parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions

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http://france.elsevier.com/direct/CRASS1/

Partial Differential Equations

Global existence of solutions to a singular

parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions

Hassan Ibrahim

^a

, Mustapha Jazar

^b

, Régis Monneau

^a

aCERMICS, École des ponts, Université Paris-Est, 6 & 8, avenue B. Pascal, 77455 Marne-la-Vallée cedex 2, France bLaMA-Liban, Lebanese University, P.O. Box 826, Tripoli, Lebanon

Received and accepted 30 July 2008

Presented by Haïm Brezis

Abstract

We study the existence of (distribution/viscosity) solutions of a singular parabolic/Hamilton–Jacobi coupled system. Our motivation stems from the study of the dynamics of dislocation densities in a crystal of finite size. The method of the proof consists in considering a parabolic regularization of the system, and then passing to the limit after obtaining some uniform bounds using in particular an entropy estimate for the densities.To cite this article: H. Ibrahim et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Résumé

Existence globale de solutions pour un système couplé parabolique/Hamilton–Jacobi singulier avec condition de Diri- chlet.Nous étudions l’existence de solutions mixtes (distribution/viscosité) pour un système couplé parabolique/Hamilton–Jacobi posé sur un interval. Notre motivation vient de l’étude de la dynamique de densités de dislocations dans un cristal de taille finie.

L’idée de la preuve consiste à considérer une régularisation parabolique appropriée, et ensuite à passer à la limite en utilisant en particulier une estimation entropique pour les densités.Pour citer cet article : H. Ibrahim et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Version française abrégée

Pour tout tempsT >0, et l’interval spatialI =(−1,1), nous étudions le système parabolique/Hamilton–Jacobi suivant :

κ_tκ_x=ρ_tρ_x surI_T :=I×(0, T ), ρ_t=ρ_xx−τ κ_x surI_T,

qui est une version intégrée du modèle de Groma, Czikor et Zaiser [4] décrivant la dynamique de densités de dislocations dans un cristal. Les solutions physiquement acceptables, correspondant à des densités positives de dislocations, sont celles vérifiant

E-mail addresses:[email protected] (H. Ibrahim), [email protected] (M. Jazar), [email protected] (R. Monneau).

doi:10.1016/j.crma.2008.07.031

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κ_x|ρ_x| dansD(I_T).

Notre résultat principal est :

Théorème 1 (Existence globale). Soit (ρ⁰, κ⁰)une donnée initiale surI satisfaisant(4),(5). Alors il existe une fonction(ρ, κ)telle que pour toutT >0,(ρ, κ)∈(C(IT))²avecρ∈C(IT), solution de(1),(6), avec les conditions initiales(2), et les conditions de Dirichlet au bord(3).

L’idée de la preuve du Théorème 1 consiste à considérer une régularization parabolique (8) du système (1), en ajoutant une petite viscositéε >0. Nous prouvons alors l’existence globale (Théorème 1.3) d’une solution au niveau ε, puis passons à la limite quandεtend vers zéro.

1. Introduction and main results

Motivated by the study of the elastoviscoplastic properties of crystals, Groma, Czikor and Zaiser [4] have proposed a model describing the dynamics of dislocation densities. Dislocations are defects in a crystal structure that move when submitted to an exterior applied stress. We consider a one dimensional framework where the crystal is modelized by the intervalI:=(−1,1), and we consider two types of dislocation defects: the positive and negative ones (according to their Burgers vector, see [5]). Letθ⁺andθ⁻represent the density of the positive and negative dislocations respectively.

Indeed, we will work with the primitives (up to a constant):

ρ^±_x =θ^±, ρ=ρ⁺−ρ⁻ and κ=ρ⁺+ρ⁻.

For a given timeT >0, andτ ∈R, a constant applied stress, we consider an integrated form of the model described in [4], namely:

κ_tκ_x=ρ_tρ_x onI_T :=I×(0, T ),

ρ_t=ρ_xx−τ κ_x onI_T, (1)

with initial and boundary conditions

ρ(x,0)=ρ⁰(x), κ(x,0)=κ⁰(x), ∀x∈I, (2)

ρ(±1, t )=0 and κ(±1, t )= ±1, ∀t∈(0, T ). (3) The non-negativity of the densitiesθ⁺andθ⁻at the initial time is interpreted in terms of the unknownsρ⁰andκ⁰by:

κ_x⁰ρ_x⁰ onI. (4)

Denote fors, r∈N,D_t^rD_x^su=_∂t^∂^sr⁺∂x^r^u^s, and denote by∂I the boundary ofI,I_T the closure ofI_T, and byD(I_T)the space of distributions overIT. We now introduce the notion of viscosity solution:

Definition 1.1 (Viscosity solution). Assume ρ ∈C¹(I_T). A function κ ∈C(I_T) such that x →κ(x, t ) is non- decreasing, is called a viscosity solution of the first equation of (1) if it satisfies∀φ∈C¹(I_T):

(i) for any local maximumX₀=(x₀, t₀)∈I_T ofκ−φ, we have:φ_t(X₀)φ_x(X₀)ρ_t(X₀)ρ_x(X₀), (ii) for any local minimumX₀=(x₀, t₀)∈I_T ofκ−φ, we have:φ_t(X₀)φ_x(X₀)ρ_t(X₀)ρ_x(X₀).

The main result of this Note is the following:

Theorem 1.2(Global existence of a mixed solution). Letρ⁰, κ⁰∈C^∞(I )¯ satisfying(4)and

D_xρ⁰, D_xκ⁰∈C₀^∞(I ). (5)

Then there exists(ρ, κ)such that for everyT >0,(ρ, κ)∈(C(I_T))²withρ∈C¹(I_T), is a solution of (1),(2)and(3).

Moreover, this solution satisfies:

κ_x|ρ_x| onD(I_T). (6)

However, the solution has to be interpreted in the following sense:

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(i) κ is a viscosity solution ofκ_tκ_x=ρ_tρ_xinI_T, (ii) ρis a distributional solution ofρ_t=ρ_xx−τ κ_xinI_T,

(iii) the initial and the boundary conditions are satisfied pointwisely.

The principal difficulty we have to face is to deal with the first equation of (1), that we can formally rewrite as κ_t =ρ_tρ_x/κ_x which is singular asκ_x vanishes. The idea is to pass to the limit asε→0 in the family of solutions (ρ^ε, κ^ε),ε >0, of the particular¹parabolic regularization of (1), namely:

⎧⎨

⎩

κ_t^ε=εκ_xx^ε +ρ_x^ερ_xx^ε

κ_x^ε −τρ_x^ε onI_T, ρ_t^ε=(1+ε)ρ_xx^ε −τ κ_x^ε onI_T,

(8) with some initial data and the same boundary conditions

ρ^ε(x,0)=ρ^ε,0(x), κ^ε(x,0)=κ^ε,0(x), ∀x∈I, (9)

ρ^ε(±1, t )=0 and κ^ε(±1, t )= ±1, ∀t∈(0, T ). (10) Concerning system (8), (9) and (10) we have the following global existence and uniqueness result:

Theorem 1.3(Global existence of smooth solutions for the regularized system, [6]). Letρ^ε,0, κ^ε,0∈C^∞(I )¯ satisfying the compatibility conditions:

(1+ε)ρ_xx^ε,0=τ κ_x^ε,0 and (1+ε)κ_xx^ε,0=τρ_x^ε,0 on∂I, (11)

and

κ_x^ε,0>ρ_x^ε,0 onI .¯ (12)

Then there exists(ρ^ε, κ^ε)∈(C^∞(I¯×(0,∞)))²unique solution of(8),(9)and(10)forT = ∞, satisfying forr, s∈N:

D^r_tD^s_xρ^ε, D^r_tD_x^sκ^ε

∈

CI¯× [0,∞)2

, 2r+s3 (13)

and

κ_x^ε>ρ^ε_x onI¯× [0,∞). (14)

The boundary conditions (11) that we have imposed on the initial data of the regularized system are natural here.

In fact, assumeρ^εandκ^εare sufficiently regular solutions of (8), (9) and (10). From (10), we know thatρ^εandκ^εare constants on∂I×(0, T )and thereforeρ_t^ε=κ_t^ε=0 on∂I×(0, T )which, using (8) at timet=0, immediately implies (11). The compatibility conditions (11), joint with the Hölder theory for parabolic equations imply the regularity (13).

To do the proof of Theorem 1.2, we will apply Theorem 1.3 with initial conditionsρ^ε,0,κ^ε,0constructed from ρ⁰,κ⁰. In fact, condition (5) is a sufficient technical condition to insure (11) and (12) for instance, with the special choice:ρ^ε,0(x)=^ρ⁰^(x)₍₁⁺₊^{ετ ψ(x)}_ε)2 ,κ^ε,0(x)=^κ⁰^(x)₁₊⁺_ε^εx, withψ (x)=_4τ¹2[1−cosτ (x²−1)]whenτ=0.

2. Sketch of the proof of Theorem 1.2

We need a framework where system (1) is stable under approximation. Roughly speaking, theC¹regularity ofρ that appears in Theorem 1.2 is expected since it satisfies a parabolic equation (the second equation of (1)). In this case, considering the first equation of (1), we see that the right-hand sideρ_tρ_xis continuous and hence, assumingκ_x0, we can interpretκas a viscosity solution. This takes us in a natural way to the framework of viscosity solutions where

1 This comes from the natural parabolic regularization for the system satisfied byθ^±, which is:

θ_t^±^,ε=εθ_xx^±^,ε±

θx^+,ε−θx^−,ε θ⁺^,ε+θ⁻^,ε −τ

θ^±^,ε

x

withθ^±^,ε=κ_x^ε±ρ_x^ε

2 . (7)

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the stability property is well satisfied (see [1, Lemma 2.3]). We want to show (asεgoes to zero) that(ρ^ε, κ^ε)→(ρ, κ) in(L^∞_loc(I_T))²,ρ_x^ε→ρ_xinL^∞_loc(I_T), andρ_t^ε→ρ_t inD(I_T)withρ_t∈C(I_T).

Step 1. (Convergence ofκ^ε). Writing down the entropy associated to system (7) S^ε(t )=

I

±

θ^±^,ε(x, t )logθ^±^,ε(x, t )dx,

we show thatS^ε(t )C(T )fort∈ [0, T], which implies that

Iκ_x^εlogκ_x^εC₁(T ), which gives theε-uniform con- trol of the modulus of continuity of κ^ε with respect to the variable x. On the other hand, remark thatκ^ε satisfies κ_t^ε−εκ_xx^ε =f^ε where we will show that f^ε is ε-uniformly bounded. Then it is possible to deduce locally the ε-uniform control of the modulus of continuity of κ^ε with respect to the variable t. Indeed, we havef^ε =^ρ_κ^x^εε

xA^ε_x withA^ε=ρ_x^ε−τ κ^εsatisfyingA^ε_t =(1+ε)A^ε_xx+^τρ_κε^x^ε

x A^ε_x. Hence, using interior estimates for parabolic equations (see [7, Proposition 7.1]), we obtain

A^ε→A and A^ε_x→Ax inL^∞_loc, (15)

and joint to (14), we conclude thatf^εisε-uniformly bounded. Finally, using the fact that κ^ε _L∞(IT)1, the convergence ofκ^εdirectly follows by Arzela–Ascoli Theorem.

Step 2. (Convergence ofρ^ε,ρ^ε_xandρ_t^ε). Using similar arguments as in Step 1, particularly (14), and the fact that ρ_t^ε−ερ_xx^ε =A^ε_x, we deduce thatρ^ε →ρ inL^∞_loc(IT). However, sinceρ^ε satisfies a linear parabolic equation (the second equation of (8)), we can write ρ_x^ε=τ κ^ε+A^ε, andρ_t^ε=ερ_xx^ε +A^ε_x, therefore using (15), we deduce, with ρt=Ax∈C(IT), thatρ_x^ε→ρxinL^∞_loc(IT), andρ_t^ε→ρt=AxinD(IT).

Step 3. (Passing to the limit and boundary conditions). We rewrite system (8) in terms ofA^ε, we get:

κ_t^εκ_x^ε=εκ_x^εκ_xx^ε +ρ_x^εA^ε_x onI_T, ρ_t^ε=ερ_xx^ε +A^ε_x onI_T.

Using Steps 1 and 2, we can pass to the limit in the above system, using in particular the stability property for viscosity solutions in order to pass to the limit in the first equation. Our result then directly follows. The only thing left is to recover the boundary conditions. This is made by the equicontinuity ofρ^εandκ^ε with respect toxnear∂I× [0, T], and the equicontinuity ofρ^εandκ^εwith respect tot nearI× {t=0}.

3. Sketch of the proof of Theorem 1.3

Step 1. (A lower bound onκ_x^ε). We have the following comparison principle for system (8) (which gives a stronger version than inequality (6) for system (1)).

Proposition 3.1(A comparison principle for system (8)). Let(ρ^ε, κ^ε)be the solution given by Theorem1.3. Choose β =β(ε, τ ) >0 large enough. Let M(x, t ):=cosh(βx){κ_x^ε(x, t )−

γ²(t )+(ρ_x^ε(x, t ))²} for (x, t )∈I_T, where γ (0)=^γ₂⁰ for someγ0∈(0,1), withκ_x^ε,0

γ₀²+(ρ^ε,0_x )²onI, and γ

γ −

c₀+ρ_xxx^ε (·, t )

L^∞(I )

, c₀=c₀(ε, β, τ ). (16)

Thenm(t ):=minIM(x, t )satisfiesm(t )γ²(t )for allt∈ [0, T]. In particular, we have κ_x^ε(x, t )

γ²(t )+

ρ_x^ε(x, t )2

onI_T. (17)

The idea of the proof of Proposition 3.1 is to write the partial differential inequality satisfied byM(x, t )derived from system (8), and to deduce thatm(t )satisfies the following ordinary differential inequality in the viscosity sense:

mtb0m+b1

for some coefficientsb₀,b₁depending in particular onγ,γandρ_xxx. On the other hand, we can show that

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γ²

tb₀γ²+b₁, and we conclude by comparison.

Comments on the strategy of the proof of Theorem 1.3. Recall that we work onIT. The termEthat will appear in the sequel may certainly vary from line to line but always has the form E=E(T )=ce^cT,c >0 is a positive constant independent of time but depending onε. By applying a fixed point argument, we can show the existence of a local smooth solution(ρ^ε, κ^ε)of (8), (9) and (10) forT >0 small enough. This solution satisfies inequality (17) of Proposition 3.1 which somehow linearizes the first equation of (8) and may leads to a set ofa prioriestimates on the solution. We remark form inequality (16) that we need to have a good control on ρ_xxx^ε (·, t ) _L∞(I )in order to prevent γ from vanishing at a finite time. Otherwise, we cannot guarantee the long time existence of(ρ^ε, κ^ε). In fact, using Hölder estimates for parabolic equations [9], we get:

ρ_xxx^ε (·, t )

L^∞(I ) E

γ (t ), (18)

which, if plugged in (16), does not preventγfrom vanishing, and here is the principal difficulty in treating system (8).

Morea prioriestimates concerning system (8) can also be obtained, namely fort∈(0, T ):

ρ_xxx^ε

BMOp(I×(0,t ))E and ρ_xxx^ε

W₂^2,1(I×(0,t )) E

γ⁴(t ), (19)

withW₂^2,1(I_T)= {u∈L²(I_T), (u_t, u_x, u_xx)∈(L²(I_T))²}, and the parabolic bounded mean oscillation space BMOp

is now recalled.

Definition 3.2 (Parabolic bounded mean oscillation space). A function u∈ L¹_loc(I_T) is said to be of bounded mean oscillation, u∈ BMOp(I_T), if the quantity: u _BMO_p_(I_T₎ =sup_Q_⊂_I_T(_|_Q¹_|

Q|u−m_Q(u)|) is finite. Here Q=Q_r(x₀, t₀)= {(x, t ); |x −x₀|< r, t₀−r²< t < t₀}withr >0, andm_Q(u)=_|_Q¹_|

Qu. The BMOp space is a Banach space whose elements are defined up to an additive constant.

Step 2. (A parabolic Kozono–Taniuchi inequality). We seek to find an estimate on ρ_xxx^ε (·, t ) L∞(I ) better than (18). In fact, the spaceL^∞lies in between the spaces BMOpandW₂^2,1, and it seems natural to estimate theL^∞norm by interpolation between these two spaces. Indeed, this is the goal of the next result.

Proposition 3.3 (A parabolic Kozono–Taniuchi inequality). Letu∈W₂^2,1(I_T), then there exists a constantE such that, for allt∈(0, T ), the following estimate holds(withlog⁺a=max(0,loga)):

u L^∞(I×(0,t ))E u BMO_p(I×(0,t ))

1+log⁺ u _W2,1 2 (I×(0,t ))

. (20)

The proof is an adaptation of the Kozono–Taniuchi inequality which is shown on the whole spaceRⁿin the elliptic case [8, Theorem 1]. It is worth mentioning that the original type of the logarithmic Sobolev inequality was found in [2], and [3]. Using inequality (20) together with (19), we obtain:

ρ_xxx^ε (·, t )

L^∞(I )E 1+log⁺ E γ⁴(t )

. (21)

Step 3. (Long time existence). Using the sharper estimate (21) on ρ_xxx^ε (·, t ) _L∞(I ), we can chooseγ solution of the following ODE:

γ

γ = −E 1+log⁺ 1 γ

,

with some new constantE=E(T ). From Proposition 3.1, we finally deduce that:

κ_x^ε(·, t )γ (t )e⁻^e^e

c(t+1)

>0 fort∈ [0, T], (22)

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wherec >0 is a positive constant independent of time. Indeed, the timeT being arbitrary, we see that (22) is true for all timet0. From (22), the followinga prioriestimates onρ^εandκ^εcan be obtained:

D_x^sρ^ε(·, t )

L^∞(I )e^e^e^c(t+1), D^s_xκ^ε(·, t )

L^∞(I )e^e^e^c(t+1), ∀s∈N, s3, ∀t0. (23)

The abovea prioriestimates (22) and (23) permit to show the long time existence by time iteration.

References

[1] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications (Berlin) (Mathematics & Applications), vol. 17, Springer-Verlag, Paris, 1994.

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[3] H. Brézis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980) 773–789.

[4] I. Groma, F.F. Czikor, M. Zaiser, Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics, Acta Mater. 51 (2003) 1271–1281.

[5] J.R. Hirth, L. Lothe, Theory of Dislocations, second ed., Kreiger Publishing Company, Florida 32950, 1982.

[6] H. Ibrahim, M. Jazar, R. Monneau, Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system, preprint, hal-00281487.

[7] H. Ibrahim, M. Jazar, R. Monneau, Dynamics of dislocation densities in a bounded channel. Part II: existence of weak solutions to a singular Hamilton–Jacobi/parabolic strongly coupled system, preprint, hal-00281859.

[8] H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Commun. Math. Phys. 214 (2000) 191–200.

[9] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967.