Global existence and uniqueness of solutions for the equations of third grade fluids

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Global existence and uniqueness of solutions for the equations of third grade fluids Valentina Busuioca and Dragos¸ Iftimieb aSection de mathematiques, Institut Bernoulli, Ecole Polytechnique Federale de Lausanne. CH–1015 Lausanne. Switzerland. Email: bIRMAR, Universite de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France. Email: Abstract. We consider the equations governing the motion of third grade fluids in Rn, n = 2, 3. We show global existence of solutions without any smallness assumption, by assuming only that the initial velocity belongs to the Sobolev space H2. The uniqueness of such solutions is also proven in dimension two. Introduction The fluids of grade n, introduced by Rivlin and Ericksen [1], are the fluids for which the stress tensor is a polynomial of degree n in the first n Rivlin-Ericksen tensors defined recursively by A1 = A = 2D, Ak+1 = d dt Ak + L tAk + AkL, where ddt = ∂t + u · ? denotes the material derivative and L = (∂jui)i,j, L t = (∂iuj)i,j, D = 1 2 (∂iuj + ∂jui)i,j. For third grade fluids, physical considerations were taken into account by Fosdick and Rajagopal [2] in order to obtain the following form for the constitutive law: T = ?pI + ?A1 + ?1A2 + ?2A 2 1 + ?|A1| 2A1,

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Global existence and uniqueness of solutions for the equations of third grade ﬂuids Valentina Busuioc a and Drago¸sIftimie b ´ a Sectiondemathe´matiques,InstitutBernoulli,EcolePolytechniqueFe´de´raledeLausanne. CH–1015 Lausanne. Switzerland. Email: adriana.busuioc@epﬂ.ch b IRMAR,Universit´edeRennes1,CampusdeBeaulieu, 35042 Rennes cedex, France. Email: iftimie@maths.univ-rennes1.fr

Abstract. We consider the equations governing the motion of third grade ﬂuids in R n , n = 2 , 3. We show global existence of solutions without any smallness assumption, by assuming only that the initial velocity belongs to the Sobolev space H 2 The uniqueness of such solutions is also proven in . dimension two.

Introduction The ﬂuids of grade n , introduced by Rivlin and Ericksen [1], are the ﬂuids for which the stress tensor is a polynomial of degree n in the ﬁrst n Rivlin-Ericksen tensors deﬁned recursively by A 1 = A = 2 D A k +1 =ddt A k + L t A k + A k L where ddt = ∂ t + u ∙ r denotes the material derivative and L = ( ∂ j u i ) i,j  L t = ( ∂ i u j ) i,j  D =12( ∂ i u j + ∂ j u i ) i,j . For third grade ﬂuids, physical considerations were taken into account by Fosdick and Rajagopal [2] in order to obtain the following form for the constitutive law: T = − pI + νA 1 + α 1 A 2 + α 2 A 12 + β | A 1 | 2 A 1  which, introduced in the equation of conservation of momentum leads to the following equation: (1) ∂ t ( u − α 1 4 u ) − ν 4 u + u ∙ r u − α 1 div( u ∙ r A + L t A + AL ) − α 2 div A 2 − β div( | A | 2 A ) = f − r p div u = 0 . Moreover, the coeﬃcients ν , α 1 , α 2 and β must satisfy the following hypotheses: ν ≥ 0  α 1 > 0  β ≥ 0 and | α 1 + α 2 | ≤ (24 νβ ) 1 / 2 . 1

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