factthat J n is a self-adjoint operator in L , in order to obtain similar cancellations we get (for a δ > t h k v ( n ) k H s + J k Q ( n ) + ˙ Q ( n ) k H s + J k ˙ Q ( n ) k H s + (cid:0) a µ − J (cid:1) k Q ( n ) k H s + L k∇ Q ( n ) k H s i + (cid:0) β − | ˜ µ | − ε ) a − δ (cid:1) k∇ v ( n ) k H s + (cid:0) µ − J − δ (cid:1) k ˙ Q ( n ) k H s + ( a ε − δ ) k Q ( n ) k H s + ( L − δ ) k∇ Q ( n ) k H s . (cid:16) k∇ v ( n ) k H s + k ˙ Q ( n ) k H s + k Q ( n ) k H s + k∇ Q ( n ) k H s (cid:17) ×× (cid:16) k v ( n ) k H s + k Q ( n ) k H s + k ˙ Q ( n ) k H s + k∇ Q ( n ) k H s (cid:17) , (3.16)Defining the functions x ( t ) and y ( t ) by x ( t ) := k∇ v ( n ) k H s + k ˙ Q ( n ) k H s + k Q ( n ) k H s + k∇ Q ( n ) k H s ,y ( t ) := k v ( n ) k H s + k Q ( n ) k H s + k ˙ Q ( n ) k H s + k∇ Q ( n ) k H s , respectively, and thanks to Lemma 5.1 and inequality (3.16), we get the following bound:sup t ∈ R + n k v ( n ) ( t ) k H s + k Q ( n ) ( t ) k H s + k ˙ Q ( n ) ( t ) k H s + k∇ Q ( n ) ( t ) k H s o ++ Z R + n k∇ v ( n ) ( t ) k H s + k ˙ Q ( n ) ( t ) k H s + k Q ( n ) ( t ) k H s + k∇ Q ( n ) ( t ) k H s o d t . k v k H s + k Q k H s + k ˙ Q k H s + k∇ Q k H s . We claim that these uniform estimates allow us to pass to the limit as, n goes to ∞ . Wefirst observe that we can obtain a uniform bound also for ∂ t Q n in L ∞ t H s . Indeedsup t ∈ R + k ∂ t Q n k H s = sup t ∈ R + k ˙ Q n − v ( n ) · ∇ Q ( n ) k H s k ˙ Q n k L ∞ t H s + k v n k L ∞ t H s k∇ Q n k L ∞ t H s . k v k H s + k Q k H s + k ˙ Q k H s + k∇ Q k H s . Thus, by classical compactness, weak convergence arguments and thanks to the Aubin-Lions lemma, there exists Q ∈ L ∞ t H s +1 ∩ L t H s +1 , v ∈ L ∞ t H s ∩ L t H s +1 , and ω ∈ L ∞ t H s ∩ L t H s , such that, up to a subsequence, we have the following convergences Q ( n ) → Q strong in L ∞ t,loc H s +1 − µloc ˙ Q ( n ) → ω strong in L ∞ t,loc H s − µloc v ( n ) → v strong in L ∞ t,loc H s − µloc ∇ v ( n ) ⇀ ∇ v weak in L t H s for any suitably small positive constant µ . iquid crystal model with inertia 23 Assuming s − µ > d/
2, we have that J n ( v ( n ) · ∇ Q ( n ) ) strongly converges to v · ∇ Q in L ∞ t,loc H s − µloc , as n → ∞ , with v · ∇ Q ∈ L ∞ t H s . Furthermore ∂ t Q = lim n →∞ ∂ t Q ( n ) = lim n →∞ (cid:16) ˙ Q ( n ) − v ( n ) · ∇ Q ( n ) (cid:17) = ω − v · ∇ Q ∈ L ∞ t H s , where the limits are considered in the distributional sense. Then, we deduce ∂ t Q ∈ L ∞ t H s and ω = ˙ Q ∈ L ∞ t H s . Finally, the order-tensor equation yields J ∂ t ˙ Q ( n ) = − J J n ( v ( n ) · ∇ ˙ Q ( n ) ) − µ ˙ Q ( n ) + µ J n [Ω ( n ) , Q ( n ) ] + L ∆ Q ( n ) + ˜ µ A ( n ) −− aQ ( n ) + b (cid:16) J n ( Q ( n ) Q ( n ) ) − tr { ( Q ( n ) Q ( n ) ) } Id d (cid:17) − cJ n ( Q ( n ) tr { ( Q ( n ) Q ( n ) ) } , hence, observing that k J n ( v ( n ) · ∇ ˙ Q ( n ) ) k H s − . k v ( n ) · ∇ ˙ Q ( n ) k H s − = k∇ · { v ( n ) ⊗ ˙ Q ( n ) }k H s − . k v ( n ) ⊗ ˙ Q ( n ) k H s . k v ( n ) k H s k ˙ Q ( n ) k H s , then ∂ t ˙ Q ( n ) belongs to L t,loc H s − , with uniformly in n bounded seminorms. Thus ∂ t ˙ Q ( n ) ⇀ ∂ t ˙ Q weakly in L t,loc H s − , up to a subsequence. Moreover, since J n ( v ( n ) ⊗ ˙ Q ( n ) ) converges weakly to v ⊗ ˙ Q in L t,loc H s ,then J n ( v ( n ) · ∇ ˙ Q ( n ) ) converges weakly to v · ∇ ˙ Q in L t,loc H s . Then, summarizing we deducethat ¨ Q ( n ) converges weakly to ¨ Q in L t,loc H s .These convergences allow us to pass to the limit in the classical solutions of (3.15), deducingthat ( u, Q ) is classical solution of system (1.1) and (1.3). Uniqueness part:
We now prove the uniqueness of the strong solutions previously ob-tained. Let us consider ( u , Q ) and ( u , Q ) to be strong solutions with same initial data.From here on we will use the following notation: δQ := Q − Q , δ ˙ Q := ˙ Q − ˙ Q , δv := v − v , δA := A − A , δ Ω := Ω − Ω . We begin the proof by considering the difference between the order-parameter equations ofthe two solutions, namely J h ( δ ˙ Q ) t + v · ∇ δ ˙ Q + δv · ∇ ˙ Q i + µ δ ˙ Q = L ∆ δQ − aδQ + b (cid:2) Q δQ + δQQ ++tr { Q δQ + δQQ } Id d (cid:3) − cδQ tr { Q } − cQ tr { δQQ } − cQ tr { Q δQ } +˜ µ δA + µ [Ω , δQ ] + µ [ δ Ω , Q ] . We multiply by δ ˙ Q , take the trace and integrate over R d to get:dd t h J k δ ˙ Q k L x + L k∇ δQ k L x + a k δQ k L x i + µ k δ ˙ Q k L x = L h ∆ δQ, v · ∇ δQ i L x ++ L h ∆ δQ, δv · ∇ Q i L x − J h v · ∇ δ ˙ Q, δ ˙ Q i L x − J h δv · ∇ ˙ Q , δ ˙ Q i L x −− a h δQ, v · ∇ δQ + δv · ∇ Q i L x + b h Q δQ + δQQ , δ ˙ Q i L x −− c h δQ tr { Q } + Q tr { δQQ } + Q tr { Q δQ } , δ ˙ Q i L x ++ ˜ µ h δA, δ ˙ Q i L x + µ h [Ω , δQ ] + [ δ Ω , Q ] , δ ˙ Q i L x . (3.17)We now estimate each term on the right-hand side. First we remark that h ∆ δQ, v · ∇ δQ i L x = h δQ αβ, jj , ( v ) i δQ αβ,i i L x = −h δQ αβ, j , ( v ) i,j δQ αβ,i i L x −h δQ αβ, j , ( v ) i δQ αβ,ij i L x | {z } =0 , where for the second equality we have integrated by parts. Then we obtain h ∆ δQ, v · ∇ δQ i L x . k∇ δQ k L x k∇ δQ k L x k∇ v k L ∞ x . k∇ v k H s k∇ δQ k L x , Similarly, we can proceed integrating by parts also for the second term, namely h ∆ δQ, δv · ∇ Q i L x = h δQ αβ,jj , δv i · ( Q ) αβ, i i L x = −h δQ αβ,j , δv i,j · ( Q ) αβ, i i L x | {z } A −h δQ αβ,j , δv i · ( Q ) αβ, ij i L x | {z } B . First, we control A using a standard estimate: A . k∇ δQ k L x k∇ δv k L x k∇ Q k L ∞ x . k∇ Q k H s k∇ δQ k L x + c β k∇ δv k L x . The term B requires a more careful analysis. First, we define the parameter θ in (0 , / / s − d/
2. Thus, since ∆ Q belongs to L ( R + , H s − ( R d )),then it belongs also to L ( R + , H θ + d/ − ( R d )). We will make use of the following Sobolevembeddings: H s − ( R d ) ֒ → H θ + d/ − ( R d ) ֒ → L d − θ ( R d ) ,H ( R d ) ֒ → L dd − − θ ) ( R d ) (3.18)Then B is bounded by B . k∇ δQ k L x k δv k L dd − − θ ) x k ∆ Q k L d − θx . k∇ δQ k L x k δv k H k ∆ Q k H θ + d − . k∇ δQ k L x k δv k L k ∆ Q k H s − + k δQ k L x k∇ δv k L k ∆ Q k H s − . k∇ Q k H s (cid:0) k δQ k L x + k δv k L x (cid:1) + c β k∇ δv k L x + c k∇ δQ k L x . Summarizing, the second term is estimated as follows: h ∆ δQ, δv · ∇ Q i L x . k∇ Q k H s (cid:0) k∇ δQ k L x + k δQ k L x + k δv k L x (cid:1) + c β k∇ δv k L x + c k∇ δQ k L x . iquid crystal model with inertia 25 Now, let us observe that h v · ∇ δ ˙ Q, δ ˙ Q i L x = 0 because of the free divergence condition of v . Moreover, still recalling the embeddings (3.18), we have h δv · ∇ ˙ Q , δ ˙ Q i L x . k δv k L dd − − θ ) x k∇ ˙ Q k L d − θx k δ ˙ Q k L x . k δv k H k∇ ˙ Q k H s − k δ ˙ Q k L x . k ˙ Q k H s k δv k L k δ ˙ Q k L x + k ˙ Q k H s k∇ δv k L k δ ˙ Q k L x . k ˙ Q k H s (cid:0) k δv k L x + k δ ˙ Q k L x (cid:1) + c β x k∇ δv k L x + c µ k δ ˙ Q k L x . The remaining terms can easily controlled by the H¨older inequality and the Sobolev em-bedding H s ֒ → L ∞ x . First the terms related to the parameter a fulfill h δQ, v · ∇ δQ i L x . k δQ k L x k v k L ∞ k∇ δQ k L x . k v k H s (cid:0) k δQ k L x + k∇ δQ k L x (cid:1) , h δQ, δv · ∇ Q i L x . k δQ k L x k δv k L x k∇ Q k L ∞ x . k∇ Q k H s (cid:0) k δQ k L x + k δv k L x (cid:1) , The terms related to b can be bounded as follows h Q δQ + δQQ , δ ˙ Q i L x . k ( Q , Q ) k L ∞ x k δQ k L x k δ ˙ Q k L x . k ( Q , Q ) k H s k δQ k L x + c µ k δ ˙ Q k L x and finally the one multiplied by c is estimated by h δQ tr { Q } + Q tr { δQQ } + Q tr { Q δQ } , δ ˙ Q i L x . k ( Q , Q ) k H s (cid:0) k δQ k L x + k δ ˙ Q k L x (cid:1) . It remains to control the terms related to µ and µ which can be handled through h δA, δ ˙ Q i L x . k δA k L x k δ ˙ Q k L x . k δ ˙ Q k L x + c β k∇ δv k L x and h [Ω , δQ ] + [ δ Ω , Q ] , δ ˙ Q i L x . (cid:0) k∇ v k H s + k Q k H s (cid:1)(cid:0) k δQ k L x + k δ ˙ Q k L x (cid:1) ++ c µ k δ ˙ Q k L x + c β k∇ δv k L x . Using all the previous estimates in the equality (3.17), we obtaindd t h J k δ ˙ Q k L x + L k∇ δQ k L x + a k δQ k L x i + µ k δ ˙ Q k L x . (cid:16) k Q k H s + k∇ v k H s + k v k H s ++ k ˙ Q k H s + k∇ Q k H s + k Q k H s (cid:17)(cid:16) k δv k L x + k δ ˙ Q k L x + k δQ k L x + k∇ δQ k L x (cid:17) ++ c β k∇ δv k L x + c µ k δ ˙ Q k L x . (3.19) Now let us consider the difference between the momentum equations of the two solutions,namely ∂ t δv + v · ∇ δv + δv · ∇ v − β δv = − L ∇ · n ∇ δQ ⊗ ∇ Q + ∇ Q ⊗ ∇ δQ o − + β ∇ · n tr { δQA } Q + tr { Q δA } Q + tr { Q A } δQ o + β ∇ · (cid:8) A δQ + δAQ (cid:9) ++ β ∇ · (cid:8) δQA + Q δA (cid:9)o + µ ∇ · n δ ˙ Q − [ δ Ω , Q ] − [Ω , δQ ] o ++ µ ∇ · (cid:8) [ δQ, ( ˙ Q − [Ω , Q ])] + [ Q , ( δ ˙ Q − [ δ Ω , Q ] − [Ω , δQ ])] o . (3.20)We proceed similarly as before, multiplying scalarly by δv and integrating everything over R d , and by parts, to obtain12 dd t k δv k L x + β k∇ δv k L x = L h∇ δQ ⊗ ∇ Q + ∇ Q ⊗ ∇ δQ, ∇ δv i L x + − β h tr { δQA } Q + tr { Q A } δQ, ∇ δv i L x + β h tr { Q δA } Q , ∇ δv i L x −− β h A δQ + δAQ , ∇ δv i L x − β h δQA + Q δA, ∇ δv i L x − µ h δ ˙ Q, ∇ δv i L x ++ µ h [ δ Ω , Q ] + [Ω , δQ ] , ∇ δv i L x − µ h [ δQ, ˙ Q ] , ∇ δv i L x − µ h [ Q , δ ˙ Q ] , ∇ δv i L x − + µ h [ Q , [ δ Ω , Q ] + [Ω , δQ ]] , ∇ δv i L x + µ h [ δQ, [Ω , Q ]] , ∇ δv ] i L x −h v · ∇ δv, δv i L x − h δv · ∇ v , δv i L x , (3.21)We proceed by estimating each term on the right-hand side. First we have h∇ δQ ⊗ ∇ Q + ∇ Q ⊗ ∇ δQ, ∇ δv i L x . (cid:16) k∇ Q k L ∞ x + k∇ Q k L ∞ x (cid:17) k∇ δQ k L x k∇ δv k L x . (cid:16) k∇ Q k H s + k∇ Q k H s (cid:17) k∇ δQ k L x + c β k∇ δv k L x , while the terms concerning β are handled by h tr { δQA } Q + tr { Q A } δQ, ∇ δv i L x .. (cid:16) k∇ u k L ∞ x k Q k L ∞ x + k∇ u k L ∞ x k Q k L ∞ x (cid:17) k δQ k L x k∇ δv k L x . (cid:16) k∇ v k H s k Q k H s + k∇ v k H s k Q k H s (cid:17) k δQ k L x + c β k∇ δv k L x , and h tr { Q δA } Q , ∇ δv i L x . k Q k L ∞ x k Q k L ∞ x k∇ δv k L x . k Q k H s k Q k H s k∇ δv k L x . Now, we bound the terms related to β and β as follows: h A δQ + δAQ , ∇ δv i L x . k∇ v k L ∞ k δQ k L x k∇ δv k L x + k Q k L ∞ k∇ δv k L x . k∇ v k H s k δQ k L x + (cid:0) c β + k Q k H s (cid:1) k∇ δv k L x , iquid crystal model with inertia 27 h δQA + Q δA, ∇ δv i L x . k δQ k L x k∇ v k L ∞ x k∇ δv k L x + k Q k L ∞ x k∇ δv k L x . k∇ v k H s k δQ k L x + (cid:0) c β + k Q k H s (cid:1) k∇ δv k L x . Now, we move on and bound the terms related to µ by h δ ˙ Q, ∇ δv i L x . k δ ˙ Q k L x + c β k∇ δv k L x , h [ δ Ω , Q ] + [Ω , δQ ] , ∇ δv i L x . k Q k L ∞ x k∇ δv k L x + k∇ v k L ∞ x k δQ k L x k∇ δv k L x . k∇ v k H sx k δQ k L x + (cid:0) c β + k Q k H sx (cid:1) k∇ δv k L x , while the terms related to µ can be handled by h [ δQ, ˙ Q ] , ∇ δv i L x . k δQ k L x k ˙ Q k L ∞ x k∇ δv k L x . k ˙ Q k H s k δQ k L x + c β k∇ δv k L x , h [ Q , δ ˙ Q ] , ∇ δv i L x . k Q k L ∞ x k δ ˙ Q k L x k∇ δv k L x . k Q k H s k δ ˙ Q k L x + c β k∇ δv k L x and also h [ Q , [ δ Ω , Q ] + [Ω , δQ ]] , ∇ δv i L x . k Q k L ∞ x k Q k L ∞ x k∇ δv k L x + k Q k L ∞ x k∇ v k L ∞ x k δQ k L x k∇ δv k L x . k Q k H sx k∇ v k H sx k δQ k L x + (cid:16) c β + k Q k H sx k Q k H sx (cid:17) k∇ δv k L x , h [ δQ, [Ω , Q ]] , ∇ δv i L x ≤ k Q k L ∞ k∇ v k L ∞ k δQ k L x k∇ δv k L x ≤ k Q k H s k∇ v k H s k δQ k L + c β k∇ δv k L Finally, let us remark that h v · ∇ δv, δv i L x = 0 and h δv · ∇ v , δv i L x . k∇ v k L ∞ x k δv k L x . k∇ v k H s k δv k L x . Thus, summarizing all the previous estimates and using them in (3.20) we get12 dd t k δv k L x + β k∇ δv k L x . n k∇ v k H s + k∇ v k H s + k∇ v k H s + k∇ Q k H s + k Q k H s + k∇ Q k H s + k∇ v k H s k Q k H s + k∇ v k H s k Q k H s + k ˙ Q k H s o × n k δv k L x + k∇ δQ k L x + k δQ k L x + k δ ˙ Q k L x o ++ n c β + k Q k H sx k Q k H sx + k Q k H sx + k Q k H sx o k∇ δv k L x (3.22) Now, defining the functions Ψ = Ψ( t ) and f = f ( t ) byΨ := 12 k δv k L x + J k δ ˙ Q k L x + L k∇ δQ k L x + a k δQ k L x f := n k Q k H s k∇ v k H s + k∇ v k H s + k∇ v k H s + k∇ Q k H s + k Q k H s ++ k∇ Q k H s + k∇ v k H s k Q k H s + k∇ v k H s k Q k H s + k ˙ Q k H s ++ k ˙ Q k H s + k Q k H s + k v k H s o , and observing that f ∈ L loc ( R + ), we finally take the sum between (3.19) and (3.22), obtainingdd t Ψ + µ k ˙ δQ k L x + β k∇ δv k L x . f Ψ + c µ k δ ˙ Q k L x ++ n c β + k Q k H sx k Q k H sx + k Q k H sx + k Q k H sx o k∇ δv k L x . Hence, assuming c β , c µ and the initial data small enough, we can absorb by the left-handside the terms related to k δ ˙ Q k L x and k∇ δv k L x on the right-hand side, so that the followinginequality is fulfilled: dd t Ψ . f Ψ . Then, since Ψ(0) = 0, the Gronwall’s inequality yields Ψ to be constantly null, especially δv = v − v = 0 and δQ = Q − Q = 0 . (3.23)This concludes the proof of Theorem 1.2. (cid:3)
4. Twist waves
In the following we consider an example of a “twist-wave” solution, that is a solution of thecoupled system for which the flow v is zero. As noted in the introduction, this amounts todetermining a solution of the Q -tensor equation (1.3) with zero flow (hence v = Ω = A = 0,˙ Q becomes ∂ t Q ) but satisfying an additional nonlinear constraint namely (1.24).Our ansatz is inspired from stationary-case studies [3], namely the“melting-hedgehog wave” obtained by taking in R d with d = 2 or d = 3 the ansatz: T ( t, x ) := f ( t, | x | ) ¯ H ( x )with f : R + × R → R a function to be determined and ¯ H the “hedgehog” function (see [3]for details about its physical significance) :¯ H ij ( x ) := x i x j | x | − δ ij d , i, j = 1 , . . . , d Let us note that in order to avoid a discontinuity at 0 for T we need to take f ( t,
0) = 0 , ∀ t ≥ iquid crystal model with inertia 29 i.e. the hedgehog “melts” at the origin.Then one can check that the equation (1.23) reduces to an equation for f only, namely: J f tt + µ f t = L (cid:18) f rr + f r d − r − dr f (cid:19) − af + b ( d − d f − c ( d − d f (4.2)Note that in order to avoid a singularity at the origin for f we need to further have that f r ( t,
0) = 0 , ∀ t ≥ G ( t, x ) = f ( t, | x | ) then (4.3) holds for G sufficiently smooth in the x variable.On the other hand in order to check that the constraint equation (1.24) holds it sufficesto check that both ∇ · T t and ∇ · ( ∇ T ⊗ ∇ T ) can be expressed as gradients .We have: ∇ · T t = (cid:18) f t ( t, | x | )( x i x j | x | − δ ij d ) (cid:19) ,j = ( d − (cid:18) d ∂ r f t ( t, r ) + f t ( t, r ) r (cid:19) x i | x | (4.4)Then (4.3) implies f tr ( t,
0) = 0 , ∀ t ≥ f there exists a function g : R + × R → R suchthat g r = ( d − (cid:16) d ∂ f∂r∂t ( t, r ) + f t ( t,r ) r (cid:17) , hence ∇ · T t = ∇ x g ( t, | x | ).Furthermore, we have: ∇ · ( T ⊗ T ) = ( T kl,i T kl,j ) ,j = T kl,ij T kl,i + T kl,i ∆ T kl As T kl,ij T kl,j = ( |∇ T | ) ,i it suffices to check that T kl,i ∆ T kl is a gradient, which in our caseamounts to checking that f r ( f rr + ( d − f r r − dfr ) x i | x | (cid:18) x k x l | x | − δ kl d (cid:19) (cid:18) x k x l | x | − δ kl d (cid:19) = d − d f r ( f rr + ( d − f r r − dfr ) x i | x | is a gradient.Thus, assuming (4.1) and (4.3) we have that there exists h : R + × R → R such that h r = d − d f r ( f rr + ( d − f r r − dfr ), hence ∇ x h ( t, | x | ) = ∇ · ( ∇ T ⊗ ∇ T ).These formal computations provide indeed a twist wave under the smoothness conditionsmentioned before. In order to make the above rigorous we continue with the proof of Propo-sition 1.3: Proof.
We proceed in several steps. We will first show that the Q -tensor equation (1.23)has global in time strong solutions for arbitrary initial data. We then prove a weak-strong since the commutator appearing in (1.24) vanishes for our ansatz uniqueness result for solutions of the Q -tensor equation (1.23) and show that the f -equation(1.25) has weak solutions and then that these weak solutions provide a global weak solutionof (1.23). We use the weak-strong uniqueness to conclude that the solutions provided by f ( t, | x | ) ¯ H ( x ) are twist-wave solutions. Step 1: Global strong solutions of (1.23)We just provide here the apriori estimates necessary for obtaining the weak and strongsolutions. The actual construction through an approximation scheme can be done similarlyas in the proof of Theorem 1.2.We first obtain the apriori boundedness of the L norm. To this end we multiply (1.23)by T t , integrate over R d and by parts , to get: ddt Z R d J | T t | + L |∇ T | + ψ B ( T ) dx + µ Z R d | T t | dx = 0 (4.6)We note that because ψ B ( T ) can be negative this does not suffice for obtaining estimateson the L norm of T . Thus we multiply (1.23) by T , integrate over R d and by parts, to get: J d dt Z R d | T | dx − J Z R d | T t | dx + µ ddt Z R d | T | dx + L Z R d |∇ T | dx = Z R d ( − a | T | + b tr( T ) − c | T | ) dx (4.7)We multiply (4.6) by J and (4.7) by µ and add them together to get: J µ d dt Z R d | T | dx + ddt Z R d J | T t | + LJ |∇ T | + J ψ B ( T ) + µ | T | dx + Lµ Z R d |∇ T | dx ≤ C Z R d | T | dx (4.8)with the large enough constant C depending just on µ , a , b and c .Integrating over [0 , t ] we obtain: J µ ddt Z R d | T | ( t, x ) dx + Z R d (cid:18) J | T t | + LJ |∇ T | + J ψ B ( T ) + µ | T | (cid:19) ( t, x ) dx ≤≤ J µ Z R d ( T t : T )(0 , x ) dx + Z R d (cid:18) J | T t | + LJ |∇ T | + J ψ B ( T ) + µ | T | (cid:19) (0 , x ) dx + C Z t Z R d | T | ( s, x ) ds dx (4.9) throughout the proof we always assume as usually that we can integrate by parts without boundaryterms iquid crystal model with inertia 31 which implies: J µ ddt Z R d | T | ( t, x ) dx + Z R d J | T t | ( t, x ) dx ≤ C + C Z R d | T ( t, x ) | dx + Z t Z R d | T | ( s, x ) ds dx (4.10)with the constants C depending only on the initial data and the constants in the equation.Thus using Gronwall inequality and Fubini (to turn the double time integral into a weightedtime integral) together with (4.6) we obtain apriori control over certain energy-level norms: T ∈ L ∞ (0 , T ; L ) ∩ L ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; L ) < C ( a, b, c, J, µ , T, k T k H , k ∂ t T k L ) (4.11) T t ∈ L ∞ (0 , T ; L ) < C ( a, b, c, J, µ , T, k T k L , k ∂ t T k L ) (4.12)In order to obtain control over the H s norm we multiply (1.23) with T t in the H s innerproduct, with s > d obtaining:12 ddt (cid:0) J k T t k H s + L k∇ T k H s (cid:1) + µ k T t k H s dx = h T t , − aT + b ( T − | T | d I d ) − cT | T | i H s ≤ C k T t k H s (cid:0) k T k H s + k T k H s (cid:1) ≤ C k T t k H s + C k T k H s + ε k T k H s ≤ C k T t k H s + C k T k H s + ε k T k s H · ε k T k s − s H s +1 (4.13)where in the last inequality we estimated H s through interpolation between H and H s +1 . Inorder to be able to control the term ε k T k s − s H s we need to have s − s ≤ s ≤ . Thususing Gronwal the previous lemma gives apriori control on k T t k H s + k T k H s +1 in L ∞ (0 , T ) forany T >
0, provided that s ≤ . We can then repeate the same estimates as above but at thelast line estimate through interpolation of H s between H and H s +1 with s ≤ . Repeatinginductively we obtain control for arbitrary s > d . Step 2: Weak-strong uniqueness
We assume that the weak solutions have regularity: T ∈ L ∞ (0 , T ; L ) ∩ L ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; L ) (4.14) T t ∈ L ∞ (0 , T ; L ) (4.15)We consider the difference of two solutions T and T of (1.23) with T being a weaksolution and T a strong solution. We denote δT := T − T , and note that In here we need d = 2 as we assumed that d < s and d = 3 would contradict the restriction s ≤ δT (0 , x ) = ∂ t δT (0 , x ) ≡ δT satisfies the equation: J δT tt + µ δT t = L ∆ δT − aδT + b (cid:18) δT + T δT + δT T − | δT | d I d − T : δTd I d (cid:19) − c (cid:2) δT | δT | + δT | T | + 2 δT ( δT : T ) + T | δT | + 2 T ( δT : T ) (cid:3) (4.16)We multiply the last relation by δT t , integrate over R d and by parts, to get: ddt Z R d J | δT t | + L |∇ δT | + ψ B ( δT ) dx + µ Z R d | δT t | dx = b Z R d ( T δT + δT T ) : δT t dx − c Z R d | T | ( δT : δT t ) + 2( δT : T )( δT : δT t ) + | δT | ( T : δT t ) + 2( δT : T )( δT t : T ) dx ≤ C (cid:18)Z R d | δT | + | δT | dx (cid:19) (cid:18)Z R d | δT t | dx (cid:19) (4.17)with C a constant depending on k T k L ∞ , b and c .On the other hand, multiplying (4.16) by δT , integrating over R d and by parts, we get: J d dt Z R d | δT | dx − J Z R d | δT t | dx + µ ddt Z R d | δT | dx + L Z R d |∇ δT | dx = Z R d ( − a | δT | + b tr( δT ) − c | δT | ) dx + b Z R d ( T δT + δT T ) : δT dx − c Z R d (cid:0) | δT | | T | + 3( δT : T ) | δT | + 2( δT : T ) (cid:1) dx ≤ C Z R d | δT | − c Z R d | δT | dx dx (4.18)with C a constant depending on k T k L ∞ , a , b and c .We multiply (4.17) by J and (4.18) by µ and add them together to get: J µ d dt Z R d | δT | dx + ddt Z R d J | δT t | + LJ |∇ δT | + J ψ B ( δT ) + µ | δT | dx + Lµ Z R d |∇ δT | dx ≤ C Z R d | δT t | + | δT | dx (4.19)with C a constant depending on k T k L ∞ , a , b and c .Integrating over [0 , t ] we obtain: iquid crystal model with inertia 33 J µ ddt Z R d | δT | ( t, x ) dx + Z R d (cid:18) J | δT t | + LJ |∇ δT | + J ψ B ( δT ) + µ | δT | (cid:19) ( t, x ) dx ≤≤ J µ Z R d ( δT t : δT )(0 , x ) | {z } =0 dx + Z R d J | δT t | + LJ |∇ δT | + J ψ B ( δT ) + µ | δT | | {z } =0 (0 , x ) dx + C Z t Z R d ( | δT t | + | δT | )( s, x ) ds dx (4.20)which implies: J µ ddt Z R d | δT | ( t, x ) dx + Z R d J | δT t | ( t, x ) dx ≤ C Z R d | δT ( t, x ) | dx + C Z t Z R d ( | δT t | + | δT | )( s, x ) ds dx (4.21)Integrating one more time and using δT t (0 , · ) = δT (0 , · ) ≡ δT t ( t, · ) = δT ( t, · ) ≡ t > Step 3: weak solutions of the f -equation (4.2)We will just provide here the apriori estimates necessary for obtaining the weak solutions.The actual construction of the approximation scheme can be done through a straightforwardmodification of the scheme used in the proof of Theorem 1.2 and it is left to the interestedreader.We assume that (4.2) has a classical solution. We multiply it by f t r and integrate over R to get ddt Z R (cid:26) J f t + L f r + h B ( f ) + d f r (cid:27) r dr + µ Z R f t r dr = 0 (4.22)where we used the reduced potential h B ( f ) = a f − b ( d − d f + c ( d − d f (4.23)On the other hand, multiplying (4.2) by f r and integrating over R we obtain: d dt Z R J f r dr − J Z R f t r dr + µ ddt Z R f r dr + L Z R f r r dr = − d Z R f dr + Z R ( − af + b ( d − d f − c ( d − d f ) r dr (4.24)We now multiply (4.22) by J and add to it (4.24) multiplied by µ to get: d dt Z R J µ f r dr + ddt Z R J (cid:26) J f t + L f r + h B ( f ) + d f r (cid:27) r dr + µ ddt Z R f r dr = − Lµ Z R f r r dr − dµ Z R f dr + Z R µ ( − af + b ( d − d f − c ( d − d f ) r dr (4.25)Integrating over [0 , t ] we get: ddt Z R J µ f ( t, r ) r dr + Z R J (cid:26) J f t + L f r + h B ( f ) + d f r (cid:27) ( t, r ) r dr + µ Z R f ( t, r ) r dr = Z R J µ f t (0 , r ) f (0 , r ) r dr + Z R J (cid:26) J f t + L f r + h B ( f ) + d f r (cid:27) (0 , r ) r dr + µ Z R f (0 , r ) r dr − µ Z t Z R (cid:18) Lf r + 2 df + af − b ( d − d f + c ( d − d f ) (cid:19) ( s, r ) r drds (4.26)Integrating one more time over [0 , t ] we further obtain: Z R J µ f ( t, r ) r dr + Z t Z R (cid:26) J (cid:18) J f t + L f r + h B ( f ) + d f r (cid:19) + µ f (cid:27) ( s, r ) r drds = Z R J µ f (0 , r ) r dr + t Z R J µ f t (0 , r ) f (0 , r ) r dr + t Z R J (cid:26) J f t + L f r + h B ( f ) + d f r (cid:27) (0 , r ) r dr + t µ Z R f (0 , r ) r dr − µ Z t Z τ Z R (cid:18) Lf r + 2 df + af − b ( d − d f + c ( d − d f ) (cid:19) ( s, r ) r drds dτ (4.27)Using the fact that for an arbitrary function a ∈ L loc ( R ; R ) we have: R t R τ a ( s ) ds dτ = R t ( t − τ ) a ( τ ) dτ and that J, µ , L, c > Z R J µ f ( t, r ) r dr ≤ C + C t + C t Z t Z R f ( s, r ) r drds (4.28)which implies for any T > f ∈ L ∞ (0 , T ; L ( R , r dr )). Using this bound and inte-grating (4.22) on [0 , T ] we also get f ∈ L ∞ (0 , T ; H ( R , r dr )) ∩ L ∞ (0 , T ; L ( R , r dr )) and f t ∈ L ∞ (0 , T ; L ( R , r dr )). Step 4: the existence of smooth twist solutions
One can easily see that the pre-viously obtained weak solution of equation of the f -equation (1.25) will provide a weaksolution of the Q -equation (1.23) through the formula T ( t, x ) = f ( t, x ) ¯ H ( x ) , ∀ t ≥ , x ∈ R d .Due to the weak-strong uniqueness we have that T ( t, x ) is also a strong solutions and thussince we can take s arbitrary we have T smooth. In particular T is continuous at 0 which, iquid crystal model with inertia 35 because of the discontinuity of ¯ H necessarily implies that f ( t, · ) is continuous on [0 , ∞ )for any t ≥ f ( t,
0) = 0 hence condition (4.1) holds. Furthermore by evaluating therepresentation formula T ( t, x ) = f ( t, | x | ) ¯ H for at the component 11 of the matrices and atthe point x = (0 , r ) we have that T ( t, , r ) = f ( t, r )( r r − ) hence f ( t, r ) = T ( t, , r ).Similarly f ( t, r ) = T ( t, , − r ). Since T is a smooth function, we have that its restrictionto the line { (0 , r ) , r ∈ R } is also a smooth function that is furthermore even. Thus we getthat f ∈ C ∞ [0 , ∞ ) with f r ( t,
0) = 0 as a consquence of the evenness of T ( t, , · ). A similarargument holds for f t providing its smoothness and f rt ( t,
0) = 0, hence conditions (4.3) and(4.5) hold. Thus, the arguments provided before the statement of Proposition 1.3 hold in arigorous sense and we have obtained a twist wave. (cid:3)
5. Some Open Problems
The study and the techniques developed in the paper generate some natural open ques-tions: • Global existence for small data and a negative The main technical assumptionthat we make for obtaining the existence of a global solution is that the coefficient a appearing in equation (1.3) is positive. This captures a physically relevant regime, inwhich the nematic state is nevertheless just a local but not global minimizer of thebulk potential. It would be interesting and technically challenging to see if one canobtain global existence for a negative.We suspect that the case a negative should be treated with different tools, as thatcase allows in the case J = 0 for a solution of the Q -equation (without flow) whose L p norms increase (which seems incompatible with our strategy of the proof of globalexistence). • Long-time behaviour, and relation to inertieless version
The usual expectation for the case of the damped wave equation is that in the longtime one has a diffusive-type behaviour and this is shown in a number of papers (seefor instance [4]). Our system has a resemblance with a damped wave equation but itis not clear if its structure allows for a similar conclusion. • The J → limit It is a natural question to try to understand the singular limit J → J = 0 has. It is natural toconjecture that after an initial boundary layer in time the solutions will convergestrongly to the formal J = 0 limit. • More twist-waves (with genuine wave structure)
We provide in the last sectionan example of a twist-wave solution that will remain a solution even if one also sets J = 0. It would be interesting to provide more such examples, and in particularto understand if there are examples which would not survive as twist-wave solutionswhen setting formally J = 0. • The stability of the twist wave solution
In [3] it was shown that in the stationarycase the melting hedgehog solution is stable. This is a crucial feature for determiningthe physical relevance of such a solution because only the stable solutions can beobserved experimentally. It would be thus very interesting to see if one has dynamicalstability of the “melting hedgehog wave” solution, i.e. if one starts with the anapproximation of the hedgehog initial data in Q and a small initial data in u will thissolution stay close to the melting hedgehog one? Will it evolve in the long-time tothe melting hedgehog wave? AppendixLemma 5.1.
Let y be a positive function in W , loc ( R + ) and x a function in L loc ( R + ) that isalmost everywhere positive. Let us assume that y ′ ( t ) + x ( t ) Cy ( t ) x ( t ) , (5.1) for almost every t in R + .There exists ε > a suitably small number such that if we take the initial datum y (0) = y ∈ (0 , ε ) , then y and x belong to L ∞ ( R + ) and L ( R + ) respectively, and moreover k y k L ∞ ( R + ) + k x k L ( R + ) y . Proof.
Assuming y / C , we define T > t > y ( t ) < / C .Then, for every t ∈ [0 , T ] we get y ′ ( t ) + 12 x ( t ) , so that, integrating from 0 to T , we deduce y ( T ) + 12 Z T x ( t ) y < C .
This yields that T = + ∞ and that k y k L ∞ ( R + ) + k x k L ( R + ) y . (cid:3) Acknowledgement
The activity of Francesco de Anna on this work was partially supported by funding fromthe European Union’s Horizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement ModCompShock, No 642768, through a three-month fel-lowship at the University of Sussex.The activity of Arghir Zarnescu on this work was partially supported by a grant of theRomanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI,project number PN-II-RU-TE-2014-4-0657 ; by the Project of the Spanish Ministry of Econ-omy and Competitiveness with reference MTM2013-40824-P; by the Basque Government iquid crystal model with inertia 37 through the BERC 2014-2017 program; and by the Spanish Ministry of Economy and Com-petitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.Both authors thank Professor Marius Paicu for inspiring discussions and also to the anony-mous referee for the careful reading and a number of remarks that helped to improve thepaper.
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