On solvability of a non-linear heat equation with a non-integrable convective term and data involving measures

Nonlinear Analysis: Real World Applications 12 (2011) 571–591

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

On solvability of a non-linear heat equation with a non-integrable convective term and data involving measures✩ Miroslav Bulíček a , Luisa Consiglieri b , Josef Málek a,∗ a

Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague, Czech Republic

b

Department of Mathematics and CMAF, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal

article

abstract

info

Article history: Received 20 December 2009 Accepted 4 July 2010 Keywords: Non-linear heat equation Convective term Non-linear convection–diffusion equation Almost everywhere convergence of gradients Lipschitz approximation of a Bochner function Weak solution Entropy solution Large data Existence

Considering a mixed boundary-value problem for a non-linear heat equation with the nonhomogeneous Neumann condition, the right-hand side and the initial condition in space of sign-measures, we establish large-data existence results even if the convective term is not integrable. In order to develop a theory under minimal assumptions on given data, we deal with two concepts of solution: weak solution (for data in measures) and entropy solution (for L1 -data). Regarding the entropy solution we identify conditions ensuring its uniqueness. Improved properties of the Lipschitz approximations of Bochner functions represent an important tool in establishing the existence of large-data solutions. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Formulation of the problem For an unknown e, we consider a scalar non-linear parabolic equation1 e,t + div g(·, e) − div q(·, e, ∇ e) = f

in Q := (0, T ) × Ω ,

(1.1)

where T > 0 is the length of time interval and Ω is a bounded open set in Rd , d ≥ 2, with the Lipschitz boundary ∂ Ω . Eq. (1.1) is completed by the initial condition e(0, x) = e0 (x)

(x ∈ Ω )

(1.2)

and by the following mixed boundary conditions q(t , x, e(t , x), ∇ e(t , x)) · n(x) = fN (t , x) e(t , x) = e˜ D (t , x)

(t , x) ∈ Γ N ,

(t , x) ∈ Γ D ,

(1.3)

✩ Miroslav Bulíček and Luisa Consiglieri thank the Jindřich Nečas Center for Mathematical Modeling, the project LC06052 financed by MSMT, for its support. J. Málek’s contribution is a part of the research project MSM 0021620839 financed by MSMT; the support of GACR 201/05/0164 and GACR 201/06/0352 is also acknowledged. ∗ Corresponding author. Tel.: +420 221913220; fax: +420 222323394. E-mail addresses: [email protected] (M. Bulíček), [email protected] (L. Consiglieri), [email protected] (J. Málek). 1 One could also use the terms non-linear heat equation with convection or non-linear convection–diffusion equation to name Eq. (1.1).

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.07.001

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where n(x) denotes the outward normal at x ∈ ∂ Ω and Γ D , Γ N are defined as

Γ D := (0, T ) × ∂ Ω D ,

Γ N := (0, T ) × ∂ Ω N ,

where ∂ Ω D ⊂ ∂ Ω and ∂ Ω N := ∂ Ω \ ∂ Ω D are the open Lipschitz parts of the boundary ∂ Ω such that ∂ Ω D ∪ ∂ Ω N = ∂ Ω . We set the following assumptions on given functions q and g. First, for q > 1, we assume that q = q(t , x, e, u) : Q × R × Rd → Rd is a Carathéodory function that fulfills q-coercivity, (q − 1) growth and strict monotonicity conditions. More precisely, we require that

• for all (e, u) ∈ R × Rd : q(·, e, u) is measurable,

(1.4)

• for almost all (t , x) ∈ Q : q(t , x, ·, ·) is continuous in R × R , • there are C1 , C2 > 0 such that for all (e, u) ∈ R × Rd d

q(·, e, u) · u ≥ C1 |u|q

and |q(·, e, u)| ≤ C2 |u|q−1 ,

• for all e ∈ R and for all u1 , u2 ∈ R , d

(1.5)

(1.6)

u1 ̸= u2

(q(·, e, u1 ) − q(·, e, u2 )) · (u1 − u2 ) > 0.

(1.7)

We also assume that g(·, e) = v(·)e div v = 0

with a given v fulfilling :

in Q

and v · n = 0

on (0, T ) × ∂ Ω .

(1.8)

Such a structure of g and q appears naturally in various convection–diffusion problems as well as in the incompressible Navier–Stokes–Fourier system describing unsteady flows of incompressible fluids with possible dependence of the heat conductivity on the temperature, its gradient and some other relevant quantities. We discuss the latter in a more detail. To be more explicit in specifying our motivation to investigate (1.1), let us consider the Navier–Stokes–Fourier system for incompressible fluids with the density uniform at any place and time. The governing equations, that are a consequence of balance equations in continuum thermodynamics, take the form div v = 0,

v,t + div (v ⊗ v) = −∇ p + div S,

 |v|2

 e+

2

 + div

e+

,t

|v|2 2

(1.9)

 

+ p v − div q = div (Sv).

(1.10)

Here, v is the velocity, p is the pressure, e represents the (specific density of the) internal energy, q is the heat flux, and S is the constitutively determined part of the Cauchy stress. Behavior of many fluids can be well characterized by the constitutive relations: S = 2µ(e, p, |D|2 ) and

q = κ(e, ∇ e)∇ e.

(1.11)

In (1.11), 2D(v) stands for ∇ v + (∇ v)T , µ is the dynamical viscosity and κ is the heat conductivity. The physical background for the viscosities of the type (1.11)1 is given in detail in [1], while models with the heat conductivities of the form (1.11)2 are discussed in [2], see also [3,4] for more comments. It is quite popular among researchers in incompressible fluid mechanics to subtract the scalar product of (1.9)2 and v from (1.10) and consider the balance of energy in its equivalent2 form e,t + div (ev) − div q = S · D(v).

(1.12)

Upon inserting (1.11) into (1.12) we end up with the equation e,t + div (ev) − div (κ(e, ∇ e)∇ e) = 2µ(e, p, |D(v)|2 )|D(v)|2 .

(1.13)

The Eqs. (1.9) together with (1.13) represent, in three spatial dimensions, a closed system of five non-linear partial differential equations (for the five unknowns e, p and v = (v1 , v2 , v3 )) that reveal to be very difficult from the point of view of long-time and large-data mathematical analysis. This is due to several reasons. We state just two of them. First, the non-linearities are not only in e and v , but also in p, ∇ v and ∇ e, and the available a priori estimates concern of e, v, p, ∇ v and ∇ e, but not higher derivatives. Thus to control spatial and temporary oscillations of p, ∇ v and ∇ e is not a simple task. Second, in the three-dimensional setting, there are only a few the a priori (long-time and large-data) estimates available. Note that these global estimates form the basic building blocks for the long-time and large-data analysis. We do not want to list all these estimates here (preferring rather to refer the interested reader to [5,1,3] for details), we however would like to mention just one of them, namely, T

∫ 0

∫ Ω

µ(e, p, |D(v)|2 )|D(v)|2 dx dt < ∞.

(1.14)

2 This equivalence is however violated if we deal with the notion of weak solution and v is not admissible test function in the weak formulation of (1.9). Then one should not use (1.12) and deal with (1.10) only. See [5] or [1] for details.

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Put another way, the right-hand side of (1.13) is due to (1.14) only L1 -integrable, which is the cause of various complications. For example, the weak limits of sequences bounded in L1 can be identified as the signed Radon measures, but not in general as L1 -functions, etc. (see also discussion below in this paper). It is known that in some cases, for example if

µ = µ(p) or µ = const.,

(1.15)

the system (1.9) and (1.14) can be uncoupled. Indeed, since the viscosities of form (1.15) do not depend on e, one can solve (1.9) for (v, p) first and then study (1.14) where v is already fixed. Thus, using the notation f := 2µ(p)|D(v)|2 we observe that due to (1.14) f ∈ L1 (Q ) and (1.13) coincides with Eq. (1.1), which is the subject of analysis in this paper. We wish to mention that we are currently completing the development of the large-data mathematical theory for (1.9)–(1.11) (see [3]). The methods that have been developed in [3] are new, but technically still crude and complicated. Since the methods lead to remarkably new results even for (1.1), we decided to present our approach to the analysis of the scalar equation (1.1) first. This is the subject of this presentation. 1.2. Main results The main goal of this paper is to establish the large-data existence theory to (1.1)–(1.8) under as weak assumptions on q, f , fN , e˜ D and v as possible or to be more precise we want to establish a long-time and large-data existence theory under those assumptions that give at least formally meaning to all terms appearing in the weak or entropy formulations of (1.1). In order to define what we mean by weak and entropy solution and in order to formulate main theorems we fix notation of function spaces. For p ∈ [1, ∞] and k ∈ N, we use the symbols Lp (Ω ), Lp (Ω ) := Lp (Ω )d , W k,p (Ω ) and Wk,p (Ω ) := W k,p (Ω )d for standard k,p scalar- and vector-valued Lebesgue and Sobolev spaces. We set WD (Ω ) := {ϕ; ϕ ∈ W k,p (Ω ), ϕ|∂ Ω D = 0} and we denote k,p

its dual (WD (Ω ))∗ by W −k,p (Ω ). If X is a Banach space then Lp (0, T ; X ), ‖ · ‖p;X stands for the standard Bochner space. For q > 1, we define E through ′





E := e ∈ L∞ (0, T ; L1 (Ω )); for all λ ∈ (0, q − 1), (1 + |e|)



q−1−λ q

 ∈ Lq (0, T ; W 1,q (Ω )) .

Next, M (Q ), M(Γ N ), M (Ω ) denote the spaces of sign-measures on the sets Q , Γ N and Ω , respectively. We also write  ′ a · b =: (a, b)A whenever a ∈ Lq (A) and b ∈ Lq (A), and we use ⟨·, ·⟩A to denote the duality pairing. A Finally, for k, δ ∈ (0, ∞) we introduce the following truncating functions Tk and their δ -mollifications Tk,δ as Tk (z ) :=



z sign(z )k

Tk,δ (z ) :=



if |z | ≤ k, if |z | > k,

z sign(z )(k + δ/2)

(1.16) if |z | ≤ k, if |z | ≥ k + δ,

(1.17)

whereas Tk,δ is defined on (k, k + δ) in such a way that Tk,δ ∈ C 2 (R), 0 ≤ Tk′,δ ≤ 1 uniformly w.r.t. δ for z ∈ R and Tk,δ is concave on R+ and convex on R− . The symbols Θk and Θk,δ denote the primitive functions to Tk and Tk,δ respectively, i.e.,

Θk (s) :=

s

∫

Tk (t ) dt ,

Θk,δ (s) :=

0

s

∫

Tk,δ (t ) dt .

(1.18)

0

These definitions suffice to formulate the assumptions on data and state main existence results. Concerning Dirichlet boundary data e˜ D we assume the existence of an ‘‘extension’’ eD such that eD ∈ L∞ (0, T ; L∞ (Ω )) ∩ Lα (0, T ; W 1,β (Ω )) for some α, β ≥ 1

(1.19)

(eD ),t ∈ Lq (0, T ; W −1,q (Ω )) and eD |Γ D = e˜ D . ′

′

Theorem 1.1. Let 3 f ∈ M (Q ),

e0 ∈ M (Ω ) and

fN ∈ M (Γ N ).

(1.20)

Assume that the parameter q that appears in (1.6) fulfills q>

2d d+1

.

(1.21)

3 All statements studied in this paper hold for f of the form f = f + f , where f ∈ M (Q ) (or f ∈ L1 (Q )) and f ∈ Lq (0, T ; W −1,q (Ω )). This easy 1 2 1 2 generalization is omitted, for simplicity. ′

′

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M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

Assume also that v ∈ Lr (0, T ; Ls (Ω ))

such that r , s ≥ 1 satisfy s >

d(q − 1) q(d + 1) − 2d

and

r′ s

<

q(d + 1) − 2d d

.

(1.22)

Let e˜ D satisfy (1.19) with α ≥ max(r ′ , q) and β ≥ max(s′ , q). Then there is a weak solution e ∈ E to (1.1)–(1.8) such that for all k > 0 the solution e satisfies Tk (e) = Tk (eD ) on Γ D and

−(e, ϕ,t )Q + (q, ∇ϕ)Q = (ev, ∇ϕ)Q + ⟨f , ϕ⟩Q + ⟨e0 , ϕ(0)⟩Ω + ⟨fN , ϕ⟩Γ N for all ϕ ∈ D (−∞, T ; C ∞ (Ω )), ϕ|Γ D = 0.

(1.23)

We will show in Section 1.4 that not only weak solution is well defined for e ∈ E with q satisfying (1.21) and v fulfilling (1.22) but also some terms in (1.23) are not meaningful for q ≤ d2d without better information on e. On the other hand, +1 inspired by [6], one can introduce the so-called entropy solution to (1.1)–(1.8) that is well defined for all q ∈ (1, ∞) and even for v ∈ L1 (0, T ; L1 (Ω )). However, this concept of solution requires that the data f , fN and e0 belong at least to L1 . The following existence result holds. Theorem 1.2. Let f ∈ L1 (Q ),

e0 ∈ L1 (Ω ) and

fN ∈ L1 (Γ N ).

(1.24)

Assume that q from (1.6) satisfies 1 < q < ∞ and v from (1.8) fulfills v ∈ Lr (0, T ; Ls (Ω ))

with 1 ≤ r , s ≤ ∞.

(1.25)

Let e˜ D satisfy (1.19) with α ≥ max(r , q) and β ≥ max(s , q). Then there is an entropy solution e ∈ E to (1.1)–(1.8) such that for 4 for all k ≥ 0 and almost all t ∈ (0, T ) ′

′

Tk (e)|Γ D = Tk (eD )|Γ D ,

⟨ϕ,t , Tk (e − ϕ)⟩Qt +

∫ Ω

Θk (e(t ) − ϕ(t )) − Θk (e0 − ϕ(0)) dx + (Tk (e − ϕ)v, ∇ϕ)Qt + (q, ∇ Tk (e − ϕ))Qt

≤ (f , Tk (e − ϕ))Qt + (fN , Tk (e − ϕ))Γ N

(1.26)

t

′ ′ for all ϕ ∈ L∞ (0, T ; L∞ (Ω )) ∩ Lα (0, T ; W 1,β (Ω )) such that ϕ,t ∈ Lq (0, T ; W −1,q (Ω )) and ϕ|Γ D = eD |Γ D . ′

′

Moreover, if q does not depend on e, i.e., q(t , x, e, u) = q(t , x, u) and v ∈ Lq (0, T ; Lq (Ω )), then e is unique in the class of all entropy solutions satisfying (1.26). In (1.26) we started to use obvious notation: Qt := (0, t ) × Ω , ΓtN := (0, t ) × ∂ Ω N and ΓtD := (0, t ) × ∂ Ω D . Results related to Theorems 1.1 and 1.2 are discussed in Section 1.3. The main difficulty in the proofs of Theorems 1.1 and 1.2 is to show that if the sequence {en } is bounded in E then ∇ en converge almost everywhere (at least for a subsequence). Since this key step is common to the both theorems we formulate such a result separately, in Theorem 1.3. In addition, as we assume the boundedness of {en } in E a priori in this key step, we can establish the almost everywhere convergence of ∇ en for much more general g. More precisely, we assume that for a given Carathéodory function g(t , x, e) there exists a sequence of Carathéodory functions gn and there are scalar functions g ∈ L1 (Q ) and a ∈ L∞ loc (R) such that for a.a. (t , x) ∈ Q and all e ∈ R

  d −    n |g (t , x, e)| +  ∂xi gi (t , x, e) ≤ g (t , x)a(e)  i=1  n

(1.27)

and (as n → ∞) en → e a.e. in Q ⇒ gn (t , x, en ) → g(t , x, e)

a.e. in Q .

(1.28)

Theorem 1.3. Let q fulfill (1.4)–(1.7) with q > 1 and g approximated by gn fulfill (1.27)–(1.28). Let {f n }∞ n=1 be bounded in is bounded in E and fulfill L1 (0, T ; L1 (Ω )). Moreover, assume that {en }∞ n=1

⟨Ta,b (en ),t , ϕ⟩Q + (q(·, en , ∇ en ), ∇(Ta′,b (en )ϕ))Q = (f n Ta′,b (en ), ϕ)Q + (gn (·, en ), ∇(Ta′,b (en )ϕ))Q , 1,∞ for all ϕ ∈ L∞ (0, T ; W0 (Ω )) and all a, b ∈ R+ .

(1.29)

4 The relation (1.26) can be formally derived from (1.1) by taking the product of (1.1) and T (e − ϕ), integrating the result over Q , adding and subtracting k t ⟨ϕ,t , Tk (e − ϕ)⟩ + (v, ∇ϕ Tk (e − ϕ)), using the fact that div v = 0 and v · n|∂ Ω = 0, and replacing the equality by inequality.

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Then there exists a subsequence {eℓ } ⊂ {en }, and e such that e ∈ E,

(1.30)

∇ eℓ → ∇ e a.e. in Q .

The structure of the paper is the following. In Section 1.3 we discuss the related results established earlier and covering some special cases of Theorems 1.1 and 1.2. We are also concerned to explain main novelties of the approach presented here. The last part of this section contains various auxiliary calculations confirming the optimality of the results. In Section 2 we prove Theorems 1.1 and 1.2 assuming that the key step of the proof stated in Theorem 1.3 holds. Section 3 is devoted to the proof of Theorem 1.3. Finally, in Appendix we introduce the Lipschitz approximations of Bochner functions and summarize their properties needed for the proof of Theorem 1.3. 1.3. Novel approaches and related former studies We start recalling a very simple fact: if sup ‖un ‖q;Lq (Ω ) ≤ M < +∞,

(1.31)

n

and lim (q(·, e, um ) − q(·, e, un ), um − un )Q = 0

(1.32)

n,m→∞

then the strict monotone property (1.7) for q implies (see Appendix, [7]) the existence of u ∈ Lq (0, T ; Lq (Ω )) and {uℓ } ⊂ {un } such that uℓ → u

a.e. in Q .

(1.33)

In order to prove Theorem 1.3 that is the key step in the problem considered here, one observes that the first natural choice for {un }, namely un = ∇ en , is not admissible due to the lack of uniform estimates (1.31). Indeed, although q(·, e, u) fulfills Lq -coercivity and (q − 1) growth conditions (1.6), the uniform estimates (1.31) are not in place as the right-hand side (and the other data) is only L1 -integrable or belongs to the space of sign-measures M (Q ). It has been observed earlier (this idea goes back to Frehse [8] and Boccardo, Murat [9]) that the sequence un = ∇ Tk (en ) fulfills (1.31). If one proves, instead of (1.32), that for some θ ∈ (0,1)

∫



lim

n,m→∞

(q(·, e, um ) − q(·, e, un )) · (um − un )

θ

dx dt = 0

(1.34)

Q

then (1.33) follows as well. Thus, it suffices to deal with un = ∇ Tk (en ) and consider (1.34) instead (1.32). It has been well documented that the truncation functions Tk and Tk,δ can be successfully incorporated towards (1.34), under certain conditions. More precisely,5 Boccardo et al. in [10] proved Theorem 1.1 provided that6 q>

2d + 1

,

d+1 div g = div ve = v · ∇ e ∈ M (Q )

(1.35) and f ∈ M (Q ),

while Prignet [6] proved Theorem 1.2 for q > 1,

(1.36)

div g = div ve = v · ∇ e ∈ L1 (Q ) and f ∈ L1 (Q ). Also, Boccardo et al. [11] proved Theorem 1.1 under hypothesis that7 v ∈ L2 (Q ),

f ≡ 0 and q(·, e, u) = A(·, e)u.

In another relevant study Blanchard and Murat [12] proved Theorem 1.2 assuming that g = Φ(e),

(1.37) 8

(1.38)

5 Almost everywhere convergence of the gradient of solutions to elliptic problems were treated for example in [7]. 6 Boccardo et al. [10] and Prignet [6] do not consider the convective term div g explicitly in their formulation. However, their methods can be without any difficulties extended to cover (1.35) and (1.36). 7 It means that q = 2 in [11]. 8 While we consider the convection term of the form div g(x, t , e), Blanchard and Murat in [12] restricted themselves to a particular subcase div g(e). Their assumption (1.38) simplifies the analysis significantly since one is allowed to take ϕ := Tm (e) in (1.26). Such ϕ s are not however admissible test function if one assumes general relationships (1.27)–(1.28) investigated in this paper.

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where Φ is a continuous function. Our aim is to replace the conditions (1.35)–(1.38) by a significantly weaker requirements, namely g ∈ L1 (Q ) or g = ve with v ∈ L1 (Q ),

(1.39)

and simultaneously f ∈ M (Q )

or f ∈ L1 (Q ).

(1.40)

To treat these cases, further methods and approaches are needed. Note that the (formal) integration by parts ⟨div g, ϕ⟩Q = −(g, ∇ϕ)Q suggests to focus on Lipschitz continuous functions. Thus, one needs to introduce suitable Lipschitz approximations of Bochner functions. The fact that such an approximation approach could work is supported by its successful development to the recent analysis of various steady (elliptic-like) problems (see [13–16]), and evolutionary (parabolic-like) problems as well (see [17–19]). In this study, the articles [14,19] are used as the points of reference. In fact, even if the authors study a totally different problem in [19], their result is closely related to the problem (1.1)–(1.8). Precisely, applying the procedure developed in [19] step by step to (1.1)–(1.8) one proves Theorem 1.1 under the assumptions g ∈ L1 (Q ) and f ∈ Lq (0, T ; W −1,q (Ω )). ′

′

(1.41)

The restriction to such f ’s seems to be essential in analysis performed in [19]. On the other hand, various important applications9 require to deal with the right-hand side (or other data) in measures or in L1 . In order to develop methods to establish large-data existence theory for (1.1) with data fulfilling (1.39) and (1.40), we proceed, roughly speaking, in the following manner: we use L∞ -truncations Tk (e) to treat L1 -data (or data in measures), to have estimates of ∇ Tk (e) in Lq (Q ), and to deal with notion of entropy solution that is suitable to cover the case g = ev with v ∈ L1 (Q ), and to these L∞ -truncated functions we apply Lipschitz approximations in order to deal with convective term g under minimal integrability as stated in (1.39). As the consequence of such an extension we also obtain the generalized version of lemma on Lipschitz approximation of Bochner functions that was firstly introduced in [18] to improve integrability of very weak solution to parabolic equation with p-Laplacian and then used in [19] to get the existence of weak solution to unsteady flows of very general class of power-law-like incompressible fluids. This appropriate composition of L∞ -truncations and W 1,∞ -approximations is the main novel tool presented here in this paper that enabled us to prove Theorems 1.1 and 1.2 under very minimal assumptions on data. Evenmore, by Theorem 1.1 for data in measures we establish long-time and large-data existence result for q > 32 in three dimensions (it seems that even in the setting investigated in [10] we are able to give the same existence result for larger range of q). In order to relax even this bound and to establish result for all q > 1 we consider g of the form g = ve, require that v ∈ L1 (Q ) and deal with the concept of entropy solution.10 Theorem 1.2 formulates the statement regarding its existence for all q > 1 and g = ve fulfilling (1.39)2 . In addition, while the most earlier existence studies treat homogeneous Dirichlet boundary data, we consider here, nonhomogeneous mixed boundary-value problem under minimal requirements on data. The reason is to provide somehow a complete picture about the problem. We hope it does not make the presentation worse accessible to the readers. Of course, the results presented here are to our best knowledge new even for homogeneous Dirichlet problem. Finally, we wish to remark that this study is extended to the analysis of the evolutionary Navier–Stokes–Fourier-like systems for incompressible fluids with the heat conductivity depending on the temperature, its gradient, and other relevant quantities in [3]. Theorem 1.3 established here is one of the key steps used in [3]. 1.4. Auxiliary calculations The aim of this subsection is to observe that if e ∈ E all terms in (1.23) and (1.26), and also Tk (e)|Γ D are meaningful. Note first that if e ∈ E , then for all k > 0 Tk (e) ∈ Lq (0, T ; W 1,q (Ω ))

Tk′ (e)|∇ e|q ∈ L1 (Q ) and

(1.42)

and thus the trace of Tk (e) on Γ is well defined and all terms in (1.26) are finite. It also follows directly from definition of E that11 D

(1 + |e|) ∇e

q−1 q

1

(1 + |e|) q

∈ L∞ (0, T ; Lq (Ω )) ∩ Lq (0, T ; W 1,q (Ω )), ′

∈ Lq (0, T ; Lq (Ω )).

(1.43) (1.44)

9 For example, in the incompressible Navier–Stokes–Fourier system (see [5,3] for example) the role of f plays the (non-negative) product T · D, where T is the Cauchy stress and D is the symmetric part of the velocity gradient, and this is in general only L1 -quantity or even (non-negative) measure. 10 Concept of entropy solution seems to be equivalent to the concept of renormalized solution: this is e ∈ E fulfilling the inequality achieved from (1.29) after taking limit n → ∞. 11 For simplicity, we assume that we can take λ = 0 in the definition of E and then derive the consequences of this limiting case. These consequences then indeed hold if we do not require their validity for the limiting case relevant to taking λ = 0.

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577



 Thus, applying, for q > d2d ⇐⇒ d(dq−−q1) > 1 if q ∈ [1, d) , the interpolation inequality (see [20]) +1  [ d(q − 1) d(q − 1) for q ≤ d and a ∈ [1, ∞) for q > d, ‖ · ‖aq′ ≤ c ‖ · ‖q1′−α ‖ · ‖α1,q with α = ′ , a ∈ 1, a (qd + q − 2d) d−q to (1 + |e|)

q−1 q

we observe that

‖e‖a ≤ ‖(1 + |e|)‖a ≤ c ‖(1 + |e|)1−1/q ‖(q1′ −α)q ‖(1 + |e|)1−1/q ‖1α,qq . ′

′

This together with (1.43) implies that T

∫

‖e‖ba ≤ c

T

∫ 0

0

‖(1 + |e|)1−1/q ‖b1α,qq ≤ c < ∞ ⇐⇒ α bq′ ≤ q. ′

Requiring that a = b and α bq′ = q leads to a = e∈L

d(q−1)+q d

d(q − 1) + q

(Q ) and

d

d(q−1)+q , d

(1.45)

which implies that

> 1 ⇐⇒ q >

2d d+1

.

(1.46)

Next, by (1.6), |q(·, e, ∇ e)| ≤ C2 |∇ e|q−1 . Thus, we need to check if e ∈ E implies that |∇ e|q−1 ∈ Lz (Q ) with some z > 1. Since

∫

|∇ e|(q−1)z =

∫ 

Q

Q

|∇ e|q (1 + |e|)

 (q−q1)z

(1 + |e|)

(q−1)z q

∫ ≤ Q

|∇ e|q 1 + | e|

 (q−q1)z ∫

(q−1)z

(1 + |e|) q−(q−1)z

 q−(qq−1)z

< ∞, (1.47)

Q

where the finiteness is a consequence of (1.44) and (1.46) provided that

(q − 1)z q − (q − 1)z

≤

d(q − 1) + q d

⇐⇒ z ≤

d(q − 1) + q , (d + 1)(q − 1)

and it is easy to check that z > 1 without any additional constraint on q. Thus q is more than L1 -integrable if q > d2d . +1 r s 1 r′ s′ Finally, if v ∈ L (0, T ; L (Ω )) then ve belongs at least to L (Q ) provided that e ∈ L (0, T ; L (Ω )). Taking a = s′ and b = r ′ in (1.45) one observes that the requirements s′ <

d(q − 1) d−q

and

α r ′ q′ < q

are equivalent to (1.22). Also note that without any further information on e it seems that we cannot define consistently weak solutions to (1.1)–(1.8) for lower value of r , s and q. 2. Proofs of Theorems 1.1 and 1.2 We proceed in the following way. First, by smoothing the convective term and data we introduce (in Section 2.1) a sequence of auxiliary smooth problems that approximate (1.1)–(1.8). Once we discuss the solvability of these approximate problems, we derive (Section 2.2) the estimates that are uniform with respect to regularizing parameters. These uniform bounds, formulated in terms of suitable Bochner spaces, form a basis for taking limit from the approximate to the original problem. This limiting process is presented in Section 2.3 using however, in an essential manner, the result stated in Theorem 1.3. In Section 2.4 we prove the uniqueness result stated in Theorem 1.2. 2.1. Definition of approximate problem Let ωn denote a standard mollifier with support in a ball of radii 1n , i.e., ωn (t , x) = nd+1 ω(nt , nx) where ω ∈ C ∞ (Rd+1 );  supp ω ⊂ {(t , x) ∈ Rd+1 ; t 2 + |x|2 ≤ 1}, Rd+1 ω dx dt = 1, symmetric. For v ∈ Lr (0, T ; Ls (Ω )), we define Hn (v) by12

Hn (v) = (χ v) ∗ ωn − ∇ηn , where

χ ( t , x) =



0 1

if dist (x, ∂ Ω ) + dist (t , {0, T }) ≤ 2/n elsewhere,

12 H (v) is divergenceless part of the Helmholtz decomposition of (χ v) ∗ ω . n n

578

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

and ηn is such that div Hn (v) = 0

for all n ∈ N.

(2.1)

It means that ηn solves

1ηn = div [(χ v) ∗ ωn ] in Ω , ∫ ∇ηn · n = 0 on ∂ Ω , ηn dx = 0. Ω

Since div v = 0 and v · n|(0,T )×∂ Ω = 0, it follows from the definition and properties of Helmholtz decomposition that for all n:

Hn (v) ∈ L∞ (0, T ; Lk (Ω )) for all k ∈ [1, ∞),

(2.2)

Hn (v) → v strongly in Lr (0, T ; Ls (Ω )).

n ∞ n ∞ 1 1 N 1 We also introduce L∞ -sequences {f n }∞ n=1 , {fN }n=1 and {e0 }n=1 that are uniformly bounded in L (Q ), L (Γ ) and L (Ω ), ∗ N respectively, and that converges weakly to f , fN and e0 in M (Q ), M (Γ ) and M (Ω ), respectively (or strongly in L1 (Q ), L1 (Γ N ) and L1 (Ω ) in case that the limit functions f , fN and e0 belong to L1 ). We also approximate eD from (1.19) ′ ′ 13 by a sequence {enD }∞ in Lα (0, T ; W 1,β (Ω )) ∩ W 1,q (0, T ; W −1,q (Ω )) and n=1 of smooth functions converging to eD strongly ∗ ∞ ∞ n weakly in L (0, T ; L (Ω )). For such regular data, we define an approximate problem: for each n ∈ N find e such that

en,t + div Hn (v)en − div q(·, en , ∇ en ) = f n





q(·, e , ∇ e ) · n = n

n

e =

n

enD

fNn

in Q ,

on Γ , N

(2.3)

on Γ , D

en (0) = en0

in Ω .

The existence of weak solution to (2.3) can be established starting from Galerkin approximations and using standard monotone operator theory. 2.2. Estimates uniform with respect to n Let en be (possibly non unique) solution to (2.3) for given Hn (v), f n , fNn , enD and en0 . We aim to derive14 estimates for {en } that are uniform w.r.t. n ∈ N, or more precisely that are bounded by a constant depending on M ∈ (0, ∞) defined as





M := sup ‖enD ‖∞;L∞ + ‖enD ‖α;W 1,β + ‖(enD ),t ‖q′ ;W −1,q′ + ‖f n ‖1;L1 + ‖fNn ‖1,L1 + ‖en0 ‖1 + ‖Hn (v)‖r ;Ls (Ω ) .

(2.4)

n

First, multiplying (2.3)1 by Tk (en − enD ) (note that this function is equal to 0 on Γ D ) and integrating the result over Qt we obtain (after using integration per parts)

⟨en,t − (enD ),t , Tk (en − enD )⟩Qt + (q(·, en , ∇ en ), ∇ Tk (en − enD ))Qt + (Hn (v)Tk (en − enD ), ∇(en − enD ))Qt = (f n , Tk (en − enD ))Qt + (fNn , Tk (en − enD ))Γ N − ⟨(enD ),t , Tk (en − enD )⟩Qt − (Hn (v)Tk (en − enD ), ∇ enD )Qt .

(2.5)

t

The right-hand side (RHS) can be estimated by means of M (see (2.4)) and k so that (denoting Qtk := {(t , x) ∈ Qt ; |en − enD | ≤ k}) RHS of (2.5) ≤ C (k, M , C1−1 ) +

C1 2

∫ Qtk

|∇ en |q dx dτ .

(2.6)

To estimate LHS of (2.5) we observe first that (2.1) implies that the third term is identically zero. To estimate the second term we use the assumption (1.6) and (1.19), the Hölder and the Young inequalities and (2.4) and deduce that

(q(·, en , ∇ en ), ∇ Tk (en − enD ))Qt ≥

C1 2

∫ Qtk

|∇ en |q dx dτ − C (k, M , C1 , C2 ).

(2.7)

Since Θk′ = Tk (see (1.18)), we finally obtain by using (2.5)–(2.7) that

∫ Ω

Θk (en (t ) − enD (t )) dx ≤ C (M , k, C1 , C2 )

(2.8)

13 In case that α or β are infinity we assume only weak∗ convergence and we assume that the strong convergence holds in arbitrary ‘‘bigger’’ space. 14 These a priori estimates are for example shown in [10,21,4] under assumption that g ≡ 0 and for homogeneous boundary data. Thus sometimes, we do not provide all details.

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591 k2

and consequently, because Θk (s) = k|s| − k = 1) that

2

579

sgn s for |s| ≥ k and that |enD | ≤ M a.e. in Q , we deduce (taking for example

‖en ‖∞;L1 (Ω ) ≤ C (M , C1 , C2 ).

(2.9) n

15

Next, in order to estimate the behavior of ∇ e we first define s+ := max(0, s),

s− := min(0, s)

(2.10)

and define ϕ± := (1 ± (e − ) ) − 1 for some λ ∈ (0, q − 1). Clearly, ϕ± |Γ D = 0 and |ϕ± | ≤ 1, i.e., ϕ± ∈ L (Q ). Hence taking L2 -scalar product of (2.3)1 and ϕ± we obtain enD ± −λ

n

∞

⟨en,t , ϕ± ⟩Q + (Hn (v)ϕ± , ∇ en )Q + (q(·, en , ∇ en ), ∇ϕ± )Q = (f n , ϕ± )Q + (fNn , ϕ± )Γ N .

(2.11)

As a consequence of the fact that |ϕ± | ≤ 1 and the definition of M (see (2.4)) we observe that

  n (f , ϕ± )Q + (f n , ϕ± )Γ  ≤ M . N N

(2.12)

Decomposing the first integral in (2.11) as

16

(en,t , ϕ± )Q = ⟨((en − enD )± ),t , ϕ± ⟩Q + ⟨(enD ),t , ϕ± ⟩Q ]T ∫ [ ±1 = (1 ± (en (t ) − enD (t ))± )1−λ − (en (t ) − enD (t ))± dx + ⟨(enD ),t , ϕ± ⟩Q , Ω 1−λ t =0

(2.13)

we estimate the first term on RHS of (2.13) by means of M (given in (2.4)) and the Young inequality as follows:

 n  ⟨(e ),t , ϕ± ⟩Q  ≤ C + C1 λ D 4

∫

(1 ± (en − enD )± )−1−λ |∇ en |q dx dt ,

(2.14)

Q±

where Q± := {(t , x) ∈ Q ; (en − enD )± ̸= 0}. Similarly,

|(Hn (v)ϕ± , ∇ en )Q |

div Hn (v)=0

=

(2.4)

|(Hn (v)ϕ± , ∇ enD )Q | ≤ C (M ).

(2.15)

Finally, using (1.6), we find

λC1

∫

(1 ± (en − enD )± )−λ−1 |∇ en |q dx dt ∫ ≤ ∓(q(·, en , ∇ en ), ∇ϕ± )Q + C2 λ (1 ± (en − enD )± )−1−λ |∇ en |q−1 |∇ enD | dx dt . Q±

Q±

Consequently, using (2.4), the Young inequality and the fact that (1 + (en − enD )± )−1−λ ≤ 1, we derive that

λ

C1 2

∫

(1 ± (en − enD )± )−λ−1 |∇ en |q dx dt ≤ ∓(q(·, en , ∇ en ), ∇ϕ± )Q + C (M , λ).

(2.16)

Q±

Hence, combining (2.11)–(2.16), we deduce that

∫

(1 ± (en − enD )± )−λ−1 |∇ en |q dx dt ≤ C .

(2.17)

Q±

Finally, having (2.9), we will show that (2.17) implies that

‖en ‖E ≤ C .

(2.18)

To do it, we first observe that it is enough to show that

∫

(1 + |en |)−1−λ |∇ en |q dx dt ≤ C .

(2.19)

Q

Secondly, we conclude that since |enD | ≤ M, (2.17) immediately implies (2.19). Hence, (2.18) holds and consequently

‖en ‖E ≤ C H⇒ ‖Tk (en )‖q;W 1,q ≤ C .

(2.20) 17

Finally, using the Eq. (2.3) we simply deduce that

‖Tk (en ),t ‖1;W −2,z ′ (Ω ) ≤ C ,

for sufficiently large z .

15 Note that s ≥ 0 and s ≤ 0. + − 16 If λ = 1 then one has to use in (2.13) appropriate primitive function, namely ln(1 ± (en − en ) ). D ± 17 We take ϕ ∈ W 2,z ∩ W 1,z as a multiplier in Eq. (2.3). D

(2.21)

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M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

2.3. Limits for n → ∞ Since k is arbitrary, using (2.20)–(2.21) and generalized version of Aubin–Lions compactness lemma, see [22], we can find a not relabeled subsequence {en } such that en → e almost everywhere in Q ,

(1 + |en |)

q−1−λ q

⇀ (1 + |e|)

q−1−λ q

(2.22) weakly in Lq (0, T ; W 1,q (Ω )),

Tk (e ) ⇀ Tk (e) weakly in L (0, T ; W n

q

1,q

(2.23)

(Ω )).

(2.24)

n

Using Fatou’s lemma and the non-negativity of |e | we finally obtain that (2.24) e ∈ L∞ (0, T ; L1 (Ω )) H⇒ e ∈ E .

Combining (2.22)–(2.24) and strong convergence (2.2)2 , one can use the interpolation inequality used in Section 1.4 and deduce that

Hn (v)en → ve strongly in L1 (0, T ; L1 (Ω )).

(2.25)

At this point, it is simple to observe that all assumptions of Theorem 1.3 are satisfied. Consequently,

∇ en → ∇ e almost everywhere in Q .

(2.26)

Thus, using (2.22)–(2.26), convergence properties of f , , and Vitali’s theorem we can let n → ∞ in (2.3) and obtain (1.23). Note that the fact that Tk (e) = Tk (eD ) on Γ D for all k is direct consequence of convergence properties of enD , (2.24) and (2.26). It remains to verify the validity of (1.26). For this purpose, we consider arbitrary ϕ ∈ L∞ (0, T ; L∞ (Ω )) ∩ Lα (0, T ; ′ ′ 1,β WD (Ω )) satisfying ϕ,t ∈ Lq (0, T ; W −1,q (Ω )), and define ϕ n := ϕ + enD . Multiplying (2.3)1 by Tk (en − ϕ n ) (note that n n (e − ϕ )|Γ D = 0), integrating it over Qt , adding and subtracting ⟨ϕ,nt , Tk (en − ϕ n )⟩ + (Hn (v), Tk (en − ϕ n )∇ϕ n )Qt , using integration by parts, and the fact that div Hn (v) = 0 we come to the equality n

fNn

en0

⟨ϕ,nt , Tk (en − ϕ n )⟩Qt + ‖Θk (en − ϕ n )(t )‖1 − ‖Θk (en − ϕ n )(0)‖1 + (Tk (en − ϕ n )Hn (v), ∇ϕ n )Qt +(q(·, en , ∇ en ), ∇ Tk (en − ϕ n ))Qt = (f n , Tk (en − ϕ n ))Qt + (fNn , Tk (en − ϕ n ))Γ N .

(2.27)

t

Letting n → ∞ in (2.27), it follows from all properties proved above, and from strong convergence of Hn (v), f n , fNn , en0 and enD in corresponding spaces, that we can take the limit in the first, third, fourth, sixth and seventh terms in (2.27) and arrive at the corresponding terms in (1.26). Since Θk (s) ≥ 0 for all s ∈ R, one can use Fatou’s lemma to deduce that for a.a. t ∈ (0, T )

∫ Ω

Θk (e(t ) − ϕ(t )) dx ≤ lim inf n→∞

∫ Ω

Θk (en (t ) − ϕ n (t )) dx.

(2.28)

We split the remaining integral as (denoting qn := q(·, en , ∇ en ))

(qn , ∇ Tk (en − ϕ n ))Qt = (qn − q(·, en , ∇ϕ n ), ∇ Tk (en − ϕ n ))Qt + (q(·, en , ∇ϕ n ), ∇ Tk (en − ϕ n ))Qt =: I1n + I2n .

(2.29)

I1n

Due to the monotone properties of q (see (1.7)), is non-negative and because of almost everywhere convergence of en , ∇ en , ϕ n and ∇ϕ n we apply Fatou’s lemma to deduce that I1 := (q(·, e, ∇ e) − q(·, e, ∇ϕ), ∇ Tk (e − ϕ))Qt ≤ lim inf I1n . n→∞

(2.30)

Since I2n is a product of strongly and weakly converging sequences we conclude that n→∞

I2n → (q(·, e, ∇ϕ), ∇ Tk (e − ϕ))Qt =: I2 . Since I1 + I2 = (q(·, e, ∇ e), ∇ Tk (e − ϕ))Qt the proof of (1.26) is complete. 2.4. Uniqueness of entropy solution Assume that e1 ∈ E is an entropy solution fulfilling (1.26). Let e2 and {en2 } be the entropy solution and its approximate sequence constructed in the proof of Theorem 1.2. Our aim is to show that e1 = e2 in Q . We follow the procedure presented in [6] (we do modifications due to the presence of the convective term and non-homogeneous boundary conditions). For m ≥ M (where M comes from (2.4)), we take ϕ := Tm,δ (en2 ) − enD + eD in (1.26) (note that due to the smoothness of solution to the approximate problem (2.3) and because |enD | ≤ M, such ϕ multiplied by a smooth cut-off function is an admissible test function), multiply (2.3)1 by Tm′ ,δ (en2 )Tk (Tm,δ (en2 ) − enD + eD − e1 ) and integrate it over Qt , and finally add the resulting equality

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

581

(coming from (2.3)) and inequality (coming from (1.26)) to obtain (setting q1 := q(·, e1 , ∇ e1 ), qn2 := q(·, en2 , ∇ en2 ), q2 := n n n q(·, e2 , ∇ e2 ) and ψm ,δ := e1 − eD + eD − Tm,δ (e2 ))

⟨(eD − enD ),t , Tk (ψmn ,δ )⟩Qt +

∫ Ω

Θk (ψmn ,δ (t )) − Θk (ψmn ,δ (0)) dx + (q1 − qn2 Tm′ ,δ (en2 ), ∇ Tk (ψmn ,δ ))Qt

− (qn2 , Tk (ψmn ,δ )Tm′′ ,δ (en2 )∇ en2 )Qt + (v − Hn (v), ∇ Tm,δ (en2 )Tk (ψmn ,δ ))Qt + (v, Tk (ψmn ,δ )∇(eD − enD ))Qt     ≤ f − f n Tm′ ,δ (en2 ), Tk (ψmn ,δ ) Q + fN − fNn Tm′ ,δ (en2 ), Tk (ψmn ,δ ) Γ N . t

(2.31)

t

We would like to study the limit δ → 0 in (2.31). This limit can be identified easily in all terms of (2.31) except for the term −(qn2 , Tk (e1 − Tm,δ (en2 ))Tm′′ ,δ (en2 )∇ en2 )Qt . The limit in this term we shall discussed in detail. We start rewriting it by means of

t

Eq. (2.3). Defining Rm,δ (t ) := 0 1 − Tm′ ,δ (s) ds we have R′′m,δ = −Tm′′ ,δ and R′m,δ (s) = 0 for all |s| ≤ m. Thus, multiplying (2.3) by R′m,δ ((en2 )± ), integrating the result over Qt , using the integration by parts (since m ≥ M we have that R′m,δ ((en2 )± )|Γ D = 0) and the facts that div Hn (v) = 0 and Hn (v) · n|∂ Ω = 0, we conclude that (recall that s± are defined in (2.10))

∫ Ω

Rm,δ ((en2 (t ))± ) − Rm,δ ((en2 (0))± ) dx − (qn2 Tm′′ ,δ ((en2 )± ), ∇ en2 )Qt = (f n , R′m,δ ((en2 )± ))Qt + (fNn , R′m,δ ((en2 )± ))Γ N . t

Consequently, since Rm,δ is non-negative and Tk,δ is concave on R+ , and Rm,δ is non-positive and Tk,δ is convex on R− we deduce that qn2

0 ≤ ∓(

, Tm,δ (( ) )∇ ) ′′

en2 ±

en2 Qt

∫ ≤± Ω

Rm,δ ((en0 )± ) dx ± (f n , R′m,δ ((en2 )± ))Qt ± (fNn , R′m,δ ((en2 )± ))Γ N , t

which implies that

∫

qn2 Tm′′ ,δ

| Qt

( ) · ∇ | dx dt ≤ en2

en2

∫

|en0 | − Tm,δ (|en0 |) dx + (f n , Tm′ ,δ ((en2 )− ) − Tm′ ,δ ((en2 )+ ))Qt   + fNn , Tm′ ,δ ((en2 )− ) − Tm′ ,δ ((en2 )+ ) Γ N . Ω

(2.32)

t

Thus, after taking the limit in (2.31) w.r.t. δ and using (2.32), we obtain

⟨(eD − enD ),t , Tk (ψmn )⟩Qt +

∫ Ω

Θk (ψmn (t )) − Θk (ψmn (0)) dx

+ (q1 − qn2 Tm′ (en2 ), ∇ Tk (ψmn ))Qt + (v − Hn (v), ∇ Tm (en2 )Tk (ψmn ))Qt + (v, Tk (ψmn )∇(eD − enD ))Qt       ≤ f − f n Tm′ (en2 ), Tk (ψmn ) Qt + fN − fNn Tm′ (en2 ), Tk (ψmn ) Γ N + k ‖en0 ‖1 − ‖Tm (en0 )‖1 t     + k f n , Tm′ ((en2 )− ) − Tm′ ((en2 )+ ) Qt + k fNn , Tm′ ((en2 )− ) − Tm′ ((en2 )+ ) Γ N ,

(2.33)

t

n where ψm := e1 − eD + enD − Tm (en2 ). Next, we let n → ∞ in (2.33). The convergence properties of enD , en0 , f n , fNn and Hn (v) imply that the first, fourth18 and fifth integral on LHS of (2.33) vanish, and we can also identify the limits in all integrals on RHS of (2.33). Since Θk is nonnegative, using Fatou’s lemma, and strong convergence of en0 one can also take lim inf in the second integral of RHS. The third term of LHS is due to (1.7) bounded from below as follows:

(q1 − qn2 Tm′ (en2 ), ∇ Tk (ψmn ))Qt ≥ (q1 − qn2 Tm′ (en2 ), Tk′ (ψmn )∇(enD − eD ))Qt . Using the convergence properties of enD and the facts that |enD |, |eD | ≤ M and that e1 , e2 ∈ E we see that RHS of last inequality tends to zero (as n → ∞). Thus, defining ψm := e1 − Tm (e2 ), we obtain for19 a.a. m ∈ R+

∫

Θk (ψm (t )) − Θk (ψm (0)) dx − k (‖e0 ‖1 − ‖Tm (e0 )‖1 )     ≤ f (1 − Tm′ (e2 )), Tk (ψm ) Q + fN (1 − Tm′ (e2 )), Tk (ψm ) Γ N

Ω

t

t

+ k(f , Tm ((e2 )− ) − Tm ((e2 )+ ))Qt + k(fN , Tm, ((e2 )− ) − Tm′ ((e2 )+ ))Γ N . ′

′

′

t

(2.34)

Finally, we would like to let m → ∞ in (2.34) to conclude that

∫ Ω

Θk (e1 (t ) − e2 (t )) ≤ 0 H⇒ e1 = e2 a.e. in Q .

(2.35)

18 Here is the key limiting procedure where we need that v ∈ Lq′ (Q ). 19 The limiting procedure n → ∞ is valid only for almost all m ∈ R because we need to characterize the limit of T ′ (en ). For details see the proof of m 2 Lemma 3.2 in [10].

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M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

To the terms on LHS of (2.34) we apply Fatou’s lemma and use the fact that e0 ∈ L1 (Ω ). It remains to show that RHS of (2.34) vanishes. This is however a consequence of the fact that f ∈ L1 (Q ) and fN ∈ L1 (ΓN ) provided that (note that integrands in all terms are zero if |e2 | ≤ m) m→∞

meas {(t , x) ∈ Q ; |e2 (t , x)| ≥ m} → 0,

(2.36)

m→∞

meas {(t , x) ∈ Γ ; |e2 (t , x)| ≥ m} → 0.

(2.37)

Clearly, (2.36) follows from the fact that e2 ∈ L (Q ). To establish (2.37), we apply the continuity of the trace operator from 1

W 1,q (Ω ) into Lq (∂ Ω ) to (1 + |e2 |)

∫ Γ

q−1−λ q

and obtain

(1 + |e2 |)q−1−λ dS dt ≤ C ‖e2 ‖E ≤ C .

Taking λ =

q −1 , 2

we have

meas {(t , x) ∈ Γ ; |e2 (t , x)| ≥ m} ≤

∫  Γ

1 + |e2 |

 q−2 1

1+m

≤

C

(1 + m)

q−1 2

,

which implies (2.37), and the LHS in (2.35) is proved. Thus, the proof of uniqueness of entropy solution is complete. 3. Proof of Theorem 1.3 The proof of Theorem 1.3 is split into three steps. First, we formulate a sufficient condition (3.2) that implies the assertion. n,m n ,m In the second step we connect this condition with another condition (integrals I13 and I14 ) having the advantage that can be characterized by means of governing Eq. (1.29) and we show that all pollution terms that appeared in this process of n n comparison of (3.2) with I13 and I14 are small. Finally, we use the properties of Lipschitz approximation of Bochner functions (formulated in Appendix), weak formulation (1.29) of the problem, and conclude the assertion. 3.1. A sufficient condition to establish (1.30) Let ζ ∈ C0∞ (Q ) denote the cut-off function of the form

ζ (t , x) := ζ1 (t )ζ2 (x), where ζ1 ∈ C0∞ (0, T ), ζ2 ∈ C0∞ (Ω ) and ζ (t , x) = 1 if dist(x, ∂ Ω ) > η > 0 and t ∈ (η, T − η). To simplify the notation we denote qn,m := q(·, en , ∇ em ) and gn := gn (·, en ). We start observing that if en → e a.e. in Q

(3.1)

and for some θ > 0 lim

n,m→∞

n ,m I0

∫

 n ,n θ ζ (q − qn,m ) · ∇(en − em ) = 0,

:= lim

n,m→∞

(3.2)

Q

then, using the assumption (1.7) saying that q is strictly monotone in ∇ e, there exists a subsequence of {en } that we do not relabel such that ∇ en → ∇ e. Thus, it remains to show (3.1) and (3.2). For arbitrary a, b > 0 and arbitrary n ∈ N, we introduce the following Carathéodory functions Gna,b (t , x, v) : Q × R → Rd and gna,b (t , x, v) : Q × R → R through Gna,b

(t , x, v) :=

v

∫

gn (t , x, s)Ta′′,b (s) ds,

0

gna,b (t , x, v) :=

d −

(3.3)

  ∂xi Gna,b i (t , x, v),

i =1

and we set := (·, en ) and gna,b := gna,b (·, en ). Using this notation, notice that (1.29) means that the following renormalized equation holds in the sense of distribution Gna,b

Gna,b

Ta,b (en ) ,t − div qn,n Ta′,b (en ) + Ta′′,b (en )qn,n · ∇ en = f n Ta′,b (en ) − div gn Ta′,b (en ) − Gna,b − gna,b .













(3.4)

Thus, having en bounded in E , using assumption on q and g that implies that gn Ta′,b (en ) − Gna,b − gna,b are for all a, b uniformly   bounded in L1 (Q ) w.r.t. n and using the Eq. (3.4), we observe that for sufficiently large z Ta,b (en ) ,t are for all a, b bounded



1 ,z



(uniformly w.r.t. n) in L1 (0, T ; (WD (Ω ) ∩ W 2,z (Ω ))∗ ). Therefore, using generalized version of Aubin–Lions compactness

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

583

lemma (see [22]) and the fact that we can choose a, b arbitrarily, we conclude that there is a subsequence (that is again not relabeled) such that en → e a.e. in Q . Thus, (3.1) is proved. Next, since en ∈ E we can simply get that (by using (1.6))

∫

n,m

(1 + |∇ en |qθ + |∇ em |qθ )

≤c

I0

Q

∫

|∇ en |qθ |en |−θ |en |−θ ≤ c + c

≤ c + 2c

∫

Q

θ

|en | 1−θ .

(3.5)

Q n ,m

θ Thus, if 1−θ < 1 ⇔ θ < 21 , we see that I0 is finite. The rest of the paper is focused on proving (3.2). To simplify the notation, we write S (n, m, ε, ρ, Λ, δ, ν, k) ≤ O if the following holds

lim lim

lim

lim

lim

lim

lim S (n, m, ε, ρ, Λ, δ, ν, k) = 0,

k→∞ ν→∞ δ→0+ Λ→∞ ρ→0+ ε→0+ n,m→∞

and S is a quantity depending on marked parameters. We also define for positive numbers k, ε, δ the quantity

ωmn := Tk+ε,δ (en ) − Tk,δ (em ). Note, that it follows from (3.1) that m,n→∞

ε→0

‖Tk+ε,δ (e) − Tk,δ (e)‖1;L1 (Ω ) → 0. (3.6) Next, subtracting (3.4) for em and a = k and b = δ from (3.4) considered for en with a = k + ε and b = δ , we obtain in ‖ωmn ‖1;L1 (Ω )

→

the sense of distribution

  (ωmn ),t − div qn,n Tk′+ε,δ (en ) − qm,m Tk′,δ (em )

= −Tk′′+ε,δ (en )qn,n · ∇ en + Tk′′,δ (em )qm,m · ∇ em + Tk′+ε,δ (en )f n − Tk′,δ (em )f m    n ′  n m ′ m n m − gnk+ε,δ − gm k,δ − div g Tk+ε,δ (e ) − g Tk,δ (e ) − Gk+ε,δ + Gk,δ .

(3.7)

If we set Qmn := qn,n Tk′+ε,δ (en ) − qm,m Tk′,δ (em ), Hmn := gn Tk′+ε,δ (en ) − gm Tk′,δ (em ) − Gnk+ε,δ + Gm k,δ , mn

G

:=

−Tk′′+ε,δ (en )qn,n

n

· ∇e +

Tk′′,δ (em )qm,m

(3.8)

· ∇ e + Tk+ε,δ (e )f − Tk,δ (e )f − m

′

n

n

′

m

m

gnk+ε,δ



−

gm k,δ



,

and use this notation in (3.7), we see that ωmn solves the equation mn ω,mn − Hmn ) = Gmn . t − div (Q

(3.9)

3.2. A link connecting Eq. (3.9) with condition (3.2) n ,m

Our aim is to use (3.9) (or (3.7)) in order to replace I0 ε > 0, we have n ,m

0 ≤ I0

∫ = Q

∫ = =: For θ <

1 , 2

n,m

I2

+

n ,m I2

{|en −em |>ε}

.

(3.10)

we have for some fixed α(θ ) > 0 (using the same procedure as in (3.5))

∫

|∇ em |θ q + |∇ en |θ q

≤C {|en −em |>ε}

 α(θ ) ≤ C meas{(t , x) ∈ Q : |en (t , x) − em (t , x)| > ε}

Hölder ineq.

1

∫

|en − em | 2

α(θ)

χ{|en −em |>ε} 1 |en − em | 2 ∫ α(θ) (3.1) α(θ) 1 ≤ C ε− 2 |en − em | 2 ≤ O, =C

. First of all, for

 n,n θ ζ (q − qn,m ) · ∇(en − em ) ∫ [· · ·]θ + [· · ·]θ

{|en −em |≤ε} n ,m I1

n ,m

in (3.2). For this purpose we start modifying I0

Q

Q

584

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591 n ,m

arguing by means of Vitali’s theorem. In order to study the integral I1 n ,m

I1

∫



= Q

∫ =

θ ζ (qn,n − qn,m ) · ∇ Tε (en − em ) ∫ n ,m n ,m [· · ·]θ + [· · ·]θ =: I3 + I4 .

{|em |>k}

{|em |≤k}

n,m

Similarly as in estimating I2 n ,m

I4

we decompose it, for k > 0, as

∫ = {|em |≤k}

n ,m

, we obtain I3

≤

 C α(θ) k

≤ O . Concerning I4n,m we observe that, for δ > 0

[ζ (qn,n Tk′+ε,δ (en ) − qn,m Tk′,δ (em )) · ∇ Tε (Tk+ε,δ (en ) − Tk,δ (em ))]θ .

Next, setting amn := |∇ Tk+ε,δ (en )| + |∇ Tk,δ (em )|, we know that

‖amn ‖q;Lq (Ω ) ≤ ‖∇ Tk+ε,δ (en )‖q;Lq + ‖∇ Tk,δ (em )‖q;Lq ≤ C (k)(‖en ‖E + ‖em ‖E ) ≤ C (k). It follows from the proof of Theorem 2.5 in [14], see lines (2.23)–(2.27), that for all ν ∈ N there exists a sequence {λmn } ⊂ ν

ν+1

(22 , 22

) such that

λqmn meas({M(amn ) > λmn }) ≤ C (k)2−ν .

(3.11)

Denoting Dmn := {(t , x) : M (amn ) > λmn }, n ,m

we decompose I4 n ,m I4

as

∫

θ

∫

n ,m

[· · ·]θ =: I5

[· · ·] +

= {|em |≤k}∩D

{|em |≤k}\D

mn

n ,m

Again, similarly to the estimates of I2

mn

n ,m

and I3

n ,m

, we observe that I5

non-negative on the set {|em | ≥ k + δ}, we see that n ,m I6

∫

θ

≤

+ I6n,m .

θ

∫

[· · ·] ≤ C

[· · ·]

({|em |≤k}∪{|em |≥k+δ})\Dmn

≤



c (k) λmn

α(θ )

≤ O . Since the integrand in I4n,m is

.

Thus, we have n ,m 1 θ

C (I6

) ≤ =:

∫

∫ [· · ·] −

Q \Dmn n ,m n ,m I7 I8

+

[· · ·] {k<|em |
.

Using (1.6) and the basic properties of M we conclude that20

|I8n,m | ≤ C λqmn meas({k < |em | < k + δ}) ≤ O . n ,m

In order to finally connect I7 n ,m

I7

n,m

with I0

n,m

we add and subtract the term qm,m Tk′,δ (em ) into the integrand of I7

and obtain

∫

[ζ (qn,n Tk′+ε,δ (en ) − qm,m Tk′,δ (em ) + qm,m Tk′,δ (em ) − qn,m Tk′,δ (em )) · ∇ Tε (ωmn )] Q \Dmn ∫ n ,m =: I9 + [ζ Tk′,δ (em )(qm,m − qn,m ) · ∇ Tε (ωmn )]. =

Q \Dmn

We observe that the last integral tends to zero as m, n → +∞ due to the assumption on continuity of q and the fact that en , em and ∇ em take values in compact sets.

20 Here, in limiting procedure we know that meas({k < |e | < k + δ}) → meas({k < |e| < k + δ}) only for almost all k, δ . This is reason why in m definition of the symbol O only lim are taken into account. For details, see the proof of Lemma 3.2 in [10]. →

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

585

Recalling definition of Gmn and Hmn given in (3.8) we set Gmn := {(t , x); M αmn (|Gmn (t , x)|) > Λ},

Hmn := {(t , x) : M αmn |Hmn (t , x)| > 1},





where the numbers αmn > 0 will be specified later, and we write n,m I9

∫

∫

=

∫

+ Q \(Dmn ∪Gmn ∪Hmn ) n,m n ,m n ,m I10 I11 I12

=:

+

+

+ (Q \Dmn )∩Gmn

(Q \Dmn )∩Hmn

.

At this point, we apply Lemma A.1 and observe, that meas Gmn ≤ C (k, δ)Λ−1 meas Hmn ≤ C ‖Hmn ‖1;L1 (Ω ) → 0

as n, m → ∞ and ε → 0.

(3.12)

Consequently, n ,m |I11 | ≤ C (k, λmn ) meas (Gmn ) ≤ O , n ,m |I12 | ≤ C (k, λmn ) meas (Hmn ) ≤ O .

Finally, we define open sets Emn as Emn := Dmn ∪ Gmn ∪ Hmn . Since Emn and Eq. (3.9) satisfy the assumptions of Lemma A.4 in Appendix, we introduce the Lipschitz truncation operator αmn n,m LEmn and rewrite I10 as n,m

I10

∫

mn ζ (qn,n Tk′+ε,δ (en ) − qm,m Tk′,δ (em )) · ∇ Tε (LαEmn (ωmn )) ∫ ∫ n ,m n ,m = − =: I13 + I14 .

=

Q \Emn

Q

Emn

n ,m

n ,m

3.3. Smallness of I13 and I14 by means of Lipschitz approximation First, we summarize the important properties of the terms in (3.9), and the Lipschitz approximations of ωmn as well. The assumptions of Theorem 1.3 and (3.1) imply that

‖ωmn ‖1;L1 (Ω ) → 0 as m, n → ∞, ε → 0+ ,

(3.13)

‖H ‖1,L1 (Ω ) → 0 as m, n → ∞, ε → 0+ ,   1 mn ‖G ‖1;L1 (Ω ) ≤ C k, , δ   1 ‖Qmn ‖q′ ;Lq′ (Ω ) ≤ C k, , δ

(3.14)

mn

(3.15)

(3.16)

and it follows from Lemma A.4 and (A.15) that αmn

‖∇ LEmn (ω )‖ mn



∞;L∞ (Ω )

1



1

≤ C k, , , λmn , Λ , δ αmn

mn ‖LαEmn (ωmn )‖1;L1 (Ω ) → 0 as m, n → ∞, ε → 0+ .

(3.17) (3.18)

n ,m

At this point, we discuss the smallness of I14 . From a priori estimates (that means from the fact that en are bounded in E ), we obtain n,m I14

≤ C (k)

∫ Emn

  ∇(Lαmn (ωmn ))q Emn

 1q

.

α

Next, let {QRi mn (Xi )} be the covering Emn from Lemma A.2 in Appendix. Then, for ρ > 0, we decompose the set Emn into two parts such that ρ

Emn,1 := {X = (t , x) ∈ Emn : ∃Ri > ρ, X ∈ QRi (Xi )}, ρ

ρ

Emn,2 := Emn \ Emn,1 .

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M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

ρ

α

α

mn If X ∈ Emn,1 then clearly there is Xi such that X ∈ QRi mn (Xi ). It follows from the definition of LEmn given in (A.14) and the properties of the covering formulated in Lemma A.2 that

   −     mn ∇x (Lαmn (ωmn ))(X ) = ∇ αmn  ψ ( X )ω j Q Emn Rj  j∈A  i − 1∫ C − ≤C |ωmn | ≤ αmn d+3 ‖ωmn ‖1;L1 (Ω ) . α mn Rj Q ρ j∈A R i

(3.19)

j

ρ

α

For X ∈ Emn,2 , we first note that X is contained in QRi mn (Xi ) for certain i (now fixed). Using again the properties of the covering stated in Lemma A.2 we have

      −  α mn mn ∇x (L (ω ))(X ) = ∇ ψj (X )ωmn Q αmn  Emn Rj  j∈A  i ∑    ψj =1     − j∈Ai   = ∇ ψj (X ) ωmn Q αmn − ωmn Q αmn + ωmn Q αmn  Rj 4Ri 4Ri   j∈Ai   −C   ωmn Q αmn − ωmn Q αmn  ≤   R 4R j i Rj j∈Ai    − C ∫  mn mn αmn dY  − ω ( Y ) − = ω Q   4Ri Rj  Q αmn  j∈Ai Rj ∫   C  mn  ≤ − ω (Y ) − ωmn Q αmn  dY . α Q4Rmn

Ri

(3.20)

4Ri

i

Applying Poincaré inequality (A.13) for ωmn to the estimate (3.20) together with the fact (Ri ≤ ρ) we have

∫   ∇x (Lαmn (ωmn ))(X ) ≤ C − Emn

α Q4Rmn

|∇ωmn | + αmn (|Qmn | + |Hmn | + ρ|Gmn |)dY .

(3.21)

i

α

α

mn Moreover, from the property (A.10) of the covering, that there exists Z ∈ Q \ Emn such that Q4Rmn (Xi ) ⊂ Q12R (Z ), thus using i i (3.21) we get

∫   ∇x (Lαmn (ωmn ))(X ) ≤ C − Emn

αmn Q12R (Z )

|∇ωmn | + αmn (|Qmn | + |Hmn | + ρ|Gmn |)dY .

(3.22)

i

Since Z ∈ Q \ Emn we can use the definition of the set Emn to estimate all integrals appearing in (3.22) to conclude that

  ∇x (Lαmn (ωmn ))(X ) ≤ C (λmn + αmn (λq−1 + 1 + ρ Λ)). mn Emn

(3.23) n,m

Finally, using (3.23) and (3.19) we can easily bound the integral I14 such that 1

n ,m

q −1 I14 ≤ C (k)(αmn ρ −d−3 ‖ωmn ‖1;L1 + λmn + αmn (λmn + 1 + ρ Λ))(meas(Dmn ) + meas(Gmn ) + meas(Hmn )) q .

For the sequence λmn satisfying (3.11), we define 2−q αmn := λmn .

(3.24) q mn

n ,m I14

It follows from (3.12), (3.13) and (3.11) (note that λmn ≤ λ as λmn ≥ 1) that ≤ O. αmn n ,m The last step is to bound the integral I13 . We use ζ Tε (LEmn (ωmn )) as a test function in (3.7) to obtain that n ,m

I13

∫ = Q

∫

mn ζ Qmn · ∇ Tε (LαEmn (ωmn ))

∫   αmn αmn mn = Q · ∇ ζ Tε (LEmn (ω )) − Qmn Tε (LEmn (ωmn )) · ∇ζ Q Q ∫ (3.9) αmn αmn mn = −⟨ω,mn , ζ T ( L (ω ))⟩ + Gmn Tε (LEmn (ωmn )) ε Q t Emn Q ∫   αmn mn + Hmn · ∇ ζ Tε (LEmn (ωmn )) − Qmn Tε (LαEmn (ωmn )) · ∇ζ . mn

Q

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

587

Using (3.13)–(3.18) we obtain that all terms except time derivative ‘‘are’’ O quantities, namely α

mn mn mn I13 ≤ O − ⟨ω,mn ))⟩Q . t , ζ Tε (LEmn (ω

(3.25)

To estimate the last term in (3.25) we use (A.18) to conclude that αmn

∫

−⟨ω , ζ Tε (LEmn (ω ))⟩Q ≤ C ‖ω ‖1;L1 + mn ,t

mn

mn

Emn

mn mn |(LαEmn (ωmn )),t | · |ωmn − LαEmn (ωmn )|.

α

Next, we estimate the time derivative in the last integral. Let X ∈ QRi mn then



αmn

−

(LEmn (ω )),t (X ) = mn

   mn mn mn ψj (X ) ω Q αmn − ω Q αmn + ω Q αmn Rj

j∈Ai

=

−

∫ (ψj ),t (X )−

4Ri

,t

ωmn (Y ) − ωmn Q αmn dY .

α QR mn

j∈Ai

4Ri

4Ri

j

Thus, using the properties of ψj and our covering we have

∫ −1 −2 mn |(LαEmn (ωmn )),t (X )| ≤ C αmn Ri −

|ωmn (Y ) − ωmn Q αmn | dY .

α Q4Rmn

(3.26)

4Ri

i

Moreover, for the second term in the integral we have

(ω

mn

αmn

− LEmn (ω ))(X ) = mn

−



ψj (X ) ω (X ) − ω mn

j∈Ai

∫ − ψj (X )− =

αmn

 mn αmn QR j

 mn  ω (X ) − ωmn (Y ) dY .

QR

j∈Ai

j

αmn

Integrating this relation w.r.t. X over QRi

, adding and subtracting ωmn Q αmn and using the property (A.11) of covering, we 4Ri

are led to the following inequality

∫ α QR mn

mn |(ωmn − LαEmn (ωmn ))(X )| dX ≤ C

∫ α Q4Rmn

i

|ωmn (Y ) − ωmn Q αmn | dY .

(3.27)

4Ri

i

Combining the estimates (3.26) and (3.27), we finally have

∫

αmn

α QR mn

αmn

αmn

|(LEmn (ω )),t | |LEmn (ω ) − ω | ≤ C αmn Ri meas(QRi mn

mn

−1 −2

mn

i

2

∫ ) −

α Q4Rmn

|ω

mn

− ωmn Q αmn | dY 4Ri

.

(3.28)

i

Thus, we obtain

∫ [· · ·] ≤ Emn

−∫ α Q4Rmn

i:Ri ≥ρ

[· · ·] +

−∫ i:Ri ≤ρ

i

α Q4Rmn

[· · ·].

(3.29)

i

The first sum in (3.29) can be estimated with help of (3.28) as

−∫ i:Ri ≥ρ

α Q4Rmn

−1 −2d−6 [· · ·] ≤ C αmn ρ meas(Emn )‖ωmn ‖1;L1 ≤ O .

(3.30)

i

To estimate the second sum we use the same trick as in (3.23), we apply Poincaré inequality to get

−∫ i:Ri ≤ρ

α Q4Rmn

[· · ·] ≤

C

−

α −1 q −1 αmn meas(QRi mn )(λmn + αmn λmn + αmn + ραmn Λ)2

i:Ri ≤ρ

i

≤

−1 2 2q−2 C meas(Emn ) αmn λmn + αmn (λmn + 1 + ρ 2 Λ2 )



(3.24)

  −q = C meas(Emn ) 2λqmn + λ2mn (1 + ρ 2 Λ2 ) .

n,m

n ,m

Similarly as for I14 , (3.12), (3.13) and (3.11) imply that I13 proof of point-wise convergence of ∇ en .

 (3.31)

≤ O . Thus, the relation (3.2) is proved, which completes the

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M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

Appendix. Lipschitz approximation This Appendix summarizes the properties of Lipschitz approximations of Bochner functions. We prove only those assertions that extend the results established in [19]. We start with the definition of the modified parabolic metric dα on Rd+1 and corresponding balls. For X , Y ∈ Rd+1 where X := (t , x), Y := (s, y), and for R > 0, α > 0, A ⊂ Rd+1 we define

 |t − s|1/2 , α 1/2   QRα (X ) := Y ∈ Rd+1 ; dα (X , Y ) < R , diam α A := sup dα (X , Y ). 

dα (X , Y ) := max |x − y|,

X ,Y ∈A

Let a ∈ [1, ∞). For 0 ≤ g ∈ La (0, ∞; La (Rd )) we introduce the parabolic maximal functions M (g ) and M α (g ) through



∫

M(g )(t , x) := sup −

0<ρ<∞ (t −ρ,t +ρ)

∫

Mα (g )(t , x) := sup −

QRα (t ,x) QRα (t ,x)

∫ sup −

0


g (s, y) dy

ds,

g (s, y) dy ds.

Note that M and M α share the following properties:

Mα (g ) ≤ M (g ) in Rd+1 ,

(A.1)

for any a ∈ (1, ∞) there is C (a, d) such that for all g ∈ L (0, T ; L (Ω )) there holds ‖Mα (g )‖a;La ≤ ‖M (g )‖a;La ≤ C (a, d)‖g ‖a;La . a

a

(A.2)

See [23] for the proof21 of (A.2). It is a straightforward consequence of (A.2) that meas{(t , x) ∈ Q ; M (g )(t , x) > Λ} ≤ C a (a, d)‖g ‖aa;La Λ−a , meas{(t , x) ∈ Q ; M α (g )(t , x) > Λ} ≤ C a (a, d)‖g ‖aa;La Λ−a .

(A.3)

The next lemma extends the relation (A.3)2 also for a = 1. The proof is very similar to the proof in Section 1 of [23]. Lemma A.1. Let Ω ⊂ Rd be an arbitrary open bounded set and α ∈ R+ . Then there exists C (Ω ) > 0 such that for all g ≥ 0, g ∈ L1 (0, T ; L1 (Ω )) there holds meas {(t , x) ∈ (0, T ) × Ω ; M α (g )(t , x) > Λ} ≤ C (Ω )‖g ‖1;L1 Λ−1 .

(A.4)

Proof. For any g ∈ L1 (0, T ; L1 (Ω )) we define on the set of its Lebesgue points the sets A := {(t , x); g (t , x) ≥ Λ},

E := {(t , x) ∈ Q ; M α (g )(t , x) > Λ} \ A. By a simple argument one observes that meas A ≤ Λ−1 ‖g ‖1;L1 and thus it is enough to prove the property (A.4) only for E. Due to strict inequality in the definition of E, it follows from the definition of M α that for any (t , x) ∈ E there exists R > 0 such that

Λ−1

∫ QRα (t ,x)

g (s, y) dy ds ≥ meas QRα (t , x).

(A.5)

In such a way, we can find a family of parabolic cubes {QRβ }, with QRβ := QRαβ (tβ , xβ ), centered at each (tβ , xβ ) ∈ E, and radius Rβ such that (A.5) holds. Obviously, {QRk } from the family {QRβ } such that



β

QRβ ⊇ E. Next, let us assume that we can extract a (finite or infinite) sequence

QRk ∩ QRj = ∅ for all k, j ∈ N such that k ̸= j,

−

meas QRk ≥ C (Ω )meas E .

k

21 In (A.2) we use the convention that g (t , x) ≡ 0 for all t ≥ T and all x ∈ Rd \ Ω .

(A.6)

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

589

Then, using (A.5) and (A.6), we deduce meas E

(A.6)2

≤ C −1 (Ω )

−

meas QRk

k (A.5)

C −1 (Ω )Λ−1

≤

−∫ k

(A.6)1

g (t , x) dx dt ≤ C ‖g ‖1;L1 Λ−1 , QR

k

and the assertion (A.4) holds. In order to complete the proof it remains to show the validity of (A.6). We proceed inductively. First, we take some QR1 ∈ {QRβ } such that diam α QR1 ≥

1

sup(diam α QRβ ). 2 β

Next, assume that we have already defined QR1 , . . . , QRk and we look for QRk+1 . We take QRk+1 such that it is mutually disjoint from QR1 , . . . , QRk and diam α QRk+1 ≥

1

sup(diam α QRγ ), 2 γ

where QRγ ∈ {QRβ }

(A.7)

and QRγ ∩ QRi = ∅ for all i = 1, . . . , k.

Clearly, the sequence {QRk } satisfies (A.6)1 and it remains to prove (A.6)2 . First, if Hence assume that

−

∑

k

meas QRk = ∞ then (A.6)2 holds.

meas QRk < ∞.

(A.8)

k

Once we show that E⊂



Q5Rk ,

(A.9)

k

we have that meas E ≤

−

meas Q5Rk ≤ C 5d+2

k

−

meas QRk ,

k



and (A.6)2 follows. Hence, it is enough to show (A.9), which clearly follows if each QRβ ⊂ k Q5Rk . Let QRβ be arbitrary. If there is some k such that QRβ = QRk then there is nothing to prove. Next, using (A.8) we see that diam α QRk → 0 as

k → ∞ and therefore since Rβ > 0 we can find22 j such that diam α QRj+1 < 12 diam α QRβ . But using the definition of QRk it immediately implies that QRβ must intersect one of the cubes QR1 , . . . , QRj . Let QRm be such a cube. Then we also have that 1 2

diam α QRβ ≤ diam α QRm . Consequently, we see that QRβ ⊂ Q5Rm that finishes the proof.



Lemma A.2 (Covering Lemma). Let E ⊂ Rd+1 be an open bounded set. Then there exists a countable family of cubes {QRαi (Xi )}i∈N and a family of smooth functions {ψi }i∈N such that ∞  i =1

QRαi /2 =

∞ 

QRαi = E

i =1

4Ri ≤ dα (Xi , ∂ E ) ≤ 8Ri , α

∀i ∈ N, with 0 < Ri < 1 α

Rj > 2Ri ⇒ QRi (Xi ) ∩ QRj (Xj ) = ∅ QRαi /4 (Xi ) ∩ QRαj /4 (Xj ) = ∅ ∀i, j ∈ N, i ̸= j α ψi ∈ C0∞ (Q2R (Xi )), i /3

∀i ∈ N

α R2i |∂t ψi | + Ri |∇ψi | ≤ C (d) in Rd+1 ∀i ∈ N ∞ −

ψ i (X ) = 1,

∀X ∈ E .

i =1

22 In case that the sequence {Q } is finite and if diam Q ≥ Rk α Rk

1 2

diam α QRβ for all k, we set QRj as the last term of the sequence {QRk }.

(A.10)

590

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

 Moreover, defining Ai :=



α α j ∈ N : Q 2R (Xi ) ∩ Q 2R (Xj ) ̸= ∅ , we have i j 3

card(Ai ) ≤ C (d), α

3

∀i ∈ N

α

QRj (Xj ) ⊂ Q4Ri (Xi ) ⊂ E ,

∀j ∈ Ai .

(A.11)

Proof. The proof can be found in [19], note that it suffices to combine all information from Lemma 3.1 in [19], Lemma C.1 in [19] together with the estimates (3.4)–(3.7) in [19].  We also introduce the notation for mean value over an arbitrary set A for an integrable function u:

∫

uA := − u dx dt . A

Lemma A.3 (Poincaré Inequality). Let u, f ∈ L1 (QRα ) and ∇ u, q ∈ L1 (QRα ) satisfying

∫ −

QRα

uφ , t =

∫

∫ QRα

q · ∇φ +

QRα

fφ

∀φ ∈ C0∞ (QRα ).

(A.12)

Then



∫

∫ QRα

|u − uQRα | ≤ CR

QRα

|∇ u| + α|q| + α R|f | .

(A.13)

Proof. The proof for Q11 is the same as the proof of Theorem B.1 in [19]. We prove the statement for general QRα . For simplicity, we assume that QRα is a cube centered at (0,0). Let u be defined on QRα . Then we define the function v on Q11 as v(t , x) := u(α R2 t , Rx). Then we can observe that

  (A.12) v,t (t , x) = α R2 ∂αR2 t u(α R2 t , Rx) = α R div x q(α R2 t , Rx) + R2 f (α R2 t , Rx) . Thus, we are in position to apply our estimate for Q11 to obtain

∫ Q11

|v(t , x) − v Q 1 | ≤ C

∫

1

Q11

(|∇v(t , x)| + α R|q(α R2 t , Rx)| + α R2 |f (α R2 t , Rx)|).

After standard substitution, we easily obtain (A.13).



Finally, let E ⊂ Q be an open set and u ∈ L (Q ). Let {QRαi } be the covering of E from Lemma A.2 and {ψi } be the corre1

α

sponding partition of unity. Then we introduce the following truncation operator LE such that

  u(t , x) ∞ α LE u(t , x) := − ψi uQRα   i

if (t , x) ∈ Q \ E , (A.14)

if (t , x) ∈ E .

i=1

It is easy to observe (see Lemma 3.11 [19]) that for all 1 ≤ a < ∞

‖LαE u‖a,La ≤ c (a)‖u‖a,La .

(A.15) α

The last lemma of this subsection deals with the most important behavior of the operator LE . ′ Lemma A.4. Let Ω be an open bounded set in Rd . Let u ∈ L∞ (0, T ; L2 (Ω )) ∩ Lr (0, T ; W 1,r (Ω )), f ∈ L1 (Q ) and q ∈ Lr (0, T ; ′ Lr (Ω )), (1 < r < ∞), be such that

u,t = div q + f in sense of distribution. Moreover, let E ⊂⊂ Q be an open set such that

Mα (|∇ u|) + α Mα (|q|) + α M α (|f |) ≤ C ∗ < +∞,

a.e. in Q \ E .

(A.16)

Then there exists C (α, d, C ∗ ) such that

‖∇ LαE u‖∞;L∞ (Ω ) ≤ C , loc

‖(LαE u),t (LαE u − u)‖1;L1

loc

(Ω )

≤ C,

(A.17)

M. Bulíček et al. / Nonlinear Analysis: Real World Applications 12 (2011) 571–591

591

and for all φ1 ∈ C0∞ (Ω ) and all φ2 ∈ C0∞ (0, T ) we have T

∫

⟨u,t , Tε (LαE u)φ1 ⟩φ2 dt = −

0

∫

Θε (LαE u)φ1 (φ2 ),t dx dt −

∫

Q

∫ Q

α

u − LE u

Q



−



α

α

Tε (LE u) ,t φ1 φ2 dx dt





α

u − LE u Tε (LE u)φ1 (φ2 ),t dx dt



(A.18)

where Tε is defined by (1.16) and Θε by (1.18). Proof. The proof can be found in [19] (see Theorem 3.21). To be correct this conclusion is proved in [19] without the truncation function Tε and without the function f ∈ L1 (Q ). But the proof of (A.17) is the same. Thus, for sake of completeness, we show formally the validity of (A.18). Indeed, using integration by parts we have

∫

Θε (LαE u)φ1 (φ2 ),t dx dt =

− Q

∫ Q

α

α

Tε (LE u)(LE u),t φ1 φ2 dx dt , Q

α

α

u − LE u Tε (LE u)φ1 (φ2 ),t dx dt =



−

∫



∫ Q

∫ +

α





α

u − LE u

Q

that directly implies (A.18).

α

u − LE u ,t Tε (LE u)φ1 φ2 dx dt



α

Tε (LE u) ,t φ1 φ2 dx dt ,







References [1] M. Bulíček, J. Málek, K.R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli, that slip at solid boundaries, SIAM J. Math. Anal. 41 (2) (2009) 665–707. [2] L.D. Landau, E.M. Lifschitz, Lehrbuch der Theoretischen Physik. Band VI: Hydrodynamik, in: Deutscher Sprache Herausgegeben von Gerhard Heber, Akademie-Verlag, Berlin, 1966. [3] M. Bulíček, L. Consiglieri, J. Málek, Slip boundary effects on unsteady flows of incompressible viscous heat conducting fluids with a non-linear internal energy/temperature and heat flux/temperature gradient relationships, Math. Models Methods Appl. Sci. (2010) (submitted for publication). [4] L. Consiglieri, Weak solutions for a class of non-Newtonian fluids with energy transfer, J. Math. Fluid Mech. 2 (3) (2000) 267–293. [5] M. Bulíček, E. Feireisl, J. Málek, A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal. RWA 10 (2) (2009) 992–1015. [6] A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L1 data, Nonlinear Anal. 28 (12) (1997) 1943–1954. [7] G. Dal Maso, F. Murat, Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems, Nonlinear Anal. 31 (3–4) (1998) 405–412. [8] J. Frehse, A refinement of Rellich’s theorem, Rend. Mat. (7) 5 (1985) 229–242. [9] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (6) (1992) 581–597. [10] L. Boccardo, A. Dall’Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1) (1997) 237–258. [11] L. Boccardo, L. Orsina, A. Porretta, Some noncoercive parabolic equations with lower order terms in divergence form, J. Evol. Equ. 3 (2003) 407–418. [12] D. Blanchard, F. Murat, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations 177 (2) (2001) 331–374. [13] J. Frehse, J. Málek, M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (5) (2003) 1064–1083. [14] L. Diening, J. Málek, M. Steinhauer, On Lipschitz truncations of Sobolev functions, with variable exponent and their selected applications, ESAIM Control Optim. Calc. Var. 14 (2008) 211–232. [15] M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda, On steady flows of an incompressible fluids with implicit power-law-like rheology, Adv. Calc. Var. 2 (2) (2009) 109–136. [16] M. Bulíček, V. Fišerová, Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index, Z. Anal. Anwend. 28 (3) (2009) 349–371. [17] J. Kinnunen, J.L. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J. 102 (2000) 253–271. [18] J. Kinnunen, J.L. Lewis, Very weak solutions of parabolic systems of p-Laplacian type, Ark. Mat. 40 (1) (2002) 105–132. [19] L. Diening, M. Růžička, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) IX (2010) 1–46. [20] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3) 13 (1959) 115–162. [21] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1) (1989) 149–169. [22] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. (4) 146 (1987) 65–96. [23] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, New Jersy, 1970.