ANALYSIS OF A PHASE FIELD NAVIER-STOKES VESICLE-FLUID INTERACTION MODEL. Qiang Du, Manlin Li and Chun Liu

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SERIES B Volume 8, Number 3, October 27 pp ANALYSIS OF A PHASE FIELD NAVIER-STOKES VESICLE-FLUID INTERACTION MODEL Qiang Du, Manlin Li an Chun Liu Department of Mathematics Pennsylvania Sate University University Park, PA 1682, USA Abstract. This paper is concerne with the ynamics of vesicle membranes in incompressible viscous fluis. Some rigorous theory are presente for the phase fiel Navier-Stokes moel propose in [7], which is base on an energetic variation approach an incorporates the effect of bening elasticity energy for the vesicle membranes. The existence an uniqueness results of the global weak solutions are establishe. 1. Introuction. The stuy of hyroynamical an rheological properties of fluis involving vesicle membranes an cells is of interest in many biological an physiological applications. Consierable research efforts have been evote to both experimental stuies [1, 11, 13] an the evelopment of mathematical moels an computational coes of various egrees of physical relevance an sophistication regaring the membrane properties/configurations an the flui constitution in recent years [2, 3, 4, 7, 14, 15, 16, 2, 21, 22, 24, 25, 26, 27]. Vesicle membranes are forme by lipi bilayers which play an essential role in biological functions. Their equilibrium shapes are often characterize by minimizing the bening elastic energy of the membrane [12, 19, 23, 28]: E = Γ k 2 (H c ) 2 S (1) where Γ is the surface of vesicle membrane, H is the mean curvature of Γ, c the spontaneous curvature an k the bening moulus. It is known that the behavior of these vesicles, in both static configurations an uner external flow fiels, ramatically iffers from that of those roplets whose shape is governe by the surface tension (with surface energy epening only on the surface area of the membranes). In this paper, we continue our earlier stuies [7] an consier the phase fiel Navier-Stokes moel for the vesicle shape ynamics, which is governe by the coupling of the hyroynamic flui flow an the bening elastic properties of the vesicle membrane. The resulting membrane configuration an the flow fiel reflect the competition an the coupling of the kinetic energy an membrane elastic energies. 2 Mathematics Subject Classification. Primary: 58F15. Key wors an phrases. Phase Fiel, Navier-Stokes, Well-poseness, Weak Solution, Existence, Uniqueness, Vesicle Membrane, Flui Interaction, Bening Elastic Energy. The research of Q. Du an M. Li is supporte in part by the NSF DMS-49297, NSF ITR an NSF CCF The research of C. Liu is supporte in part by the NSF DMS an DMS

2 54 Q. DU, M. LI AND C. LIU In the phase fiel Navier-Stokes moel, the escription of the membrane is given in terms of a phase fiel function φ (see [8] an the subsequent works [5, 6, 7, 9, 1, 32] for etails). The phase fiel function φ, roughly speaking, is a labeling function efine on computational omain. The function φ takes value nearly +1 insie the vesicle membrane an 1 outsie, with a thin transition layer of with characterize by a small positive parameter ǫ. The zero level surface of φ represents the surface of vesicle membrane. The avantage of introucing such a labeling function is to formulate the original Lagrangian escription of the membrane evolution in the Eulerian (observer s) coorinates. As in [8], we will approximate the elastic bening energy (1) by E ǫ (φ) = k ( ǫ φ + ( 1 2 2ǫ ǫ φ + c 2)(1 φ )) 2 x. For illustration purposes, the fluis both insie an outsie the vesicle are taken to be an incompressible viscous Newtonian flui an the elastic energy associate with the vesicle eformation mainly comes from the bening energy. We want to point out that it is easy to incorporate other physical consierations of the fluis an the membrane into our energetic variational approach. The vesicle eformation an the flui velocity fiel are then regare as the result of the competition between vesicle membrane bening energy an flui kinetic energy, subject to the constraints that the volume an surface area of the vesicle are preserve. The equations governing the ynamics of the phase fiel function φ an the flui velocity fiel u can be obtaine via the energetic variation approach [7]. To enforce the two constraints, one may either aopt the Lagrange multiplier approach or use a penalty formulation [7]. Here, we focus on the latter, that is, we a two penalty terms to the elastic bening energy E ǫ (φ) to enforce the volume an surface area constraints respectively. As in [9] an [7], the moifie energy is given by where E(φ) = E ǫ (φ) M 1(A(φ) α) M 2(B(φ) β) 2, (2) A(φ) = φx, B(φ) = ǫ 2 φ ǫ (φ2 1) 2 x. We efine the following corresponing action functional which illustrate the competition between ifferent part of the energies. T 1 A[x(t, X)] = 2 x t(t, X) 2 x E(φ(t, x(t, X)))t (3) where x(t, X) can be thought as the incompressible flui trajectory in the Lagrangian coorinate an u being the flui velocity fiel. The Least Action principle yiels the actual force balance (linear momentum) equation [7]. We are thus le to the following phase fiel Navier-Stokes equation for φ an u: u t + u u = p + µ u + φ in [, T], iv u = in [, T], φ t + u φ = γ in [, T], (4) u(, x) = ũ(x) in, φ(, x) = φ(x) in

3 where PHASE FIELD NAVIER-STOKES EQUATIONS 541, the so-calle chemical potential, enotes the variational erivative of E(φ) in the variable φ (its precise form is escribe later). The above equation is complemente by bounary conitions (BC). The particular BC consiere in this paper is of the Dirichlet type for the phase fiel function φ an the no-slip bounary conition for the velocity fiel u: u = on, (5) φ = 1, φ = on. (6) The main objective of this paper is to provie a rigorous mathematical founation to the above couple phase fiel Navier-Stokes (PFNS) equation. In particular, we present the proof of existence an uniqueness of weak solution to (4). Our results inicate that we can essentially control the coupling between the velocity fiel an the phase fiel so that the natural (energy) solution spaces for the PFNS equations remain the same as that for the ecouple conventional incompressible Navier- Stokes equation an the simple phase fiel graient flow for the bening elastic energy. We elect to only focus on the case c = in this paper, though the proof can be reaily extene to the non-zero spontaneous curvature case. We note that some formal analysis has been given in [7] on the sharp interface limit of the couple PFNS moel as the interfacial with parameter ǫ. In particular, it is seen that uner a general ansatz assumption, the extra term φ in the momentum equation leas to the well-known Willmore force acting between the backgroun flui an the membrane surface (see also [3]), though the corresponing well-poseness results for the limiting system are still open an uner investigation. 2. Main results an formal estimates. In this section, we state our main results concerning the well-poseness of the couple PFNS moel an the properties of their weak solutions. Throughout the iscussion, we use the space H () to enote the space of ivergence free vector fiels in H 1(), an L2 () for the closure of ivergence free subset of Cc () in L2 (). H 1 () enotes the ual space of H (). For notational convenience, for any given time T, we also use spaces like L p (, T; L q ()) for functions of both the time an space variables as efine in [29]. In aition, we use <, > to enote the inner prouct in (an uality pairing with respect to) L 2 (), an we also use the following trilinear form: B(u, v, w) = u v w x. (7) The main results of this paper are the following existence an uniqueness theorems. Theorem 1. Existence of Weak Solution. Let be an open, boune subset of R 3 either having a smooth bounary or being a convex polyhera. There exists a pair of functions φ an u with 1. u L 2 (, T; H ()) W 1, 4 3 (, T; H 1 ()) 2. φ L 2 (, T; H 2 ()) H 1 (, T; L 2 ()) which is a weak solution to equation (4) with bounary conition (5-6), that is,

4 542 Q. DU, M. LI AND C. LIU 1. for any δ(x) H (), ξ(x) L 2 (), an a.e. t [, T], we have < u t, δ > +B(u, u, δ) = µ < u, δ > + φ δ x, (8) < φ t, ξ > +B(u, φ, ξ) = γ <, ξ >. 2. u(, x) = ũ(x), φ(, x) = φ(x) where ũ L 2 () an φ + 1 H 2 (). Theorem 2. Uniqueness of Weak Solution. For the weak solutions to equation (4) iscusse in the previous existence theorem, if in aition we have the solution satisfying u L 8 (, T; L 4 ()), then the weak solution is unique. A few remarks are first in orer. First, we aopt suitable assumptions on the omain so that we can obtain the H 2 regularity for the Laplace operator with homogeneous bounary conition. Secon, ue to the stanar theory for the conventional Navier-Stokes equations without the membrane stress [29] an the simple L 2 graient flow of the elastic bening energy without the flui transport [31], it is easy to see that the main task at han is to analyze the coupling terms in the PFNS equation, which has similar spirits as that in the stuy of couple systems for flui an liqui crystal irector [17]. Therefore, we nee to consier (control) the contribution to the momentum equation of the aitional stress tensor ue to the membrane eformation an the contribution of the convection term to the phase fiel evolution. With the energy law establishe below, it turns out that the solution space L 2 (, T; H ()) W 1, 4 3 (, T; H 1 ()) for the velocity fiel u remains the same as that for the conventional three imensional incompressible Navier- Stokes equations. This reflects the fact that the membrane stress tensor oes not pose any extra limitation on the regularity of the weak solution of the velocity fiel. Meanwhile, the solution space L 2 (, T; H 2 ()) H 1 (, T; L 2 ()) for φ also coincies with the natural space for the solution of the simple L 2 graient flow of the elastic bening energy, again showing that the effect of flui transport on the phase fiel function can also be properly controlle. Thir, ue to the limite regularity in u, the issue of uniqueness of the weak solution remains open even for the conventional Navier-Stokes equations in three imension without the membrane effect, thus, we o not have the proof of uniqueness of the global weak solutions for the couple PFNS equation in general. However, with a better regularity assumption on the weak solutions, as in the case of the conventional Navier-Stokes equations [29], the uniqueness can be assure. Before we turn to the proofs, let us also mention that we have chosen to work with the penalty formulation to incorporate the volume an surface area conservation of the vesicle membrane in time. The results presente here are for given penalty constants, so the constraints are satisfie only approximately. Careful examination of our proofs inicates that much of the estimates erive in the paper can be mae to be inepenent of the penalty constants, thus allowing one to extract a suitable limit as the penalty constants approach infinity. Such a limiting process woul lea the existence of solutions in the Lagrange multiplier formulation with the constraints exactly satisfie. However, a etaile account of the epenence an inepenence of the estimates on the penalty constants woul require teious bookkeeping notation-wise. To avoi complication, we elect to ignore such epenence in the later presentation. The etaile proofs of the above theorems are ivie into several parts which are given in the later sections. The main steps inclue establishing solution estimates

5 PHASE FIELD NAVIER-STOKES EQUATIONS 543 from the energy law an passing to the limit via a moifie Galerkin proceure. For a etaile account of similar techniques, we refer to the iscussion on the conventional incompressible Navier-Stokes equation given in [29]. We note that though we have use the bounary conition (5-6), much of our analysis can be carrie out in other cases as well, such as the Neumann an perioic bounary conitions, an in particular, the case of an inhomogeneous velocity profile on the bounary which is often use in the stuy of cell eformation in shear an/or extensional flows Formal erivation of the energy law. The issipation of the kinetic energy is one of the most basic property of the conventional incompressible Navier-Stokes equations, an it can be easily erive. It is thus interesting to note that a similar energy law hols for the couple PFNS equation, with the membrane bening elastic energy being ae to the kinetic energy to prouce the total energy. For convenience, let us enote f(φ) = ǫ φ + 1 ǫ (φ2 1)φ, g(φ) = f(φ) + 1 ǫ 2 (3φ2 1)f(φ). Then, we may rewrite the energy as E(φ) = k f(φ) 2 x + 1 2ǫ 2 M 1(A(φ) α) M 2(B(φ) β) 2. The irect computation shows = kg(φ) + M 1(A(φ) α) + M 2 (B(φ) β)f(φ). (9) We now give a formal erivation of the energy law for smooth solutions u an φ of (4). Multiply u to the first equation in (4) an to the secon equation, then integrate over, we get the following issipative energy law 1 t 2 u 2 x + E(φ) = µ u 2 x γ 2 x. (1) Immeiately one can conclue that if u an φ are the solutions of the PFNS, we have the uniform bouns (with respect to any T > ) of the following type: u L (, T; L 2 ()) L 2 (, T; H ()); φ L (, T; H 2 ()); f(φ) L (, T; L 2 ()); L 2 ((, T) ()). For weak solutions of (4), we can erive a weaker version of the energy law rigorously via a Galerkin proceure outline below Formal estimates of u t H 1 () an φ t L2 (). Base on the bouns from the energy law an the PFNS equation, we may euce better estimates on the time erivatives. 1. Let v L 2 () an v L 2 () 1, from equation (4), φ t v x u φv x + γ v x.

6 544 Q. DU, M. LI AND C. LIU An, u φv x C v L 2 () φ L3 () u L6 () C φ H2 () u H 1 (), Therefore, φ t L 2 () C vx L 2 () v L 2 (). ( φ H 2 () u H 1 () + ) L 2 (). 2. We now take v H (). Also from equation (4), we get u t v x u u v x + φ v x +µ u L2 () v L2 (), u t H 1 () u u H 1 () + µ u L 2 () + φ H 1 (). By a well known interpolation result [29], we have, u u H 1 () C u 1 2 L 2 () u 3 2 L 2 (). As for φ H 1 (), we consier for any v H 1 (), with v H 1 () 1. By the Sobolev inequality v L 6 () C v H 1 (), we get φ v x L 2 () φ L 3 () v L 6 () K 1 L 2 () φ L 3 () K 2 L 2 () φ H 2 (). Hence, ( u t H 1 () C u 3 2 L2 () u 1 2 L2 () + u L 2 () + ) L 2 () φ H 2 (). 3. Proof the existence of weak solutions. This section is evote to the proof of existence of weak solutions of the couple PFNS equation (4) with bounary conition (5-6). We first outline a moifie Galerkin approximation, then we consier its weak limit an verify it as a weak solution of (4).

7 PHASE FIELD NAVIER-STOKES EQUATIONS A moifie Galerkin approximation. Let us choose {ω n } L 2 () to be the eigenfunctions of Stokes operator, such that {ω n } forms an orthonormal basis for L 2 (). Set W n = Span{ω 1, ω 2,..., ω n }. Apply the Galerkin approximation to the velocity fiel u, one can get the approximate equation for u W n an φ H 2 (): u t + P n (u u) = µ u + P n ( φ), φ t + u φ = γ, (11) u() = P n ũ(x), φ() = φ(x) where P n is the L 2 projection operator to W n. The following Lemma asserts the existence of a solution to the approximate equation (11). It also provies a uniform energy estimate on the solution (with respect to the imension n). Lemma 1. Existence of An Approximate Solution There exists a pair of functions u(t, x) W n, φ(t, x) H 2 () satisfying < u t, w > +B(u, u, w) = µ < u, w > + φ w x, < φ t, v > +B(u, φ, v) = γ <, v >, (12) u() = P n ũ(x) φ() = φ(x), w W n an v L 2 () for almost all t [, T]. Furthermore, for a.e. ˆT [, T], ˆT µ u 2 + γ 2 xt + with a constant M inepenent of W n. 1 2 u( ˆT, x) 2 x + E(φ( ˆT, x)) M, Proof. 1. Apply Galerkin approximation to φ Denote by {υ 1, υ 2,...} the eigenfunctions of operator uner the homogeneous Dirichlet bounary conition. They consist of an orthonormal basis of L 2 (). By the assumption on the omain, we have that the eigenfunctions of the Laplace operator have H 2 regularity. Set V m = Span{υ 1, υ 2,...,υ m }. Apply a moifie Galerkin metho to φ, we get an approximate equation to equation (12) as follows: fin u m (x, t) an φ m (x, t) of the form n m u m (x, t) = i (t)ω i (x) W n an φ m (x, t) + 1 = h j (t)υ j (x) V m i=1 such that for k = 1, 2,...,n an l = 1, 2,..., m, < u m, ω k > +B(u m, u m, ω k ) = µ < u m, w k > + π m ( δe(φm) ) φ m ω k x, δe(φ m) υ l x, < φ m, υ l > +B(u m, φ m, υ l ) = γ j=1 u m (, x) = P n ũ(x), φ m (, x) = π m ( φ(x) + 1) 1. (13) Here means ifferentiating in time, an π m enotes a L 2 (also H 2 if applicable) projection to V m. The solutions have natural epenence on the inex n, but for convenience, we have suppresse this epenence in the notation of u m an φ m.

8 546 Q. DU, M. LI AND C. LIU It is easy to see that the above finite imensional ODE system has a solution local in time. 2. Energy estimate In equation (13), replace w k with u m, υ l with π m ( δe(φm) ), we have, 1 u m 2 x = µ < u m, u m > + π m ( δe(φ m) ) φ m u m x t 2 < φ m, π m( δe(φ m) ) > + π m ( δe(φ m) ) φ m u m x Since = γ δe(φ m ) π m ( δe(φ m) )x. < φ m, π m( δe(φ m) ) >=< π m (φ m ), δe(φ m) >=< φ m m, δe(φ m) >= m t E(φ m), the summation of the two expressions above gives the energy law, t 1 2 It implies, u m 2 x + E(φ m ) = µ 1 2 u m 2 x γ ˆT u m ( ˆT, x) 2 x + E(φ m ( ˆT, x)) + π m( δe(φ m) 2 ) x. (14) µ u m 2 +γ π m( δe(φ 2 m) ) xt 1 u m (, x) 2 x + E(φ m (, x)). 2 We know that u m (, x) L 2 () ũ(x) L 2 (). By the construction of φ m (, x), we also have that φ m (, x) converges to φ(x) in H 2 () as m. Hence there exists some constant M inepenent of W n, such that, 1 2 ˆT + ˆT u m ( ˆT, x) 2 x + E(φ m ( ˆT, x)) + γ π m( δe(φ m) 2 ) xt M. µ u m 2 xt Note that such an energy law essentially gives the existence of local solutions for all time. 3. Compactness of {u m } an {φ m } The energy law ensures that u m (t) L 2 () an φ m (t) L 2 () are uniformly boune in m an t [, T]. Thus the solution of ODE (13) actually exists global in time. Furthermore, the energy law also inicates u m is uniformly boune in L (, T; L 2 ()) L 2 (, T; H ()). φ m is uniformly boune in L (, T; H 2 ()) (thus in L 2 (, T; H 2 ()))

9 PHASE FIELD NAVIER-STOKES EQUATIONS 547 π m ( δe(φm) ) is uniformly boune in L 2 ((, T) ). Similar to the previous formal erivation, we can rigorously obtaining estimates on u m H 1 () an φ m L 2 () : u m is uniformly boune L 4 3(, T; H 1 ()). φ m is uniformly boune L 2 ((, T) ). Therefore, using the Aubin-Lions type compact embeing results [29], there exist some φ, u, such that, u m has a subsequence u mk converging to u weakly in L 2 (, T; H ()) an strongly in L 2 (, T; L 2 ()). φ m has a subsequence φ ml converging to φ weakly in L 2 (, T; H 2 ()) an strongly in L 2 (, T; W 1,p ()) for 1 p < 6. For convenience, if there is no ambiguity, from now on we will ientify u m an φ m with their subsequences. 4. Passing weak limits of {u m } an {φ m } Choose w(t, x) = α(t)δ(x), v(t, x) = α(t)ξ(x) where α C([, T]), δ W n, an ξ V m, we have T < u m, w > +B(u m, u m, w)t = T In which, T T < φ m, v > +B(u m, φ m, v) = γ < π m ( δe(φ m) ) φ m, w > t T δe(φ m ) = k{ f(φ m ) + 1 ǫ 2 (3φ2 m 1)f(φ m )} µ < u m, w > t < δe(φ m), v > t. +M 1 (A(φ m ) α) + M 2 (B(φ m ) β)f(φ m ) = k{ [ ǫ φ m + 1 ǫ (φ2 m 1)φ m] + 1 ǫ 2 (3φ2 m 1)f(φ m)} +M 1 (A(φ m ) α) + M 2 (B(φ m ) β)f(φ m ) (15) = ǫk 2 φ m + L(φ m ) where L(φ m ) enotes the lower orer term. Note that the L(φ m ) L2 ((,T) ) is uniformly boune by the uniform boun on φ m in L (, T; H 2 ()). Then, π m ( δe(φ m) ) = π m (ǫk 2 φ m ) + π m (L(φ m )) = ǫk 2 φ m + π m (L(φ m )). Together with the energy estimate (14), we have ǫk 2 φ m L 2 ((,T) ) π m ( δe(φ m) ) L2 ((,T) ) + π m (L(φ m )) L2 ((,T) ) π m ( δe(φ m) ) L 2 ((,T) ) + L(φ m ) L 2 ((,T) ).

10 548 Q. DU, M. LI AND C. LIU Therefore, δe(φ m) L2 ((,T) ) π m ( δe(φ m) ) L2 ((,T) ) + 2 L(φ m ) L2 ((,T) ). (a) Weak limit of { δe(φm) }. We now prove that (a subsequence of) { δe(φm) } converges to weakly in L 2 ((, T) ). By the energy law (14), f(φ m ) is uniformly boune in L (, T; L 2 ()), hence uniformly boune in L 2 ((, T) ). It is enough to show for g C ([, T] ), T lim m T f(φ m )g xt = f(φ m )g xt = T T f(φ)g xt. [ ǫ φ m + 1 ǫ (φ2 m 1)φ m]g xt. It is sufficient to check on the nonlinear terms only. Since φ m is uniformly boune in L (, T; H 2 ()), we have φ m (t) C, 1 2 () C φ m(t) H2 () M for t [, T]. Hence, φ m (t, x) is uniformly boune in [, T]. Furthermore, φ m has as subsequence converging to φ strongly in L 2 (, T, W 1,p ) for 1 p < 6. Then a subsequence of φ m converges to φ almost everywhere in [, T]. By the Lebesgue-Dominate Theorem, lim n T φ 3 mg xt = Similarly, we nee to verify that T δe(φ m ) lim g xt = m T T T φ 3 g xt. g xt. (16) To give more etails, we follow (15) term by term. First, we have T T 2 φ m g xt = φ m g xt φ g xt = T 2 φg xt as m. Next, we have T (φ 3 m φ 3 )g xt = 3 T (φ 2 m φ m φ 2 φ) g xt T C (φ 2 m φ 2 ) φ m g xt + C T φ 2 ( φ φ m ) g xt C φ 2 m φ 2 L 2 ((,T) ) φ m L 2 ((,T) ) + C φ φ m L 1 ((,T) ).

11 PHASE FIELD NAVIER-STOKES EQUATIONS 549 Hence, as m, T (φ 3 m φ3 )g xt. T Now, consier (φ 2 mf(φ m ) φ 2 f(φ))g xt T (f(φ m ) f(φ))φ 2 g xt T + (φ 2 φ 2 m )gf(φ m)xt = I 1 + I 2. We have I 1 since f(φ m ) f(φ) weakly in L 2 ((, T) ). In aition, I 2 g L ((,T) ) f(φ m ) L 2 ((,T) ) φ 2 φ 2 m L 2 ((,T) ). It is also easy to show, T lim m B(φ m )f(φ m )g xt = T B(φ)f(φ)g xt. (b) Verifying the approximate equation. Choose g L 2 ((, T) ). Since δe(φm) weakly converges to π m (g) converges strongly to g in L 2 ((, T) ), we have T T = T, < π m ( δe(φ m) ), g > t, g >)t < δe(φ m), π m (g) > < < δe(φ m), g > t + T < δe(φ m), π m (g) g > t an as m, We can conclue that π m ( δe(φm) in L 2 ((, T) ). Now, one can let m to recover ) converges to weakly T T < u t, w > +B(u, u, w)t = T + < φ t, v > +B(u, φ, v) = γ T T µ < u, w > t φ w xt, <, v > t.

12 55 Q. DU, M. LI AND C. LIU An, ˆT ( 1 u( 2 x) 2 x + E(φ( ˆT, x)) + µ u 2 + γ xt M. Since α = α(t) is arbitrarily chosen in C([, T]), one can conclue for any δ W n an ξ V m, < u t, δ > +B(u, u, δ) = µ < u, δ > + φ δ x, < φ t, ξ > +B(u, φ, ξ) = γ ξ x. (17) By ensity argument, it is also true for any δ W n an ξ L 2 (). Setting α() = 1 an α(t) =, one can show that u(, x) = P n ũ(x) an φ(, x) = φ(x). Finally, since for i > n T u m (t, x)ζ(t)ω i (x)xt = for any ζ(t) C([, T]), we have by taking m that u(t, x)ω i (x)x = for almost all t [, T] when i > n. Therefore u W n. This completes the proof of this lemma Proof of the existence theorem. We now wrap up the proof of the existence theorem. Accoring to Lemma 1, for any positive time ˆT (, T), an for each W n, the equation (11) has a solution u n an φ n, such that ˆT ( µ u n 2 + γ δe(φ ) n) 2 xt+ n where M is inepenent of W n. Hence, Also, 1 2 u n( ˆT, x) 2 x+e(φ n ( ˆT, x)) M (18) u n is uniformly boune in L (, T; L 2 ()) L 2 (, T; H ()); φ n is uniformly boune in L (, T; H 2 ()) ( thus in L 2 (, T; H 2 ()) ); is uniformly boune in L 2 ((, T) ). δe(φn) n u n is uniformly boune L 3(, 4 T; H 1 ()); φ n is uniformly boune L2 (, T; L 2 ()). Therefore, there exist some φ an u, such that, u n has a subsequence u nk converging to u weakly in L 2 (, T; H ()) an strongly in L 2 ((, T) ); φ n has a subsequence φ nl converging to φ weakly in L 2 (, T; H 2 ()) an strongly in L 2 (, T; W 1,p ()) for 1 p < 6.

13 PHASE FIELD NAVIER-STOKES EQUATIONS 551 Similar to the claim in Lemma 1, (a subsequence of ) δe(φn) n converges weakly to in L 2 ((, T) ). Choose w(t, x) = α(t)δ(x), v(t, x) = α(t)ξ(x) where α C([, T]), δ W n, an ξ C(), then consier T < u n, w > +B(u n, u n, w)t = Let n, we get, T T < u t, w > +B(u, u, w)t = T T < δe(φ n) φ n, w > t n < φ n, v > +B(u n, φ n, v) = γ T T < T φ, w > t < φ t, v > +B(u n, φ, v)t = γ T T µ < u n, w > t <, v > t. µ < u, w > t, <, v > t. Because α(t) is an arbitrarily chosen function in C([, T]), one can conclue for any δ W n, ξ C(), < u t, δ > +B(u, u, δ) = µ < u, δ > + < φ t, ξ > +B(u, φ, ξ) = γ <, ξ > φ δ x, By a ensity argument, it is also true for any δ H 1 () an ξ L2 (). Set α() = 1 an α(t) =, one can show u(, x) = ũ(x), an φ(, x) = φ(x). This conclues the proof of the main existence theorem. 4. Uniqueness of weak solution. We now provie the proof of the uniqueness of the weak solution, uner the aitional regularity assumption on the velocity fiel. We first introuce a few notations to simplify the later iscussion. Define G(φ) = 1 (kǫ φ 2 + kǫ ) 2 φ 2 + φ 2 x, then it is easy to check that 1 C φ 2 H 2 () G(φ) C φ 2 H 2 (). Define also M(φ) = δg(φ) = kǫ 2 φ k φ + φ, N(φ) = ǫ M(φ). Assume u i, φ i an p i (i = 1, 2) are two weak solutions to equation (4) satisfying the assumptions given in the uniqueness theorem. Let û = u 1 u 2, ˆφ = φ 1 φ 2 an ˆp = p 1 p 2. We erive a Gronwall type inequality for û an ˆφ to prove the uniqueness Derivation of a Gronwall inequality. First, we have, û + û u 1 + u 2 û = ˆp + µ û + (M(φ 1 ) + N(φ 1 )) φ 1 (M(φ 2 ) + N(φ 2 )) φ 2, ˆφ + u 1 φ 1 u 2 φ 2 = γ(m(ˆφ) + N(φ 1 ) N(φ 2 )). (19)

14 552 Q. DU, M. LI AND C. LIU Multiply û to the first equation in (19) an M(ˆφ) to the secon one, integrate in space, we get 1 t 2 û 2 x + û u 1 ûx = µ û 2 x + (M(φ 1 ) φ 1 M(φ 2 ) φ 2 )û + (N(φ 1 ) φ 1 N(φ 2 ) φ 2 )ûx, t G(ˆφ) + (u 1 φ 1 u 2 φ 2 )M(ˆφ)x = γ M(ˆφ) 2 x Hence, γ M(ˆφ)(N(φ 1 ) N(φ 2 ))x 1 t 2 û 2 x + G(ˆφ) + γ M(ˆφ) 2 x + µ û 2 x + B(û, u 1, û) = (M(φ 1 ) φ 1 M(φ 2 ) φ 2 )û (u 1 φ 1 u 2 φ 2 )M(ˆφ)x + (N(φ 1 ) φ 1 N(φ 2 ) φ 2 )ûx γ Recall B(u, v, w) = u v wx. Since B(û, u 1, û) = B(û, û, u 1 ), we have B(û, u 1, û) = B(û, û, u 1 ) û L4 () û L2 () u 1 L4 () û 1/4 L 2 () û 7/4 L 2 () u 1 L 4 () M(ˆφ)(N(φ 1 ) N(φ 2 ))x û 2 L 2 () + C( ) û 2 L 2 () u 1 8 L 4 () where is an arbitrary small positive number. Direct calculation shows, (M(φ 1 ) φ 1 M(φ 2 ) φ 2 )û (u 1 φ 1 u 2 φ 2 )M(ˆφ)x an = (M(φ 1 ) ˆφ + M(ˆφ) φ 2 )û (u 1 ˆφ + û φ 2 )M(ˆφ)x = M(φ 1 ) ˆφ ûx u 1 ˆφ M(ˆφ)x = J 1 J 2, J 1 M(φ 1 ) L2 () ˆφ L4 () û L4 () û 2 H 1 () + C M(φ 1) 2 L 2 () ˆφ 2 H 2 (),

15 PHASE FIELD NAVIER-STOKES EQUATIONS 553 J 2 M(ˆφ) L 2 () u 1 L 4 () ˆφ L 4 () M(ˆφ) 2 L 2 () + C u 1 2 H 1 () ˆφ 2 H 2 (). with Moreover, (N(φ 1 ) φ 1 N(φ 2 ) φ 2 )ûx = (N(φ 1 ) ˆφ ûx + = K 1 + K 2, (N(φ 1 ) N(φ 2 )) φ 2 ûx K 1 N(φ 1 ) L 2 () ˆφ L 4 () û L 4 () C N(φ 1 ) L 2 () ˆφ H 2 () û H 1 () an û 2 C H 1 () + N(φ 1) 2 L 2 () ˆφ 2 H 2 (), K 2 N(φ 1 ) N(φ 2 ) L2 () φ 2 L4 () û L4 () Similarly, M(ˆφ)(N(φ 1 ) N(φ 2 ))x C N(φ 1 ) N(φ 2 ) L2 () φ 2 H2 () û H 1 () û 2 C H 1 () + N(φ 1) N(φ 2 ) 2 L 2 () φ 2 2 H 2 () û 2 C H 1 () + N(φ 1) N(φ 2 ) 2 L 2 (). We now use a claim (to be verifie later): M(ˆφ) L 2 () N(φ 1 ) N(φ 2 ) L 2 () M(ˆφ) 2 L 2 () N(φ 1) N(φ 2 ) 2 L 2 (). N(φ 1 ) N(φ 2 ) L 2 () C φ 1 φ 2 H 2 () = C ˆφ H 2 (). (2) Recall that ˆφ 2 H 2 () CG(ˆφ). Using (2) an putting everything together, we have 1 t 2 û 2 x + G(ˆφ) C 1 2 û 2 x + G(ˆφ) ) ( M(φ 1 ) 2 L 2 () + u 1 2H 1() + u 1 8 L 4 () + N(φ 1) 2 L 2 () + 1. Using the estimates alreay erive an the extra assumption on the velocity fiel, we have ( M(φ 1 ) 2 L 2 () + u 1 2 H 1() + u 1 8 L 4 () + N(φ 1) 2 L 2 () )

16 554 Q. DU, M. LI AND C. LIU is integrable in time. This implies if û(t) = an ˆφ(t) = at t =, then û = an ˆφ = for all time, which proves the uniqueness theorem Proof of claim (2). We now verify the claim (2) use in the above proof. By (9) an the efinitions of M an N, we get N(φ) = k ǫ φ3 + 2k 3k φ + ǫ ǫ 2 φ2 f(φ) k ǫ 2 f(φ) φ +M 1 (A(φ) α) + M 2 (B(φ) β)f(φ). This leas to N(φ 1 ) N(φ 2 ) L 2 () C{ φ 3 1 φ3 2 L 2 () + φ 1 φ 2 L 2 () + ˆφ L2 () + φ 2 1 f(φ 1) φ 2 2 f(φ 2) L2 () + f(φ 1 ) f(φ 2 ) L2 () + A(ˆφ) L 2 () + B(φ 1 )f(φ 1 ) B(φ 2 )f(φ 2 ) L 2 ()} Accoring to the energy estimate erive in the proof of existence theorem, we have φ i L (, T; H 2 ()), i = 1, 2. Therefore, for i, j = 1, 2, φ i (t, x) L ((,T) ) M, (φ i (t, x)φ j (t, x)) L ((,T);L 6 ()) M, (φ i (t, x)φ j (t, x)) L ((,T);L 2 ()) M for some constant M. Now we carefully estimate the iniviual terms respectively. A generic time-inepenent constant C is use. Let φ = φ φ 1φ 2 + φ 2 2, then φ 3 1 φ3 2 L 2 () ˆφ φ L 2 () + ˆφ φ L 2 () +2 ˆφ φ L2 () C ˆφ H2 (). f(φ 1 ) f(φ 2 ) L 2 () C( ˆφ L 2 () + ˆφ L 2 () + φ 3 1 φ3 2 L 2 ()) C ˆφ H2 (). φ 2 1 f(φ 1) φ 2 2 f(φ 2) L2 () (φ 2 1 φ2 2 )f(φ 1) L2 () + φ 2 2 (f(φ 1) f(φ 2 )) L2 () C ˆφ H2 (). B(φ 1 ) B(φ 2 ) C( (φ 1 + φ 2 ) L2 ()) ˆφ L2 () +C (φ φ2 2 2)(φ 1 + φ 2 )ˆφx C ˆφ H 2 (). B(φ 1 )f(φ 1 ) B(φ 2 )f(φ 2 ) L2 () B(φ 1 )(f(φ 1 ) f(φ 2 )) L2 () + (B(φ 1 ) B(φ 2 ))f(φ 2 ) L 2 () C ˆφ H 2 (). Summing together, the claim (2) is verifie.

17 PHASE FIELD NAVIER-STOKES EQUATIONS Conclusion. In this paper, a system of couple phase fiel Navier-Stokes equations moeling the eformation an evolution of three imensional vesicle membranes in a flui fiel is analyze. A major characteristic of the current moel stuie here, in comparison of many other similar moels stuie previously in the literature on membrane flui interactions, even in the phase fiel context, is the inclusion of the bening elastic energy which is crucial for vesicle bilayers. The resulting interaction mechanism is thus base on the competition between the elastic bening energy of the membrane, with prescribe bulk volume an surface area, an the kinetic energy in the surrouning flui velocity fiels. The variation of the elastic bening energy leas to an extra stress in the Navier-Stokes equation, which involves a nonlinear combination of higher orer spatial erivatives of the phase fiel function. Our main results illustrate that these aitional contributions can be properly controlle ue to the establishment of the energy law. Such an energy issipation mechanism is intrinsic in our erivation of the couple system. The results provie a rigorous mathematical founation to the couple phase-fiel Navier-Stokes equations an their numerical simulations [7]. The results can be extene to cases with the Neumann an perioic bounary conitions, as well as inhomogeneous Dirichlet conition for the velocity fiel. We note that our analysis largely relies on the amping term in the evolution of the phase fiel function an can not be reaily extene to the case of a pure transport. Such cases were treate in [18] for the systems for viscoelastic materials. However, the nonlinear coupling terms involve higher erivatives in this paper. In aition, the various estimates erive in the paper are not uniform with respect to the small interfacial with parameter ǫ an thus cannot be use in the stuy of the sharp interface limits. Such issues pose interesting challenges for future stuies. Extensions to interactions of vesicles with other types of fluis (possibly with ifferent types insie an outsie the vesicle) may also be consiere. Finally, it will be also interesting to stuy the existence of classical solutions of the system in this paper, in particular, the existence for the large viscosity situations. REFERENCES [1] M. Abkarian, C. Lartigue, an A. Viallat, Tank Treaing an Unbining of Deformable Vesicles in Shear Flow: Determination of the Lift Force, Phys. Rev. Lett., 88 (22), 6813 [2] J. Beaucourt, F. Rioual, T. Sion, T. Biben, an C. Misbah, Steay to unsteay ynamics of a vesicle in a flow, Phys. Rev. E, 69 (24), 1196 [3] T. Biben, K. Kassner an C. Misbah, Phase-fiel approach to three-imensional vesicle ynamics, Phys. Rev. E, 72 (25), [4] T. Biben an C. Misbah, Tumbling of vesicles uner shear flow within an avecte-fiel approach, Phys. Rev. E, 67 (23), 3198 [5] Q. Du, C. Liu, R. Ryham an X. Wang, A phase fiel formulation of the Willmore problem, Nonlinearity, 18 (25), , [6] Q. Du, C. Liu, R. Ryham an X. Wang, Moeling the Spontaneous Curvature Effects in Static Cell Membrane Deformations by a Phase Fiel Formulation, Communications in Pure an Applie Analysis, 4 (25), [7] Q. Du, C. Liu, R. Ryham an X. Wang, Moeling Vesicle Deformations in Flow Fiels via Energetic Variational Approaches, preprint, 26. [8] Q. Du, C. Liu, an X. Wang, A phase fiel approach in the numerical stuy of the elastic bening energy for vesicle membranes, Journal of Computational Physics, 198 (24), [9] Q. Du, C. Liu, an X. Wang, Retrieving Topological Information For Phase Fiel Moels, SIAM Journal on Applie Mathematics, 65 (25),

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