STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS

Transcription

1 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS LARS DIENING, MICHAEL RŮŽIČKA Abstract We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p structure p = corresponds to the Newtonian case) We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from p > 5 3 see [6]) to p > 7 Moreover, for 5 7 < p we improve the regularity of the velocity field and show that 5 u C[0, T ], W,6p ) ε div )) for all ε > 0 Within this class of regularity, we prove uniqueness for all p > 7 5 We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as 7 < inf pt, x) sup pt, x) 5 Introduction In this paper we show existence of strong solutions to the following system describing the motion of generalized Newtonian fluids : ) t u divsdu)) + u )u + π = f, on I, div u = 0, on I, u0) = u 0, on We consider the three dimensional space periodic case, ie let be the three dimensional torus and let I = [0, T ] with T > 0 The functions u, π, and f denote the velocity, the pressure, and the external force The function u 0 is a given initial value By Du we denote the symmetric part of the gradient u, ie Du = u + u) ) We assume that the extra stress S is induced by a p potential F as defined below Standard examples for S are SDu) + Du ) p Du, SDu) + Du ) p Du, with < p < We compensate the missing boundary conditions by restricting the solutions to ones with mean value zero This ensures that the Poincaré inequality remains valid Mathematical results for the system ) can be found in [3, 5, 9,, 6, 8, 4] We want to mention that the existence of global strong solutions to the problem 99 Mathematics Subject Classification 76A05, 35D0, 35D05, 46B70 Key words and phrases Non Newtonian fluid flow, Regularity of generalized solutions of PDE, Existence of generalized solutions of PDE, Electrorheological fluids, Parabolic interpolation We refer to [6] for an extensive discussion on such fluids In the monograph [6] the reader can find a detailed discussion of the problem cf [8, 3, 0,, 7, 7] for more recent results)

2 LARS DIENING, MICHAEL RŮŽIČKA ), ie u L I, V p ) L I, W, )), t u L I, L )), satisfies for almost all t I, ) t u, ϕ + SDu), Dϕ + u u, ϕ = f, ϕ ϕ V p, is established for p 3d+ d+ cf [6, 9, ]), where d is the dimension of The existence of a local in time strong solution for arbitrary data and the existence of a global strong solution for small data is proved in the case p > 3d 4 d cf [8, 6]) Moreover, in [6] it has been shown that for p > 5 3 there exists strong solutions for short time and large data 3) u L I,L )) + t u L I,L )) C, where I Φ u) + Du )p u dx solution further satisfies 4) I Φ u) L I) C, t u L I,L )) + J Φ u) L I) C, It has been shown in [0] that this where J Φ u) + Du )p t u dx In this paper 3 we prove the short time existence of strong solutions for large data for all 7 5 < p Moreover, we show that the solution satisfies additionally to 3) and 4) 5) I Φ u) L 5p 6 p I) C We refer to theorem 7 for the precise statement of this result From 3) and 4) we deduce that u C[0, T ], W,6p ) ε div )) for all ε > 0, especially u CI, W, 5, 5 div )) We will show that every weak solution v CI, Wdiv )) of ) satisfies v = u, especially u is unique within its class of regularity see theorem 9) System ) is usually studied under the assumption that p is constant, with < p < Motivated by the model for the motion of electrorheological fluids in [, ] which has been further studied in [5], we are also interested in the case, where p is a function in space and time Electrorheological fluids are a special type of smart fluids which change their material properties due to the application of an electric field In the model in [] p is not a constant but a function of the electric field E, ie p = p E ) The electric field itself is a solution to the quasi static Maxwell equations and is not influenced by the motion of the fluid Therefore it is possible to consider ) for a given function p : I, ) In this case we speak of a time and space dependent potential Due to the nature of the Maxwell equations it is reasonable to consider smooth p We define p := inf I p and p + := sup I p In [4] it has been proven that there exists a strong solution to ) for large time and data as long as p W, I ) and 4 5 < p p + < p The extra condition p + < p is due to the use of classical Sobolev spaces The reason is that the energy Du p dx cannot be fully expressed in terms of classical Sobolev spaces if p is non constant In that case the generalized Sobolev spaces 3 This paper is based on the PhD thesis [4] of L Diening 4 The case of Dirichlet boundary conditions is treated in [5]

3 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 3 W k,p ) provide the right setting See [, 5, 5] and the references therein for a definition of these spaces and applications to fluid mechanics In section 8 we generalize our results on short time existence to the case, where p is non constant, ie we prove short time existence of strong solutions for large data as long as 7 5 < p p + Moreover, we show that the solution satisfies 3) and 4), while 5) will be replaced by I Φ u) 5p 6 C L p I) See theorem and corollary for the precise statement As before we deduce that the solution is unique within its class of regularity cf theorem 3) Finally we also extend these results to the system governing the motion of electrorheological fluids The Potential and the Extra Stress We assumte that the extra stress tensor S is induced by a p potential In this section we give a definition of a p potential and derive basic properties of it We will consider only the case p constant The case p non constant will be covered by section 8 Since we are dealing with functions from R n n to R, we will distinguish the partial derivatives by i and jk, a single index means a partial derivative with respect to the i-th space coordinate, while a double index represents a partial derivative with respect to the j, k)-component of the underlying space of n n- matrices By we denote the space gradient, while n n denotes the matrix consisting of the partial derivatives with respect to the space of matrices In a few cases we use d i instead of i to indicate a total derivative Note that by B sym we denote the symmetric part of a matrix B R n n, ie B sym = B + B ) Further let Rsym n n be the subspace of Rn n consisting of the symmetric matrices Moreover we use C as a constant which is generic but does not depend on the ellipticity constants For the notation of the function spaces see section 4 We will assume that < p throughout the whole paper Definition Let < p and let F : R 0 R 0 be a convex function, which is C on R 0, such that F 0) = 0, F 0) = 0 Assume that the induced function Φ : R n n R 0, defined through ΦB) = F B sym ), satisfies 6) jk lm Φ)B)C jk C lm γ + B sym ) p C sym, 7) jklm n n Φ)B) γ + B sym ) p for all B, C R n n with constants γ, γ > 0 Such a function F, resp Φ, is called a p-potential and the corresponding constants γ, γ are called the ellipticity constants of F, resp Φ We define the extra stress S induced by F, resp Φ, by SB) := n n ΦB) = F B sym ) Bsym B sym for all B R n n \{0} We will see in remark that S can be continuously extended by S0) = 0 Note that SB) does only depend on the symmetric part of B

4 4 LARS DIENING, MICHAEL RŮŽIČKA Standard examples are F s) = s 0 + a ) p a da and F s) = Remark Observe that for all B R n n \ {0} jk Φ)B) = F B sym ) Bsym jk jk lm Φ)B) = F B sym ) B sym, δ sym ) jk,lm B sym Bsym jk Bsym lm B sym 3 s 0 + a) p a da + F B sym ) Bsym jk where δ sym jk,lm := δ jlδ km + δ jm δ kl ) Hence jk lm Φ)B)B jk B lm = F B sym ) B sym jklm So by 6) and 7) we conclude that for all B R n n \ {0} 8) γ + B sym ) p F B sym ) γ + B sym ) p B sym B sym lm B sym, Since F C R 0 ), this estimate also holds for B = 0 From the formula above for jk Φ)B), the continuity of F at zero with F 0) = 0, and the boundedness of / Bsym in R n n \ {0}, we deduce B sym jk 9) lim S jkb) = lim jkφ)b) = 0 B 0 B 0 Remark 3 Let B, C R n n Due to ΦB) = F B sym ), we have ΦB) = ΦB sym ), thus the jk lm Φ are symmetric in j, k and l, m and j, k), l, m) This implies that 0) jk lm Φ)B)C jk C lm = jk lm Φ)B sym )C sym Csym ) ) jklm jklm n n Φ)B) = n n Φ)B sym ), n nφ)b) = n nφ)b sym ) Thus it suffices to verify 6), 7) for all symmetric matrices Since later we will mostly deal with symmetric matrices, we will in some cases leave out the symmetrization of the matrices, ie we will use B instead of B sym and restrict the admitted matrices to the symmetric ones jk lm, As in [6], one can deduce from 6) and 7) the following properties of S Theorem 4 There exist constants c, c > 0 independent of γ and γ such that for all B, C R n n sym there holds S0) = 0, 3) 4) 5) S ij B) S ij C))B ij C ij ) c γ + B + C ) p B C, ij ij S ij B)B ij c γ + B ) p B, SB) SC) c γ + B + C ) p B C, SB) c γ + B ) p B

5 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 5 The same inequalities hold true for all B, C R n n if B and C is replaced on the right-hand side by B sym and C sym 3 Special Energies Later in our a priori estimates we will encounter the following two important expressions I Φ t, u) := αβ jk Φ) Dut) ) 6) r D αβ ut), r D jk ut), 7) r jkαβ J Φ t, u) := αβ jk Φ) Dut) ) t D αβ ut), t D jk ut), jkαβ where Φ is a p potential and u denotes a sufficiently smooth function over the space time cylinder The brackets, denote integration over the space domain These two expressions will arise when we are going to test the equation of motion with u, resp t u Since I Φ and J Φ are very similar, it is useful to introduce another functor G Φ by 8) G Φ t, w, v) := jkαβ αβ jk Φ)Dwt))D αβ vt), D jk vt), where w : I R d and v : I R d or v : I R d d ) are sufficiently smooth functions Mostly we will simply write I Φ u), J Φ u), and G Φ w, v) instead I Φ t, u), J Φ t, u), and G Φ t, w, v) We have 9) I Φ u) = G Φ u, u), J Φ u) = G Φ u, t u) Due to the properties of Φ we estimate 0) G Φ w, v) γ + Dw ) p Dv dx The expression + Dw ) will appear quite often in all the chapters, so it is very useful to introduce the shortcut ) As a consequence ) 3) Note that 4) Dw := + Dw ) I Φ u) C γ Du) p Du dx, J Φ u) C γ Du) p t Du dx j k u m = j D km u + k D mj u m D jk u which implies u 3 Du 6 u Thus, Du can always be replaced by u and vice versa) by increasing the multiplicative constant Closely connected to the quantities I Φ u) and J Φ u) is the function Du) p, which will be important when examining the regularity of solutions

6 6 LARS DIENING, MICHAEL RŮŽIČKA Lemma 5 Let Φ be a p potential Then there exists C > 0, such that for all sufficiently smooth) u and almost all times t I there holds ) p ) γ Dut) 5) C I Φt, u) t ) p ) γ Dut) 6) C J Φ t, u) Proof Observe that Du) ) p p 4 = p Du) 7) Djk u) D jk u) jk Raising this to the power of two and integrating over proves the first inequality If we replace in the calculations above by t, we get the result for J Φ u) In the following we will derive more useful estimates for G Φ w, v), I Φ u) and J Φ u): Lemma 6 Let Φ be a p potential Then for all sufficiently smooth) u and almost all times t I there holds 8) 9) γ ut) p p C I Φ t, u) + γ Dut) p p γ t Dut) p p C J Φt, u) + γ Dut) p p Proof Note that for all q [, ], a 0, b there holds 30) a q a b q + b q Indeed, there is nothing to prove if q =, so let q < In this case < q <, and Young s inequality gives Now 30) implies a q = a b q ) q b q)q ) Young a b q + b q u p Du) p u + Du) p Since in general u 3 Du see 4)) we deduce u p p Du) p u dx + Du p p 9 Du) p Du dx + Du p p ) C γ I Φ u) + Du p p The estimate for t Du follows analogously Lemma 7 Let Φ be a p potential Then for all sufficiently smooth) u and v, for all q, and almost every t I there holds: 3) where Dvt) q C γ γ G Φ t, w, v) ) p Dwt)) q, q q q = for q =

7 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 7 Proof Observe that q < and < q ) = q Further for q < Dv q q = Dw) p Dv ) q p)q Dw) dx Dw) p Dv dx = Dw) p Dv dx q q p)q Dw) p) Dw) q q q q This and 0) prove the lemma for q < The case q = is similar Note that this lemma is applicable to I Φ u) = G Φ u, u) and as well to J Φ u) = G Φ u, t u) Analogously we have Lemma 8 Let Φ be a p potential Then for all sufficiently smooth) u and v and for all q there holds: Du v) q C SDu) SDv), Du v) p p Du) + Dv) q, q where q q = for q = Proof Analogously to the proof of lemma 7 Du v) q q = Du + Dv) p Du v) ) q Du p)q + Dv) dx Du+ Dv) p Du v) dx q p)q Du+ Dv) 4) C γ SDu) SDv), Du v) q Du) p) + This proves the lemma for q < The case q = is similar q p) Dv) q q q Note that the following estimates for I Φ and J Φ will depend on the dimension of the underlying space, which is in our case three dimensional Lemma 9 For all sufficiently smooth) u and almost every t I there holds γ Dut) p 3p I C Φ t, u) + Dut) ) 3) p p Proof By lemma 5 and the embedding W, ) L 6 ) there holds γ Du p 3p = γ Du) p 6 C γ Du) p ) + Du) p ) C I Φ u) + γ Du p p) This proves the lemma

8 8 LARS DIENING, MICHAEL RŮŽIČKA Lemma 0 For all sufficiently smooth) u with u, = 0 and almost every t I there holds 33) 34) 35) ut) p C I, 3p Φ t, u) + ), p+ t ut) p C J, 3p Φ t, u) p p+ Proof From lemma 7 we deduce This implies Du 3p C I Φ u) Du) p 6p p+ p IΦ t, u) + ) p, CJ Φ t, u) + I Φ t, u) + ) C I Φ u) p Du 3p ) C I Φ u) p + Du 3p ) C I Φ u) p + C Du 3p, since u, = 0 p+ Du p 3p C I Φ u) + ) p+ Since u 3 Du see 4)) and u, = 0, we get u p C I, 3p Φ u) + ) p+ Analogously we can use lemma 7 to get t Du 3p C J Φ u) p Du) 6p p+ p C J Φ u) + C Du 3p p+ 33) C J Φ u) + C IΦ u) + ) p + IΦ u) ) p p C J Φ u) ) p, since u, = 0 ) p Now u, = 0 and Korn s inequality imply t u, 3p C t Du 3p C J Φ u) + IΦ u) ) p p, p+ p+ which proves 34) The rest is an application of Young s inequality 4 Galerkin Approximation - The Case 3 < p Let us introduce the spaces, which we will need later As before let be the three dimensional torus By L q ), q ), resp W k,q ), k,q ), we denote the classical Lebesgue and Sobolev spaces For a Banach space X we denote by L q [0, T ], X) the Bochner space with q integrability and values in X We will also make use of the space CI, X) of continuous functions with values in X Due to the constraint div u = 0 of incompressibility we introduce spaces of divergence-free functions For < q < and k N 0 let V := {ϕ C per) : div ϕ = 0, ϕ, = 0}, L q div ) := V, q), W k,q div ) := V, k,q),

9 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 9 where V, ) is the closure of V with respect to the given norm and f, g f g dx is the scalar product with respect to the space In order to prove theorem 7 we will use a Galerkin approximation and derive some a priori estimates for the approximative solutions u N In section 6 we will pass to the limit N showing the existence of a desired solution u of system ) Let {ω r } denote the set consisting of the eigenvectors of the Stokes operator denoted by A Let λ r be the corresponding eigenvalues and X N = span{ω,, ω N } Note that ω r, = 0 Define P N u = N r= u, ωr ω r Then 36) λ r ω r, u N = A ω r, u N = ω r, u N and P N : W s, X N, s, ) are uniformly continuous for all 0 s 3 See [6, 4] for a proof) Let us define u N t, x) = N r= cn r t)ωr x) and f N = P N f, where the coefficients c N r t) solve the Galerkin system for all r N) t u N, ω r + SDu N ), Dω r + u N )u N, ω r = f N, ω r, 37) u N 0) = P N u 0 We will show that this Galerkin approximation has a short time solution u N, which will converge to a solution of ) Since the matrix ω j, ω k with j, k =,, N is positive definite, the Galerkin system 37) can be rewritten as a system of ordinary differential equations This in turn fulfills the Carathéodory conditions and is therefore solvable locally in time, ie on a small time interval I = [0, T ) In theorem 7 we assume that f L I, W, )) and t f L I, L )) and thus f N = P N f L I, W, )) and t f N = P N t f) L I, L )) This implies c N r, tc N r, t cn r L I ) Thus u N, t u N, t un L I, X N ) Note that the norms may depend on N) To ensure solvability for large times at least for this finite dimensional problem we have to establish a first a priori estimate Since u N, t u N, t un L I, X N ), we can test the Galerkin system 37) with u N and get d t u N + SDuN ), Du N = f N, u N Note that u N )u N, u N = 0 due to div u N = 0 The coercivity of S see 4)), the continuity of P N on L ), and Gronwall s inequality imply This implies max [0,T ] un + T T Du N p p dt C f dt + u 0 CT, f, u 0) 0 c N r L I ) CT, f, u 0 ), r N As a consequence we can iterate Carathéodory s theorem to push the solvability of the Galerkin system 37) up to any fixed time interval I = [0, T ) Hence, independently of N 38) u L I,L )) + u Lp I,W,p )) C, where we have used Korn s inequality We got the first a priori estimate by using u N as a test function To derive our second a priori estimate we want to use Au N as a test function The special choice 0

10 0 LARS DIENING, MICHAEL RŮŽIČKA of base functions ω r ensures that we do not leave X N, the space of admissible test functions More explicitly we multiply the r-th equation of the Galerkin system 37) by λ r c N r and use 36) to obtain t u N, Au N ) SDu N ), DAu N u N )u N, Au N = f N, u N Due to the periodicity we have A =, so d t u N SDu N ), D u N u N )u N, u N = f N, u N Using partial integration and the properties of S we deduce cf [6]): un )u N, u N = 39) k u N i iu N j ku N j dx, 40) Thus we have 4) SDu N ), D u N = ijklm ijk ij kl Φ)Du N ) m D ij u N m D kl u N dx 4) C I Φ u N ) d t u N + c I Φ u N ) u N f N, u N, If p > 5 one can show that un 3 3 C ε u N p p un + εi Φu N ) see [6]), which enables us to apply Gronwall s inequality after absorbing εi Φ u N ) on the left hand side This would gives us a global estimate If p > 5 3 we can show that u N 3 3 C ε u N p p un R + ε I Φu N ) for some constant < R < and thereafter absorb εi Φ u N ) on the left hand side and apply a local version of Gronwall s inequality see lemma 4) Instead of using Gronwall s inequality it is also possible to divide the inequality by + u N ) R as was done in [6] and derive the same local estimates This in turn implies enough regularity for u N to justify all the later testing of the Galerkin system with t u N and t u N t Nevertheless we will not make use of these facts, since we are also interested in smaller values of p than 5 3 What we do is, we test immediately with tu N t to get in addition to 4) another estimate Then we will use the resulting two estimates at the same time to derive quite strong a priori estimates for u N for values up to p > 3 in this section and up to p > 7 5 in the next section Let us take the time derivative of the Galerkin system 37): t u N, ω r + t SDu N ), Dω r + t u N )u N ), ω r = t f N, ω r, for r N Since u N W, I, X n ), this makes sense and we can even test with t u N W, I, X n ): d t t u N + t SDu N )), t Du N + t u N )u N ), t u N = t f N, t u N Once again the second term on the left-hand side has a sign, namely t SDu N )), t Du N = ij kl Φ)Du N ) t D ij u N, t D kl u N ikl 4) c J Φ u N )

11 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS This yields 4) Recall that d t t u N + c J Φ u) t u N )u N ), t u N + t f N, t u N d t u N + c I Φu N ) u N f N, u N 43) By a first view we have gained nothing We have to control one more bad term, namely t u N )u N ), t u N, but we only got more information about the time derivative of u N But the critical term u N 3 3, which gave the lower bound for p has no time derivatives The next lemma shows that J Φ u N ) reveals indeed more information Lemma Let < q <, then for almost every t I ) d t q Dut) q q C JΦ t, u) q p Dut) q p 44) q p εj Φ t, u) + C ε Dut) q p, where Du + Du ) Proof Note that Hence t Du) q ) = q Du) q D jk u) t D jk u) d t Du q q) q = q Du) q t Du dx ) p Du) t Du Du) q p dx q C J Φ u) q p) Du q p, q p where Du q p Du) q p dx even if q p < The rest is an implication of Young s inequality This lemma enables us to produce d t Du N q q) on the left-hand side of 4) if we add C Du N q p q p to the right-hand side We have three critical terms to control: u N 3 3, t u N )u N ), t u N, C Du N q p q p The first and the second one will be easier to estimate for large q, but the third one for small q The problem now is to find the optimal choice for q We start by examining which values of q are needed for the first and the second term In the view of lemma 4, we will be able to control arbitrary powers of Du N q q and t u N Note that we will skip the index N of un to keep the notations simple Lemma Let q > 9 3p, then there exists a constant R = R p) > q, such that u 3 3 C ε Du R q + εi Φ u) + ε Proof If q 3, then there is nothing to prove, so assume q < 3 We can interpolate L 3 ) = [L q ), L 3p )] with ) = + 3 q 3p = 3 q)p qp ), = 3p q 3p q

12 LARS DIENING, MICHAEL RŮŽIČKA Therefore If 3 < p, there exists an δ > such that u 3 3 u 3 ) q u 3 3p u 3 3 C ε u 3 )δ q C ε u 3 )δ q + ε u p 3p + C ε u p, 3p p+ C ε u 3 )δ q + εc I Φ u) + ), where we have used lemma 0 So by Korn s inequality u 3 3 C 3 )δ ε Du q + ε I Φ u) + ε We still have to verify 3 < p, but this is equivalent to 33 q)p 3p q which holds due to the assumptions on q < p 9 3p Lemma 3 Let q > 9 3p, then there exist constants R = R p) > and R 3 = R 3 p) > q such that < q, t u )u, t u εj Φ u) + C ε t u R R3 + Du q + ) Proof Note that lemma 7 q 45) t Du q p+q Furthermore W, p+q ) L 6q 6 3p+q q q p+q ) implies C J Φ u) Du) p q p C J Φ u) p Du q 6q we can use the interpolation L q This and Korn s inequality implies 6 3p+q ) Since 9 3p < q is equivalent to q q ) = [L ), L 6 3p+q )] 6q q < t u )u, t u t u q u q q C t u ) t u 6q u q 6 3p+q C t u ) t u q u q p+q J Φ u) p Du 45) C t u ) q ) u q εj Φ u) + C ε t u R R3 + Du q + ) It is indeed interesting that both terms t u N )u N ), t u N and u N 3 3 require the same bound for q, which is q > 9 3p Now we have to find the upper bound for q, in order to control u N q p q p Unfortunately this requires extensive calculations, so we will postpone this to the next section Since the calculations for p > 3 are a lot simpler, we will finish this section by outlining how to proceed in this simpler case

13 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 3 So let us assume for the rest of this section that p > 3 3+p Set q :=, then q p 3, so u N q p q p C un ) That means that u N q p q p can be controlled if un 3 3 can be controlled But the choice of q and p > 3 9 3p ensures that q > Hence by lemma, lemma 3, Korn s inequality, and the above calculations we get d t u N ) + c IΦ u N ) C + Du N R q + f N, u N ), d t t u N ) + dt Du N q q) + c J Φ u N ) C + t u N R + Du N R3 q + t f N, t u N ) The remaining terms involving f N are easy to control: Overall f N, u N P N f, u N C f, u N C f, + C Du N q, t f N, t u N P N t f) t u N C t f t u N C t f + C tu N d t u N ) + dt t u N ) + dt Du N q q) + c I Φ u N ) + c J Φ u N ) C + Du N max{r,r3,} q + t u N max{r,} + f, + tf ) Now lemma 4 ensures that for small times, ie T is small, we get boundedness of the following expressions uniformly in N): t u N L I,L )), tu N L I,L )), un q L I,L q )), I Φ u N ) L I ), J Φ u N ) L I ), where I = [0, T ] Later in section 6 will see that these a priori estimates are sufficient to pass to the limit N to get a solution u of our original problem ) But beforehand we will show in the next section how to derive similar a priori estimates in the more general case 7 5 < p 5 The case p > 7 5 If p is smaller than 3, we have to do more subtle calculations just add 43) and 4) in order to get control of Du N q p We cannot q p Recall that we in order to control the terms u N 3 3 and t u )u, t u But need q > 9 3p this implies q p > 3 So Du N q p q p is worse than un 3 3 Since Du N q p q p grows with respect to q a lot faster than Du N q q, the term Du N q p q p requires a preferably small choice of q But since we cannot control u N 3 3 for q = 9 3p we certainly cannot control the worse term Du N q p q p, for q = 9 3p and thus for no q 9 3p Hence we must proceed in a different way The central idea is that we have not made use of the term d t u N Since it contains less information than d t Du N q q, there is no need to extract information

14 4 LARS DIENING, MICHAEL RŮŽIČKA out of it So we try to transfer d t u N in its original form tu N, u N to the right-hand side of 43) This gives 46) I Φ u N ) C u N f N, u N + t u N, u N ), The disadvantage is that we have to control one extra term, but the advantage is that we can raise this inequality to the r-th power This gives, as long as we can control the right-hand side, information on I Φ u N ) r, which can be used to Du q p q p control for higher values of q Before we calculate the maximal allowed r and the resulting q we will reduce 46) to a more suitable form: Lemma implies that for q > 9 3p there holds Since this reduces to 47) I Φ u N ) C + Du N R q + f N, u N + t u N, u N ) f N L I,W, )) = P N f L I,W, )) C f L I,W, )) C, I Φ u N ) C + Du N R q + t u N, u N ) Since we can control arbitrary powers of Du N q by the local Gronwall s lemma 4, we see that the convective and the force terms do not raise difficulties for q > 9 3p, even if we raise the inequality to the r-th power The following lemma gives control of the remaining term t u N, u N Lemma 4 For < p there holds 4p ) 3p t u, u C t u J Φ u) p 3p ) IΦ u) + ) p+ 3p ) Proof With the help of lemma 0 we conclude with Therefore = p 3p lemma t u, u t u 3p u p, 3p p+ C t u 3p I Φ u) + ) p p t u t u I, 3p Φ u) + ) p +p t u JΦ u) IΦ u) + ) p p p 3p and = 4p 4 3p This lemma and 47) imply I Φ u N ) C + Du N R q Thus by Young s inequality for p > 6 5 = + 3p ) IΦ u) + ) p p and p + p = p+ 3p ) This proves the + t u N 4p ) 3p J Φ u N ) p 3p ) IΦ u N ) + ) p+ ) 3p ) 48) I Φ u N ) C + Du N R q + t u N 8p ) 5p 6 J Φ u N ) ) p 5p 6

15 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 5 We are finally at the point where we can raise the inequality to the r-th power I Φ u N ) r Cr) + Du N rr q + t u N r 8p ) 5p 6 J Φ u N ) r ) p 5p 6 As long as r < 5p 6 p, the last term can be broken up into a large power of tu N and J Φ u N ) We summarize Lemma 5 Let 4 9 3p 3 < p, q >, and r < 5p 6 p, then there exist constants R 4 = R 4 p), R 5 = R 5 p), such that I Φ u N ) r C ε + Du N R4 q + t u N ) R5 + εjφ u N ) As in the case p > 3 we calculate from 4) by using lemma 3 for q > 9 3p d t t u N ) + c JΦ u N ) C + t u N R + Du N R3 q + t f ), where we have used t f N = P N t f) C t f Hence by lemma and lemma 5 for q > 9 3p and r < 5p 6 p 49) I Φ u N ) r C ε + Du N R4 q d t t u N ) + dt Du N q q) + c JΦ u N ) + t u N ) R5 + εjφ u N ), C + t u N R + Du N R3 q + Du N q p q p + tf ) Different from the case p> 3 we can use I Φu N ) r to control Du N q p q p Lemma 6 Let < p, p < q < min{ 3pr+) 3+r, p} and r, then there exists a constant R 6 >, such that q p Du q p C R6 ε Du q + ε I Φ u) r + ) Proof From the assumptions we know that p < q < p, which implies q < q p < 3p Hence we can interpolate L q p ) = [L q ), L 3p )] with = = qp q) q p)3p q) Now we can estimate q p Du q p )q p) q p) Du q Du 3p 33) Du )q p) q IΦ u) + ) q p) p 3pq p) q p)3p q) and If q p) p < r, then we can use Young s inequality with δ := pr q p) > to obtain the desired result q p Du q p C )q p)δ ε Du q + ε I Φ u) r + ) Still we have to verify the condition q p) p q p) p < r 3q p) 3p q which holds due to the assumptions on q < r For this note that 3pr + ) < r q <, 3 + r This lemma and 49) imply d t t u N ) + dt Du N q ) q + c JΦ u N ) + c I Φ u N ) r 50) C + t u N max{r,r5} + Du N max{r3,r4,r6} q + t f )

16 6 LARS DIENING, MICHAEL RŮŽIČKA as long as 5) max { p, 9 3p } { < q < min 3pr+) 3+r, p } and r < 5p 6 p This is the crucial estimate, which will on the one hand provide us with the desired a priori estimate and on the other hand restrict the value of admissible p It is therefore of importance to determine the exact range of p for which we can find suitable q, r which fulfill 5) We will do so now: Since 3pr+) 3+r is increasing in r we can always find a suitable r if and only if p, q fulfill max { p, 9 3p } < q < min { 6p ), p } The existence of a suitable q is in turn equivalent to p > 7 5 Hence we have shown that for all p > 7 5 we can find suitable r and q, such that 5) is valid, eg q = 5 and r = 5 3 Before we can apply lemma 4 to ensure uniform a priori estimates for un, we have to take a look at the initial data, namely u N )0) q and t u N )0) The first one is easily bounded by u N )0) q = P N u 0 q C P N u 0,p C P N u 0, C u 0, C To bound t u N )0) let ϕ L ) with ϕ, then t u N, ϕ = t u N 0, P N ϕ = div SDu N 0 )) + u N 0 )u N 0 f N 0), P N ϕ SDu N 0 ) + C u 0, + f N 0) C Du N 0 )p Du N 0 + C u 0, + f N 0) C u 0, + u 0, + f0) ) C, since p Here we have used that f L I, W, )) and that t f L I, L )) implies f CI, L )) Thus t u N )0) C So we can apply lemma 4 to 50) and get for small times I = [0, T ] 5) 53) t u N L I,L )) + u N L I,L q )) + I Φ u N ) L r I ) + J Φ u N ) L I ) C We use 48) to get rid of the r dependence: 54) I Φ u N ) L 5p 6 p I ) C, where 5p 6 p = if p = In the next section we will show that these a priori estimates are by far enough to pass to the limit N 6 Passage to the Limit Theorem 7 Let 7 5 < p Let Φ be a p potential with induced extra stress S, ie S = n n Φ Assume that f L I,W, )) + t f L I,L )) + u 0 W, ) K div

17 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 7 Then there exists a constant T = T K) with 0 < T < T, such that the system ) has a strong solution u on I = [0, T ] satisfying 55) t u L I,L )) + u L I,W, 5 )) + J Φ u) L I ) + I Φ u) 5p 6 C L p I ) Proof In sections 4 and 5 we have proven the existence of approximative solutions u N, which solve 37) and satisfy 56) t u N L I,L )) + u N L I,L q )) + I Φ u N ) L 5p 6 p I ) + J Φu N ) L I ) C Since q = 5 and r = 5 3 was an admissible choice within the derivation of the a priori estimates, we can assume q 5 Estimate 56) especially implies I Φu N ) L I ) C, so by lemma 6 we get u N p p,i C and therefore un L p I,W,p )) C, since u N, = 0 Overall we can pick a subsequence still denoted by u N ) with 57) u N u in L p I, W,p )), 58) u N u in L I, W, 5 )), 59) t u N t u in L I, L )), where we have used that the weak limit of distributions on I is unique Since W,p ) W, ) for p > 7 5, the lemma of Aubin Lions implies the existence of a subsequence, such that 60) u N u in L I ) As a consequence we get convergence of the convective term 6) Observe that 6) u N )u N u )u in L 4 3 I ) SDu N ) L I ) 5) C Du N ) p L I ) C + u N L I )) C On the other hand by 60) Du N Du ae in I, so 63) SDu N ) SDu) ae in I due to the continuity properties of S Now Vitali s theorem, 6), and 63) imply 64) SDu N ) SDu) ae in L I ) Choose ω r and ϕ C0 I ), then we can conclude from 37), 59), 6), and 64) that t ϕ u, ω r + SDu), Dω r + u )u, ω ) r dt = ϕ f, ω r dt I Furthermore u fulfills t u L I ) + SDu) L I ) + u )u L 4 3 I ) C I

18 8 LARS DIENING, MICHAEL RŮŽIČKA Since {ω, ω, } is dense in W s, s,, div ) and Wdiv ) Wdiv ) for s > 5, we deduce that t ϕ u, ω + SDu), Dω + u )u, ω ) dt = ϕ f, ω dt I I is fulfilled for all ω W s, div ), especially for all ω V Note that t u, ω, SDu), Dω, u )u, ω, f, ω L I ) so 65) t u, ω + SDu), Dω + u )u, ω = f, ω for all ω V and ae t I It remains to show that u0) = u 0 But this follows from the parabolic embedding P N u 0 u0) = u N 0) u0) 66) C u N u L I,L )) t u N t u L I }{{},L )) 0 }{{} 0 C Since P N u 0 u 0 in L ) we get u0) = u 0 Overall we have shown by 65) and 66) that u satisfies ) in the weak sense It remains to prove the norm estimates for u, I Φ u) and J Φ u) First of all, from 58) and 59) there follows t u L I,L )) + u L I,W, 5 )) C Define H : I R d d R d d R by Ht, x, y, z) := αβ jk Φ)t, x, y) z αβ z jk, then jkαβ a) H 0, b) H is measurable in t, x) for all y, z, c) H is continuous in z and y for almost every t, x) I, d) H is convex in z for all y and almost every t, x) I Furthermore 67) J A Φ u N ) L I,L )) = HDu N, t Du N ), L I ) 68) I Φ A u N ) L I,L )) = d HDu N, k Du N ) L I ) k= Due to lemma 0 and I Φ u N ) L I) + J Φu N ) L I) C we have t u N Lp I,W, 3p C p+ )) Thus we can pass to a subsequence still denoted by u N ) with 69) t u N t u in L p I, L 3p p+ ))

19 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 9 Note that 69), 57), and 60) imply u N u t u N t u u N u in L I ), in L I ), in L I ) Thus from the semicontinuity theorem of De Giorgi [9], pg 3), 56), and 67) follows 70) J Φ A u) L I ) C Furthermore H, Du N, k Du N fulfill all the requirements of corollary 6 Thus we deduce from 56) and corollary 6 7) This proves the theorem I Φ u) L 5p 6 p I ) C The next corollary shows what regularity for u can be deduced from 55) This justifies that we call u a strong solution Corollary 8 Let u be the solution of theorem 7, then u L p5p 6) p I 3p,, W p+ )), t u L I, W, div )) ), Du) p CI, L p ) p, 4p ) p ) Lorentz space) For all s < 6p ) there holds u CI, W,s )) Furthermore there exists a pressure π with such that 7) ae in I π L 5p 6) p I, L )) t u divsdu)) + u )u + π = f Proof From 55) and lemma 5 we deduce that Thus by theorem 35 with = Du) p 5p 6) L p I, W, )), t Du) p ) L I, L )) p 4p ) we get Du) p CI, [W, ), L )], ) = CI, B 5p 6 4p ), 4p ) p )) Besov Space CI, L p ) p, 4p ) p )) Lorentz Space For more details regarding Besov spaces and Lorentz spaces see Bergh, Löfström [3] and Triebel [6] Let s < 6p ), then Du) p CI, L p ) p, 4p ) p )) CI, L s p ))

20 0 LARS DIENING, MICHAEL RŮŽIČKA CI ), so Du CI, L s )) From Korn s inequal- As a consequence Du) p s p ity we deduce From u )u u, 5 u CI, W,s )) and the choice s := 5 From 55) and lemma 0 we deduce that Further note that so u )u CI, L )) u p5p 6) 3p C L p I,,W p+ )) SDu)) C Du) p Du, SDu)) C I Φ u) < 6p ) we deduce Thus SDu) L p5p 6) p I, W, )) We have shown that all the terms t u, divsdu)), and u )u in 65) are in L 5p 6) p I, L )) Thus De Rahm s theorem ensures the existence of a pressure π with π L 5p 6) p I, L )) From and lemma 5 we deduce t SDu)) C Du) p t Du t SDu)) C J Φ u) This and 7) prove t u L I, W, div )) ) This proves the corollary 7 Uniqueness Theorem 9 Let 7 5 < p and let u and v be weak solutions of ) with Then u = v u, v CI, W, 5 )) Proof Let e := u v We take the difference of the equations of u and v and use e as a test function, then t e, e + SDu) SDv), Du Dv + u )u v )v, e = 0 This reduces to 73) d t e + SDu) SDv), Du Dv e )u, e Since p > 7 5, there exists q > 8 5 with Thus Lemma 8 implies p q q < 5 p p Du) + Dv) q q L I) De q C SDu) SDv), Du Dv

21 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS Korn s inequality implies e 3q 3 q d t e + c e 3q 3 q C e,q C De q So by 73) e )u, e e 4 u 7, C e Since q > 8 3q 5 there holds < 3 q < 4 7 and L 4 7 ) = [L ), L 3q 3 q )] with 0 < < Thus for δ > 0 d t e + c e 3q 3 q C e ) e 3q 3 q Gronwall s inequality implies e = 0, ie u = v C δ e + δ e 3q 3 q Note that for p > 7 5 we have derived for small times the existence of a strong solution u with u CI, W,s )) for all s < 6p ) Especially this solution satisfies u CI, W, 5 )) Theorem 9 above ensures that this solution is unique within the class of strong solutions in CI, W, 5 )) It is interesting to observe that the uniqueness as proven above exactly holds up to the same bound p > 7 5 for which we have derived the existence of such solutions 8 Space and Time Dependent Potentials In the previous sections we have assumed that S is induced by a p potential, where p is constant with 7 5 < p In the study of electrorheological fluids it is necessary to admit p, which may vary in space and time We are especially interested in the model studied in [,, 4, 5], ie 74) 75) div E = 0, curl E = 0, ρ 0 t u div S + ρ 0 u )u + π = ρ 0 f + χ E [ E]E, div u = 0, where E is the electric field, P the polarization, ρ 0 the constant density, u the velocity, π the pressure, f the mechanical, χ E the dielectric susceptibility and the extra stress S is given by 76) S = α + D ) p ) E E + α 3 + α 33 E ) + D ) p D + α 5 + D ) p DE E + E DE) Moreover, p is function of the electric field, ie p = p E ) Note that 74) decouples from 75) Thus, while solving 75) we can assume that E and p are given functions Due to the nature of the Maxwell equations 74) it is reasonable to consider smooth E and p We define p := inf I p and p + := sup I p Here we study the simplified model 77) with t u div SDu) ) + u )u + π = f, SDu) = + Du ) p Du, div u = 0, where p is a given function with p W, I ) We will show that all the results of the previous sections may be transfered to this model Especially, we will prove short time existence of strong solutions of 77) for large data as long as 7 5 < p p + We will also show, that this solution is unique within its class

22 LARS DIENING, MICHAEL RŮŽIČKA of regularity Instead of repeating all the steps as in the case where p is constant, we will indicate all necessary changes in the calculations Definition 0 Let p : I, ] be a W, I ) function with < p := inf p p + := sup p Let F : I R 0 R 0 be such that for ae t, x) I the function F t, x, ) is a pt, x) potential see definition ) and the ellipticity constants do not depend on t, x), ie the function Φ : R n n R 0 defined through Φt, x, B) = F t, x, B sym ) satisfies 78) jk lm Φ)t, x, B)C jk C lm γ + B sym ) pt,x) C sym, jklm 79) n n Φ)t, x, B) γ + B sym ) pt,x) for all B, C R n n with constants γ, γ > 0 Further we assume that F is continuously differentiable with respect to t and x and that t F )t, x, ) : R 0 R 0, resp j F )t, x, ) : R 0 R 0, is a C function on R 0, resp a C function on R >0, for all t I, x Moreover, assume that for j =,, d 80) t F )t, x, 0) = 0, t F )t, x, R) > 0 for all R > 0, j F )t, x, 0) = 0, j F )t, x, R) > 0 for all R > 0, t n n Φ)t, x, B) γ3 + B sym ) pt,x) ln + B sym ), n n Φ)t, x, B) γ3 + B sym ) pt,x) ln + B sym ), with γ 3 > 0 Such a function F, resp Φ, is called a time and space dependent p-potential and the corresponding constants γ, γ and γ 3 are called the ellipticity and growth constants of F, resp Φ The function p is called the exponent of the potential As in definition we define the extra stress S by S := n n Φ The standard examples of such S are SDu) + Du ) p Du, SDu) + Du ) p Du, where p W, I ), especially 77) is included Assume for the rest of this section that F, resp Φ is a time and space dependent p potential and S := n n Φ Then remark, remark 3, and theorem 4 still hold true We will now transfer the estimates of section to the case of time and space dependent p potentials The involved generic constants C may depend on p W, I ) We define the functionals I Φ, J Φ, and G Φ as in 6), 7), and 8) Then ) and 3) hold true without change, while 5) and 6) must be modified to γ Du) ) p C I Φ u) + Du) p ln Du) ) 8) dx, 8) p ) γ t Du) C J Φ u) + Du) p ln Du) ) dx In order to explain why there appears the logarithmic term, we will deduce 8) explicitly: Note that p ) Du) p 83) C Du) u + p Du) p ln Du)

23 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 3 Now 83), p L I ) C, and ) prove 8) Furthermore, 8) and 9) hold true, if p p is exchanged by ρ p ), where ρ p, g) : I R 0 is defined by ρ p t, g) := gt, x) pt,x) dx Especially 84) 85) γ ρ p t, u) C I Φ t, u) + γ ρ p t, Du) γ ρ p t, t Du) C J Φ t, u) + γ ρ p t, Du) Mostly we will simply write ρ p g) instead of ρ p t, g) Moreover, 3) and lemma 8 hold true without change The estimates of lemma 9 and 0 will in our case be modified to and 86) 87) γ ρ3p Du) ) 3 u p, 3p p + t u p, 3p p + C I Φ u) + C I Φ u) + ), C J Φ u) p Du p ln Du) dx + ρ p Du) ) IΦ u) + ) p CJ Φ u) + I Φ u) + ) With all these estimates above it is possible to transfer all calculations of section 4 and 5 to the case of a time and space dependent p potential Indeed, the test function u can be applied without change Especially we get 88) u N L I,L )) + Du N px,t) dx dt C I The estimates for the test functions u and t u involve additional terms of lower order, which can always be estimated As an example we will in analogy to 40) consider the nonlinear main part divsdu)) when tested with u, ie SDu N ), D u N = ij kl Φ)Du N ) m D ij u N m D kl u N dx ijklm + m kl Φ)Du N ) m D kl u N dx klm 6),80) I Φ u N ) C I Φ u N ) ε Du N ) p ln + Du N ) u N dx Du N ) p u N dx C ε Du N ) p ln + Du N ) dx ) c I Φu N ) C ε Du N ) p + ln Du N ) ) dx

24 4 LARS DIENING, MICHAEL RŮŽIČKA Thus we can derive an analogy to 47) with an additional logarithmic term on the right hand side Repeating the calculations of section 5 we get compare with 47)) I Φ u N ) C + Du N R q + t u N, u N + Du N ) p +ln Du N ) ) ) 89) dx Note that for every δ > 0 there holds Du N ) p + ln Du N ) ) C δ Du N ) p+δ Especially, if we choose p + δ q, then we can hide the logarithmic term in 89) in Du N q q, ie 90) I Φ u N ) C + Du N R q + t u N, u N ) This proves that 47) holds even in the case of a space and time dependent potential The rest of the calculactions in section 5, eg testing with t u, can be carried out in the same way, that is all logarithmic terms will be hiden in Du N q q Overall, this proves that the crucial estimates 50) and 5) remain valid, where p in 5) has to be replaced by p, the lower bound of p Thus, we have the following result for short time existence for the system 77): Theorem Let p W, I ) with 7 5 < p p + Let S be induced by a space and time dependent p potential Φ, ie S = n n Φ Let 7 5 < p p + and f L I,W, )) + t f L I,L )) + u 0 W, div ) K Then there exists a constant T = T K) with 0 < T < T, such that the system 77) has a strong solution u on I = [0, T ] Further 9) t u L I,L )) + u CI,W, 5 )) + J Φ u) L I ) + I Φ u) 5p 6 C L p I ) Corollary Let u be the solution of theorem, then u L p 5p 6) p I 3p,, W p + )), t u L I, W, div )) ), Du) p CI, L p ) p, 4p ) p ) Lorentz space) For all s < 6p ) there holds u CI, W,s )) Furthermore there exists a pressure π with such that 9) ae in I π L 5p 6) p I, L )) t u divsdu)) + u )u + π = f

25 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 5 Proof Using 9) we proceed as in corollary 8 The logarithmic terms that appear due to the space and time dependency of p see 8) and 8)) can easily be controlled due to u CI,W, < 5 )) The following theorem shows that the solutions are unique within their class of regularity Theorem 3 Let p, S be as in theorem Let u and v be weak solutions of 77) with Then u = v u, v CI, W, 5 )) We want to remark that the results of theorem, corollary,, and theorem 3 can easily be generalized to the case of electrorheological fluids: In the model 74) and 75) the equation of motion 75) is the crucial one The equations for E namely 74) decouple, ie E is the solution of the quasi static Maxwell equations These equations are well studied and we assume therefore that a smooth unique solution E of 74) is given and we want to solve 75) We further assume that the dependence of p = p E ) on E is sufficiently smooth, such that p W, ) Since ρ 0 is a constant, we can simplify 75) to 77), which is exaclty the system we have studied above The only difference is that the extra stress defined in 76) is not induced by a time and space dependend p potential Φ as defined in defintion 0 Nevertheless under suitable conditions on α, α 3, α 33, and α 5 see [5]) the extra stress S still satisfies the monotonicity conditions of theorem 4 Moreover, if we generalize the definitions of I Φ, J Φ, and G Φ by I Φ t, u) := αβ S jk ) Dut) ) r D αβ ut), r D jk ut), r jkαβ J Φ t, u) := αβ S jk ) Dut) ) t D αβ ut), t D jk ut), jkαβ G Φ t, w, v) := αβ S jk )Dwt))D αβ vt), D jk vt), jkαβ then it can easily be shown that 8) till 87) still hold These estimates allow to generalize theorem, corollary, and 3 to system 75) for electrorheological fluids 9 Appendix Lemma 4 local version of Gronwall s lemma) Let T, α, c 0 > 0 be given constants and let h L 0, T ) with h 0 ae in [0, T ], 0 f C [0, T ]) satisfy 93) f t) ht) + c 0 ft) +α Let t 0 [0, T ] be such that α c 0 Ht 0 ) α t 0 <, where t Ht) := f0) + 0 hs) ds Then for all t [0, t 0 ] there holds ft) Ht) + Ht) α c0 Ht) α t ) ) α

26 6 LARS DIENING, MICHAEL RŮŽIČKA Proof Let δ > 0 be small such that α c 0 Ht 0 )+δ) α t 0 < Let t [0, t 0 ] be fixed Then H δ := Ht ) + δ satisfies α c 0 Ht ) + δ) α t 0 < Define y : [0, t ] R 0 by yt) := t 0 fs) +α ds Then y C [0, t ]), y0) = 0, y 0, and for all t [0, t ] 94) y t) = ft) +α = t f0) + H δ + c 0 yt) ) +α Further, let g C [0, t ]) be given by Then 0 ) +α 93) f s) ds Ht) + c 0 yt) ) +α gt) := H δ + c 0 yt) ) α g t) = α c 0 Hδ + c 0 yt) ) α y t) 94) α c 0 Thus gt) g0) + α c 0 t for t [0, t ] and therefore Since α c 0 H α δ t <, we estimate H δ + c 0 yt) ) α H α δ + α c 0 t and H δ + c 0 yt) H α δ α c 0 t ) α H yt) c α 0 δ α c 0 t ) ) α H δ = c 0 H δ H α δ α c 0 t ) α ) The limit δ 0 implies for the choice t = t yt ) c 0 Ht ) Ht ) α ) ) α c 0 t α From 93) we deduce ft ) Ht ) + c 0 yt ) Since t [0, t 0 ] was arbitrary, this proves the lemma Lemma 5 Let be a domain in R d and I := [0, T ] R, T > 0 Further let F : I R m R n R with a) F 0, b) F measurable in t, x for aa y, z, c) F t, x,, ) continuous for aa t, x, d) F t, x, y, ) be convex for all y and aa t, x If w N w in L loc I ) and wn w in L loc I ), then for all r, s with r <, < s < or r = s = there holds F t, x, w, w) lim inf F t, x, w N, w N ) L r I,L s )) N L r I,L s ))

27 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 7 Proof If r = s = the result follows immediately from theorem De Giorgi) in [9], pg 3 So let us assume r <, < s < Since L s ) is reflexive, the dual of L r I, L s )) is L r I, L s )) see [6]) and for all f L r I, L s )) f L r I,L s )) = sup ϕ L r I,L s )) f, ϕ For ϕ L r I, L s )) with ϕ 0 and ϕ L r I,L s ) let F ϕ : I R m R n R be defined by F ϕ t, x, y, z) := F t, x, y, z)ϕt, x) Then F ϕ fulfills a) F ϕ 0, b) F ϕ measurable in t, x for aa y, z, c) F ϕ t, x,, ) continuous for aa t, x, d) F ϕ t, x, y, ) be convex for all y and aa t, x Hence by theorem De Giorgi) from [9], pg 3 F t, x, w, w)ϕt, x) dx dt 95) I lim inf N lim inf N F t, x, w N, w N )ϕt, x) dx dt I F t, x, w N, w N ) Lr I,L s )) Since F 0, the norm formula for F reduces to F t, x, w, w) L r I,L s )) = sup ϕ L r I,L s )) F t, x, w, w)ϕt, x) dx dt ϕ 0 This and 95) proves the lemma I Corollary 6 Let be a domain in R n and I := [0, T ] R, T > 0 Let F : I R m R n R and G : I R m R n n with n F t, x, y, z) = G αβ t, x, y)z α z β Moreover, assume G satisfies a) G 0, ie is positive semidefinite, b) G measurable in t, x for aa y, c) Gt, x, ) continuous for aa t, x α,β= If w N w in L loc I ) and wn w in L loc I ), then for all r, s with r <, s < there holds F t, x, w, w) Lr I,L s )) lim inf N F t, x, w N, w N ) Lr I,L s ))

28 8 LARS DIENING, MICHAEL RŮŽIČKA Proof For < q < define F q by F q t, x, y, z) := F t, x, y, z)) q then F q fulfills the requirements of lemma 5 Thus F t, x, w, w) = Fq t, x, w, w) Lr I,L s )) This proves the corollary lim inf N = lim inf N q L r q I,L s q )) F q t, x, w N, w N q ) F t, x, w N, w N ) L r I,L s )) L r q I,L s q )) We will now prove an interpolation theorem, which is very useful in deriving estimates for solutions of parabolic problems pointwise in time Suppose that u is a solution to a parabolic problem, such that u, resp t u, are in the Bochner spaces L p0 I, A 0 ), resp L p I, A ), where A 0 and A are Banach spaces Then we will see that u is with respect to the time a continuous function with values in the real interpolation space A,q := [A 0, A ],q, where = p 0, p ) and q = qp 0, p ) The proof will be quite standard and we will mainly compile results which can be found in [4], [3] and [] Nevertheless to the knowledge of the authors there exists no exact statement of this result in literature Since this interpolation result plays a fundamental role in this paper it is indispensable to prove it in some detail Definition 7 Let A 0 A be two Banach spaces Let p j <, α j R and η j = α j + p j for j = 0, Let X denote the space XA 0, A ) = {u : u L loc R 0, A 0 ), u L loc R 0, A )} We shall work with the Banach spaces V = V p 0, α 0, A 0 ; p, α, A ) with V = {u : u XA 0, A ), u V < }, u V = max { t α0 u L p 0 R 0,A 0), t α u L p R 0,A )} These spaces have been introduced by J L Lions and J Peetre [4], but J Bergh and J Löfström have defined similar Banach spaces Ṽ A, p, η) see [3], corollary 33) The interconnection of these spaces is that both Ṽ A, p, η) and V p 0, α 0, A 0 ; p α, A ) are norm equivalent, if we choose η j = α j + p j with j = 0, see [3], remark 34) Lemma 8 Let u XA 0, A ) then there exists b A 0 + A = A such that t ut) = b + u τ)dτ, ae in R >0 Especially u is continuous on 0, ) with values in A Proof The proof of this lemma is standard We refer to Adams [], lemma 7, for a similar result Hence every function u X has a limit u0 + ) in A 0 + A = A The lemma still holds true if R >0 is replaced by an intervall I = [0, T ]

29 STRONG SOLUTIONS FOR GENERALIZED NEWTONIAN FLUIDS 9 Definition 9 By T = T p 0, α 0, A 0 ; p, α, A ) we denote the space of traces u of functions v V = V p 0, α 0, A 0 ; p, α, A ) equipped with the quotient norm u T = inf v V v0 + )=u Then T p 0, α 0, A 0 ; p, α, A ) is a Banach space Theorem 30 Let p j <, α j R, and η j = α j + p j with j = 0, Further let and p be given by = If η 0 > 0 and η <, then η 0 η 0 + η, T p 0, α 0, A 0 ; p, α, A ) = A,p p = p 0 + p Proof J Bergh and J Löfström have shown that their trace space T A, p, ) is under the stated conditions η 0 > 0 and η < equivalent to the trace space T p 0, α 0, A 0 ; p, α, A ) of J L Lions and J Peetre see [3], remark 34) Further J Bergh and J Löfström showed that T A, p, ) = A,p see [3], pg 7-75) Remark 3 Let p 0 <, < p < and α 0 = α = 0, then η 0 = p 0 > 0 and η = p < Hence the requirements of theorem 30 are automatically fulfilled for T p 0, 0, A 0 ; p, 0, A ) This case will later be of great importance Hence we define: Definition 3 For p j < with j = 0,, let V p 0, A 0 ; p, A ) := V p 0, 0, A 0 ; p, 0, A ), T p 0, A 0 ; p, A ) := T p 0, 0, A 0 ; p, 0, A ) Theorem 33 Let p j < with j = 0, and p 96) =, p + p p 0 p 0 then 97) and Furthermore 98) T p 0, A 0 ; p, A ) = A, V p 0, A 0 ; p, A ) L R 0, A, ) CR 0, A, ) u L R 0,A, ) u V, 99) u L R 0,A, ) C u L p 0R 0,A 0) u L p R 0,A ) Proof In the notation of theorem 30 we have η 0 = p 0 > 0 and η = p < and = η 0 η 0 + η = p 0 p 0 + = p p p + p p 0 p 0