Reduction of the oscillation in rescaled cylinders

Một phần của tài liệu Figueiredo i rodrigues j santos l (eds ) free boundary problems theory and applications (Trang 240 - 246)

Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)

3. Reduction of the oscillation in rescaled cylinders

Fix (x0,0)× {0}, and takeR >0 such thatK2R(x0)Ω. By translation, we may assume thatx0= 0. Introduce the cylinder

Q(Rp,2R) :=K2R×(0, Rp) and define

à+= ess sup

Q(Rp,2R)

θ ; à = ess inf

Q(Rp,2R)

θ ; ω= ess osc

Q(Rp,2R)

θ=à+−à .

236 E. Henriques and J.M. Urbano Construct the cylinder

Q(a0Rp, R) =KR×(0, a0Rp) , a0= ω

2m 2p

,

where m > 1 is to be chosen. Without loss of generality, we may assume that

ω

2m 1 so that the following relations hold:

Q(a0Rp, R)⊂Q(Rp,2R) and ess osc

Q(a0Rp,R)

θ≤ω .

The proof of Theorem 1.2 is a well-known consequence of the following iter- ative argument.

Proposition 3.1. There exist constants σ (0,1), and C, m > 1, that can be determined a priori only in terms of the data, such that constructing the sequences

ω0=ω ωn+1=σ ωn

and R0=R

Rn+1= CRn and the family of boxes

Qn= (anRpn, Rn) , an= ωn

2m 2p

, we have

Qn+1⊂Qn and ess osc

Qn

θmax ωn,2 ess osc

KRn θ0

, (3.1)

for alln= 0,1,2, . . ..

To prove Proposition 3.1, assume first that both inequalities à+−ω

4 ≤à+0 := ess sup

KR

θ0 and à+ω

4 ≥à0 := ess inf

KR

θ0 (3.2) hold. Subtracting the second inequality from the first one we get

ω

2 ≤à+0 −à0 = ess osc

KR

θ0 . and the proposition is trivially proved.

Without loss of generality, assume that the second inequality in (3.2) fails.

Then the levels k = à+ 2ωs, for s 2, verify k à0 and, consequently, the energy and logarithmic estimates (2.5) and (2.6), respectively, hold for (θ−k). The next result has a double scope: it determines the parametermthat defines the height of the constructed initial cylinder and defines a level such that the subset ofQ

a0Rp,R2

whereθ is below that level is small.

Lemma 3.2. For all ν (0,1), there exists m > 3, depending only on the data, such that

(x, t)∈Q

a0Rp,R 2

: θ(x, t)< à+ ω 2m

< ν Q

a0Rp,R 2

.

Proof.Consider estimate (2.6) written for (θ−k), with k=à+ω4, and for a cutoff function 0≤ξ≤1, defined inKR, and verifying

ξ≡1 inKR

2 ; ξ≡0 on|x|=R ; |∇ξ| ≤ 2

R .

Takem >3 sufficiently large so that 0< c= 2ωm < Hk. The logarithmic function Ψ is well-defined and, sinceHkω4, the following inequalities hold

Ψ(m−2) ln 2 and

Ψ2p ω 2m

p2

. Then, from (2.6), we get for allt∈(0, a0Rp), the estimate

KR×{t}

ψ2

ξp≤C(m−2) KR 2

. Next, integrate over the smaller set

x∈KR

2 :θ(x, t)< à+ ω 2m

, ∀t∈(0, a0Rp)

where ξ = 1 and Ψ (m−3) ln 2, since Hk ω4. Consequently, for all t (0, a0Rp),

x∈KR

2 : θ(x, t)< à+ ω 2m

≤C m−2 (m−3)2

KR 2

.

The proof is complete if we choosemso large thatC (mm3)22 < ν.

The next lemma provides a uniform lower bound for θ within a smaller cylinder, through a specific choice of the valueν that appears in Lemma 3.2.

Lemma 3.3. There existsν0(0,1), depending only on the data, such that if Q

a0Rp,R 2

: θ(x, t)≤à+ ω 2m

≤ν0 Q

a0Rp,R 2

then

θ(x, t)≥à+ ω

2m+1 , a.e.(x, t)∈Q

a0Rp,R 4

. Proof.Consider the decreasing sequences of real numbers

Rn =R 4 + R

2n+2 ; kn =à+ ω

2m+1 + ω

2m+1+n , n= 0,1, . . . and, in the energy estimates (2.5), takeϕ=(θ−kn)ξnp, where 0≤ξn1 are smooth cutoff functions, defined inKRn, and verifying

ξ≡1 inKRn+1 ; ξ≡0 on|x|=Rn ; |∇ξn| ≤ 2n+3 R . Introduce the level

k¯n =kn+kn+1

2 .

238 E. Henriques and J.M. Urbano Since

KRn×{t}

(θ−kn)2ξnp=

KRn×{t}

(θ−kn)p(θ−kn)2pξpn

(kn−k¯n)2p

KRn×{t}

(θ−k¯n)pξnp

=a02(n+3)p

KRn×{t}

(θ¯kn)pξnp and (θ−kn)p ω

2m

p

, the referred energy estimates take the form sup

0<t<a0Rp

KRn×{t}

(θ¯kn)pξnp+ 1 a0

2(n+3)p

Q(a0Rp,Rn)

(θ¯kn)p ξnp

≤C(p) ω

2m

p 22pn Rp

1

a0 Q(a0Rp,Rn)

χ[(θkn)>0] . Introducing the change of variablez=at

0, defining the new functions θ¯(x, z) =θ(x, a0z) ; ξ¯n(x, z) =ξn(x, a0z), and denotingVp=L(Lp)∩Lp(W1,p), we arrive at

θ−k¯n)p

Vp(Q(Rp,Rn+1))≤C(p)22pn Rp

ω 2m

p

An , where

An:=

Rp 0

|An(z)| dz , An(z) :=

x∈KRn: (¯θ−kn)>0 . Since

ω 2m

p

2(n+3)pAn+1

Q(Rp,Rn+1)

θ−k¯n)p

≤C A1+

N+pp

nθ¯kn)p

Vp(Q(Rp,Rn+1)) , using Corollary 3.1 of [3, page 9], we conclude

An+1≤C 23pn Rp A1+

p N+p

n

and, consequently,

Yn+1≤C23pnA1+

N+pp

n , for Yn := An

|Q(Rp, Rn)| .

IfY0≤CNp+p23(N+p)2p then, by Lemma 4.1 of [3, page 12],Yn 0 when n→ ∞which completes the proof. Observe that, by the hypothesis,

Y0= (x, z)∈Q(Rp, R) : ¯θ(x, z)< à+2ωm

|Q(Rp, R)| ≤ν0 so we just have to take

ν0≡CN+pp23(N+p)2p .

Now we can finally conclude the first iteration step in the proof of Proposition 3.1. Indeed, takingν =ν0 from Lemma 3.3, and determining the corresponding valuemwith the help of Lemma 3.2, we arrive at

θ(x, t)≥à+ ω

2m+1 , a.e.(x, t)∈Q

a0Rp,R 4

, and then we conclude that

ess osc

Q(a0Rp,R4) θ

1 1

2m+1

ω=σ ω . TakingC= 4 in Proposition 3.1, we get Q1⊂Q

a0Rp,R4

, and then ess osc

Q1 θ ess osc

Q(a0Rp,R4) θ≤σ ω=ω1. We can now repeat the whole process starting fromQ1.

Remark 3.4. Observe that we do not get a reduction on the t-direction since the cutoff functionsξare independent oft.

Remark 3.5. The regularity result can be further extended; one can obtain conti- nuity up to the lateral boundary Σ using a reasoning similar to the one presented in [2] and [8].

References

[1] DiBenedetto, E.,Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4)130(1982), 131–176.

[2] DiBenedetto, E., A boundary modulus of continuity for a class of singular parabolic equations, J. Differential Equations63(1986), 418–447.

[3] DiBenedetto, E., Degenerate Parabolic Equations, Universitex, Springer-Verlag, New York, 1993.

[4] DiBenedetto, E., Urbano, J.M. and Vespri, V.,Current issues on singular and degen- erate evolution equations, Handbook of Differential Equations, Evolutionary Equa- tions, vol.1, pp. 169–286, Elsevier/North-Holland, Amsterdam, 2004.

[5] Henriques, E. and Urbano, J.M., On the doubly singular equation γ(u)t = ∆pu, Comm. Partial Differential Equations30(2005), 919–955.

[6] Henriques, E. and Urbano, J.M., Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J., to appear.

240 E. Henriques and J.M. Urbano

[7] Urbano, J.M., Continuous solutions for a degenerate free boundary problem, Ann.

Mat. Pura Appl. (4)178(2000), 195–224.

[8] Urbano, J.M.,A singular-degenerate parabolic problem: regularity up to the Dirichlet boundaryin: FBPs: Theory and Applications I, pp. 399–410, GAKUTO Int. Series Math. Sci. Appl.13, 2000.

Eurica Henriques

Departamento de Matem´atica

Universidade de Tr´as-os-Montes e Alto Douro Quinta dos Prados, Apartado 1013

5000-911 Vila Real, Portugal e-mail:eurica@utad.pt Jos´e Miguel Urbano

Departamento de Matem´atica Universidade de Coimbra 3001-454 Coimbra, Portugal e-mail:jmurb@mat.uc.pt

Fast Reaction Limits and Liesegang Bands

D. Hilhorst, R. van der Hout, M. Mimura and I. Ohnishi

1. Introduction

The purpose of this study is to start understanding from a mathematical viewpoint experiments in which regularized structures with spatially distinct bands and rings of precipitated material were exhibited, with clearly visible scaling properties. This phenomenon has been originally observed by Liesegang [1] in 1896, after whom the name “Liesegang bands/rings” has been coined. Since then there have been a large number of contributions to the understanding of such precipitated pattern formation from experimental as well as theoretical viewpoints. However, as far as we know, there has not been any mathematical study of this problem apart from numerical simulations. In this note we introduce a one-dimensional reaction diffusion system which is a simplified model of the supersaturation model proposed by Keller and Rubinow [2] in 1981 and study the occurrence of precipitated bands in this system, by means of singular limit analysis.

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