[精品论文]胶体杆状沉积物动力学模型的数学分析.doc

精品论文胶体杆状沉积物动力学模型的数学分析陈秀卿1，李晓龙1，刘建国21 北京邮电大学理学院，北京 1008762 杜克大学物理系与数学系，durham nc 27708 美国 摘要：本文研究了胶体杆状沉积物动力学模型。该模型是由不可压 (navier-)stokes 方程和 fokker-planck 方程耦合的方程组，在 2011 年由 helzel, otto and tzavaras 导出。对于含有 stokes 方程的情形，得到了二维问题整体弱解的存在性和唯一性。 关键词：应用数学，stokes 方程，fokker-planck 方程，整体弱解，唯一性中图分类号： o175.2analysis of kinetic model for sedimentation of colloidal rodschen xiu-qing1, li xiao-long1, liu jian-guo21 school of sciences, beijing university of posts and telecommunications, beijing, 1008762 department of physics and department of mathematics, duke university, durham, nc27708 usaabstract: we investigate a kinetic model for the sedimentation of colloidal rods, deduced by helzel, otto and tzavaras (2011), which couples the impressible (navier-)stokes equation with the fokker-planck equation. with no-ux boundary condition for distribution function, we establish the existence and uniqueness of global weak solution to the two dimensional model involving stokes equation.key words: stokes equation, fokker-planck equation, global weak solution, uniqueness0introductionrod-like particle suspensions in uid is common in nature, such as bacteria swimming in the water and liquid crystal molecules moves in a solvent. the dilute suspensions of passive rod-like particles can be eectively modeled by a coupled microscopic fokker-planck equation and macroscopic (navier-)stokes equation, known as doi model (see doi 1; doi and edwards 2). we refere to 3 for the doi model for suspensions of active rod-like particles without基金项目： national science foundation of china (grant 11101049), the research fund for the doctoral program ofhigher education of china (grant 20090005120009); national science foundation (nsf) of the usa, grant dms 10-11738. 作者简介： chen xiuqing (1977-)，male, associate professor, major research direction：partial dierential equations, email: . li xiaolong，male，graduate student, email: . liu jian-guo, male,professor, major research direction: numerical analysis, partial dierential equations, computational fluid dynamics, email:- 43 -considering the eects of gravity. recently an extended model under gravity was introduced by hezel, otto and tzavaras4, which readst f + x (u f ) n f + n (idn n)x un f =x (id + n n)e2 f + x (id + n n)x f (0.1) =sd1(dn n id) f dn,(0.2)ret u + (u x )u x u + x p =x (sd1)f dne2 ,(0.3)x u =0,(0.4)where (t, x, n) 0, ) sd1 , rd is a bounded domain with of class c1 and sd1 rdis the unit sphere; is a stress tensor, p is the pressure, e2 is the unit vector in the upwarddirection; n and n denote the tangential divergence and laplace-beltrami operator on sd1 ,respectively. in this model, f (t, x, n) is a distribution function which represents the congurationof a suspension of rod-like particles and u(t, x) is the uid velocity induced by the other particles in the suspension.the coecients re 0, 0 and 0 are constants. re is a reynolds number; measures the relation between the rate of work of buoyancy vs. the rate of work of viscousforce; measures the relation between the rate of work of elastic forces vs. the rate of work of buoyancy (see 4, remark 2.1-2.2). for convenience, if re = 0, the model including a stokes equation is called stokes-type.the rst term on the right hand side of (0.1) represents the eects of gravity on rod-like particles. taking this particular form are due to the fact that the frictional coecient in the tangential direction is twice as large as that in the normal direction. this result comes from classical slender body theory (see5, 6, 7). in fact, particles move sideways with a slight concentration in 45 degrees with a horizontal orientation sedimentation slower than particles with a vertical orientation. one of the most peculiar phenomenon in sedimentation of the rod- like particle is the packet formation and alignment in gravity direction (see 4). an anisotropic diusivity in the second term on the right hand side of (0.1) is also due to the fact that inhomogeneity of the frictional coecients in the tangential direction and normal direction.thatdening the concentration density := sd1f dn and integrating (0.1) over sd1 , we deducet + u x = x (e2 + x +sd1n n f dn e2 + x sd1)n n f dn.(0.5)the last two terms account for the anisotropic eects of gravity and diusivity. as a result, we do not have l and l2 estimates for density .while in contrast, in fene and hookean model with centre-of-mass diusion, the density satises a convection diusion equationt + u x dx = 0,where the maximum principle holds (see 8-9); in the doi model for active rod-like particlesuspensions (see 10),t + u x + x (sd1)(n f ) dn dx = 0,where has l2 -estimate. the l or l2 -estimate for is the foundation to establish the globalentropy weak solution in 8-9 and 10.it can be shown that this model with boundary conditions(id + n n)(e2 f + x f ) v| = 0;u| = 0(0.6)has the following entropy estimate (see 4)d ( ( f ln f f + f x e2 )dn + re)2|u| dx + 4 nf 2 dndxdt sd12sd112+ sd1(i + n n)(e2 f + x f ) (e2 f + x f )dndx +f|x u| dx = 0, (0.7)which reveals that the total energyis dissipated.e(u, f ) := ( sd1( f ln f f + f x e2 )dn +r e2)2|u|dxfor this model, without l or l2 -estimate of density , the entropy estimate is not enoughto establish the global weak solutions. we need the further l2 -estimate for distribution f . infact, multiplying (0.1) by f and integrating on sd1 , one has1 d 2| f | dndx +2|n f | dndx + (id + n n)x f x f dndx2 dt= sd1sd11 (id + n n)e2 f x f dndx +sd1(id n n)x un n f 2 dndx.(0.8)sd12 sd1since id + n n is a positive denite matrix with the smallest eigenvalue 1, we have2|x f | (id + n n)x f x f .(0.9)this and cauchy-schwartz inequality yield that1 d 22| f | dndx +|n f 2 +2| |x f | dndx2 dt csd1| f | dndx +sd11 2(id n n)x un n f 2 dndx.(0.10)sd12 sd1integrating by parts, we havei = :sd1(id n n)x un n f 2 dndx22=sd1(dn n id) f 2 : x udndx cx ul2 () f l4 (;l2 (sd1 ).(0.11)if d = 2, we have from gagliardo-nirenberg and cauchy-schwartz inequality (see (2.42)-(2.43)thattherefore, 4i sd1|x f | dndx + c(x ul2 () + 1) f l2 (sd1 ) .(0.12)d 1 2 ( 2 2 ) 2 2| f | dndx +|n f | +|x f |dndx c (x ul2 () + 1) f l2 ( sd 1 ) . (0.13)dt 2sd1sd14it follows from (0.13), gronwalls inequality and entropy estimate (0.7) that t ( 2 )22 f l2 (sd1 ) ce0 x ul2 () +1 d s c(t).(0.14)while for d 3, we cannot perform similar estimates as (0.12)-(0.14).2with the entropy estimate and two dimensional (2d) l2 -estimate at hand, in this paper weaim to prove the existence of global weak solution to the 2d stokes-type model with boundary conditions (0.6) and initial conditionf |t=0 = f0 on sd1 .(0.15)there are many works about the mathematical analysis of doi model for passive particlesuspensions in the literature (see 11, 12, 13, 14, 15, 16, 4, 17, 18), in which the density satises a transport equation with and without diusion, and hence the maximum principle holds. in the doi model for active particle suspensions (see 10), the density has l2 -estimate. the estimates of are common foundations of their proofs. however, this model lacks such good estimates for as mentioned before, which is the main dierence in contrast with other doi related models. a combination of entropy estimate and l2 -estimate solves this problem.the paper is organized as follows. section 2 collected some preliminaries. in section 3, we prove the existence of the global weak solution for the 2d stokes-type model, where a semi- implicit scheme is used to construct the approximate problem and compactness is shown. then in section 4, we prove the uniqueness of the weak solutions.for conciseness in presentation, we set = = 1 in the rest of this paper.1preliminariesthe following notations will be used in this paper.lp () = lp (, rd ), hm () = hm (, rd ), c () = c (, rd ), v = u c () : x u = 0,0 0 00h = u l2 () : x u = 0, u v| = 0, v = u h1 () : x u = 0, vm = v hm (), where v is dense in h, v and vm . we also use the notations: s := s1 ; id rdd denotes the unit matrix. a b (or a b) denotes a is continuously (or compactly) embedded in b.f ( or ) f in a denotes a sequence f0 a converges strongly (weakly or weaklystar) to f in a as 0. c(a, b, ) denotes a constant only dependent on a, b, .to tackle the coupling term in (0.1), we need the following lemma of integration-by-parts. lemma 1. (10, lemma 1.1 or 17, appendix ii) let f w 1,1 (sd1 ) and x rdd be a constantmatrix with tr(x) = 0. thensd1(id n n)xn n f dn =sd1(dn n id) f : xdn.dene z(s) := s(logs 1) + 1, s 0, ) and some cut-o functions below. these cut-o functions will be used in the approximate problem, the entropy estimate and the l2 estimatein section 3.denition 1. let l 1. denee l (s) := 0, if s 0,s, if 0 s l,l, if s l;zl (s) := s(lns 1) + 1, 0 s l,s l 2 2s2gl (s) := 2l + s(lnl 1) + 1, s l;l 22 ,s l,2 + l(s l),s l.with some elementary computations, one could verify the following properties (also seebarrett and suli 8-9 for some of them). lemma 2. let l 1. thene l c0,1 (r); gl c1,1 (r); zl c2,1 (r+ ) c(0, ).(1.1)s2(gl ) (s) = e l (s), gl (s) 2 , s 0, ).(1.2)zl (s) z(s), s 0, ).(1.3) (zl ) (s) = e l (s)1 s1 , s r+ .(1.4) (zl ) (s + ) 1 , (0, 1), s 0, ).(1.5)ls 0, ), lim e l (s) = s. (1.6)2zl (e l (s) + ) +2+ z(s + ), (0, 1), s 0, ).(1.7)the global weak solutions to the 2d stokes-type model are dened as below.denition 2. suppose f0 l1 ( s) and f0 0 a.e. on s. a pair of measurable functions(u, f ) is called a global weak solution to the 2d stokes-type model with boundary conditions(0.6) and initial condition (0.15) ifu l2 (0, ; v), f 0 a.e. on 0, ) s, (1.8)f l (0, ; l1 ( s), f l2 (0, ; h1 ( s),(1.9)0and for any v c (0, ) ) with x v = 0, x u : x vdxdt = (2n n id) f : x vdndxdt f e2 vdndxdt; (1.10)0 0s0 s0for any c (0, ) s), 0sf t dndxdt 0s(u f ) x dndxdt +0sn f n dndxdt =(id n n)x un f n dndxdt0s (id + n n)(e2 f + x f ) x dndxdt +f0 (x, n)(0, x, n)dndx.(1.11)0ssremark 1. it follows from gagliardo-nirenberg inequality that f f 1/2 f 1/2 l4 (;l2 (s) c h1 (;l2 (s) l2 (;l2 (s)and hence by hlder inequality that f 1/2c f f 1/2.(1.12) l4 (0,;l4 (;l2 (s) l2 (0,;h1 (s) l (0,;l2 (s)this and (1.9) yield f l2 (0, ) ; l1 (s). therefore x un f related integral makes sense.2existence of global weak solutionin this section, we apply barrett and sli 8-9s cut-o techniques with some improve- ments and follow the usual procedure in proving the existence of global weak solution.first we use a semi-implicit scheme to construct a sequence of approximate solutions, where the leray-schauder xed point theorem and cut-o techniques are used. in the construction of approximate solutions, our cut-o function is motivated by but dierent from that of barrett and sli 8-9. first barrett and sli 8-9 used a cut-o only from top by l 1, then theyused another cut-o from below by 0. they established the uniform estimates for andtook the limit 0. it seems that their whole process is quite involved. however, we used acut-o function by chopping o from above by l 1 and from below by 0 for the drag-term (seedenition 1). this single cut-o function is sucient for the proof of existence for approximate solutions.then we use compactness to show that these constructed approximate solutions converge to a weak solution. in order to apply the time-space compactness theorems with assumptions on derivatives (such as aubin-lions-simon lemma, see 19, theorem 5; dubinski lemma, see 20, theorem 1), the traditional rothe method for evolutionary pdes (see 21 and 22) is necessary and requires the construction of linear interpolation functions (also known as rothe functions). however, the approach of the rothe functions is fairly indirect and sometimes tedious, requiring more estimates and sometimes even more regularity assumptions on the initial data. in contrast, our approach is to apply theorem 1 in dreher-jngel 23, a time- space compactness theorem with simple piecewise-constant functions of t, instead of the more complicated rothe functions.now we state our main result.theorem 2.1. suppose f0 l2 ( s) and f0 0 a.e. on s. then the initial-boundaryproblem of 2d stokes-type model has a global weak solution (u, f ) which satisesu l (0, ; v) l2(0, ; v2 ),(2.1)locloclocf l (0, ; l2 ( s) l2(0, ; h1 ( s),(2.2)loclocand for a.e. t 0, ),f h1 (0, ; (h1 ( s) ); (2.3)s tz( f (t)dndx +0x u(s)l2 () d s + 222 t ( x 02( ) f s+ l2(s) n 2 )( ) f sd s l2 (s)sz( f0 )dndx + c f0 l1 (s) .(2.4)2.1approximate problemin the construction of the approximate problem, a cut-o function chopping o above by some l 1 and chopping o below by 0 is used to ensure the boundedness of the linear functional (2.9) for the discrete fokker-planck equation required by the lax-milgram theorem, and the boundedness estimates for the existence of xed-point solutions needed by the leray- schauder xed point theorem. using this eective cut-o, we obtain the existence of weakmsolutions in v h1for the approximate problem, then by applying the standard method forthe resulting elliptic equation we get the nonnegativity of approximate distribution functions.for any xed 0 1 and for any k n, given f k1 , the approximate problem for 2dstokes-type model with cut-o readsx uk : x vdx = s(2n n id) f k : x vdndx sf k e2 vdndx, v v; (2.5)sf k f k1dndx s(uk f k ) x dndx +sn f k n dndx=s1 (id n n)x uk n e 4 ( f k ) n dndx 1 s(id + n n)e2 e4 ( f k ) + x f k x dndx, h1 ( s),(2.6)1.in which e 4 is the cut-o function given by denition 1.denition 3.z := f l2 ( s) : f 0 a.e.on s(2.7)proposition 1. let f k1 z. then there exists (uk , f k ) v (z h1 ( s) which solves(2.5)-(2.6).proof. step 1. let f l2 ( s). we claim that there exists a unique element u v such thata(u, v) = a( f)(v), v v,(2.8)where a(u, v) = x u : x vdx, u, v v anda( f)(v) = s(2n n id) f : x vdndx sfe2 vdndx, v v.then thanks to poincar inequality and s (2n n id) fdnl2 () c f 2, we havel (s)that a(, ) is a bounded, coercive bilinear functional on v and a( f) v . hence by lax-milgramtheorem, we nish the proof of step 1.step 2. we prove that for suchf l2 ( s) and solution u v in (2.8), there exists aunique element f h1 ( s) such thatb(u)( f , ) = b( f, u)(), h1 ( s),(2.9)whereb(u)( f , ) =sf dndx + s(id + n n)x f x + n f n dndx (u f ) x dndx, f , h1 ( s);b(