Hindawi Function Space Volume 2017, Article ID 7916730, 8 page http://doi.org/10.1155/2017/7916730 Reearch Article A New Kind of Weak Solution of Non-Newtonian Fluid Equation Huahui Zhan 1 and Bifen Xu 2 1 School of Applied Mathematic, Xiamen Univerity of Technology, Xiamen 361024, China 2 School of Teacher Education, Jimei Univerity, Xiamen 361021, China Correpondence hould be addreed to Huahui Zhan; huahuizhan@163.com Received 5 April 2017; Accepted 15 June 2017; Publihed 13 July 2017 Academic Editor: Alberto Fiorenza Copyright 2017 Huahui Zhan and Bifen Xu. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. If the non-newtonian fluid equation with a diffuion coefficient i degenerate on the boundary, the weak olution lack the regularity to define the trace on the boundary. By introducing a new kind of weak olution, the tability of the olution i etablihed without any boundary condition. 1. Introduction and the Main Reult The quailinear parabolic equation u div (a (x) u p 2 u) b i (x) D i u+c(x, t) u =f(x, t), (x, t) Q T = (0, T) come from a hot of applied field uch a the theory of non-newtonian fluid, the tudy of water infiltration through porou media, and combution theory; one can refer to [1 4] and the reference therein. Here p>1, D i = / x i, a C(), R N i a bounded domain with the appropriately mooth boundary.ifa(x)>c>0, the equation with the type of (1) have been extenively tudied; one can refer to [5 7] and the reference therein. If a(x) 0, one want to obtain the well-poedne of the equation; the initial value (1) u (x, 0) =u 0 (x), x (2) i invariably impoed. But the boundary value condition u (x, t) =0, (x, t) (0, T) (3) may be overdetermined. Yin and Wang [8] made a more important devoting to the problem; they claified the boundary into three part: the nondegenerate boundary, the weakly degenerate boundary, and the trongly degenerate boundary, by mean of a reaonable integral decription. The boundary value condition hould be upplemented definitely on the nondegenerate boundary and the weakly degenerate boundary. On the trongly degenerate boundary, they formulated a new approach to precribe the boundary value condition rather than defining the Fichera function a treating the linear cae. Moreover, they formulated the boundary value condition on thi trongly degenerate boundary in a much weak ene ince the regularity of the olution i much weaker nearthiboundary.inaword,inteadofthewholeboundary condition (3), only a partial boundary condition u (x, t) =0, (x, t) Σ p (0, T) (4) iimpoedin[8],whereσ p. In our paper, for implim, we aume that a(x), b i (x),and c(x, t) are C 1 function, and a (x) >0, a (x) =0, x, x ; the equation i degenerate on the boundary. In our previou work [9, 10], we have hown that uch degeneracy may reult in the fact that the weak olution of the equation lack the regularity to define the trace on the boundary. Accordingly, how to contruct a uitable function, which i independent of the boundary value condition, to obtain the tability of theweakolution,becomeformidable.themainaimofthe paper i to olve the correponding problem by introducing a new kind of the weak olution. (5)
2 Function Space Definition 1. Function u(x, t) i aid to be a weak olution of (1) with the initial value (2), if u L (Q T ), u t L 2 (Q T ), a (x) u p L 1 (Q T ), and for any function g() C 1 (R), g(0) = 0, φ 1 C 1 0 (), φ 2 L (0, T; W 1,p loc ()), Q T [u t g(φ 1 φ 2 )+a(x) u p 2 u g (φ 1 φ 2 ) +u(b ixi (x) g(φ 1 φ 2 )+b i (x) g xi (φ 1 φ 2 )) c(x, t) ug (φ 1 φ 2 )+f(x, t) g(φ 1 φ 2 )] =0. Theinitialvalueiatifiedintheeneofthat (6) (7) t 0 u (x, t) u 0 (x) dx=0. (8) The exitence of the olution can be proved in a imilar wayathatin[8];weomitthedetailhere.inourpaper,we mainly are concerned about the tability of the weak olution without any boundary value condition. Theorem 2. Let u, V betwoweakolutionof(1)withtheinitial value u 0 (x), V 0 (x),repectively;uppoep>1and If then b i (x) ca(x). (9) a (p 1) (x) dx c, (10) u (x, t) V (x, t) dx c u 0 (x) V 0 (x) dx, i true without any boundary value condition. a.e.t [0, T) (11) Theorem 3. Let u, V be two nonnegative olution of (1) with the initial value u 0, V 0,repectively.If1<p 2and a 1/(p 1) (x) dx<, (12) then the tability of the weak olution i true in the ene of (11). Let u give a comparion between Theorem 2 and 3. To ee that, we pecially aume that =B R (0) ={x R N : x <R}, a (x) = (R x ) α +, α > 0. (13) Thenitieaytoknowthatifα<1/(p 1), then condition (10) i atified; while α<p 1, then condition (12) i true. Thu if 1<p<2,then 1 >p 1, (14) p 1 which implie that when R 1, R x <1, a (p 1) (x) dx < a 1/(p 1) (x) dx. (15) In thi cae, Theorem 2 cannot include Theorem 3; Theorem 3 ha it independent ene. Certainly, if p 2, Theorem2ha it ole important ignificance. At the ame time, intead of condition (10) (or (12)), we have the following reult in the tability or the local tability. Theorem 4. Let u, V be two nonnegative olution of (1) with the initial value u 0, V 0,repectively.Ifp>1, a(x) atifie (12), and, for mall enough >0, u(x) and V(x) atify that 1 ( a (x) u p (p 1)/p c, \ 1 ( a (x) V p (p 1)/p c, \ then tability (11) i true. Here ={x :a(x)>}. (16) Theorem 5. Let u, V be two olution of (1) with the differential initial value u 0 (x), V 0 (x), repectively. Then there exit a poitive contant β 1uch that a β u (x, t) V (x, t) 2 dx c a β u 0 (x) V 0 (x) 2 dx. In particular, for any mall enough contant δ>0, δ u (x, t) V (x, t) 2 dx (17) (18) cδ β u 0 (x) V 0 (x) 2 dx. If u 0 = V 0, by the arbitrarine of δ, one can ee that u(x, t) = V(x, t), a.e.(x, t) Q T ; the uniquene of the olution i true. We have ued ome technique in [9]. But there are many eential improvement in our paper. The main reult of my previou work [9] were etablihed on the aumption of that u (x) cd(x), (19) V (x) cd(x), where d(x) i the ditance function from the boundary. Condition(19)imuchtrongerthantheuualhomogeneou boundary value condition (3), o the concluion in [9] are not perfect. But in my new paper, we have introduced the new kind of the weak olution (Definition 1); alo we can etablih the tability of the weak olution without any boundary value condition.
Function Space 3 2. The Proof of Theorem 2 Since a(x) c in,we have For mall η>0,let S η () = h η (τ) dτ, 0 h η () = 2 η Obviouly h η () C(R),and (1 η ). + h η () 0, (20) a β (u V) S η (aβ (u V))( u p 2 u V p 2 V)dx = a (p 1)/p a β (u V) S {:a β η (aβ (u V)) u V <η} a (p 1)/p ( u p 2 u V p 2 V)dx (24) h η () 1, S η () 1; S η () = gn, η 0 (21) ( a (p 1)/p a β (u V) {:a β u V <η} S 1/p p η (aβ (u V)) ( a (x) {:a β u V <η} η 0 S η () =0. ( u p + V p (p 1)/p ). ProofofTheorem2.Let u, V be two olution of (1) with the initial value u 0 (x), V 0 (x).wecanchooes η (a β (u V)) a the tet function. Then S η (a β (u V)) Thu dx + a β+1 (x) ( u p 2 u V p 2 V) (u V) S η (aβ (u V))dx + a (x) ( u p 2 u V p 2 V) a β (u V) S η (aβ (u V))dx + D i (b i (x)) (u V) S η (a β (u V)) +a β b i (x)(u V) (u V) xi S η (aβ (u V))dx + b i (x)(u V) a β x i (u V) S η (aβ (u V))dx + c (x, t)(u V) S η (a β (u V))dxdt =0. S η (a β (u V)) η 0 dx = d dt u V 1, a β+1 (x) ( u p 2 u V p 2 V) (u V) S η (aβ (u V))dx 0. (22) (23) If {x : u V =0}ha 0 meaure, ince conequently a (p 1) (x) dx<, (25) a (p 1)/p a β (u V) S p {:a β η (aβ (u V)) u V <η} dx c, ( a (x) ( u p + V p (p 1)/p ) η 0 {:a β u V <η} =( a (x) ( u p + V p (p 1)/p ) {: u V =0} =0. If {x : u V =0}ha a poitive meaure, obviouly, ( η 0 a (p 1)/p a β (u V) {:a β u V <η} S 1/p p η (aβ (u V)) =( a (p 1)/p a β (u V) {: u V =0} S 1/p p η (aβ (u V)) =0. (26) (27) By (21) and condition (10), uing the Lebegue dominated convergence theorem, in both cae, we have η 0 a β (u V) S η (aβ (u V)) ( u p 2 u V p 2 V)dx =0. (28)
4 Function Space Then While, by (9), b i (x) ca(x), b i (x)(u V) a β x i (u V) S η (aβ (u V))dx b i (x)(u V) S η (aβ (u V)) a β (u V) aβ x i a β (u V) a β a β (u V) dx. dx c S η (aβ (u V)) b i (x) a dx c a S η (aβ (u V)) η 0 b i (x)(u V) a β x i (u V) S η (aβ (u V))dx =0. Moreover, by b i (x) ca(x), b i (x) a β (x)(u V) S η (aβ (u V)) (u V) xi dx c( a (x) ( u p + V p 1/p ) ( aβ (x)(u V) S p/(p 1) (p 1)/p η (aβ (u V)) ). (29) (30) (31) Corollary 6. Let u, V be two weak olution of (1) with the initial value u 0 (x), V 0 (x), repectively.if(9)itrueanditi uppoed that then the tability u p 1 dx c, V p 1 dx c. u (x, t) V (x, t) dx c u 0 (x) V 0 (x) dx, i true without any boundary value condition. a.e.t [0, T) (35) (36) Proof. If (35) i true, then (30) i true by (21). Thu the corollarycanbeprovedinaimilarwayathatoftheorem2 3. The Proof of Theorem 3 and 4 Proof of Theorem 3. By Definition 1, for any φ 1 C0 1(), φ 2 L (0, T; W 1,p loc ()),wehave [ (φ Q T 1 φ 2 ) +a(x) ( u p 2 u V p 2 V) (φ 1 φ 2 ) + (u V) (b ixi (x) (φ 1 φ 2 )+b i (x) (φ 1 φ 2 ) xi ) c(x, t)(u V) (φ 1 φ 2 )] = 0. (37) Therefore, we have η 0 b i (x)(u V) S η (aβ (u V)) (u V) xi a β (x) dx =0. η 0 D i (b i (x)) (u V) S η (a β (u V))+ c (x, t) (u V) S η (a β (u V))dx c u V 1. Now, let η 0in (22). Then It implie that (32) d dt u V 1 c u V 1. (33) u (x, t) V (x, t) dx c(t) u 0 V 0 dx, Theorem 2 i proved. t [0, T). (34) For a mall poitive contant >0, ={x :a(x)> } a before, let { 1, if x, φ (x) = { 1 a (x), { if x \ (38). Now, we can chooe φ 1 =φ (x), φ 2 =χ [τ,] S η (u V),and integrate them over Q T ; accordingly, φ (x) S η (u V) τ + φ (x) τ a(x) ( u p 2 u V 2 V) (u V) S η (u V) + a (x) ( u p 2 u τ V p 2 V) φ (x) S η (u V) + (u V)(b ixi (x) φ (x) S η (u V) τ +b i (x) (φ (x) S η (u V)) xi )dxdt c (x, t)(u V) φ (x) S η (u V) = 0. τ (39)
Function Space 5 Clearly, φ (x) a (x) ( u p 2 u V p 2 V) (u V) (40) S η (u V) dx 0, a (x) ( u p 2 u V 2 V) φ (x) S η (u V) dx \ a (x) ( u p 2 u V 2 V) φ (x) S η (u V) dx \ a (x) ( u p 2 u V 2 V) φ (x) dx c [ \ a (x) u p 1 a dx (41) Thu a (x) u c( \ p (p 1)/p a (x) V +c( \ p (p 1)/p. (44) 0 a (x) ( u p 2 u V p 2 V) (45) φ (x) S η (u V) dx =0. There i one more point that I hould touch on i that, by that a 1/(p 1) (x)dx < c, uing (21) and the Lebegue dominated convergence theorem, η 0 φ b i (x)(u V) S η (u V)(u V) x i dx + \ a (x) V p 1 a dx]. Since 1<p 2, a c, then \ a p dx c c p 1, (42) c ( a (x) a p 1/p \ c ( a p 1/p c. \ By (41)-(43), uing the Holder inequality, a (x) ( u p 2 u V 2 V) φ (x) S η (u V) dx (43) η 0 ( a 1/p S η (u V) p/(p 1) (p 1)/p (u V) ( a (x) ( u p + V p 1/p ) =0, 0 φ xi b i (x)(u V) S η (u V) dx c 0 c a dx \ 0 ( a (x) \ a p 1/p a ( \ 1/(p 1) (p 1)/p (x) =0, (46) c [ \ a (x) u p 1 a dx + \ a (x) V p 1 a dx] c ( a a p 1/p \ a (x) u ( \ p (p 1)/p + c ( a (x) a p 1/p \ a (x) V ( \ p (p 1)/p by (43) and a 1/(p 1) (x)dx < c,while i obviouly true. By (46)-(47), η 0 (u V) b ixi (x) φ (x) S η (u V) dx u V dx η 0 (u V)(b ixi (x) φ (x) S η (u V) +b i (x) (φ (x) S η (u V)) xi )dx u V dx. (47) (48)
6 Function Space At lat, η 0 0 τ φ (x) S η (u V) (49) d = τ dt u V L 1 () dt. Now, after letting 0,letη 0in (37). Then, uing (40), (45) (49), and by the Gronwall inequality, we have u (x, t) V (x, t) dx c u 0 V 0 dx. (50) Proof of Theorem 4. In the firt place, imilar to the proof of Theorem 3, we have (39) (41). There i one more point that we hould touch on that ince u(x) and V(x) atify (16), uing the Holder inequality, we have a (x) ( u p 2 u V 2 V) Q T (u V) [b ixi (φ 1 φ 2 )+b i (x) (φ 1 φ 2 ) xi ]dxdt + Q T (u V) c (x, t) (φ 1 φ 2 )dxdt. In particular, we chooe φ 1 =a β, φ 2 =χ [τ,] (u V), (52) (53) where χ [τ,] i the characteritic function on [τ, ] and the contant β 1.DenotingQ τ = [τ,],then Q τ a (x) ( u p 2 u V p 2 V) [(u V) a β ]dxdt φ (x) S η (u V) dx c ( a a p 1/p \ = Q τ a 1+β ( u p 2 u V p 2 V) (u V) + Q τ a (x) (54) a (x) u ( \ p (p 1)/p + c ( a (x) a p 1/p \ a (x) V ( \ p (p 1)/p a a c( \ p 1/p a (x) a +c( \ p 1/p, (51) whichgoetozeroa 0ince that a(x) C 1 ().Thuwe have (45) too. Lat but not the leat, ince a 1/(p 1) (x)dx <, imilar to the proof of Theorem 3, we have (46) (49). So, a the proof of Theorem 3, we know that tability (11) i true. 4. The Local Stability ProofofTheorem5.Let u, V be two olution of (1) with the initial value u 0 (x), V 0 (x), repectively. From the definition of the weak olution, if g() =, foranyφ 1 C0 1(), φ 2 L (0, T; W 1,p loc ()), φ 1 φ 2 = a (x) Q T Q T ( u p 2 u V p 2 V) (φ 1 φ 2 )dxdt Clearly, ( u p 2 u V p 2 V) (u V) a β. Q τ a 1+β ( u p 2 u V p 2 V) (u V) 0. (55) For the econd term on the right-hand ide of (54), ince a c, (u V) a (x) Q τ ( u p 2 u V p 2 V) a β u Q τ V a (x) ( u p 1 + V p 1 ) aβ c( a (x) ( u p + V p (p 1)/p )dxdt) τ p ( a (x) τ aβ u V p 1/p ) c( a (x) ( u p + V p (p 1)/p )dxdt) τ ( a 1+p(β 1) u V p 1/p ) τ c( a 1+p(β 1) u V p 1/p ). τ (56)
Function Space 7 Now, ince β 1implie 1+p(β 1) β,wehave (u V) a( u Q p 2 u V p 2 V) a β τ If p 2, c( a β u V p 1/p ). τ ( a β u V p 1/p ) τ c( a β u V 2 1/p ). τ If 1<p<2,byHolder inequality, ( a β u V p 1/p ) τ c( a β u V 2 1/2 ). τ Moreover, by β 1, u, V L (Q T ),weeailyeethat b i (x)(u V)[(u V) a Q β ] xi τ (57) (58) (59) = a β [u (x, ) V (x, )] 2 dx a β [u (x, τ) V (x, τ)] 2 dx. By (54) (61), we let 0in (52). Then ρ β [u (x, ) V (x, )] 2 dx a β [u (x, τ) V (x, τ)] 2 dx c( ρ β u (x, t) V (x, t) 2 q ), 0 where q<1.by(62),weeailyhowthat a β u (x, ) V (x, ) 2 dx a β u (x, τ) V (x, τ) 2 dx. Thu, by the arbitrarine of τ,wehave (61) (62) (63) a β u (x, ) V (x, ) 2 dx a β u 0 V 0 2 dx. (64) The proof i complete. = b i (x)(u V) Q 2 a β x i τ + Q b i (x)(u V)(u V) xi a β c u V a τ β 1 dx 1/p +( a (β (1/p))p u V p ) τ ( a (x) ( u p + V p 1/p )dxdt) τ c( a β u V 2 1/2 ) τ dt)1/p +c( a β u V 2 dx. τ At lat, it i eaily to deduce that (b ixi (x)(u V) Q 2 +c(x, t)(u V) 2 )a β τ c( a β u V 2 1/2 ), τ (u V) a β Q τ (60) Conflict of Interet The author declare that they have no conflict of interet. Acknowledgment The paper i upported by Natural Science Foundation of Fujian Province (no. 2015J01592), upported by Science Foundation of Xiamen Univerity of Technology, China. Reference [1] R. Ari, The Mathematical Theory of Diffuion and Reaction in PermeableCatalyt,I,II,Clarendon,Oxford,UK,1975. [2] E. C. Child, An Introduction to the Phyical Bai of Soil Water Phenomena, Wiley, London, UK, 1969. [3] Z.Wu,J.Zhao,J.Yin,andF.Li,Nonlinear Diffuion Equation, World Scientific Publihing, Toh Tuck Link, Singapore, 2001. [4] E. DiBenedetto, Degenerate Parabolic Equation, Springer, New York, NY, USA, 1993. [5] E. Nabana, Uniquene for poitive olution of p-laplacian problem in an annulu, Annale de la Faculté de Science de Touloue,vol.8,no.1,pp.143 154,1999. [6] K.Lee,A.Petroyan,andJ.L.Vazquez, Large-timegeometric propertie of olution of the evolution p-laplacian equation, Differential Equation, vol.229,no.2,pp.389 411, 2006. [7] J. N. Zhao, Exitence and nonexitence of olution for u t div( u p 2 u) = f( u, u, x, t), Mathematical Analyi and Application,vol.172,no.1,pp.130 146,1993.
8 Function Space [8] J. Yin and C. Wang, Evolutionary weighted p-laplacian with boundary degeneracy, Differential Equation, vol. 237, no. 2, pp. 421 445, 2007. [9] H. Zhan, The tability of the olution of an equation related to the p-laplacian with degeneracy on the boundary, Boundary Value Problem,vol.2016,article178,2016. [10] H. Zhan, The olution of a hyperbolic-parabolic mixed type equation on half-pace domain, Differential Equation,vol.259,no.4,pp.1449 1481,2015.
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