https://doi.org/10.1007/s10231-017-0673-0
Homogenization of an evolution problem with L log L
data in a domain with oscillating boundary
Antonio Gaudiello1 · Olivier Guibé2
Received: 3 January 2017 / Accepted: 12 May 2017 / Published online: 1 June 2017
© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017
Abstract We consider a N -dimensional structure, N∈ N\{1}, with a very rough boundary.
Precisely, in 3D the structure consists in a box with the upper side covered withε-periodically distributed asperities having fixed height and size depending onε. In this structure, we study the asymptotic behavior, asε vanishes, of an evolution Neumann problem with source term and initial data having L log L a priori estimates. We identify the limit problem.
Keywords Homogenization of rough boundaries· Evolution problem · Zygmund space Mathematics Subject Classification 35B27· 35j25 · 35R05
1 Introduction
Let N ∈ N \ {1}. A generic point of RN−1is denoted by x. A generic point ofRNis denoted
by(x, xN).
Letε be a parameter taking values in a sequence of positive numbers converging to zero. InRN we consider a forest+ε of cylinders with fixed height and small cross section of size of orderε, ε-periodically distributed on a box −. Precisely, letω ⊂⊂]0, 1[N−1and ⊂ RN−1be two open bounded connected sets with Lipschitz boundary, h+ and h−be two positive numbers, and introduce the following subsets ofRN (see Figs.1,2).
B
Antonio Gaudiello [email protected] Olivier Guibé1 DIEI, Università degli Studi di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043 Cassino (FR), Italy
2 Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, Avenue de l’Université, BP.12 Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France
Fig. 1 ε Fig. 2 ⎧ ⎪ ⎨ ⎪ ⎩ + ε = {k∈ZN−1: εω+εk⊂⊂} (εω + εk) × [0, h+[, −= ×] − h−, 0[, ε= + ε ∪ −, += ×]0, h+[, = × {0}, = +∪ ∪ −= ×] − h−, h+[. In what follows,χ+
ε denotes the characteristic function of+ε in+andχ+ε∩ the char-acteristic function of+ε ∩ in .
Note that+is the Hausdorff limit of the sequence{+ε}ε, and χ+
ε |ω| weak in L∞(+), (1.1)
χ+ε∩ |ω| weak in L∞(). (1.2)
In what follows, T ∈]0, +∞[, and v denotes the zero extension to (resp. ]0, T [×) of any functionv defined in D (resp. ]0, T [×D), with D subset of .
Boundary value problems in a domain with numerous asperities arise in many fields of biology, physics, and engineering sciences. For instance, for understanding the motion of ciliated microorganisms, the flow in a channel with rugose boundary, heat transmission through winglets, propagation of electromagnetic waves in regions with rough boundaries, air flow through compression system in turbo machine such as a jet engine, the vibrations of foundations of buildings (for instance, see [4,13,30,45], and [51]). It is often impossible
to approach these problems directly with numerical methods, because the rough boundary requires a large number of mesh points in its neighborhood. Thus, the computational cost associated with such a problem grows rapidly whenε gets smaller. Moreover, it can occur that the required discretization step becomes too small for the machine precision. Then, the goal is to approach the original problem, when the periodicityε gets smaller, with a problem in which can be numerically solved.
Let A: (t, x) ∈]0, T [× → A((t, x)) = (Ai j((t, x)))i, j∈{1,...,N}∈ RN×N be a N×
N-matrix function such that
A∈ (L∞(]0, T [×))N×N,
∃α ∈]0, +∞[ : A((t, x))ξξ ≥ α|ξ|2, ∀ξ ∈ RN, a.e. (t, x) ∈]0, T [×; (1.3)
a:]0, T [× →]0, +∞[ be a function such that
a∈ L∞(]0, T [×),
∃β ∈]0, +∞[ : a((t, x)) ≥ β, a.e. (t, x) ∈]0, T [×; (1.4) fε:]0, T [×ε→ R be a source term such that
⎧ ⎪ ⎨ ⎪ ⎩ fε∈ L2(]0, T [×ε), ∃c ∈]0, +∞[ : T 0 ε (1 + | fε|) ln(1 + | fε|)dxdt ≤ c, ∀ε; (1.5)
u0ε: ε→ R be an initial data such that
⎧ ⎪ ⎨ ⎪ ⎩ u0ε∈ L2(ε), ∃c ∈]0, +∞[ : ε (1 + |u0 ε|) ln(1 + |u0ε|)dx ≤ c, ∀ε; (1.6)
andν be the unit outer normal on ∂ε. This paper is devoted to studying the asymptotic behavior, asε tends to zero, of the weak solution of the following evolution problem
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂tuε− div(ADuε) + auε= fε, in ]0, T [×ε, A Duε· ν = 0, on ]0, T [×∂ε, uε(0, ·) = u0ε(·), a.e. in ε. (1.7)
The novelty of this paper lies in L log L a priori estimates assumed on the source term and on the initial data [see the second line in (1.5) and (1.6)]. Usually, in all previous studies on the parabolic case, only L2 a priori estimates on the source term and on the initial data were assumed, or sometimes stronger assumptions were made (for instance; see [31,34], and [39]).
The weak formulation of (1.7) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈ L20, T ; H1(ε) ∩ C[0, T ]; L2(ε) , ∂tuε∈ L2 0, T ;H1(ε) , T 0 < ∂tuε, ψ >(H1(ε)),H1(ε)dt+ T 0 ε (ADuεDψ + auεψ) dxdt = T 0 ε fεψdxdt, ∀ψ ∈ L20, T ; H1(ε) , uε(0, ·) = u0 ε, a.e. in ε, (1.8)
admits a unique solution, and it is equivalent to the following one (for instance, see [46] and [53]) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈ L20, T ; H1(ε) ∩ C[0, T ]; L2(ε) , T 0 ε −uεvϕ+ ADuεDvϕ + auεvϕ dxdt = T 0 ε fεvϕdxdt, ∀v ∈ H1(ε), ∀ϕ ∈ C∞0 (]0, T [) , uε(0, ·) = u0ε, a.e. in ε. (1.9)
To describe the limit problem, the matrix function A is needed to be decomposed in the following way A= A B1 B2 AN N , (1.10) where A= (Ai j)i, j=1,...,N−1, B1= (Ai N)i=1,...,N−1, B2= (AN j)j=1,...,N−1.
Note that Ais invertible, since it is coercive with coefficientα given in (1.3). Then, one can define
ahom: (t, x) ∈]0, T [× → AN N((t, x)) − B2((t, x))(A((t, x)))−1B1((t, x)). (1.11) An easy computation shows that ahom ∈ L∞(]0, T [×) and
ahom(t, x) ≥ α, a.e. in ]0, T [×. The following theorem states the main result of this paper.
Theorem 1.1 For everyε, let uεbe the unique solution of (1.8), under assumptions (1.3), (1.4), (1.5), and (1.6), and set u+ε = uε|
+ε, u −
ε = uε|−. Moreover, let A, B1 and ahom
be defined in (1.10) and (1.11), respectively. Then, there exists a subsequence of{ε}, still denoted by{ε}, and (depending on the subsequence) f ∈ L1(]0, T [×), u0∈ L1(), and
u= (u+, u−) ∈UN = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u+, u−) ∈ L1(]0, T [×+) × LNN+2+1 0, T ; W1,NN+2+1(−) : ∂xNu+∈ L 1(]0, T [×+), u+ |(t, ·) = u−|(t, ·), a.e. t ∈]0, T [ , if N ≥ 3, (u+, u−) ∈ L1(]0, T [×+) × Lp0, T ; W1,p(−) ∀p ∈ [1,4 3[: ∂xNu+∈ L1(]0, T [×+), u+|(t, ·) = u−|(t, ·), a.e. t ∈]0, T [ , if N = 2, (1.12)
where u+|(t, ·) and u−|(t, ·) denote the trace of u+(t, ·) and u−(t, ·) on , respectively, such that fε f weakly in L1(]0, T [×), (1.13) u0 ε u0weakly in L1(), (1.14) u+ε |ω|u+weakly in L1(]0, T [×+), (1.15) ∂xNu+ε = ∂xNu+ε |ω|∂xNu+weakly in L 1(]0, T [×+), (1.16) Dxu+ε −|ω|∂xNu+(A)−1B1weakly in(L 1(]0, T [×+))N−1, (1.17) u−ε u−weakly in LNN+2+1 0, T ; W1,NN+2+1(−) , if N ≥ 3, Lp0, T ; W1,p(−) , ∀p ∈ [1,4 3[, if N = 2, (1.18) asε tends to zero, and u is a solution of the following problem:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u= (u+, u−) ∈UN, |ω| T 0 + −u+vϕ+ ahom∂xNu+∂xNvϕ + au+vϕ dxdt + T 0 − −u−vϕ+ ADu−Dvϕ + au−vϕ dxdt = T 0 fvϕdxdt, ∀v ∈ W 1,∞(), ∀ϕ ∈ C∞ 0 (]0, T [), u+(0, ·) = 1 |ω|u0|+, a.e. in +, u−(0, ·) = u0|−, a.e. in −. (1.19)
The main difficulty in the proof of Theorem1.1relates to obtain a priori estimates on the solution, independently ofε (see Sect.2). Note that we cannot use the Boccardo–Gallouët estimates (see [19]), since the domain+ε depends onε, and we do not have any control on the constants in the Sobolev embedding Theorem. Then, the first task is devoted to proving that
{u+ε}εand{|Du+ε|}εare bounded in L1(]0, T [×+) and equi-integrable. (1.20) To this aim, using
ln(1 + |uε((t, ·))|) sign(uε((t, ·))) χ]0,s[(t),
with s∈ [0, T ], as test function in (1.8), and a variant of the Young’s inequality [see (2.5)], we are able to prove that
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ {uε}ε is bounded in L∞(0, T ; L log L()) , |Duε|2 1+ |uε| ε is bounded in L1(]0, T [×), (1.21)
independently ofε. These estimates imply (1.20).
The second task is devoted to obtaining sharper a priori estimates on the solution in− (the reason of this sharp estimate will be explained later). Evaluating
T
0
1+ |uε((t, ·))| sign(uε((t, ·)))2
and using (1.21), and the Sobolev embedding theorem in−(note that the Sobolev embed-ding constant does not depend onε, since the domain does not depend on ε) provide that
u−εεis bounded in L1
0, T ; LNN−2(−)
, independently ofε, if N ≥ 3. By using this esti-mate, we prove thatu−εεis bounded in LNN+2(]0, T [×−), independently of ε, if N ≥ 3. Finally, combining this result and the second estimate in (1.21) gives thatDu−εεis bounded in
LNN+2+1(]0, T [×−)
N
, independently ofε, if N ≥ 3. Of course, in −we obtain sharper estimates if N= 2.
Assumptions (1.5) and (1.6), and previous estimates imply (1.13), (1.14), (1.15), (1.16), (1.18), and
Dxu+ε dweakly in(L1(+))N−1
for a subsequence. The method of oscillating test functions introduced by L. Tartar (see [56]) allows us to identify dand limit problem (1.19) satisfied by(u+, u−) (see Sect.3). Finally, we recover the initial condition in (1.19). Note that for proving that
u+|(t, ·) = u−|(t, ·), a.e. t ∈]0, T [, we pass to the limit in
T 0 u+εϕdxdt= T 0 χ+ε∩u − εϕdxdt, ∀ϕ ∈ C0∞(]0, T [×), ∀ε. asε tends to zero. Then, for passing to the limit in the right-hand side, the weak convergence of u−ε in L1(]0, T [×−) is not enough. For this reason we looked for sharper estimates in −.
We cannot expect the uniqueness of the solution in problem (1.19) due to the weak regularity of the solution (u is solution in the sense of distribution), and for instance, see the counterexample in [55] for the elliptic case.
Note that by using more technical test functions, it is possible to weaken the L log L assumption in (1.5) and (1.6) and to deal with the case L(log L)α with 1/2 < α ≤ 1 (see Remark3.1).
As we explained above, the novelty of this paper lies in L log L a priori estimates assumed on the source term and on the initial data. For the properties of L log L we refer to [2,21,41] and [54].
In the case of L1data, i.e., assuming fε f weak in L1(]0, T [×) and u0
ε u weak
in L1(), and without any L log L assumptions, the method developed in the present paper cannot handle the asymptotic analysis of problem (1.8). Indeed the main difficulty is to prove some relative compactness in L1(]0, T [×) of u
εand Duε. Since+ε varies inε, we cannot use the Boccardo–Gallouët estimates, and because f belongs to L1(]0, T [×), there is no reason to expect a bound on0Tε fεlog(1+|uε|)dxdt. To our knowledge this case remains
an open problem. The suitable setting to approach the problem would be the framework of renormalized solution (for instance, see [11,17,18,28,36,37,50], and the references therein) as in [43] for the stationary case, but it is beyond the scope of this paper.
A large bibliography on the homogenization of very rough boundaries developed in the last 20 years. The first paper was [23] (see also [22]), where the authors derived the limit problem for the Laplace equation with the homogeneous Neumann boundary condition and with the right-hand side in L2. As pioneering paper, we also recall [40] where an Helmholtz equation posed in two half-planes communicating through a random set of channels was
considered. The problem treated in [22] was revisited in [47] and [29]. A corresponding spectral problem was studied in [49], a corresponding problem with a contrasting diffusivity in [44], while a corresponding nonlinear problem in [12] and [16]. Evolution problems were treated in [31] and [39]. Control problems were considered in [32–35,38], and [51]. Stokes flows were the object of the analysis in [4] and [30]. Linear elastic problems were considered in [9,13–15,52] (this last one deals also with the Maxwell equation), while a nonlinear elastic problem in [6]. A thick multi-level junctions with nonlinear conditions of the Signorini type was treated in [48]. Teeth with vanishing height were considered in [1,3,5,7,8,10,20,24–26] and [27]. In all these papers, classical estimates are assumed on the source term and initial data, or sometimes also stronger assumptions are made. A source term having L log L a priori estimates was studied only in [42] for the corresponding elliptic problem.
2 A priori estimates
This section is devoted to proving the following a priori estimates for the solution of (1.8).
Proposition 2.1 Let{ fε}εand{u0
ε}εbe two sequences satisfying (1.5) and (1.6), respectively. Moreover, for everyε, let uεbe the unique solution of (1.8), under assumptions (1.3), (1.4), (1.5), and (1.6), and set u+ε = uε|
+ε, u −
ε = uε|−. Then
{fε}εis bounded in L1(]0, T [×) and equi-integrable, {u0
ε}εis bounded in L1() and equi-integrable,
(2.1) ⎧
⎨ ⎩
{u+ε}εis bounded in L1(]0, T [×+) and equi-integrable, {|Du+ε|}εis bounded in L1(]0, T [×+) and equi-integrable,
(2.2) and ⎧ ⎨ ⎩ ∃c1∈]0, +∞[ : u− ε LN+2 N+1 0,T ;W1, N+2N+1(−) ≤ c1, ∀ε, if N ≥ 3, ∀p ∈ [1,4 3[ ∃Cp∈]0, +∞[ : u−ε Lp(0,T ;W1,p(−)) ≤ Cp, ∀ε, if N = 2. (2.3)
Proof The proof of this proposition will be developed in several steps. Step 1. The first step is devoted to proving (2.1).
Indeed, (1.5) and (1.6) imply ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A|fε|dtdx = A∩{(t,x)∈]0,T [×ε:| fε((t,x))|≤k} | fε|dtdx + A∩{(t,x)∈]0,T [×ε:| fε((t,x))|>k} | fε|dtdx ≤ k|A| +ln(1 + k)c , ∀A ⊆]0, T [× measurable, ∀k ∈ N, ∀ε, B |u0 ε|dx = B∩{x∈ε:|u0ε(x)|≤k} |u0 ε|dx + B∩{x∈ε:|u0ε(x)|>k} |u0 ε|dx ≤ k|B| + c ln(1 + k), ∀B ⊆ measurable, ∀k ∈ N, ∀ε, which gives (2.1).
Step 2. The second step is devoted to proving ∃c2∈]0, +∞[ : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ sup s∈[0,T ] ε (1 + |uε(s, x)|) ln(1 + |uε(s, x)|)dx ≤ c2, T 0 ε |Duε|2 1+ |uε|dxdt≤ c2, ∀ε. (2.4)
For any s∈ [0, T ], function
t∈]0, T [→ ln(1 + |uε((t, ·))|) sign(uε((t, ·))) χ]0,s[(t)
belongs to L20, T ; H1(ε) , whereχ]0,s[denotes the characteristic function of]0, s[. Then, choosing this function as test function in (1.8) gives
s 0 < ∂tuε, ln(1 + |uε|) sign(uε) >(H1(ε)),H1(ε)dt + s 0 ε A Duε Duε 1+ |uε|+ a|uε| ln(1 + |uε|) dxdt = s 0 ε fεln(1 + |uε|) sign(uε)dxdt, ∀s ∈ [0, T ], ∀ε,
from which, by virtue of assumptions (1.3), (1.4), and the following Young’s inequality [see [2], Sect. 8, formula (2)] |rs| ≤ (1 + |r|) ln(1 + |r|) + e|s|− 1, ∀r, s ∈ R, (2.5) it follows that s 0 < ∂tuε, ln(1 + |uε|) sign(uε) >(H1(ε)),H1(ε)dt + s 0 ε α |Duε|2 1+ |uε|+ β|uε| ln(1 + |uε|) dxdt ≤ s 0 ε ((1 + | fε|) ln(1 + | fε|) + |uε|) dxdt, ∀s ∈ [0, T ], ∀ε. (2.6)
Now, denoting with z the antiderivative of ln(1 + |t|) sign(t), i.e., z: t ∈ R → (1 + |t|) ln(1 + |t|) − |t|, the first integral in (2.6) can be rewritten as
s 0 < ∂tuε, ln(1 + |uε|) sign(uε) >(H1(ε)),H1(ε)dt = s 0 < ∂t uε, z(uε) >(H1(ε)),H1(ε)dt= s 0 ∂t ε z(uε)dx dt = ε z(uε(s, x))dx − ε z(uε(0, x))dx, ∀s ∈ [0, T ], ∀ε. (2.7)
Combining (2.6) with (2.7), and using assumptions (1.5) and (1.6) imply ε (1 + |uε(s, x)|) ln(1 + |uε(s, x)|)dx + α s 0 ε |Duε|2 1+ |uε|dxdt+ β s 0 ε |uε| ln(1 + |uε|)dxdt ≤ ε |uε(s, x)|dx + s 0 ε |uε|dxdt + 2c, ∀s ∈ [0, T ], ∀ε,
where c is the constant satisfying (1.5) and (1.6). Consequently, one has ε (1 + |uε(s, x)|) ln(1 + |uε(s, x)|)dx + α s 0 ε |Duε|2 1+ |uε|dxdt+ β s 0 ε |uε| ln(1 + |uε|)dxdt ≤ (e2− 1)|| + {x∈ε:|uε(s,x)|>e2−1}|uε(s, x)|dx + (eβ2 − 1)||T + (t,x)∈]0,s[×ε:|uε(t,x)|>e 2 β−1|uε|dtdx + 2c ≤ (e2− 1)|| +1 2 {x∈ε:|uε(s,x)|>e2−1}(1 + |uε(s, x)|) ln (1 + |uε(s, x)|) dx + (eβ2 − 1)||T + β 2 (t,x)∈]0,s[×ε:|uε(t,x)|>e 2 β−1|uε| ln(1 + |uε|)dtdx + 2c ≤ (e2− 1)|| +1 2 ε (1 + |uε(s, x)|) ln (1 + |uε(s, x)|) dx + (eβ2 − 1)||T + β 2 s 0 ε |uε| ln(1 + |uε|)dxdt + 2c, ∀s ∈ [0, T ], ∀ε, which implies (2.4).
Step 3. By using estimate (2.4), the third step is devoted to proving (2.2). Indeed, estimates in (2.4) and the Hölder inequality imply
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A |u+ε|dtdx = A∩{(t,x)∈]0,T [×+ε:|u+ε((t,x))|≤k} |u+ε|dtdx + A∩{(t,x)∈]0,T [×+ε:|u+ε((t,x))|>k} |u+ ε|dtdx ≤ k|A| +ln(1 + k)c2T , A |Du+ε|dtdx ≤ T 0 ε |Du+ε|2 1+ |u+ε|dxdt A (1 + |u+ε|)dtdx ≤√c2 (k + 1)|A| + c2T ln(1 + k), ∀A ⊆]0, T [×+measurable, ∀k ∈ N, ∀ε, which gives (2.2).
Step 4. By using estimate (2.4) again, the fourth step is devoted to proving the following estimates in−. ⎧ ⎨ ⎩ ∃c3∈]0, +∞[ :u−ε L1 0,T ;LNN−2(−) ≤ c3, ∀ε, if N ≥ 3, ∀p ∈ [1, +∞[ ∃Ap ∈]0, +∞[ :u−εL1(0,T ;Lp(−)) ≤ Ap, ∀ε, if N = 2. (2.8)
Indeed, estimates in (2.4) imply ε |uε((t, x))|dx ≤ || +ln 2c2 , ∀t ∈ [0, T ], ∀ε, (2.9) ε |Duε((t, x))|2 1+ |uε((t, x))|dx< +∞, t a.e. in ]0, T [, ∀ε. Consequently, 1+ |uε((t, ·))| sign(uε((t, ·))) ∈ H1(ε), a.e. t ∈]0, T [, ∀ε, and 1+ |uε((t, ·))| sign(uε((t, ·)))2 H1(ε) = 1 4 ε |Duε((t, x))|2 1+ |uε((t, x))|dx+ ε (1 + |uε((t, x))|) dx ≤ 1 4 ε |Duε((t, x))|2 1+ |uε((t, x))|dx+ 2|| + c2 ln 2, a.e. t ∈]0, T [, ∀ε. Then, using the Sobolev embedding theorems in−, one has
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1+ |u−ε((t, ·))|2 LN2N−2(−) ≤ s2 N 1 4 ε |Du− ε((t, x))|2 1+ |u−ε((t, x))|dx+ 2|| + c2 ln 2 , a.e. t ∈]0, T [, ∀ε, if N ≥ 3, ∀p ∈ [1, +∞[, 1+ |u−ε((t, ·))|2 Lp(−) ≤ s2 2,p 1 4 ε |Du− ε((t, x))|2 1+ |u−ε((t, x))|dx+ 2|| + c2 ln 2 , a.e. t ∈]0, T [, ∀ε, if N = 2, (2.10) where sN denotes the Sobolev embedding constant of H1(−) in L
2N
N−2(−) if N ≥ 3,
and s2,pthe Sobolev embedding constant of H1(−) in Lp(−), p ∈ [1, +∞[, if N = 2. Finally, integrating (2.10) over]0, T [, and using again the second estimate in (2.4), one obtains (2.8).
Step 5. From (2.8) and (2.9), we deduce the following estimates ⎧ ⎨ ⎩ ∃c4 ∈]0, +∞[ :u−ε LNN+2(]0,T [×−)≤ c4, ∀ε, if N ≥ 3, ∀p ∈ [1, 2[ ∃Bp∈]0, +∞[ :u−εLp(]0,T [×−)≤ Bp, ∀ε, if N = 2. (2.11)
Indeed, the Hölder inequality and (2.9) imply ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −|u − ε((t, x))| N+2 N dx= −|u − ε((t, x))|1+ 2 Ndx ≤ −|u − ε((t, x))| N N−2dx N−2 N −|u − ε((t, x))|dx 2 N ≤ −|u − ε((t, x))| N N−2dx N−2 N || + c2 ln 2 2 N , ∀t ∈ [0, T ], ∀ε, if N ≥ 3, −|u − ε((t, x))|1+θdx≤ −|u − ε((t, x))| 1 1−θdx 1−θ −|u − ε((t, x))|dx θ ≤ −|u − ε((t, x))| 1 1−θdx 1−θ || + c2 ln 2 θ , ∀t ∈ [0, T ], ∀θ ∈ [0, 1[, ∀ε, if N = 2.
Then, integrating these inequalities over[0, T ] and using (2.8), one obtains (2.11). Step 6. The sixth step is devoted to proving
⎧ ⎪ ⎨ ⎪ ⎩ ∃c5∈]0, +∞[ : Du− ε LNN+2+1(]0,T [×−)N ≤ c5, ∀ε, if N ≥ 3, ∀p ∈ [1,4 3[ ∃Cp∈]0, +∞[ : Du−ε (Lp(]0,T [×−))2 ≤ Cp, ∀ε, if N = 2. (2.12)
Indeed, Hölder inequality combined with the second estimate in (2.4) gives T 0 −|Du − ε|qdxdt≤ T 0 − |Du− ε|2 1+ |u−ε|dxdt q 2 T 0 −(1 + |u − ε|) q 2−qdxdt 2−q 2 ≤ cq2 2 T 0 −(1 + |u − ε|) q 2−qdxdt 2−q 2 , ∀q ∈ [1, 2[, which imply (2.12), by virtue of (2.11).
Finally, (2.3) follows from (2.11) and (2.12).
3 Proof of Theorem
1.1
This section is devoted to proving Theorem1.1. Proof This proof will be developed in several steps.
Step 1. The first step is devoted to proving a first convergence result.
Combining Proposition2.1and Dunford-Pettis Theorem ensure the existence of a sub-sequence of {ε}, still denoted by {ε}, and f ∈ L1(]0, T [×), u0 ∈ L1(), u+ ∈ L1]0, T [×+ , u−∈ LNN+2+1
0, T ; W1,NN+2+1(−)
if N ≥ 3, u−∈ Lp0, T ; W1,p(−) , for any p∈ [1,43[, if N = 2, and d= (d1, . . . , dN−1) ∈ (L1(]0, T [×+))N−1(depending on the subsequence) such that (1.13), (1.14) (1.15), (1.16), and (1.18) hold true, and
Dxu+ε dweakly in(L1(]0, T [×+))N−1, (3.1)
asε tends to zero. Note that to prove (1.18) when N = 2, one can use a diagonal argument with the sequence
ph= 43− h+21
Step 2. The second step is devoted to proving
u+|(t, ·) = u−|(t, ·), a.e. t ∈]0, T [, (3.2) which ensures that u∈UN.
Relation (3.2) will be proved only in the case N≥ 3, the proof being similar when N = 2. At first note that u+ε(t, ·), u−ε(t, ·), u+(t, ·) and u−(t, ·) have a trace on , for a.e. t ∈]0, T [,
since ⎧ ⎨ ⎩ u+ε(t, ·) ∈ L2(+), ∂xNu + ε(t, ·) ∈ L2(+), u−ε(t, ·) ∈ H1(−), u+(t, ·) ∈ L1(+), ∂xNu +(t, ·) ∈ L1(+), u−(t, ·) ∈ W1,NN+2+1(−), for a.e. t∈]0, T [.
Moreover, one has T 0 u+εϕdxdt= T 0 χ+ε∩u − εϕdxdt, ∀ϕ ∈ C0∞(]0, T [×), ∀ε. (3.3) As far as the left-hand side of (3.3) is concerned, (1.15) and (1.16) imply
− T 0 u+εϕdxdt = T 0 +∂xN u+εϕdxdt + T 0 + u+ε∂xNϕdxdt → |ω| T 0 +∂xNu +ϕdxdt + |ω| T 0 +u +∂ xNϕdxdt = −|ω| T 0 u +ϕdxdt, ∀ϕ ∈ C0∞(]0, T [×), (3.4) asε tends to zero.
As far as the right-hand side of (3.3) is concerned, (1.18) implies T 0 u−ε(t, ·)ϕ(t, ·)dt T 0 u−(t, ·)ϕ(t, ·)dt weakly in W1,NN+2+1(−), ∀ϕ ∈ C∞ 0 (]0, T [×), asε tends to zero. Consequently, one has
T 0 u−ε(t, ·)ϕ(t, ·)dt → T 0 u−(t, ·)ϕ(t, ·)dt strongly in LNN+2+1(), ∀ϕ ∈ C∞0 (]0, T [×), (3.5)
asε tends to zero, from which, taking also into account (1.2), one deduces T 0 χ+ε∩u − εϕdxdt = χ+ε∩ T 0 u−εϕdt dx → |ω| T 0 u−ϕdt dx= |ω| T 0 u −ϕdxdt, ∀ϕ ∈ C∞ 0 (]0, T [×), (3.6) asε tends to zero.
Finally (3.2) follows from (3.3), (3.4) and (3.6). Step 3. The third step is devoted to proving that
Ad+ |ω|∂xNu+B1= 0 a.e. in ]0, T [×+, (3.7) and to identify the limit equation satisfied by u.
For every h in{1, . . . , N − 1}, let {wεh}ε⊂ W1,∞(+) be a sequence such that wh
ε → 0 strongly in L∞(+), (3.8)
asε tends to zero, and ∂xiw
h
ε = −δi,ha.e. in+ε, ∀i ∈ {1, . . . , N},
whereδi,h is the Kronecker delta. The existence of such sequences is proved in [23] (see [22], too). Then, choosingv = wh
εζ with ζ ∈ C∞0 (+) in (1.9), one has T 0 + ⎛ ⎝−uεwhεζ ϕ+ ADuεDζ whεϕ − N j=1 Ah j∂xjuεζ ϕ + a uεζ w h εϕ ⎞ ⎠ dxdt = T 0 +fεζ w h εϕdxdt, ∀ζ ∈ C0∞(+), ∀ϕ ∈ C0∞(]0, T [) ∀h ∈ {1, . . . , N − 1}. (3.9) Passing to the limit, asε vanishes, in (3.9) and using (1.13), (1.15), (1.16), (3.1) and (3.8), give T 0 + ⎛ ⎝N −1 j=1 Ah jdj+ Ah N|ω|∂xNu+ ⎞ ⎠ ζϕdxdt = 0, ∀ζ ∈ C∞ 0 (+), ∀ϕ ∈ C0∞(]0, T [), ∀h ∈ {1, . . . , N − 1}, which implies T 0 + ⎛ ⎝N −1 j=1 Ah jdj+ Ah N|ω|∂xNu+ ⎞ ⎠ φdxdt = 0, ∀φ ∈ C∞ 0 (]0, T [×+), ∀h ∈ {1, . . . , N − 1}, which provides (3.7).
By virtue of assumption (1.3), Ais invertible. Consequently system (3.7) admits a unique solution
d= −|ω|∂xNu+(A)−1B1a.e. in]0, T [×+, (3.10) which, combined with (3.1), gives (1.17).
Finally, passing to the limit, asε vanishes, in (1.9) withv ∈ W1,∞() and ϕ ∈ C0∞(]0, T [), and using (1.13), (1.15), (1.16), (1.17), (1.18), (3.7) and (3.10), one obtains that u satisfies the equation in (1.19).
Step 4. The last step is devoted to recovering the initial condition in (1.19): u+(0, ·) = 1
|ω|u0|+, a.e. in
+, u−(0, ·) = u0
|−, a.e. in −. (3.11) In (1.8) choosingψ = vϕ, with v ∈ C0∞(−) and ϕ ∈ C∞([0, T ]) such that ϕ(0) = 1 andϕ(T ) = 0, one has
T 0 − −uεvϕ+ ADuεDvϕ + auεvϕ dxdt− −u 0 εvdx = T 0 − fεvϕdxdt, ∀v ∈ C0∞(−), ∀ϕ ∈ C∞([0, T ]) : ϕ(0) = 1, ϕ(T ) = 0. (3.12) Passing to the limit, asε vanishes, in (3.12), and using (1.13), (1.14), and (1.18) give
T 0
−
−uvϕ+ ADuDvϕ + auvϕ dxdt− −u 0vdx = T 0 − fvϕdxdt, ∀v ∈ C0∞(−), ∀ϕ ∈ C∞([0, T ]) : ϕ(0) = 1, ϕ(T ) = 0. (3.13) Similarly, one proves that
T
0
−
−uvϕ+ ADuDvϕ + auvϕ dxdt= T
0
− fvϕdxdt,
∀v ∈ C0∞(−), ∀ϕ ∈ C0∞(]0, T [) . (3.14)
which implies that
−u(·, x)v(x)dx belongs to W
1,1(]0, T [). Thus, integrating by parts in (3.14) gives T 0 ∂t −uvdx ϕdt + T 0 −(ADuDvϕ + auvϕ) dxdt = T 0 − fvϕdxdt, ∀v ∈ C∞ 0 (−), ∀ϕ ∈ C0∞(]0, T [) , (3.15)
which implies (see Lemma4.1in Appendix) T 0 ∂t −uvdx ϕdt + T 0 −(ADuDvϕ + auvϕ) dxdt = T 0 − fvϕdxdt, ∀v ∈ C0∞(−), ∀ϕ ∈ C∞([0, T ]) : ϕ(0) = 1, ϕ(T ) = 0. (3.16) Integrating by parts (3.16) gives
T
0
−
−uvϕ+ ADuDvϕ + auvϕ dxdt− −u(0, x)vdx = T 0 − fvϕdxdt, ∀v ∈ C ∞ 0 (−), ∀ϕ ∈ C∞([0, T ]) : ϕ(0) = 1, ϕ(T ) = 0. (3.17) Finally, comparing (3.13) with (3.17) insures that (3.11) holds true in−. Similarly, one
proves that (3.11) holds true in+.
Remark 3.1 Note that by using more technical test functions, it is possible to weaken the L log L assumption in (1.5) and (1.6) and to deal with the case L(log L)αwith 1/2 < α ≤ 1, namely
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fε∈ L2(]0, T [× ε), ∃c ∈]0, +∞[ : T 0 ε (1 + | fε|) ln(1 + | fε|)αdxdt≤ c, ∀ε; and ⎧ ⎪ ⎨ ⎪ ⎩ u0 ε ∈ L2(ε), ∃c ∈]0, +∞[ : ε (1 + |u0 ε|) ln(1 + |u0ε|)αdx ≤ c, ∀ε.
The main difficulties are to obtain estimates on uεand Duεwhich are sufficient to deduce thatuε and Duεare bounded in L1(]0, T [×) and equi-integrable. To this aim the proof of Proposition2.1(the a priori estimates) should be modified. In particular, the appropriate and technical test function log(1 + |uε|)αsign(uε((t, ·))) together with generalized Young
inequalities allows one to prove that
∃c2 ∈]0, +∞[ : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ε (1 + |uε(s, x)|) ln(1 + |uε(s, x)|)αdx ≤ c2, s 0 ε |Duε|2 1+ |uε|(log(1 + |uε)) α−1dxdt≤ c2, ∀s ∈ [0, T ], ∀ε.
Since 1/2 < α ≤ 1 and at the expense of additional computations, Step 3 of Proposition2.1 holds, while Steps 4 and 5 (the study on−) remain unchanged.
Acknowledgements The first author is a member of (and his scientific activity is partially supported by) the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) (Italy), and by project “MICINN MTM 2013-44883-P” (Spain). The second author is a member of (and his scientific activity is partially supported by) the M2NUM project, which is co-financed by the European regional development fund (FEDER-FSE/IEJ HN0002081) of the European Union and by the Conseil Régional de Normandie (France). The authors wish to thank “Laboratoire de Mathématiques Raphaël Salem - Université de Rouen”, where the first author was invited as visiting professor in June 2015 and where this research started. The authors also wish to thank The “Fields Institute for Research in Mathematical Sciences” (Toronto, Canada) were the first author was invited as visiting professor in June 2016.
4 Appendix
Lemma 4.1 Letϕ ∈ C∞([0, T ]). Then,
∃{ϕn}n∈N⊂ C∞0 (]0, T [) : ϕn ϕ weak in L∞(]0, T [). (4.1)
Proof Let{ζn}n∈N⊂ C0∞(]0, T [, [0, 1]) be such that ζn → 1 a.e. in ]0, T [.
Then,ϕn = ζnϕ satisfies (4.1).
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