- Email: [email protected]
Buoyancy-driven viscous flow with L1-data
Nonlinear Analysis 46 (2001) 737 – 755
www.elsevier.com/locate/na
Buoyancy-driven viscous "ow with L1-data Jind%rich Ne%casa; b; ∗ , Tom(as% Roub(+c% ekb; c a Department
of Mathematical Sciences, Northern Illinois University, De Kalb, IL 60115 –2888, USA Institute, Charles University, Sokolovsk&a 83, CZ-186 75 Praha 8, Czech Republic c Institute of Information Theory and Automation, Academy of Sciences, Pod vod& arenskou v0e z0 &1 4, CZ-182 08 Praha 8, Czech Republic b Mathematical
Received 14 February 2000; received in revised form 30 April 2001
Keywords: Non-Newtonean "uids; Heat equation; Dissipative heat; Adiabatic heat; L1 heat sources
1. Introduction, problem formulation This paper deals with buoyancy-driven "ow of heat-conductive non-Newtonean incompressible "uids. There are various models appearing in the literature, cf. e.g. [2,9,16] for a genesis of various possibilities. The starting point is always the complete compressible "uid system of n + 2 conservation laws for mass, impulse, and energy; n denotes the spatial dimension. Then, the so-called incompressible limit represents a small perturbation around a stationary homogeneous state, i.e. around constant mass density, constant temperature, and zero velocity. It should be emphasized that, though the original full system is thermodynamically consistent, the incompressible limit system of n + 1 equations, in general, violates both the energy conservation law and the Clausius–Duhem inequality, except special cases as Remark 2 below. We consider a bounded Lipschitz domain in Rn ; n ¿ 2; I := [0; T ] a time interval, and denote Q := × I; := @ × I . To cover various possibilities, we consider the
This research has been carried out during a stay of the second author at the Northern Illinois University, % % partly supported also by the grants A 107 5707 (GA AV CR) and CEZ J13=98113200007 (CR). Besides, this author is thankful to Josef M(alek, Jan Mal(y , and Michael RuH z% i%cka for helpful discussions. ∗ Corresponding author. Mathematical Institute, Charles University, Sokolovsk( a 83, CZ-186 75 Praha 8, Czech Republic. E-mail addresses: [email protected] (J. Ne%cas), [email protected] (T. Roub(+c% ek).
c 2001 Elsevier Science Ltd. All rights reserved. 0362-546X/01/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 6 7 6 - 9
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
738
following fairly general system of equations: @u + (u · ∇)u − div (e(∇u)) + ∇ = g(1 − 0 ); @t
(1.1a)
e(∇u) = 12 ∇u + 12 (∇u)T ;
(1.1b)
div u = 0;
(1.1c)
@ + u · ∇ − kO = 1 (e(∇u)) : e(∇u) + 2 g · u + h; (1.1d) @t n n where [ij ] : [eij ] = i=1 j=1 ij eij ; k is the heat conductivity, 0 is the linearized relative mass density variation with respect to temperature, 1 re"ects dissipation eLects, 2 expresses adiabatic heat eLects, (e) is the viscous stress, g an external (e.g. gravitational) force, and h = h(x; t) is the external heat source. For notational simplicity, we normalized the mass density and the heat capacity to 1. For a rigorous derivation of a system like (1:1) we refer to Kagei et al. [9, System (16)] who showed how the coeQcient 1 depends on Ostrach’s dissipation number, while the coeQcient 2 also depends on the Reynolds and the Prandtl numbers. In [9] the Navier–Stokes case (i.e. (e) = e) has been considered and, for smooth data h and 0 , existence of a regular solution local in time or, in case of small data, global in time has been shown. Let us remark that our existence results, though holding globally for large and irregular data, unfortunately do not cover [9] because our needed assumption p ¿ n ¿ 2 just excludes the Navier–Stokes case. The system should be completed by boundary and initial conditions. For simplicity, we will consider an initial-value problem with no-slip boundary condition for velocity and perfect isolation for temperature, i.e. u(·; 0) = u0 ; u = 0;
(·; 0) = 0
on ;
@ = 0 on @
(1.2a) (1.2b)
with denoting the unit normal to the boundary @ of . Remark 1. Often, a simpler, so-called Oberbeck–Boussinesq model is used for buoyancydriven "ow of heat-conductive incompressible "uids. This model neglects both the dissipative and the adiabatic heat sources, i.e. 1 = 2 = 0, and usually considers (e) = e which turns (1.1a) – (1.1c) into the Navier–Stokes system, cf. e.g. [7] or [16], and sometimes it is combined with other phenomena as solidiUcation, see [17]. For a non-Newtonean model coupled with the heat equation we refer to M(alek et al. [14], where omitting the heat sources allowed to consider p-power law "uids even with p less than n, and to Rodriguez and Urbano [18] who allowed the viscosity to depend also on temperature. Besides, some models include the dissipative heat but not the adiabatic heat sources (i.e. our model (1:1) with 1 ¿ 0 but 2 = 0), cf. [11, Section 50] or also, e.g. [8].
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
739
Remark 2. Though perhaps formal, it is worth observing that, for 0 = 2 := and for 1 = 1, model (1:1) admits a consistent thermodynamical justiUcation. The departure point is the choice of the speciUc Helmholtz free energy as (; q) := (1 − )’(q) − ln ;
(1.3)
where q stands for the coordinate of the volume in question (i.e. u = dq=dt with d=dt standing for the material derivative in time); in other words, q is just the Lagrange coordinate. The Urst term is then the speciUc potential energy at the position q. This is proportional to the mass density, which however depends on the temperature due to the thermal expansion with the coeQcient . Without not much loss of generality, we assume here the force g to have a potential, which is just ’. In most applications, g is a constant gravitational acceleration so that ’(q) := g · q: Then one deUnes the standard speciUc entropy @ = ’(q) + (1 + ln ) @
s(; q) := −
(1.4)
and the speciUc internal energy !(; q) := (; q) + s(; q) = ’(q) + :
(1.5)
Then we postulate the kinetic energy Tkin , the dissipation potential R, and the total free energy, respectively, as 1 Tkin (u) := |u|2 d x; (1.6a) 2 1 T(e(∇u)) d x with T(e) := (te) : e dt; (1.6b) R(u) :=
#(; q) :=
0
(; q) d x
(1.6c)
provided we assume additionally smoothness and symmetry of , namely @ij =@ekl = |e|2 @kl =@eij . For example, in the special case ij = %(|e|2 )eij one has T(e) = 12 0 %(t) dt, cf. [13, Section 1:1:5]. The Hamilton variation principle adapted for nonconservative systems with holonomic constraints C(q) = 0 (cf. also [3, Section 1:3]) says that the T integral 0 Tkin (q)−U ˙ (q)+F; q+C(q) dt = 0 is stationary with U = #(; ·) being the potential energy and F = R (q) ˙ is a nonconservative force. This gives the momentum equation Tkin
du + R (u) + #q (; q) = C ∗ ; dt
Cq = 0;
(1.7)
where the apostrophe indicates the Gˆateaux diLerential, C = div forms the constraint and C ∗ = − ∇ is the adjoint operator, with playing the role of the Lagrange multiplier to the constraint Cq = 0, cf. also [3, Section 3:1:1]. Note that (1.7) is just
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
740
(1.1a) – (1.1c) because Cq = 0 implies also (d=dt)Cq = C(d=dt)q = Cu = 0 which is just (1.1c). Requiring the conservation of the total energy Tkin (u)+ !(; q) d x, the standard procedure gives the energy balance equation
@s + div j = )(u); @t
(1.8)
where j is a heat "ux and )(u) a dissipation rate; in our case (1.6b) we have )(u) = (e(∇u)) : e(∇u). Substituting (1.4) into (1.8) then gives (1.1d) if one puts j = − k∇u, i.e. the classical Fick law in isotropic media. This choice of j satisUes the Clausius–Duhem inequality j j · ∇ )(u) d + div − dx ¿ 0 (1.9) s(; q) d x = dt 2 provided the isolation condition (1.2b) is considered and ¿ 0, which is actually satisUed for any (even distributional) solution to (1.1d) and (1:2) if 0 ¿ 0 and also if the heat source h does not cool too much, certainly if h ¿ 0, cf. Proposition 1 below.
2. Distributional solution to (1:1) and (1:2) We want to treat system (1:1) in as much generality as possible (but still physical) situations. The heat transfer (1.1d) has a natural L1 -structure, which encourages us to consider the heat sources 0 ∈ L1 () and h ∈ L1 (Q), or even as measures. Then the concept of weak solution is no longer relevant, and one must speak in terms of distributive solutions or of integral solutions in the sense of Benilan [4], based on accretivity of the stationary part of (1.1d). For the convective term with u time-independent and smooth, this accretivity was observed already by Rulla [21]. Here we will use the former option, the latter technique being exploited for proofs only. We use the following standard notation for functions spaces: Lp (; Rn ) denotes the Lebesgue space of measurable functions → Rn whose p-power is integrable, W01; p (; Rn ) is the Sobolev space of functions whose gradient is in Lp (; Rn×n ) and p (; Rn ) = {v ∈ W01; p (; Rn ); div v = 0 in the sense whose trace on @ vanishes, W0;1;DIV 1; p of distributions}, and W −1; p (; Rn ) ∼ = W0 (; Rn )∗ . Likewise, W k; p indicates all kth derivatives belonging to the Lp space. Also, “rca” will denote the regular countably additive set functions with respect to a Borel ,-algebra in question, also called Radon Z Z measures. Furthermore, we denote by L∞ w (I ; rca()) the Banach space of C()-weakly Z Z the measurable essentially bounded functions : I → rca(), i.e., for any v ∈ C(), function t → Z v(x)(d x; t) is measurable and | Z v(x)(d x; t)| 6 CvC() Z for a.a. t ∈ I (with the null set possibly depending on v) for some C ¡ + ∞ independent of Z has a separable predual, namely C(), Z we can consider the norm v. Since rca() ∞ Z ∼ L∞ Z = ess supt∈I (t)rca() Z , cf. [6, Lemma 12:2:2]. It holds Lw (I ; rca()) = w (I ;rca()) 1 ∗ ∞ Z , see [6, Theorem 12:2:11]. Let us emphasize that Lw (I ; rca()) Z = L (I ; C()) Z L∞ (I ; rca()), see [6, Example 12:2:8].
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
741
We will assume the following data qualiUcation: |(e)| 6 C(1 + |e|p−1 );
(0) = 0;
(2.1a)
((e1 ) − (e2 )) : (e1 − e2 ) ¿ .|e1 − e2 |p ;
(2.1b)
Z Rn )); g ∈ L∞ (I ; C(;
(2.1c)
k ¿ 0;
Z 0 ∈ rca();
Z h ∈ rca(Q);
u0 ∈ L2 (; Rn )
(2.1d)
with . ¿ 0. Note that (2:1) ensures all integrals in (2.2) – (2.4) below to have a good sense provided p ¿ n, as we will have to assume anyhow. Let us also recall that (2.1b) ensures ((e(∇u1 )) − (e(∇u2 ))) : e(∇u1 − ∇u2 ) d x ¿ cu1 − u2 p 1; p (2.2) n W0 (;R )
with some c = c(.; ) ¿ 0 due to Korn’s inequality. By using Green’s formula once for (1.1a) – (1.1c) and twice for (1.1d) and by-parts integration in time, one gets the following deUnition: p Z (; Rn )) and ∈ L∞ Denition . We will call u ∈ Lp (I ; W0;1;DIV w (I ; rca()) a distributional solution to (1:1) and (1:2) if T @v u − ((u · ∇)u) · v − (e(∇u)) : e(∇v) + g · v d x 0 @t g · v (d x; t) dt = − u0 (x) · v(x; 0) d x (2.3) − 0 Z
for any v ∈ Lp (I ; W01; p (; Rn ))∩W 1; p (I ; L1 (; Rn )) with v(·; T ) = 0 and p = p=(p−1), and T @v + u · (∇v + 2 gv) + kOv (d x; t) @t 0 Z + 1 (e(∇u)) : e(∇u) v d x dt = − vh (d x dt) − v(x; 0) 0 (d x)
QZ
Z
(2.4)
for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on . 3. Existence of the distributional solution We will prove the existence rather nonconstructively by using the Schauder Uxed point theorem.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
742
First, we deUne the mapping
A : # → u : Lp (I ; W −1; p ()) → Lp (I ; W01; p (; Rn ))
(3.1)
with p denoting the conjugate exponent to p, i.e. p = p=(p−1), by u being the weak solution to @u + (u · ∇)u − div (e(∇u)) + ∇ = g(1 − 0 #); @t div u = 0;
u(·; 0) = u0 ;
u(@; ·) = 0:
(3.2)
Lemma 1. Let (2:1) be valid with p ¿ n. Then A is continuous with respect to norm topologies indicated in (3:1).
Proof. Take #k → # in Lp (I ; W −1; p ()) and denote by uk the weak solution to (3.2) corresponding to #k in place of #; for the existence of uk , even under weaker condition p ¿ 1 + 2n=(n + 2) we refer e.g. to [10] or [12, Sections II.5.2 and III.2.1]. By testing with uk , we get a standard a priori estimate uk L∞ (I ;L2 (;Rn ))∩Lp (I ;W 1; p (;Rn ))∩W 1; p (I ;W −1; p (;Rn )) 6 C:
(3.3)
0
Then we can take a weakly convergent subsequence in the space indicated in (3.3). It is a standard procedure to show that this subsequence converges to a weak solution u to (3.2) which is determined uniquely, so that even the whole sequence {uk } converges weakly to u, cf. [10,12, Section II:5], or, if generalized for # ∈ Lp (I ; W −1; p ())\L2 (Q) under the condition p ¿ n, also [13, Section 5:3]. Let us prove the strong convergence: subtracting (3.2) with u and uk , testing by T uk − u, and using Korn’s inequality (2.2) and 0 [@(uk − u)=@t](uk − u) d x dt ¿ 0, we get T cuk − upp 1; p 6 ((e(∇uk )) − (e(∇u)) : e(∇uk − ∇u) d x dt n L (I ;W0 (;R ))
0
6
T 0
((uk · ∇)uk − (u · ∇)u) · (uk − u)
+ 0 (#k − #)g · (uk − u) d x dt := I1k + I2k : We have I1k =
n T n 0
j=1
i=1
@ uki @xi
ukj −
n i=1
@ ui @xi
T n @ (uki ukj − ui uj )(ukj − uj ) d x dt = @x i 0 i; j=1
uj
(ukj − uj ) d x dt
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
=−
=−
T n i; j=1
0
T 0
(uki ukj − ui uj )
743
@ (ukj − uj ) d x dt @xi
(uk ⊗ uk − u ⊗ u) : ∇(uk − u) d x dt;
where, hopefully without confusion, the indices i; j = 1; : : : ; n indicate the respective components of the vectors u or uk while k indexes the members of the sequence {uk }k∈N . As indicated in (3.3), we have also an estimate on @uk =@t. Due to the compact imbedding W 1; p () ⊂ L∞ (), we can use the classical Aubin theorem [1] to see that uk → u strongly in Lp (I ; L∞ (; Rn )). As uk is also bounded in L∞ (I ; L2 (; Rn )), we get by standard interpolation that uk − uLp+2 (Q;Rn ) 6 uk − uLp (I ;L∞ (;Rn )) uk − uL1− ∞ (I ;L2 (;Rn )) → 0
(3.4)
with = p=(p + 2) ¿ 0, cf. e.g. [12, Section III.2.1]. Hence uk → u strongly also in Lp+2 (Q; Rn ). As also ∇uk → ∇u weakly in Lp (Q) and certainly p ¿ 2, we get I1k → 0. Also, the term I2k converges to zero because #k → # in Lp (I ; W −1; p ()) and 1; p uk → u in Lp (I ; W0 (; Rn )). Furthermore, let us consider the mapping ∞ Z Z B : (u; #) → : Lp (I ; W01; p (; Rn )) × L∞ w (I ; rca()) → Lw (I ; rca())
(3.5)
with being the distributional solution to @ + u · ∇ − kO = f; @t
(·; 0) = 0 ;
@ (@; ·) = 0 @
with f = 1 (e(∇u)) : e(∇u) + 2 g · u# + h; Z i.e. ∈ L∞ w (I ; rca()) satisUes the following identity: T @v v f(d x dt) + u · ∇v + kOv (d x; t) dt + @t 0 QZ Z + v(x; 0) 0 (d x) = 0 Z
(3.6)
(3.7)
for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on . Let us still point out that, p Z Z as W 1; p () is separable, by Fattorini [6] we have L∞ w (I ; rca()) ⊂ Lw (I ; rca()) ⊂ p −1; p p −1; p Lw (I ; W ()) ∼ ()), cf. also the proof of Lemma 3 below. = L (I ; W Lemma 2. Let (2:1) be valid. Then the mapping B is well de
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
744
Z with f! smooth and 0! → 0 weakly∗ Proof. Let us take f! → f weakly∗ in rca(Q) Z with 0! smooth, and also u! → u in Lp (I ; W 1; p (; Rn )) with u! smooth. in rca() 0 Denote by ! the weak solution to the auxiliary linear problem @ @ (@; ·) = 0; (3.8) + u! · ∇ − kO = f! ; (·; 0) = 0! ; @ @t the existence of ! can be proved by the standard energy method by testing (3.6) by and by @=@t provided particular u! ’s are bounded in L∞ (Q) as we can certainly suppose. Let us denote by sign2 : R → [ − 1; 1] the regularization of the multivalued function “signum” deUned as r=|r| if |r| ¿ 2; sign2 (r) = (3.9) r=2 otherwise: Moreover, we denote by | · |2 the primitive function to sign2 such that |0|2 = 0. Then |r| − 12 2 6 |r|2 6 |r|. Then we can test (3.8) with = ! by sign2 (! ), which gives d |! |2 d x − ((u! · ∇! )sign2 (! ) − k∇! · ∇sign2 (! )) d x dt (3.10) = f! (·; t) sign2 (! ) d x 6 |f! (·; t)| d x:
Using the facts that (u! · ∇! )sign2 (! ) d x = u! · ∇|! |2 d x = − (div u! )|! |2 d x = 0
(3.11)
and that ∇! · ∇sign2 (! ) ¿ 0, and integrating (3.10) in time over [0; t], we get 1 |! (·; t)| d x − 2 meas() 6 |! (·; t)|2 d x 2 t 6 |0! |2 d x + |f! | d x dt 0
6
|0! |d x +
0
T
|f! | d x dt:
(3.12)
As (3.12) holds for any 2 ¿ 0 and any t ∈ [0; T ], we eventually get the a priori estimate ! L∞ (I ;L1 ()) 6 f! L1 (Q) + 0! L1 () :
(3.13)
Z 0! → This estimate allows us to pass to the limit with f! → f weakly in rca(Q), Z and u! → u in Lp (I ; W 1; p (; Rn )). Then, by Banach–Alaoglu– 0 weakly∗ in rca(), 0 Bourbaki theorem, we can choose a subsequence such that ! → weakly∗ in Z L∞ w (I ; rca()). By passing to the limit in the integral identity @v + u! · ∇v + kOv ! + vf! d x dt + v(·; 0)0! d x = 0; (3.14) @t Q ∗
we obtain just identity (3.7); note that u! ! converges weakly because u! → u in Z Rn )) ⊂ L1 (I ; C(; Z Rn )) and ! converges weakly∗ in Lp (I ; W01; p (; Rn )) ⊂ Lp (I ; C(;
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
745
Z ∼ 1 Z ∗ L∞ w (I ; rca()) = L (I ; C()) . Thus, we proved existence of solving (3.6) for u and # Uxed. Also, from (3.13) we get the estimate L∞ Z 6 frca(Q) Z + 0 rca() Z : w (I ;rca())
(3.15)
Z Moreover, the solution ∈ L∞ w (I ; rca()) to the linear problem (3.6) is unique, which means that, if the identity T @v + u · ∇v + kOv (d x; t) d x dt = 0 (3.16) @t Z 0 holds for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on , then necessarily = 0. This fact follows from the surjectivity of the adjoint operator to (3.6), more Z there is a smooth solution v to the problem precisely that, for any f ∈ L1 (I ; C()), @ @v + u · ∇v + kOv = f; v(·; T ) = 0; v = 0 on : (3.17) @ @t Z e.g. for In fact, it suQces to prove this for any f from a dense subset of L1 (I ; C()), Z f ∈ L2 (I ; C()), which is possible by a standard monotonicity technique. Moreover, from (3.6) we have also the a priori estimate
T
@!
k! Ov + ! (u! · ∇v) + f! v d x dt = sup
@t
−3; p rca(I ;W
v
())
6
v
61 3;p C(I ;W ()) 0
sup
0
(k! L1 (Q) OvL∞ (Q) + f! L1 (Q) vL∞ (Q)
61 3;p C(I ;W ()) 0
+ ! L∞ (I ;L1 ()) u! Lp (I ;L∞ (;Rn )) ∇vLp (I ;L∞ ()) ) 6 kTN1; p ! L∞ (I ;L1 ()) + N3; p f! L1 (Q)
+ N1; p N2; p T p (f! L1 (Q) + 0! L1 () )u! Lp (I ;W 1; p (;Rn )) ; (3.18) where Nk; p for kp ¿ n denotes the norm of the imbedding W k; p () ⊂ L∞ (). In the limit, we thus get
@
6 kTN1; p L∞ Z + N3; p frca(Q) Z
@t
w (I ;rca()) −3; p rca(I ;W
())
+ N1; p N2; p T p (frca(Q) Z )uLp (I ;W 1; p (;Rn )) : Z + 0 rca() (3.19)
To prove continuity of B, let us take #k → # in Lp (I ; W −1; p ()), the sequence {#k } p 1; p Z being supposedly bounded in L∞ (; Rn )), and w (I ; rca()), and uk → u in L (I ; W denote by k the distributional solution to (3.6) corresponding to uk and #k in place of u and #, respectively. Note that the right-hand side in (3.6), i.e. fk := 1 (e(∇uk )) : e(∇uk ) Z cf. also (3.33) below. Thus the distributional + 2 g · uk #k + h, is bounded in rca(Q), Z solution k does exist and is bounded in L∞ w (I ; rca()) and also @k =@t is bounded in
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
746
rca(I ; W −3; p ()). Then, by Banach–Alaoglu–Bourbaki theorem, we can assume that, possibly up to a subsequence, Z (3.20) k → weakly∗ in L∞ w (I ; rca()): Z V2 = By the generalized Aubin theorem (see Lemma 3 below with V1 = rca(), −3; p −1; p (), V3 = W ()), we have also W
strongly in Lp (I ; W −1; p ()):
(3.21) Z We also know that fk converges to f from (3.6) weakly in rca(Q); in fact, this convergence is even strong. Then we can make the limit passage in the integral identity (2.4) which reads here as T @v + uk · ∇v + kOv k (d x; t) dt + v fk (d x dt) + v(x; 0) 0 (d x) = 0: @t QZ Z 0 Z (3.22) k →
∗
Note that certainly the term k uk converges to u (even strongly) because {k } con verges strongly in Lp (I ; W −1; p ()) and {uk } also strongly in Lp (I ; W01; p (; Rn )). Thus = B(u; #) and even the whole sequence {k } converges because of the already proved uniqueness of . Furthermore, for %0 ; %1 ¿ 0, we deUne the convex sets Z (I ; rca( )); ess sup #(·; t) 6 % S%0 := # ∈ L∞ Z 0 ; w rca() t∈I
@#
# ∈ S%0 ;
6 %1 @t rca(I ;W −3; p ())
(3.23a)
S%0 ;%1 :=
(3.23b)
considered as a subset of Lp (I ; W −1; p ()). Proposition 1. Let 0 ¿ 0; h ¿ 0; (2:1) be ful
C1 ; C2 from (3:30) below:
(3.24)
Then (1:1) and (1:2) has at least one distributional solution (u; ) which; moreover; satis
C(#) := B(A(#); #):
(3.25)
Note that any Uxed point of C satisUes = B(u; ) and u = A(), so that the pair (u; ) is the distributional solution to (1:1) and (1:2). We will show that # ∈ S%0 ⇒ C(#) ∈ S%0 ;%1
(3.26)
provided %0 and %1 are chosen appropriately. Obviously, (u; ) = (A(#); C(#)) solves the decoupled system (3.2) and (3.6). Then we test (3.2) by u and add (3.6) tested
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
747
by ; sign with a constant ; ¿ 0; this latter test must be rigorously made by a regularization similarly as (3.13) and then passing to the limit. This eventually gives t 1 2 up 1:p dt u(·; t)L2 (;Rn ) + ; |(d x; t)| + (1 − 1 ;)c W0 (;Rn ) 2 Z 0 t 6 g · u d x + (0 + ;2 ) |#(d x; t)| |g · u| dt 0
Z
+ ;hrca(Q) Z +
2 1 2 u0 L2 (;Rn )
+ ;0 rca() Z ;
(3.27)
where c comes from (2.2) and | · | denotes the total variation when the argument is a measure. Note that we used also the estimate (1 − 1 ; sign )(e(∇u)) : e(∇u) d x ¿ (1 − 1 ;)cup 1; p . We can further estimate the right-hand side of (3.27) by n W0 (;R )
Young’s inequality as p |#(d x; t)| |g · u| 6 C! #(·; t)prca() Z + !u(·; t)W 1; p (;Rn ) ; Z
(3.28)
where C! depends on ! and also on gL∞ (Q;Rn ) and on p through the constant N1; p , namely C! = N1;pp (p!)1=(1−p) gpL∞ (Q;Rn ) =p . Then we can simply put (but perhaps not optimally) ; = 12 1−1 and ! = 14 c(0 + ;2 )−1 and from (3.27) and (3.28) we get again by Gronwall’s inequality uL∞ (I ;L2 (;Rn )) bounded and
p p L∞ Z 6 C1 + C2 #Lp (I ;rca()) Z 6 C1 + C2 T #L∞ (I ;rca()) Z w (I ;rca()) w
(3.29)
with some constants C1 and C2 , e.g. one can take 2 C1 = eT=(2;) (u0 2L2 (;Rn ) + 0 rca() Z + gL2 (Q) ); Z + hrca(Q)
C2 = eT=(2;) ;−1 |0 + ;2 |p gpL∞ (Q;Rn ) N1;pp (p!)1=(1−p) =p :
(3.30a) (3.30b)
If T is not too large, we will be able to Und %0 ¿ 0 such that
C1 + C2 T%p0 6 %0 ;
(3.31)
1−p
indeed, when taking %0 = (p C2 T ) , condition (3.24) ensures that (3.31) is satisUed. Moreover, for # ∈ S%0 bounded in p −1p Z ()) ∼ L∞ = Lp (I ; W −1; p ()); w (I ; rca()) ⊂ Lw (I ; W we get by testing (3.2) itself by u also the estimate .upLp (I ;W 1; p (;Rn )) 6 g(1 − 0 #) · urca(Q) Z 6 N1; p gLp (I ;C(;Rn )) 1 − 0 #L∞ Z uLp (I ;W 1; p (;Rn )) w (I ;rca()) with . from (2.1b), from which we can conclude that uLp (I ;W 1; p (;Rn )) 6 K%0
(3.32)
for a suitable K%0 . Furthermore, f from (3.6) can be estimated as p−1 )|∇u| L1 (Q) + 2 g · u#rca(Q) frca(Q) Z 6 C(1 + |∇u| Z + hrca(Q) Z
6 C meas(Q) + 2CK%p0 + 2 N1; p gLp (I ;C(;Rn )) K%0 %0 +hrca(Q) Z = : L %0 ;
(3.33)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
748
where C comes from (2.1a). Eventually, from (3.19) we have the estimate
@
6 kTN1; p %0 + N1; p N2; p T p (L%0 + 0 rca() Z )K%0
@t
rca(I ;W −3; p ()) +N3; p L%0 = : %1 : This choice of %1 will guarantee (3.26). We will endow S%0 ;%1 by the norm topology of Lp (I ; W −1; p ()). We can use the generalized Aubin theorem (see Lemma 3 below) in the same setting as in the proof of Lemma 2, which shows that S%0 ;%1 is sequentially compact in Lp (I ; W −1; p ()). By Lemmas 1 and 2 and by (3.26), C maps S%0 ;%1 continuously into itself, so by the Schauder theorem it has a Uxed point on S%0 ;%1 . The continuity of t → u(·; t) : I → L2 (; Rn ) follows standardly from the proved fact that u ∈ Lp (I ; W01; p (; Rn )) ∩ W 1; p (I ; W −1; p (; Rn )). It remains to prove nonnegativity of arbitrary Uxed point of C. We will test (1.1d) 1 1 by sign− 2 (): = 2 sign 2 (+2)− 2 with sign 2 deUned by (3.9); more precisely, one should Urst regularize (1.1d) similarly as in the proof of Proposition 3 below (thus can be considered as a weak solution) and after showing nonnegativity of one should pass to the limit. In such a way, speaking in terms of weak solutions, we obtain by using a corresponding modiUcation of (3.11) and (3.12) the estimate d − − [] d x 6 2 g · u sign2 () d x 6 2 |g · u|([]− (3.34) 2 + 2) d x; dt 2 − − where [ · ]− 2 denotes the primitive function to sign 2 such that [0]2 = 0. Integrating (3.34) over the time interval [0; t], one gets − (·; t) d x 6 [(·; t)]− 2 dx
6
6 2
[0 ]− 2 dx + 2 t 0
t 0
|g · u| []− 2 d xdt + 2t meas()
|g · u| − d xdt + 2t meas()
(3.35)
with − = 12 || − 12 denoting the negative part of ; we used − (·; 0) = 0− = 0, as assumed. As (3.35) holds for any 2 ¿ 0, we eventually obtain the estimate t − − |g · u| − d x dt (·; t)L1 () = (·; t) d x 6 2
= 2
0
0
t
[g · u](·; t)L∞ () − (·; t)L1 () dt;
Z recall that g · u ∈ Lp (I ; C()). From this we get − (·; t)L1 () 6 0 for all t ¿ 0 by Gronwall’s inequality, so that ¿ 0 follows. To obtain global existence at least in the case 0 1 6 2 , we need to assume a bit of regularity of the right-hand side f. This gives a certain additional continuity of ,
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
749
which allows us to speak about values of at current time level t needed to prolong the solution. Proposition 2. Let 0 ¿ 0; h ¿ 0, and (2:1) be ful
kO ∈ L∞ (I ; W −3; p ()) ⊂ L1 (I ; W −3; p ());
u · ∇ ∈ Lp (I ; W −2; p ()) ⊂ L1 (I ; W −3; p ());
(3.36a) (3.36b)
(e(∇u)) : e(∇u) ∈ L1 (Q) ⊂ L1 (I ; W −3; p ());
(3.36c)
Z ⊂ L1 (I ; W −3; p ()); g · u ∈ Lp (I ; rca())
(3.36d)
Z ⊂ L1 (I ; W −3; p ()); h ∈ L1 (I ; rca())
(3.36e)
from (1.1d) we obtain @ = kO − u · ∇ + 1 (e(∇u)) : e(∇u) + 2 g · u + h ∈ L1 (I ; W −3; p ()): @t (3.37) Z This gives ∈ C(I ; W −3; p ()). The proof of ∈ L∞ w (I ; rca()) in fact says more, Z From this, ∈ C(I ; (rca(); Z weak ∗ )) namely that t → (·; t) is also bounded to rca(). follows by standard arguments. Now it remains to prove global existence for the case 0 1 6 2 . We will do it by a successive prolongation, dividing the a priori given bounded time interval (0; T ) on a Unite number of suQciently short sub-intervals. Let us assume, for a moment, that we indeed have some solution to the coupled system (1.1) and (1.2). Testing (1.1a) by u and (1.1d) by a constant ; ¿ 0 and also using nonnegativity of proved in Proposition 1 (so that one has (d x; t) = (·; t)rca() Z ), we get t 1 2 up 1:p dt u(·; t)L2 (;Rn ) + ;(·; t)rca() Z + (1 − 1 ;)c W0 (;Rn ) 2 0
6 ;h + g · u + (;2 − 0 )g · urca(×[0; Z t]) + 12 u0 2L2 (;Rn ) + ;0 rca() Z :
(3.38)
Now we put ; = 0 =2 ; note that (;2 − 0 ) vanishes and (1 − 1 ;) ¿ 0 because 1 t 0 1 6 2 is assumed. Using the estimate ;h + g · urca(×[0; Z Z + 2 0 u t]) 6 ;hrca(Q) (·; t)2L2 (;Rn ) dt + 12 g2L2 (Q;Rn ) , by (a generalization of) Gronwall’s inequality, we get
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
750
constants M := L∞ Z ¡ + ∞ and Mu := uL∞ (I ;L2 (;Rn )) ¡ + ∞. Let us now w (I ;rca()) take a time step > ¿ 0 such that (3.24) holds with > instead of T and with (3.30a) using M and Mu instead of 0 rca() Z and u0 L2 (;Rn ) , respectively. We may also assume T=> integer. Let us now divide the time interval I = (0; T ) on T=> sub-intervals of the length >. Then, on each sub-interval one can use Proposition 1 (with T := >) but now even modiUed by allowing possibly test functions v not vanishing at terminal time so that the corresponding integral identities will contain additionally the terms u(x; t) · v(x; t) d x and Z v(x; t)(d x; t) at terminal time instances t = l>. This means we get for l = 1; 2; : : : ; T=> the following recursive integral identities l> @v u − ((u · ∇)u) · v − (e(∇u)) : e(∇v) + g · v d x (l−1)> @t g · v (d x; t) dt − u(x; l>) · v(x; l>) d x −0 Z
=−
ul−1 (x) · v(x; (l − 1)>) d x
(3.39)
for any v ∈ Lp (Il ; W01; p (; Rn )) ∩ W 1; p (Il ; L1 (; Rn )) with Il := [(l − 1)>; l>], and l> @v + u · (∇v + 2 gv) + kOv (d x; t) @t (l−1)> Z + (e(∇u)) : e(∇u) v d x + v h(d x; t) dt
−
Z
v(x; l>) (d x; l>) = −
Z
Z
v(x; (l − 1)>) l−1 (d x)
(3.40)
for any v smooth with (@=@ )v = 0 on . Note that the additional terms have a good Z weak ∗ )). Besides, all arguments sense because u ∈ C(I ; L2 (; Rn )) and ∈ C(I ; (rca(); in the proofs of Lemmas 1 and 2 can be modiUed appropriately. For e.g., in (3.14) one gets additionally the term Z v(x; T )! (d x; T ) which converges to Z v(x; T )(d x; T ) because, thanks to (3.18), @! =@t converges weakly∗ in rca(I ; W −3; p ()) to the limit @=@t belonging however to L1 (I ; W −3; p ()), cf. (3.37), so that t t @! @ ! (·; t) = 0 + (·; ) d → 0 + (·; ) d = (·; t) 0 @t 0 @t Z owing to weakly in W −3; p (), and this convergence holds even weakly∗ in rca() Z the a priori estimate of ! (·; t) in rca(). Limit passage in the term u(x; t) · v(x; t) d x p is even more standard because the mapping u → u(·; t) : Lp (I ; W0;1;DIV (; Rn )) ∩ W 1; p (I ; W −1; p (; Rn )) → L2 (; Rn ) is (norm; norm)-continuous, hence (weak; weak)continuous, as well. Of course, now we glue the solution on the particular sub-intervals simply by putting
ul = u(·; l>)
and
l = (·; l>)
for l = 1; 2; : : : ; T=>:
(3.41)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
751
Then, by summing (3.39) and (3.40) for l = 1; 2; : : : ; T=> we get just (2.3) and (2.4), respectively. Note that the initial and terminal terms on particular sub-intervals are mutually killed except the terms at t = 0 and T , and the terminal term with t = T eventually vanishes if tested as in (2.3) and (2.4). We can even establish certain more regularity in terms of absolute continuity of (·; t) if h is still a bit more regular and also if is absolutely continuous: Proposition 3. Let; in addition to assumptions of Proposition 2; also 0 ∈ L1 (); and h ∈ L1 (Q). Then (1:1) and (1:2) has at least one distributional solution (u; ) with ∈ L∞ (I ; L1 ()). Proof. The distributional solution (u; ) satisUes, in particular, @ + u · ∇ − kO − a = f @t with
(3.42)
f := 1 (e(∇u)) : e(∇u) + h ∈ L1 (Q); a := 2 g · u ∈ Lp (I ; L∞ ()) and (·; 0) = 0 ∈ L1 (): We now consider a and f Uxed. Taking regularized data f! ∈ L2 (Q) and 0! ∈ L2 (), we get by standard energy method a weak solution to (3.42), denoted by ! ∈ L2 (I ; W 1; 2 ()) ∩ L∞ (I ; L2 ()). Now we may assume f! → f in L1 (Q) and 0! → 0 in L1 (). Subtracting (3.42) for some !1 ; !2 ¿ 0 and testing it by sign2 (!1 − !2 ), we obtain similarly as in (3.10) and (3.12) the estimate |!1 (·; ) − !2 (·; )| d x
6
|0!1 − 0!2 |d x +
6 0!1 − 0!2 L1 () +
0
0
1 |a| |!1 − !2 | + |f!1 − f!2 | d x dt + 2 meas() 2
a(·; t)L∞ () !1 (·; t) − !2 (·; t)L1 ()
1 + f!1 (·; t) − f!2 (·; t)L1 () dt + 2 meas(): 2 From this we get by Gronwall’s inequality !1 − !2 L∞ (I ;L1 ()) 6 Cf!1 − f!2 L1 (Q) + C0!1 − 0!2 L1 ()
(3.43) (3.44)
with C depending on T and aL1 (I ;L∞ ()) . Thus one can see that {! }!¿0 is a Cauchy sequence in L∞ (I; L1 ()) which is a complete space. Simultaneously, this sequence Z converges to , the solution of (3.42), weakly∗ in L∞ w (I ; rca()), which shows that Z (I ; rca( )) to the linear problem belongs to L∞ (I ; L1 ()) because the solution ∈ L∞ w
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
752
(3.42) is unique. This uniqueness can be shown by the surjectivity of the operator v → (@=@t)v + u · ∇v + kOv + av likewise the uniqueness for (3.6) in the proof of Lemma 2, cf. (3.16) and (3.17). Remark 3. For u = 0, we even know that ! ∈ C(I ; L1 ()) by identifying ! with the so-called integral solution of (regularized) Eq. (3.42) in the sense of B(enilan [4]. This was generalized for u smooth and constant in time by Rulla [21]. It seems only a p technical matter to generalize it for u ∈ Lp (I ; W0;1;DIV (; Rn )). Then we could immediately strengthen the assertion of Proposition 3 to ∈ C(I ; L1 ()) because C(I ; L1 ()) is complete in its norm topology. Remark 4. The boundary condition (1:2b) might be generalized. One could consider a 1 nonhomogeneous Neumann condition @=@ = h1 with a prescribed heat "ux 1h1 ∈ L ( ). Then in identity (2.4) an additional term h1 v dS dt would appear and the L -technique (3.10) would have to be generalized, cf. [20]. 4. Supplement: Aubin theorem revisited A fundamental ingredient for evolution inUnite-dimensional systems is a compactness theorem by Aubin [1] for functions whose time derivative is bounded in Lp (I ; V3 ) with values in a re"exive Banach space V3 , see also [12, Section I:5:2]. Later, this result was generalized e.g. by Dubinski\+ [5] and Simon [22] for p = 1 and V3 an arbitrary Banach space, cf. also [19] for another generalization, namely V3 a HausdorL locally convex space. Here, we still needed a bit further generalization: Lemma 3. Let V1 , V2 , and V3 be Banach spaces having preduals V1 , V2 , and V3 , respectively; i.e. Vi = (Vi )∗ for i = 1; 2; 3. Moreover, let V1 ⊂ V2 compactly, V2 ⊂ V3 continuously, V1 and V2 be separable, 1 6 p ¡ + ∞. Then d ∞ ∈ rca(I ; V3 ) ⊂ Lp (I ; V2 ) (4.1) W := ∈ Lw (I ; V1 ); dt compactly in the sense that bounded sets in W are relatively sequentially compact in Lp (I ; V2 ). ∼ Proof (Sketch). Take {k } a bounded sequence in W. In particular, as L∞ w (I ; V1 ) = L1 (I ; V1 )∗ (cf. [6, Theorem 12:2:11]) and L1 (I ; V1 ) is separable, by Banach–Alaoglu– Bourbaki theorem we have (up to a subsequence) k *
∗ in L∞ w (I ; V1 ) weakly :
L∞ w (I ; V1 )
(4.2)
⊂ Lpw (I ; V2 ) ∼ = Lp (I ; V2 ), where the last every V2 -weakly measurable mapping is
Lpw (I ; V1 )
⊂ equality Also, we have uses the fact that, if V2 is separable, strongly measurable (or, more precisely, some elements of the equivalence classes enjoy this measurability, cf. [6, Examples 5:0:39 and 12:9:6]). By a contradiction argument, one can prove that ∀! ¿ 0 ∃a ¿ 0 ∀v ∈ V1 :
vpV2 6 !vpV1 + avpV3 ;
(4.3)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
753
see e.g. [5, Lemma 1] or [12, Section I:5:2, Lemma 5:1] or [22, Lemma 8] for details. We may consider = 0 in (4.2) without loss of generality. Take ! ¿ 0 Uxed. Putting v := k (t) into (4.3) and integrating it over I , we get k pLp (I ;V2 ) 6 ! sup k pLp (I ;V1 ) + ak pLp (I ;V3 ) = : T1k + T2k : k∈N
(4.4)
The term T1k 6 !T supk∈N k L∞ can be made arbitrarily small by taking ! ¿ 0 w (I ;V1 ) small enough. Then our goal is to show that limk→∞ T2k = 0. T T=2 Obviously, k pLp (I ;V3 ) = 0 k (t)pV3 dt + T=2 k (t)pV3 dt and we may investigate only, say, the Urst term. Take 2 ¿ 0, we can assume 2 6 T=2. For t ∈ I=2 := [0; T=2] we can decompose k = S2 k +
2
∗ %k
(4.5)
with the smoothening operator S2 and the convolution kernel 2 deUned by t 1 2 − 1 for 0 6 t 6 2; [S2 ](t) := (t + ) d; 2 2 (t) = 2 0 0 otherwise;
(4.6)
where “∗” denotes the convolution being deUned by the identity ∀v ∈ V3 :
[
2
∗ %](t); v = %;
2 (·
− t)v
(4.7)
with % ∈ L∞ (I ; V3 )∗ . If V3 would be Unite dimensional, % can be identiUed as a measure % ∈ vba(I ; V3 ) ∼ = L∞ (I ; V3 )∗ , the abbreviation “vba” indicating “bounded additive set functions vanishing on Lebesgue-zero-measure sets”, i.e. a certain Unitely additive measures, cf. [23]; then one could write 2 ( − ·)%(d) = − 1 %( + ·)(d): 2 ∗%= 2 0 R The concrete identiUcation of functionals from L∞ (I ; V3 )∗ however, is not important. Here, %k ∈ L∞ (I ; V3 )∗ is a suitable extension of dk =dt having the same norm, i.e. %k L∞ (I ;V3 )∗ = dk =dtrca(I ;V3 ) . (Note that some such extension always exists due to the Hahn–Banach theorem.) Here, we take %k as a weak∗ cluster point in L∞ (I ; V3 )∗ of a sequence {%k; ! }!¿0 ⊂ C(I ; V3 ) created by a molliUcation of the measure dk =dt. As the molliUer operator commutes with the time derivative, we get %k; ! = (d=dt)k; ! and k; ! (t) → k (t) strongly in V1 for a.a. t ∈ I , cf. [6, Theorem 5:0:23(b)]. Identity (4.5) can then be obtained by a limit passage from the classical by-parts integration identity: 2 d −1 k; ! (t + ) d [ 2 ∗ %k; ! ](t) = 2 d 0 2 2 1 = − − 1 k; ! (t + ) k; ! (t + ) d 2 =0 0 2 = −[S2 k; ! ](t) + k; ! (t) for all t ∈ I for which k; ! (t) → k (t) in V1 . T=2 T=2 T=2 (1) (2) Then 21−p 0 k (t)pV3 dt 6 0 S2 k (t)pV3 dt + 0 2 ∗ %k pV3 dt = : Ik2 + Ik2 . The convolution satisUes Young’s inequality g ∗ hLp (R) 6 gLp (R) hL1 (R) , cf. e.g.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
754
[6, Theorem 5:0:22]. This holds for h’s V3 -valued, as well. By a continuous extension from L1 (R; V3 ) to L∞ (R; V3 )∗ , we eventually get ∀g ∈ L∞ (R) : For h = %k and g = (2) Ik2
= %k ∗
g ∗ hLp (R;V3 ) 6 gLp (R) hL∞ (R;V3 )∗ : 2,
we get
p 2 Lp (I=2;V3 )
6 %k pL∞ (I=+2;V )∗ 2 pLp (R) 3
(4.8)
dk
p
6
2 pLp (R) dt
rca(I ;V3 )
with 2 Lp (R) = (p + 1)−p 21=p so that, by boundedness of dk =dt in rca(I ; V3 ), we (2) (2) have Ik2 can be made arbitrarily small if 2 ¿ 0 is small = O(2). In particular, Ik2 enough. Let us now take 2 ¿ 0 Uxed. By (4.2) with = 0, we have S2 k (t) * 0 in V1 for every t, hence also S2 k (t) → 0 in V3 because of the compactness of V1 ⊂ V3 . Thus C 2 S2 k (t)V3 6 CS2 k (t)V1 6 k (t + )V1 d 6 Ck L∞ ; w (I ;V1 ) 2 0 where C is the norm of the imbedding V1 ⊂ V3 . Thus S2 k (t) is bounded in V3 indepen T=2 (1) dently of k and t. By Lebesgue dominated convergence theorem, Ik2 := 0 S2 k (t) pV3 dt → 0 for k → ∞. In view of (4.4), the assertion is proved. Remark 5. Let us remark that assertion similar to Lemma 3 has been proved by Mielke et al. [15] who took, in our notation, p = 1, V1 = V3 = L1 (), V2 = W −1; 1 (), and assumed (t) a priori bounded in a compact subset of W −1; 1 (). They exploited necessary and suQcient condition [22, Theorem 1] for relative compactness of subsets of Lp (I ; B), B a Banach space. We preferred a direct proof of Lemma 3 to show the Une aspects about weak measurability and time derivatives in measures. References [1] J.-P. Aubin, Un th(eor]eme de compacit(e, C.R. Acad. Sci. 256 (1963) 5042–5044. [2] B.J. Bayly, C.D. Levermore, T. Passot, Density variations in weakly compressible "ows, Phys. Fluids A 4 (1992) 945–954. [3] A. Bedford, Hamilton’s Principle in Continuum Mechanics, Pitman, Boston, 1985. [4] P. Benilan, Solutions int(egrales d’(equations d’(evolution dans un espace de Banach, C.R. Acad. Sci. Paris 274 (1972) 47–50. [5] Yu.A. Dubinski\+ , Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik 67 (109) (1965) 609 – 642 (in Russian). [6] H.O. Fattorini, InUnite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. [7] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport, Hemisphere Publ., Washington, 1988. [8] Y. Kagei, Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function, Hiroshima Math. J. 25 (1995) 251–311. [9] Y. Kagei, M. RuH z% i%cka, G. Th^ater, Natural convection with dissipative heating, Comm. Math. Phys. 214 (2000) 287–313. [10] O.A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
755
[11] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959. [12] J.L. Lions, Quelques M(ethodes de R(esolution des Probl(emes aux Limites non lin(eaires, Dunod, Paris, 1969. [13] J. M(alek, J. Ne%cas, M. Rokyta, M. RuH z% i%cka, Weak and Measure-Valued Solutions to the Evolutionary PDE’s, Chapman & Hall, London, 1996. [14] J. M(alek, M. RuH z% i%cka, G. Th^ater, Fractal dimension, attractors and Boussinesq approximation in three dimensions, Act. Appl. Math. 37 (1994) 83–98. [15] A. Mielke, F. Theil, V.I. Levitas, Mathematical formulation of quasistatic phase transformation with friction using an extremum principle. preprint nr.A8 (VW-Stiftung I-70-284), Universit^at Hannover, September 1998. [16] K.R. Rajagopal, M. RuH z% i%cka, A.R. Srinivasa, On the Oberbeck–Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996) 1157–1167. [17] J.F. Rodriguez, A steady-state Boussinesq-Stefan problem with continuous extraction, Ann. Mat. Pura Appl. IV 144 (1986) 203–218. [18] J.F. Rodriguez, J.M. Urbano, On a three-dimensional convective Stefan problem for a non-Newtonian "uid, in: A. Sequiera et al. (Eds.), Nonlinear Applied Analysis, Plenum Press, New York, 1999, pp. 457–468. % [19] T. Roub(+c% ek, A generalization of the Lions-Temam compact imbedding theorem, Casopis P%e st. Mat. 115 (1990) 338–342. [20] T. Roub(+c% ek, Nonlinear heat equation with L1 -data, Nonlinear DiLerential Equations Appl. 5 (1998) 517–527. [21] J. Rulla, Weak solutions to Stefan problems with prescribed convection, SIAM J. Math. Anal. 18 (1987) 1784–1800. [22] J. Simon, Compact sets in the space Lp (0; T ; B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [23] K. Yosida, E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952) 46–66.
www.elsevier.com/locate/na
Buoyancy-driven viscous "ow with L1-data Jind%rich Ne%casa; b; ∗ , Tom(as% Roub(+c% ekb; c a Department
of Mathematical Sciences, Northern Illinois University, De Kalb, IL 60115 –2888, USA Institute, Charles University, Sokolovsk&a 83, CZ-186 75 Praha 8, Czech Republic c Institute of Information Theory and Automation, Academy of Sciences, Pod vod& arenskou v0e z0 &1 4, CZ-182 08 Praha 8, Czech Republic b Mathematical
Received 14 February 2000; received in revised form 30 April 2001
Keywords: Non-Newtonean "uids; Heat equation; Dissipative heat; Adiabatic heat; L1 heat sources
1. Introduction, problem formulation This paper deals with buoyancy-driven "ow of heat-conductive non-Newtonean incompressible "uids. There are various models appearing in the literature, cf. e.g. [2,9,16] for a genesis of various possibilities. The starting point is always the complete compressible "uid system of n + 2 conservation laws for mass, impulse, and energy; n denotes the spatial dimension. Then, the so-called incompressible limit represents a small perturbation around a stationary homogeneous state, i.e. around constant mass density, constant temperature, and zero velocity. It should be emphasized that, though the original full system is thermodynamically consistent, the incompressible limit system of n + 1 equations, in general, violates both the energy conservation law and the Clausius–Duhem inequality, except special cases as Remark 2 below. We consider a bounded Lipschitz domain in Rn ; n ¿ 2; I := [0; T ] a time interval, and denote Q := × I; := @ × I . To cover various possibilities, we consider the
This research has been carried out during a stay of the second author at the Northern Illinois University, % % partly supported also by the grants A 107 5707 (GA AV CR) and CEZ J13=98113200007 (CR). Besides, this author is thankful to Josef M(alek, Jan Mal(y , and Michael RuH z% i%cka for helpful discussions. ∗ Corresponding author. Mathematical Institute, Charles University, Sokolovsk( a 83, CZ-186 75 Praha 8, Czech Republic. E-mail addresses: [email protected] (J. Ne%cas), [email protected] (T. Roub(+c% ek).
c 2001 Elsevier Science Ltd. All rights reserved. 0362-546X/01/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 6 7 6 - 9
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
738
following fairly general system of equations: @u + (u · ∇)u − div (e(∇u)) + ∇ = g(1 − 0 ); @t
(1.1a)
e(∇u) = 12 ∇u + 12 (∇u)T ;
(1.1b)
div u = 0;
(1.1c)
@ + u · ∇ − kO = 1 (e(∇u)) : e(∇u) + 2 g · u + h; (1.1d) @t n n where [ij ] : [eij ] = i=1 j=1 ij eij ; k is the heat conductivity, 0 is the linearized relative mass density variation with respect to temperature, 1 re"ects dissipation eLects, 2 expresses adiabatic heat eLects, (e) is the viscous stress, g an external (e.g. gravitational) force, and h = h(x; t) is the external heat source. For notational simplicity, we normalized the mass density and the heat capacity to 1. For a rigorous derivation of a system like (1:1) we refer to Kagei et al. [9, System (16)] who showed how the coeQcient 1 depends on Ostrach’s dissipation number, while the coeQcient 2 also depends on the Reynolds and the Prandtl numbers. In [9] the Navier–Stokes case (i.e. (e) = e) has been considered and, for smooth data h and 0 , existence of a regular solution local in time or, in case of small data, global in time has been shown. Let us remark that our existence results, though holding globally for large and irregular data, unfortunately do not cover [9] because our needed assumption p ¿ n ¿ 2 just excludes the Navier–Stokes case. The system should be completed by boundary and initial conditions. For simplicity, we will consider an initial-value problem with no-slip boundary condition for velocity and perfect isolation for temperature, i.e. u(·; 0) = u0 ; u = 0;
(·; 0) = 0
on ;
@ = 0 on @
(1.2a) (1.2b)
with denoting the unit normal to the boundary @ of . Remark 1. Often, a simpler, so-called Oberbeck–Boussinesq model is used for buoyancydriven "ow of heat-conductive incompressible "uids. This model neglects both the dissipative and the adiabatic heat sources, i.e. 1 = 2 = 0, and usually considers (e) = e which turns (1.1a) – (1.1c) into the Navier–Stokes system, cf. e.g. [7] or [16], and sometimes it is combined with other phenomena as solidiUcation, see [17]. For a non-Newtonean model coupled with the heat equation we refer to M(alek et al. [14], where omitting the heat sources allowed to consider p-power law "uids even with p less than n, and to Rodriguez and Urbano [18] who allowed the viscosity to depend also on temperature. Besides, some models include the dissipative heat but not the adiabatic heat sources (i.e. our model (1:1) with 1 ¿ 0 but 2 = 0), cf. [11, Section 50] or also, e.g. [8].
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
739
Remark 2. Though perhaps formal, it is worth observing that, for 0 = 2 := and for 1 = 1, model (1:1) admits a consistent thermodynamical justiUcation. The departure point is the choice of the speciUc Helmholtz free energy as (; q) := (1 − )’(q) − ln ;
(1.3)
where q stands for the coordinate of the volume in question (i.e. u = dq=dt with d=dt standing for the material derivative in time); in other words, q is just the Lagrange coordinate. The Urst term is then the speciUc potential energy at the position q. This is proportional to the mass density, which however depends on the temperature due to the thermal expansion with the coeQcient . Without not much loss of generality, we assume here the force g to have a potential, which is just ’. In most applications, g is a constant gravitational acceleration so that ’(q) := g · q: Then one deUnes the standard speciUc entropy @ = ’(q) + (1 + ln ) @
s(; q) := −
(1.4)
and the speciUc internal energy !(; q) := (; q) + s(; q) = ’(q) + :
(1.5)
Then we postulate the kinetic energy Tkin , the dissipation potential R, and the total free energy, respectively, as 1 Tkin (u) := |u|2 d x; (1.6a) 2 1 T(e(∇u)) d x with T(e) := (te) : e dt; (1.6b) R(u) :=
#(; q) :=
0
(; q) d x
(1.6c)
provided we assume additionally smoothness and symmetry of , namely @ij =@ekl = |e|2 @kl =@eij . For example, in the special case ij = %(|e|2 )eij one has T(e) = 12 0 %(t) dt, cf. [13, Section 1:1:5]. The Hamilton variation principle adapted for nonconservative systems with holonomic constraints C(q) = 0 (cf. also [3, Section 1:3]) says that the T integral 0 Tkin (q)−U ˙ (q)+F; q+C(q) dt = 0 is stationary with U = #(; ·) being the potential energy and F = R (q) ˙ is a nonconservative force. This gives the momentum equation Tkin
du + R (u) + #q (; q) = C ∗ ; dt
Cq = 0;
(1.7)
where the apostrophe indicates the Gˆateaux diLerential, C = div forms the constraint and C ∗ = − ∇ is the adjoint operator, with playing the role of the Lagrange multiplier to the constraint Cq = 0, cf. also [3, Section 3:1:1]. Note that (1.7) is just
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
740
(1.1a) – (1.1c) because Cq = 0 implies also (d=dt)Cq = C(d=dt)q = Cu = 0 which is just (1.1c). Requiring the conservation of the total energy Tkin (u)+ !(; q) d x, the standard procedure gives the energy balance equation
@s + div j = )(u); @t
(1.8)
where j is a heat "ux and )(u) a dissipation rate; in our case (1.6b) we have )(u) = (e(∇u)) : e(∇u). Substituting (1.4) into (1.8) then gives (1.1d) if one puts j = − k∇u, i.e. the classical Fick law in isotropic media. This choice of j satisUes the Clausius–Duhem inequality j j · ∇ )(u) d + div − dx ¿ 0 (1.9) s(; q) d x = dt 2 provided the isolation condition (1.2b) is considered and ¿ 0, which is actually satisUed for any (even distributional) solution to (1.1d) and (1:2) if 0 ¿ 0 and also if the heat source h does not cool too much, certainly if h ¿ 0, cf. Proposition 1 below.
2. Distributional solution to (1:1) and (1:2) We want to treat system (1:1) in as much generality as possible (but still physical) situations. The heat transfer (1.1d) has a natural L1 -structure, which encourages us to consider the heat sources 0 ∈ L1 () and h ∈ L1 (Q), or even as measures. Then the concept of weak solution is no longer relevant, and one must speak in terms of distributive solutions or of integral solutions in the sense of Benilan [4], based on accretivity of the stationary part of (1.1d). For the convective term with u time-independent and smooth, this accretivity was observed already by Rulla [21]. Here we will use the former option, the latter technique being exploited for proofs only. We use the following standard notation for functions spaces: Lp (; Rn ) denotes the Lebesgue space of measurable functions → Rn whose p-power is integrable, W01; p (; Rn ) is the Sobolev space of functions whose gradient is in Lp (; Rn×n ) and p (; Rn ) = {v ∈ W01; p (; Rn ); div v = 0 in the sense whose trace on @ vanishes, W0;1;DIV 1; p of distributions}, and W −1; p (; Rn ) ∼ = W0 (; Rn )∗ . Likewise, W k; p indicates all kth derivatives belonging to the Lp space. Also, “rca” will denote the regular countably additive set functions with respect to a Borel ,-algebra in question, also called Radon Z Z measures. Furthermore, we denote by L∞ w (I ; rca()) the Banach space of C()-weakly Z Z the measurable essentially bounded functions : I → rca(), i.e., for any v ∈ C(), function t → Z v(x)(d x; t) is measurable and | Z v(x)(d x; t)| 6 CvC() Z for a.a. t ∈ I (with the null set possibly depending on v) for some C ¡ + ∞ independent of Z has a separable predual, namely C(), Z we can consider the norm v. Since rca() ∞ Z ∼ L∞ Z = ess supt∈I (t)rca() Z , cf. [6, Lemma 12:2:2]. It holds Lw (I ; rca()) = w (I ;rca()) 1 ∗ ∞ Z , see [6, Theorem 12:2:11]. Let us emphasize that Lw (I ; rca()) Z = L (I ; C()) Z L∞ (I ; rca()), see [6, Example 12:2:8].
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
741
We will assume the following data qualiUcation: |(e)| 6 C(1 + |e|p−1 );
(0) = 0;
(2.1a)
((e1 ) − (e2 )) : (e1 − e2 ) ¿ .|e1 − e2 |p ;
(2.1b)
Z Rn )); g ∈ L∞ (I ; C(;
(2.1c)
k ¿ 0;
Z 0 ∈ rca();
Z h ∈ rca(Q);
u0 ∈ L2 (; Rn )
(2.1d)
with . ¿ 0. Note that (2:1) ensures all integrals in (2.2) – (2.4) below to have a good sense provided p ¿ n, as we will have to assume anyhow. Let us also recall that (2.1b) ensures ((e(∇u1 )) − (e(∇u2 ))) : e(∇u1 − ∇u2 ) d x ¿ cu1 − u2 p 1; p (2.2) n W0 (;R )
with some c = c(.; ) ¿ 0 due to Korn’s inequality. By using Green’s formula once for (1.1a) – (1.1c) and twice for (1.1d) and by-parts integration in time, one gets the following deUnition: p Z (; Rn )) and ∈ L∞ Denition . We will call u ∈ Lp (I ; W0;1;DIV w (I ; rca()) a distributional solution to (1:1) and (1:2) if T @v u − ((u · ∇)u) · v − (e(∇u)) : e(∇v) + g · v d x 0 @t g · v (d x; t) dt = − u0 (x) · v(x; 0) d x (2.3) − 0 Z
for any v ∈ Lp (I ; W01; p (; Rn ))∩W 1; p (I ; L1 (; Rn )) with v(·; T ) = 0 and p = p=(p−1), and T @v + u · (∇v + 2 gv) + kOv (d x; t) @t 0 Z + 1 (e(∇u)) : e(∇u) v d x dt = − vh (d x dt) − v(x; 0) 0 (d x)
QZ
Z
(2.4)
for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on . 3. Existence of the distributional solution We will prove the existence rather nonconstructively by using the Schauder Uxed point theorem.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
742
First, we deUne the mapping
A : # → u : Lp (I ; W −1; p ()) → Lp (I ; W01; p (; Rn ))
(3.1)
with p denoting the conjugate exponent to p, i.e. p = p=(p−1), by u being the weak solution to @u + (u · ∇)u − div (e(∇u)) + ∇ = g(1 − 0 #); @t div u = 0;
u(·; 0) = u0 ;
u(@; ·) = 0:
(3.2)
Lemma 1. Let (2:1) be valid with p ¿ n. Then A is continuous with respect to norm topologies indicated in (3:1).
Proof. Take #k → # in Lp (I ; W −1; p ()) and denote by uk the weak solution to (3.2) corresponding to #k in place of #; for the existence of uk , even under weaker condition p ¿ 1 + 2n=(n + 2) we refer e.g. to [10] or [12, Sections II.5.2 and III.2.1]. By testing with uk , we get a standard a priori estimate uk L∞ (I ;L2 (;Rn ))∩Lp (I ;W 1; p (;Rn ))∩W 1; p (I ;W −1; p (;Rn )) 6 C:
(3.3)
0
Then we can take a weakly convergent subsequence in the space indicated in (3.3). It is a standard procedure to show that this subsequence converges to a weak solution u to (3.2) which is determined uniquely, so that even the whole sequence {uk } converges weakly to u, cf. [10,12, Section II:5], or, if generalized for # ∈ Lp (I ; W −1; p ())\L2 (Q) under the condition p ¿ n, also [13, Section 5:3]. Let us prove the strong convergence: subtracting (3.2) with u and uk , testing by T uk − u, and using Korn’s inequality (2.2) and 0 [@(uk − u)=@t](uk − u) d x dt ¿ 0, we get T cuk − upp 1; p 6 ((e(∇uk )) − (e(∇u)) : e(∇uk − ∇u) d x dt n L (I ;W0 (;R ))
0
6
T 0
((uk · ∇)uk − (u · ∇)u) · (uk − u)
+ 0 (#k − #)g · (uk − u) d x dt := I1k + I2k : We have I1k =
n T n 0
j=1
i=1
@ uki @xi
ukj −
n i=1
@ ui @xi
T n @ (uki ukj − ui uj )(ukj − uj ) d x dt = @x i 0 i; j=1
uj
(ukj − uj ) d x dt
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
=−
=−
T n i; j=1
0
T 0
(uki ukj − ui uj )
743
@ (ukj − uj ) d x dt @xi
(uk ⊗ uk − u ⊗ u) : ∇(uk − u) d x dt;
where, hopefully without confusion, the indices i; j = 1; : : : ; n indicate the respective components of the vectors u or uk while k indexes the members of the sequence {uk }k∈N . As indicated in (3.3), we have also an estimate on @uk =@t. Due to the compact imbedding W 1; p () ⊂ L∞ (), we can use the classical Aubin theorem [1] to see that uk → u strongly in Lp (I ; L∞ (; Rn )). As uk is also bounded in L∞ (I ; L2 (; Rn )), we get by standard interpolation that uk − uLp+2 (Q;Rn ) 6 uk − uLp (I ;L∞ (;Rn )) uk − uL1− ∞ (I ;L2 (;Rn )) → 0
(3.4)
with = p=(p + 2) ¿ 0, cf. e.g. [12, Section III.2.1]. Hence uk → u strongly also in Lp+2 (Q; Rn ). As also ∇uk → ∇u weakly in Lp (Q) and certainly p ¿ 2, we get I1k → 0. Also, the term I2k converges to zero because #k → # in Lp (I ; W −1; p ()) and 1; p uk → u in Lp (I ; W0 (; Rn )). Furthermore, let us consider the mapping ∞ Z Z B : (u; #) → : Lp (I ; W01; p (; Rn )) × L∞ w (I ; rca()) → Lw (I ; rca())
(3.5)
with being the distributional solution to @ + u · ∇ − kO = f; @t
(·; 0) = 0 ;
@ (@; ·) = 0 @
with f = 1 (e(∇u)) : e(∇u) + 2 g · u# + h; Z i.e. ∈ L∞ w (I ; rca()) satisUes the following identity: T @v v f(d x dt) + u · ∇v + kOv (d x; t) dt + @t 0 QZ Z + v(x; 0) 0 (d x) = 0 Z
(3.6)
(3.7)
for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on . Let us still point out that, p Z Z as W 1; p () is separable, by Fattorini [6] we have L∞ w (I ; rca()) ⊂ Lw (I ; rca()) ⊂ p −1; p p −1; p Lw (I ; W ()) ∼ ()), cf. also the proof of Lemma 3 below. = L (I ; W Lemma 2. Let (2:1) be valid. Then the mapping B is well de
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
744
Z with f! smooth and 0! → 0 weakly∗ Proof. Let us take f! → f weakly∗ in rca(Q) Z with 0! smooth, and also u! → u in Lp (I ; W 1; p (; Rn )) with u! smooth. in rca() 0 Denote by ! the weak solution to the auxiliary linear problem @ @ (@; ·) = 0; (3.8) + u! · ∇ − kO = f! ; (·; 0) = 0! ; @ @t the existence of ! can be proved by the standard energy method by testing (3.6) by and by @=@t provided particular u! ’s are bounded in L∞ (Q) as we can certainly suppose. Let us denote by sign2 : R → [ − 1; 1] the regularization of the multivalued function “signum” deUned as r=|r| if |r| ¿ 2; sign2 (r) = (3.9) r=2 otherwise: Moreover, we denote by | · |2 the primitive function to sign2 such that |0|2 = 0. Then |r| − 12 2 6 |r|2 6 |r|. Then we can test (3.8) with = ! by sign2 (! ), which gives d |! |2 d x − ((u! · ∇! )sign2 (! ) − k∇! · ∇sign2 (! )) d x dt (3.10) = f! (·; t) sign2 (! ) d x 6 |f! (·; t)| d x:
Using the facts that (u! · ∇! )sign2 (! ) d x = u! · ∇|! |2 d x = − (div u! )|! |2 d x = 0
(3.11)
and that ∇! · ∇sign2 (! ) ¿ 0, and integrating (3.10) in time over [0; t], we get 1 |! (·; t)| d x − 2 meas() 6 |! (·; t)|2 d x 2 t 6 |0! |2 d x + |f! | d x dt 0
6
|0! |d x +
0
T
|f! | d x dt:
(3.12)
As (3.12) holds for any 2 ¿ 0 and any t ∈ [0; T ], we eventually get the a priori estimate ! L∞ (I ;L1 ()) 6 f! L1 (Q) + 0! L1 () :
(3.13)
Z 0! → This estimate allows us to pass to the limit with f! → f weakly in rca(Q), Z and u! → u in Lp (I ; W 1; p (; Rn )). Then, by Banach–Alaoglu– 0 weakly∗ in rca(), 0 Bourbaki theorem, we can choose a subsequence such that ! → weakly∗ in Z L∞ w (I ; rca()). By passing to the limit in the integral identity @v + u! · ∇v + kOv ! + vf! d x dt + v(·; 0)0! d x = 0; (3.14) @t Q ∗
we obtain just identity (3.7); note that u! ! converges weakly because u! → u in Z Rn )) ⊂ L1 (I ; C(; Z Rn )) and ! converges weakly∗ in Lp (I ; W01; p (; Rn )) ⊂ Lp (I ; C(;
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
745
Z ∼ 1 Z ∗ L∞ w (I ; rca()) = L (I ; C()) . Thus, we proved existence of solving (3.6) for u and # Uxed. Also, from (3.13) we get the estimate L∞ Z 6 frca(Q) Z + 0 rca() Z : w (I ;rca())
(3.15)
Z Moreover, the solution ∈ L∞ w (I ; rca()) to the linear problem (3.6) is unique, which means that, if the identity T @v + u · ∇v + kOv (d x; t) d x dt = 0 (3.16) @t Z 0 holds for any v smooth with v(·; T ) = 0 on and (@=@ )v = 0 on , then necessarily = 0. This fact follows from the surjectivity of the adjoint operator to (3.6), more Z there is a smooth solution v to the problem precisely that, for any f ∈ L1 (I ; C()), @ @v + u · ∇v + kOv = f; v(·; T ) = 0; v = 0 on : (3.17) @ @t Z e.g. for In fact, it suQces to prove this for any f from a dense subset of L1 (I ; C()), Z f ∈ L2 (I ; C()), which is possible by a standard monotonicity technique. Moreover, from (3.6) we have also the a priori estimate
T
@!
k! Ov + ! (u! · ∇v) + f! v d x dt = sup
@t
−3; p rca(I ;W
v
())
6
v
61 3;p C(I ;W ()) 0
sup
0
(k! L1 (Q) OvL∞ (Q) + f! L1 (Q) vL∞ (Q)
61 3;p C(I ;W ()) 0
+ ! L∞ (I ;L1 ()) u! Lp (I ;L∞ (;Rn )) ∇vLp (I ;L∞ ()) ) 6 kTN1; p ! L∞ (I ;L1 ()) + N3; p f! L1 (Q)
+ N1; p N2; p T p (f! L1 (Q) + 0! L1 () )u! Lp (I ;W 1; p (;Rn )) ; (3.18) where Nk; p for kp ¿ n denotes the norm of the imbedding W k; p () ⊂ L∞ (). In the limit, we thus get
@
6 kTN1; p L∞ Z + N3; p frca(Q) Z
@t
w (I ;rca()) −3; p rca(I ;W
())
+ N1; p N2; p T p (frca(Q) Z )uLp (I ;W 1; p (;Rn )) : Z + 0 rca() (3.19)
To prove continuity of B, let us take #k → # in Lp (I ; W −1; p ()), the sequence {#k } p 1; p Z being supposedly bounded in L∞ (; Rn )), and w (I ; rca()), and uk → u in L (I ; W denote by k the distributional solution to (3.6) corresponding to uk and #k in place of u and #, respectively. Note that the right-hand side in (3.6), i.e. fk := 1 (e(∇uk )) : e(∇uk ) Z cf. also (3.33) below. Thus the distributional + 2 g · uk #k + h, is bounded in rca(Q), Z solution k does exist and is bounded in L∞ w (I ; rca()) and also @k =@t is bounded in
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
746
rca(I ; W −3; p ()). Then, by Banach–Alaoglu–Bourbaki theorem, we can assume that, possibly up to a subsequence, Z (3.20) k → weakly∗ in L∞ w (I ; rca()): Z V2 = By the generalized Aubin theorem (see Lemma 3 below with V1 = rca(), −3; p −1; p (), V3 = W ()), we have also W
strongly in Lp (I ; W −1; p ()):
(3.21) Z We also know that fk converges to f from (3.6) weakly in rca(Q); in fact, this convergence is even strong. Then we can make the limit passage in the integral identity (2.4) which reads here as T @v + uk · ∇v + kOv k (d x; t) dt + v fk (d x dt) + v(x; 0) 0 (d x) = 0: @t QZ Z 0 Z (3.22) k →
∗
Note that certainly the term k uk converges to u (even strongly) because {k } con verges strongly in Lp (I ; W −1; p ()) and {uk } also strongly in Lp (I ; W01; p (; Rn )). Thus = B(u; #) and even the whole sequence {k } converges because of the already proved uniqueness of . Furthermore, for %0 ; %1 ¿ 0, we deUne the convex sets Z (I ; rca( )); ess sup #(·; t) 6 % S%0 := # ∈ L∞ Z 0 ; w rca() t∈I
@#
# ∈ S%0 ;
6 %1 @t rca(I ;W −3; p ())
(3.23a)
S%0 ;%1 :=
(3.23b)
considered as a subset of Lp (I ; W −1; p ()). Proposition 1. Let 0 ¿ 0; h ¿ 0; (2:1) be ful
C1 ; C2 from (3:30) below:
(3.24)
Then (1:1) and (1:2) has at least one distributional solution (u; ) which; moreover; satis
C(#) := B(A(#); #):
(3.25)
Note that any Uxed point of C satisUes = B(u; ) and u = A(), so that the pair (u; ) is the distributional solution to (1:1) and (1:2). We will show that # ∈ S%0 ⇒ C(#) ∈ S%0 ;%1
(3.26)
provided %0 and %1 are chosen appropriately. Obviously, (u; ) = (A(#); C(#)) solves the decoupled system (3.2) and (3.6). Then we test (3.2) by u and add (3.6) tested
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
747
by ; sign with a constant ; ¿ 0; this latter test must be rigorously made by a regularization similarly as (3.13) and then passing to the limit. This eventually gives t 1 2 up 1:p dt u(·; t)L2 (;Rn ) + ; |(d x; t)| + (1 − 1 ;)c W0 (;Rn ) 2 Z 0 t 6 g · u d x + (0 + ;2 ) |#(d x; t)| |g · u| dt 0
Z
+ ;hrca(Q) Z +
2 1 2 u0 L2 (;Rn )
+ ;0 rca() Z ;
(3.27)
where c comes from (2.2) and | · | denotes the total variation when the argument is a measure. Note that we used also the estimate (1 − 1 ; sign )(e(∇u)) : e(∇u) d x ¿ (1 − 1 ;)cup 1; p . We can further estimate the right-hand side of (3.27) by n W0 (;R )
Young’s inequality as p |#(d x; t)| |g · u| 6 C! #(·; t)prca() Z + !u(·; t)W 1; p (;Rn ) ; Z
(3.28)
where C! depends on ! and also on gL∞ (Q;Rn ) and on p through the constant N1; p , namely C! = N1;pp (p!)1=(1−p) gpL∞ (Q;Rn ) =p . Then we can simply put (but perhaps not optimally) ; = 12 1−1 and ! = 14 c(0 + ;2 )−1 and from (3.27) and (3.28) we get again by Gronwall’s inequality uL∞ (I ;L2 (;Rn )) bounded and
p p L∞ Z 6 C1 + C2 #Lp (I ;rca()) Z 6 C1 + C2 T #L∞ (I ;rca()) Z w (I ;rca()) w
(3.29)
with some constants C1 and C2 , e.g. one can take 2 C1 = eT=(2;) (u0 2L2 (;Rn ) + 0 rca() Z + gL2 (Q) ); Z + hrca(Q)
C2 = eT=(2;) ;−1 |0 + ;2 |p gpL∞ (Q;Rn ) N1;pp (p!)1=(1−p) =p :
(3.30a) (3.30b)
If T is not too large, we will be able to Und %0 ¿ 0 such that
C1 + C2 T%p0 6 %0 ;
(3.31)
1−p
indeed, when taking %0 = (p C2 T ) , condition (3.24) ensures that (3.31) is satisUed. Moreover, for # ∈ S%0 bounded in p −1p Z ()) ∼ L∞ = Lp (I ; W −1; p ()); w (I ; rca()) ⊂ Lw (I ; W we get by testing (3.2) itself by u also the estimate .upLp (I ;W 1; p (;Rn )) 6 g(1 − 0 #) · urca(Q) Z 6 N1; p gLp (I ;C(;Rn )) 1 − 0 #L∞ Z uLp (I ;W 1; p (;Rn )) w (I ;rca()) with . from (2.1b), from which we can conclude that uLp (I ;W 1; p (;Rn )) 6 K%0
(3.32)
for a suitable K%0 . Furthermore, f from (3.6) can be estimated as p−1 )|∇u| L1 (Q) + 2 g · u#rca(Q) frca(Q) Z 6 C(1 + |∇u| Z + hrca(Q) Z
6 C meas(Q) + 2CK%p0 + 2 N1; p gLp (I ;C(;Rn )) K%0 %0 +hrca(Q) Z = : L %0 ;
(3.33)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
748
where C comes from (2.1a). Eventually, from (3.19) we have the estimate
@
6 kTN1; p %0 + N1; p N2; p T p (L%0 + 0 rca() Z )K%0
@t
rca(I ;W −3; p ()) +N3; p L%0 = : %1 : This choice of %1 will guarantee (3.26). We will endow S%0 ;%1 by the norm topology of Lp (I ; W −1; p ()). We can use the generalized Aubin theorem (see Lemma 3 below) in the same setting as in the proof of Lemma 2, which shows that S%0 ;%1 is sequentially compact in Lp (I ; W −1; p ()). By Lemmas 1 and 2 and by (3.26), C maps S%0 ;%1 continuously into itself, so by the Schauder theorem it has a Uxed point on S%0 ;%1 . The continuity of t → u(·; t) : I → L2 (; Rn ) follows standardly from the proved fact that u ∈ Lp (I ; W01; p (; Rn )) ∩ W 1; p (I ; W −1; p (; Rn )). It remains to prove nonnegativity of arbitrary Uxed point of C. We will test (1.1d) 1 1 by sign− 2 (): = 2 sign 2 (+2)− 2 with sign 2 deUned by (3.9); more precisely, one should Urst regularize (1.1d) similarly as in the proof of Proposition 3 below (thus can be considered as a weak solution) and after showing nonnegativity of one should pass to the limit. In such a way, speaking in terms of weak solutions, we obtain by using a corresponding modiUcation of (3.11) and (3.12) the estimate d − − [] d x 6 2 g · u sign2 () d x 6 2 |g · u|([]− (3.34) 2 + 2) d x; dt 2 − − where [ · ]− 2 denotes the primitive function to sign 2 such that [0]2 = 0. Integrating (3.34) over the time interval [0; t], one gets − (·; t) d x 6 [(·; t)]− 2 dx
6
6 2
[0 ]− 2 dx + 2 t 0
t 0
|g · u| []− 2 d xdt + 2t meas()
|g · u| − d xdt + 2t meas()
(3.35)
with − = 12 || − 12 denoting the negative part of ; we used − (·; 0) = 0− = 0, as assumed. As (3.35) holds for any 2 ¿ 0, we eventually obtain the estimate t − − |g · u| − d x dt (·; t)L1 () = (·; t) d x 6 2
= 2
0
0
t
[g · u](·; t)L∞ () − (·; t)L1 () dt;
Z recall that g · u ∈ Lp (I ; C()). From this we get − (·; t)L1 () 6 0 for all t ¿ 0 by Gronwall’s inequality, so that ¿ 0 follows. To obtain global existence at least in the case 0 1 6 2 , we need to assume a bit of regularity of the right-hand side f. This gives a certain additional continuity of ,
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
749
which allows us to speak about values of at current time level t needed to prolong the solution. Proposition 2. Let 0 ¿ 0; h ¿ 0, and (2:1) be ful
kO ∈ L∞ (I ; W −3; p ()) ⊂ L1 (I ; W −3; p ());
u · ∇ ∈ Lp (I ; W −2; p ()) ⊂ L1 (I ; W −3; p ());
(3.36a) (3.36b)
(e(∇u)) : e(∇u) ∈ L1 (Q) ⊂ L1 (I ; W −3; p ());
(3.36c)
Z ⊂ L1 (I ; W −3; p ()); g · u ∈ Lp (I ; rca())
(3.36d)
Z ⊂ L1 (I ; W −3; p ()); h ∈ L1 (I ; rca())
(3.36e)
from (1.1d) we obtain @ = kO − u · ∇ + 1 (e(∇u)) : e(∇u) + 2 g · u + h ∈ L1 (I ; W −3; p ()): @t (3.37) Z This gives ∈ C(I ; W −3; p ()). The proof of ∈ L∞ w (I ; rca()) in fact says more, Z From this, ∈ C(I ; (rca(); Z weak ∗ )) namely that t → (·; t) is also bounded to rca(). follows by standard arguments. Now it remains to prove global existence for the case 0 1 6 2 . We will do it by a successive prolongation, dividing the a priori given bounded time interval (0; T ) on a Unite number of suQciently short sub-intervals. Let us assume, for a moment, that we indeed have some solution to the coupled system (1.1) and (1.2). Testing (1.1a) by u and (1.1d) by a constant ; ¿ 0 and also using nonnegativity of proved in Proposition 1 (so that one has (d x; t) = (·; t)rca() Z ), we get t 1 2 up 1:p dt u(·; t)L2 (;Rn ) + ;(·; t)rca() Z + (1 − 1 ;)c W0 (;Rn ) 2 0
6 ;h + g · u + (;2 − 0 )g · urca(×[0; Z t]) + 12 u0 2L2 (;Rn ) + ;0 rca() Z :
(3.38)
Now we put ; = 0 =2 ; note that (;2 − 0 ) vanishes and (1 − 1 ;) ¿ 0 because 1 t 0 1 6 2 is assumed. Using the estimate ;h + g · urca(×[0; Z Z + 2 0 u t]) 6 ;hrca(Q) (·; t)2L2 (;Rn ) dt + 12 g2L2 (Q;Rn ) , by (a generalization of) Gronwall’s inequality, we get
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
750
constants M := L∞ Z ¡ + ∞ and Mu := uL∞ (I ;L2 (;Rn )) ¡ + ∞. Let us now w (I ;rca()) take a time step > ¿ 0 such that (3.24) holds with > instead of T and with (3.30a) using M and Mu instead of 0 rca() Z and u0 L2 (;Rn ) , respectively. We may also assume T=> integer. Let us now divide the time interval I = (0; T ) on T=> sub-intervals of the length >. Then, on each sub-interval one can use Proposition 1 (with T := >) but now even modiUed by allowing possibly test functions v not vanishing at terminal time so that the corresponding integral identities will contain additionally the terms u(x; t) · v(x; t) d x and Z v(x; t)(d x; t) at terminal time instances t = l>. This means we get for l = 1; 2; : : : ; T=> the following recursive integral identities l> @v u − ((u · ∇)u) · v − (e(∇u)) : e(∇v) + g · v d x (l−1)> @t g · v (d x; t) dt − u(x; l>) · v(x; l>) d x −0 Z
=−
ul−1 (x) · v(x; (l − 1)>) d x
(3.39)
for any v ∈ Lp (Il ; W01; p (; Rn )) ∩ W 1; p (Il ; L1 (; Rn )) with Il := [(l − 1)>; l>], and l> @v + u · (∇v + 2 gv) + kOv (d x; t) @t (l−1)> Z + (e(∇u)) : e(∇u) v d x + v h(d x; t) dt
−
Z
v(x; l>) (d x; l>) = −
Z
Z
v(x; (l − 1)>) l−1 (d x)
(3.40)
for any v smooth with (@=@ )v = 0 on . Note that the additional terms have a good Z weak ∗ )). Besides, all arguments sense because u ∈ C(I ; L2 (; Rn )) and ∈ C(I ; (rca(); in the proofs of Lemmas 1 and 2 can be modiUed appropriately. For e.g., in (3.14) one gets additionally the term Z v(x; T )! (d x; T ) which converges to Z v(x; T )(d x; T ) because, thanks to (3.18), @! =@t converges weakly∗ in rca(I ; W −3; p ()) to the limit @=@t belonging however to L1 (I ; W −3; p ()), cf. (3.37), so that t t @! @ ! (·; t) = 0 + (·; ) d → 0 + (·; ) d = (·; t) 0 @t 0 @t Z owing to weakly in W −3; p (), and this convergence holds even weakly∗ in rca() Z the a priori estimate of ! (·; t) in rca(). Limit passage in the term u(x; t) · v(x; t) d x p is even more standard because the mapping u → u(·; t) : Lp (I ; W0;1;DIV (; Rn )) ∩ W 1; p (I ; W −1; p (; Rn )) → L2 (; Rn ) is (norm; norm)-continuous, hence (weak; weak)continuous, as well. Of course, now we glue the solution on the particular sub-intervals simply by putting
ul = u(·; l>)
and
l = (·; l>)
for l = 1; 2; : : : ; T=>:
(3.41)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
751
Then, by summing (3.39) and (3.40) for l = 1; 2; : : : ; T=> we get just (2.3) and (2.4), respectively. Note that the initial and terminal terms on particular sub-intervals are mutually killed except the terms at t = 0 and T , and the terminal term with t = T eventually vanishes if tested as in (2.3) and (2.4). We can even establish certain more regularity in terms of absolute continuity of (·; t) if h is still a bit more regular and also if is absolutely continuous: Proposition 3. Let; in addition to assumptions of Proposition 2; also 0 ∈ L1 (); and h ∈ L1 (Q). Then (1:1) and (1:2) has at least one distributional solution (u; ) with ∈ L∞ (I ; L1 ()). Proof. The distributional solution (u; ) satisUes, in particular, @ + u · ∇ − kO − a = f @t with
(3.42)
f := 1 (e(∇u)) : e(∇u) + h ∈ L1 (Q); a := 2 g · u ∈ Lp (I ; L∞ ()) and (·; 0) = 0 ∈ L1 (): We now consider a and f Uxed. Taking regularized data f! ∈ L2 (Q) and 0! ∈ L2 (), we get by standard energy method a weak solution to (3.42), denoted by ! ∈ L2 (I ; W 1; 2 ()) ∩ L∞ (I ; L2 ()). Now we may assume f! → f in L1 (Q) and 0! → 0 in L1 (). Subtracting (3.42) for some !1 ; !2 ¿ 0 and testing it by sign2 (!1 − !2 ), we obtain similarly as in (3.10) and (3.12) the estimate |!1 (·; ) − !2 (·; )| d x
6
|0!1 − 0!2 |d x +
6 0!1 − 0!2 L1 () +
0
0
1 |a| |!1 − !2 | + |f!1 − f!2 | d x dt + 2 meas() 2
a(·; t)L∞ () !1 (·; t) − !2 (·; t)L1 ()
1 + f!1 (·; t) − f!2 (·; t)L1 () dt + 2 meas(): 2 From this we get by Gronwall’s inequality !1 − !2 L∞ (I ;L1 ()) 6 Cf!1 − f!2 L1 (Q) + C0!1 − 0!2 L1 ()
(3.43) (3.44)
with C depending on T and aL1 (I ;L∞ ()) . Thus one can see that {! }!¿0 is a Cauchy sequence in L∞ (I; L1 ()) which is a complete space. Simultaneously, this sequence Z converges to , the solution of (3.42), weakly∗ in L∞ w (I ; rca()), which shows that Z (I ; rca( )) to the linear problem belongs to L∞ (I ; L1 ()) because the solution ∈ L∞ w
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
752
(3.42) is unique. This uniqueness can be shown by the surjectivity of the operator v → (@=@t)v + u · ∇v + kOv + av likewise the uniqueness for (3.6) in the proof of Lemma 2, cf. (3.16) and (3.17). Remark 3. For u = 0, we even know that ! ∈ C(I ; L1 ()) by identifying ! with the so-called integral solution of (regularized) Eq. (3.42) in the sense of B(enilan [4]. This was generalized for u smooth and constant in time by Rulla [21]. It seems only a p technical matter to generalize it for u ∈ Lp (I ; W0;1;DIV (; Rn )). Then we could immediately strengthen the assertion of Proposition 3 to ∈ C(I ; L1 ()) because C(I ; L1 ()) is complete in its norm topology. Remark 4. The boundary condition (1:2b) might be generalized. One could consider a 1 nonhomogeneous Neumann condition @=@ = h1 with a prescribed heat "ux 1h1 ∈ L ( ). Then in identity (2.4) an additional term h1 v dS dt would appear and the L -technique (3.10) would have to be generalized, cf. [20]. 4. Supplement: Aubin theorem revisited A fundamental ingredient for evolution inUnite-dimensional systems is a compactness theorem by Aubin [1] for functions whose time derivative is bounded in Lp (I ; V3 ) with values in a re"exive Banach space V3 , see also [12, Section I:5:2]. Later, this result was generalized e.g. by Dubinski\+ [5] and Simon [22] for p = 1 and V3 an arbitrary Banach space, cf. also [19] for another generalization, namely V3 a HausdorL locally convex space. Here, we still needed a bit further generalization: Lemma 3. Let V1 , V2 , and V3 be Banach spaces having preduals V1 , V2 , and V3 , respectively; i.e. Vi = (Vi )∗ for i = 1; 2; 3. Moreover, let V1 ⊂ V2 compactly, V2 ⊂ V3 continuously, V1 and V2 be separable, 1 6 p ¡ + ∞. Then d ∞ ∈ rca(I ; V3 ) ⊂ Lp (I ; V2 ) (4.1) W := ∈ Lw (I ; V1 ); dt compactly in the sense that bounded sets in W are relatively sequentially compact in Lp (I ; V2 ). ∼ Proof (Sketch). Take {k } a bounded sequence in W. In particular, as L∞ w (I ; V1 ) = L1 (I ; V1 )∗ (cf. [6, Theorem 12:2:11]) and L1 (I ; V1 ) is separable, by Banach–Alaoglu– Bourbaki theorem we have (up to a subsequence) k *
∗ in L∞ w (I ; V1 ) weakly :
L∞ w (I ; V1 )
(4.2)
⊂ Lpw (I ; V2 ) ∼ = Lp (I ; V2 ), where the last every V2 -weakly measurable mapping is
Lpw (I ; V1 )
⊂ equality Also, we have uses the fact that, if V2 is separable, strongly measurable (or, more precisely, some elements of the equivalence classes enjoy this measurability, cf. [6, Examples 5:0:39 and 12:9:6]). By a contradiction argument, one can prove that ∀! ¿ 0 ∃a ¿ 0 ∀v ∈ V1 :
vpV2 6 !vpV1 + avpV3 ;
(4.3)
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
753
see e.g. [5, Lemma 1] or [12, Section I:5:2, Lemma 5:1] or [22, Lemma 8] for details. We may consider = 0 in (4.2) without loss of generality. Take ! ¿ 0 Uxed. Putting v := k (t) into (4.3) and integrating it over I , we get k pLp (I ;V2 ) 6 ! sup k pLp (I ;V1 ) + ak pLp (I ;V3 ) = : T1k + T2k : k∈N
(4.4)
The term T1k 6 !T supk∈N k L∞ can be made arbitrarily small by taking ! ¿ 0 w (I ;V1 ) small enough. Then our goal is to show that limk→∞ T2k = 0. T T=2 Obviously, k pLp (I ;V3 ) = 0 k (t)pV3 dt + T=2 k (t)pV3 dt and we may investigate only, say, the Urst term. Take 2 ¿ 0, we can assume 2 6 T=2. For t ∈ I=2 := [0; T=2] we can decompose k = S2 k +
2
∗ %k
(4.5)
with the smoothening operator S2 and the convolution kernel 2 deUned by t 1 2 − 1 for 0 6 t 6 2; [S2 ](t) := (t + ) d; 2 2 (t) = 2 0 0 otherwise;
(4.6)
where “∗” denotes the convolution being deUned by the identity ∀v ∈ V3 :
[
2
∗ %](t); v = %;
2 (·
− t)v
(4.7)
with % ∈ L∞ (I ; V3 )∗ . If V3 would be Unite dimensional, % can be identiUed as a measure % ∈ vba(I ; V3 ) ∼ = L∞ (I ; V3 )∗ , the abbreviation “vba” indicating “bounded additive set functions vanishing on Lebesgue-zero-measure sets”, i.e. a certain Unitely additive measures, cf. [23]; then one could write 2 ( − ·)%(d) = − 1 %( + ·)(d): 2 ∗%= 2 0 R The concrete identiUcation of functionals from L∞ (I ; V3 )∗ however, is not important. Here, %k ∈ L∞ (I ; V3 )∗ is a suitable extension of dk =dt having the same norm, i.e. %k L∞ (I ;V3 )∗ = dk =dtrca(I ;V3 ) . (Note that some such extension always exists due to the Hahn–Banach theorem.) Here, we take %k as a weak∗ cluster point in L∞ (I ; V3 )∗ of a sequence {%k; ! }!¿0 ⊂ C(I ; V3 ) created by a molliUcation of the measure dk =dt. As the molliUer operator commutes with the time derivative, we get %k; ! = (d=dt)k; ! and k; ! (t) → k (t) strongly in V1 for a.a. t ∈ I , cf. [6, Theorem 5:0:23(b)]. Identity (4.5) can then be obtained by a limit passage from the classical by-parts integration identity: 2 d −1 k; ! (t + ) d [ 2 ∗ %k; ! ](t) = 2 d 0 2 2 1 = − − 1 k; ! (t + ) k; ! (t + ) d 2 =0 0 2 = −[S2 k; ! ](t) + k; ! (t) for all t ∈ I for which k; ! (t) → k (t) in V1 . T=2 T=2 T=2 (1) (2) Then 21−p 0 k (t)pV3 dt 6 0 S2 k (t)pV3 dt + 0 2 ∗ %k pV3 dt = : Ik2 + Ik2 . The convolution satisUes Young’s inequality g ∗ hLp (R) 6 gLp (R) hL1 (R) , cf. e.g.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
754
[6, Theorem 5:0:22]. This holds for h’s V3 -valued, as well. By a continuous extension from L1 (R; V3 ) to L∞ (R; V3 )∗ , we eventually get ∀g ∈ L∞ (R) : For h = %k and g = (2) Ik2
= %k ∗
g ∗ hLp (R;V3 ) 6 gLp (R) hL∞ (R;V3 )∗ : 2,
we get
p 2 Lp (I=2;V3 )
6 %k pL∞ (I=+2;V )∗ 2 pLp (R) 3
(4.8)
dk
p
6
2 pLp (R) dt
rca(I ;V3 )
with 2 Lp (R) = (p + 1)−p 21=p so that, by boundedness of dk =dt in rca(I ; V3 ), we (2) (2) have Ik2 can be made arbitrarily small if 2 ¿ 0 is small = O(2). In particular, Ik2 enough. Let us now take 2 ¿ 0 Uxed. By (4.2) with = 0, we have S2 k (t) * 0 in V1 for every t, hence also S2 k (t) → 0 in V3 because of the compactness of V1 ⊂ V3 . Thus C 2 S2 k (t)V3 6 CS2 k (t)V1 6 k (t + )V1 d 6 Ck L∞ ; w (I ;V1 ) 2 0 where C is the norm of the imbedding V1 ⊂ V3 . Thus S2 k (t) is bounded in V3 indepen T=2 (1) dently of k and t. By Lebesgue dominated convergence theorem, Ik2 := 0 S2 k (t) pV3 dt → 0 for k → ∞. In view of (4.4), the assertion is proved. Remark 5. Let us remark that assertion similar to Lemma 3 has been proved by Mielke et al. [15] who took, in our notation, p = 1, V1 = V3 = L1 (), V2 = W −1; 1 (), and assumed (t) a priori bounded in a compact subset of W −1; 1 (). They exploited necessary and suQcient condition [22, Theorem 1] for relative compactness of subsets of Lp (I ; B), B a Banach space. We preferred a direct proof of Lemma 3 to show the Une aspects about weak measurability and time derivatives in measures. References [1] J.-P. Aubin, Un th(eor]eme de compacit(e, C.R. Acad. Sci. 256 (1963) 5042–5044. [2] B.J. Bayly, C.D. Levermore, T. Passot, Density variations in weakly compressible "ows, Phys. Fluids A 4 (1992) 945–954. [3] A. Bedford, Hamilton’s Principle in Continuum Mechanics, Pitman, Boston, 1985. [4] P. Benilan, Solutions int(egrales d’(equations d’(evolution dans un espace de Banach, C.R. Acad. Sci. Paris 274 (1972) 47–50. [5] Yu.A. Dubinski\+ , Weak convergence in nonlinear elliptic and parabolic equations, Mat. Sbornik 67 (109) (1965) 609 – 642 (in Russian). [6] H.O. Fattorini, InUnite Dimensional Optimization and Control Theory, Cambridge Univ. Press, Cambridge, 1999. [7] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport, Hemisphere Publ., Washington, 1988. [8] Y. Kagei, Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function, Hiroshima Math. J. 25 (1995) 251–311. [9] Y. Kagei, M. RuH z% i%cka, G. Th^ater, Natural convection with dissipative heating, Comm. Math. Phys. 214 (2000) 287–313. [10] O.A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.
J. Ne0cas, T. Roub&1c0 ek / Nonlinear Analysis 46 (2001) 737 – 755
755
[11] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959. [12] J.L. Lions, Quelques M(ethodes de R(esolution des Probl(emes aux Limites non lin(eaires, Dunod, Paris, 1969. [13] J. M(alek, J. Ne%cas, M. Rokyta, M. RuH z% i%cka, Weak and Measure-Valued Solutions to the Evolutionary PDE’s, Chapman & Hall, London, 1996. [14] J. M(alek, M. RuH z% i%cka, G. Th^ater, Fractal dimension, attractors and Boussinesq approximation in three dimensions, Act. Appl. Math. 37 (1994) 83–98. [15] A. Mielke, F. Theil, V.I. Levitas, Mathematical formulation of quasistatic phase transformation with friction using an extremum principle. preprint nr.A8 (VW-Stiftung I-70-284), Universit^at Hannover, September 1998. [16] K.R. Rajagopal, M. RuH z% i%cka, A.R. Srinivasa, On the Oberbeck–Boussinesq approximation, Math. Models Methods Appl. Sci. 6 (1996) 1157–1167. [17] J.F. Rodriguez, A steady-state Boussinesq-Stefan problem with continuous extraction, Ann. Mat. Pura Appl. IV 144 (1986) 203–218. [18] J.F. Rodriguez, J.M. Urbano, On a three-dimensional convective Stefan problem for a non-Newtonian "uid, in: A. Sequiera et al. (Eds.), Nonlinear Applied Analysis, Plenum Press, New York, 1999, pp. 457–468. % [19] T. Roub(+c% ek, A generalization of the Lions-Temam compact imbedding theorem, Casopis P%e st. Mat. 115 (1990) 338–342. [20] T. Roub(+c% ek, Nonlinear heat equation with L1 -data, Nonlinear DiLerential Equations Appl. 5 (1998) 517–527. [21] J. Rulla, Weak solutions to Stefan problems with prescribed convection, SIAM J. Math. Anal. 18 (1987) 1784–1800. [22] J. Simon, Compact sets in the space Lp (0; T ; B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [23] K. Yosida, E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952) 46–66.