Study of a family of higher order non-local degenerate parabolic equations: from the porous medium equation to the thin film equation

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Study of a family of higher order non-local degenerate parabolic equations: from the porous medium equation

to the thin film equation

Rana Tarhini

To cite this version:

Rana Tarhini. Study of a family of higher order non-local degenerate parabolic equations: from the

porous medium equation to the thin film equation. 2015. �hal-01134369�

Study of a family of higher order non-local degenerate parabolic equations: from the porous medium equation to

the thin film equation

Rana Tarhini

March 23, 2015

Abstract

In this paper, we study a nonlocal degenerate parabolic equation of order α+ 2 for α∈(0,2). The equation is a generalization of the one arising in the modeling of hydraulic fractures studied by Imbert and Mellet in 2011. Using the same approach, we prove the existence of solutions for this equation for 0 < α < 2 and for nonnegative initial data satisfying appropriate assumptions. The main difference is the compactness results due to different Sobolev embeddings. Furthermore, forα >1, we construct a nonnegative solution for nonnegative initial data under weaker assumptions.

1 Introduction

In this paper, we study the following problem







∂tu+∂x(uⁿ∂xI(u)) = 0 forx∈Ω, t >0,

∂xu= 0, uⁿ∂xI(u) = 0 forx∈∂Ω, t >0, u(0, x) =u0(x) forx∈Ω,

(1)

where Ω = (a, b) is a bounded interval inR,nis a positive real number andIis a nonlocal elliptic negative operator of orderα defined as the α/2 power of the Laplace operator with Neumann boundary conditionsI=−(−∆)^α2 whereα∈(0,2); this operator will be defined below by using the spectral decomposition of the Laplacian.

The case α = 1 was studied by Imbert and Mellet [15] who proved the existence of non- negative solutions for nonnegative initial data with appropriate conditions. In this case, when n = 3 the equation designs the physical KGD model developed by Geertsma and de Klerk [9]

and Khristianovich and Zheltov [21]. It represents the influence of the pressure exerted by a viscous fluid on a fracture in an elastic medium subject only to plane strain. This equation is derived from the conservation of mass for the fluid inside the fracture, the Poiseuille law and an appropriate pressure law (see [15, section 3] and [14] for further details). In [15], weak solutions are constructed by passing to the limit in a regularized problem. The necessary compactness estimates are obtained from appropriate energy estimates.

∗Universit´e Paris Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees UMR 8050, 61 avenue du G´en´eral de Gaulle, 94010 Cr´eteil, France.

The equation under consideration

ut+∂x(uⁿ∂xI(u)) = 0 (2)

is a nonlocal degenerate parabolic equation of orderα+ 2.

Whenα= 2, this equation coincides with the thin film equation (TFE for short)

ut+∂x(uⁿ∂_xxx³ u) = 0. (3)

This is a fourth order nonlinear degenerate parabolic equation originally studied by Bernis and Friedman [3]. This equation arises in many applications like spreading of a liquid film over a solid surface (n= 3) and Hele-Shaw flows (n= 1) (see [10, 11, 12, 16, 6, 5, 2]). TFE is derived also from a conservation of mass, the Poiseuille law (derived from a lubrication approximation of the Navier-Stokes equations for thin film viscous flows) and various pressure laws. The parameter n ∈(0,3] models various boundary conditions at the liquid-solid interface. The case n > 3 is mainly of mathematical interest [13]. In [3] weak solutionsuare exhibited in a bounded interval under appropriate boundary conditions. In addition, they proved that u is nonnegative if u0

is also so, and that the support of the solution u(t, .) increases witht if u0 is nonnegative and n≥4.

Forα= 0, the porous medium equation (PME for short) is recovered

ut−∂x(uⁿ∂xu) = 0. (4)

This is a nonlinear degenerate parabolic equation. The simple PME model describes the modeling of the motion of a gas flow through a porous medium [20]. In this case, the PME is derived from mass balance, Darcy’s law which describes the dynamics of flows through porous media, and a state equation for the pressure [20]. PME also arises in heat transfer [18] and groundwater flow [19] and was originally proposed by Boussinesq. It took many years to prove that PME is well posed and the famous source type solutions were found by Zel’dovich, Kompanyeets and Barenblatt [20]. The questions of existence, uniqueness, stability, smoothness of solutions together with dynamical properties and asymptotic behavior are well represented in [20] where two main problems are studied. First, the domain space isR^d and the initial conditionu0has a compact support so the solutionu(t, x) vanishes for all positive timest >0 outside a compact set that changes with time. Secondly, if the initial data has a hole in the support then the solution has a possibly smaller hole fort >0.

Note that TFE can be seen as a fourth order version of the classical PME [13]. Furthermore, both equations are parabolic in divergence form. In both cases, there are compactly supported source type solutions (n > 1 for PME [20] and 0 < n < 3 for TFE [4]) [8]. The most famous common properties are finite speed of propagation and the waiting time phenomenon. Similar properties are expected in our case. Self-similar solutions are constructed in [14] but other properties are still not proved. One striking difference between TFE and PME is the lack of a maximum principle for TFE [8].

The case α ∈(−2,0) corresponds to the fractional porous medium equation studied in [7].

Explicit self-similar solutions are exhibited and, under appropriate conditions, weak solutions are constructed.

In this paper, we will generalize the result of [15] to the cases 0< α61 and 1< α <2. We prove a result of existence with the same approach as that in the caseα= 1 but by modifying the compactness results. Consequently all casesα∈[0,2] are now covered.

In the caseα >1 we get the local uniform convergence of approximate solutions due to the following embedding in dimension 1

H^α2(Ω)֒→C^0,α−21(Ω).

This convergence allows one to pass to the limit in the nonlinear term and then allows us to construct nonnegative solutions for nonnegative initial data merely inH^α2(Ω).

In the caseα <1 because of the following embedding H^α2(Ω)֒→L^p(Ω) for allp < 2

1−α,

we can get a compactness result inL^p(Ω) only forp < _1−α² and not for allp <∞as in the case α= 1. Neverthless, we recover a compactness result for the termI(u) which allows us to pass to the limit and conclude.

In both cases, we prove that the solution is strictly positive under a condition on n.

Integral inequalities

Assume that Ω =R, ifuis a solution of (2) then it satisfies the energy inequality

− Z

Ω

u(t)I(u(t))dx+ 2 Z T

0

Z

Ω

uⁿ∂xI(u)²dxdt6− Z

Ω

u0I(u0)dx.

Observe that −R

uI(u) is the homogeneousH^α2 norm. Let Gbe a nonnegative function such thatG^′′(s) =_s¹n. Then the positive solution satisfies

Z

Ω

G(u(t))dx− Z T

0

Z

Ω

∂xu∂xI(u)dxdt6 Z

Ω

G(u0)dx.

Note that−R

∂xu∂xI(u) is the homogeneousH_N^α2+1norm (it is in fact a Neumann-Sobolev space, see below). We see that the energy inequality controls theL^∞(0, T;H^α2(Ω)) norm of the solution.

For the functionGmentioned above, we can take G(s) =

Z s 1

Z r 1

1

tⁿdtdr (5)

so thatGis a nonnegative convex function satisfyingG(1) =G^′(1) = 0,G(s) =∞for alls <0 and fors >0, we have

G(s) =















slns−s+ 1 whenn= 1

−_{(2−n)(n−1)}^s2−n +_n−1^s +_2−n¹ when 1< n <2

ln¹_s+s−1 whenn= 2

1 (n−2)(n−1)

1

sⁿ⁻² +_n−1^s −_n−2¹ whenn >2.

Main results

In this work, we prove three main results. We first prove the existence of nonnegative weak solutions for the problem with 0< α≤1 for nonnegative initial data with apropriate conditions.

Secondly, forα >1, we construct nonnegative solutions for nonnegative initial data in H^α2(Ω).

Finally, we prove the strict positivity of solutions for largen^′s.

Theorem 1.1 (Existence of solutions for 0 < α 6 1). Let n > 1 and α ∈ (0,1]. For any nonnegative initial conditionu0∈H^α2(Ω)such that

Z

Ω

G(u0)dx <∞ (6)

where G is a nonnegative function such that G^′′(s) =_s¹n, there exists a nonnegative function u∈L^∞(0, T;H^α2(Ω))∩L²(0, T;H_N^α2+1(Ω))

which satisfies onQ= (0, T)×Ω Z Z

Q

u∂tϕdtdx− Z Z

Q

nuⁿ⁻¹∂xuI(u)∂xϕdxdt− Z Z

Q

uⁿI(u)∂_xx² ϕdxdt=− Z

Ω

u0ϕ(0, .)dx (7) for allϕ∈ D([0, T)×Ω)¯ satisfying∂xϕ= 0on (0, T)×∂Ω.

Furthermore u satisfies for almost every t∈(0, T) Z

Ω

u(t, x)dx= Z

Ω

u0(x)dx (8)

and

ku(t, .)k^2.

H

α 2(Ω)+ 2

Z T 0

Z

Ω

g²dxds≤ ku0k^2.

H

α

2(Ω) (9)

where the function g∈L²(Q)satisfies g=∂x(uⁿ²I(u))−ⁿ₂uⁿ⁻²²∂xuI(u)inD^′(Ω), and Z

Ω

G(u(t, x))dx+ Z t

0 kuk^2.

H

α 2+1

N (Ω)ds≤ Z

Ω

G(u0)dx. (10)

Remark 1.1. The weak formulation (7) comes after two integrations by parts of the equation (2).

We recall that the functionG:R₊→R₊ is given by (5). Note that the spaceH_N^s(Ω) is defined via spectral decomposition of−(−∆)^s2 (see below).

Theorem 1.2(Existence of solutions for 1< α <2). Letn>1andα >1. For any nonnegative initial conditionu0∈H^α2(Ω), there exists a nonnegative function

u∈C

α−1 2(α+2),^α−₂¹

t,x (Q)

such that

∂xI(u)∈L²_loc(Q+) (11)

and that satisfies Z Z

Q

u∂tϕdtdx+ Z Z

Q+

uⁿ∂xI(u)∂xϕdxdt=− Z

Ω

u0ϕ(0, .)dx (12)

whereQ+ ={u >0}∩Q, for allϕ∈ D([0, T)×Ω)¯ satisfying∂xϕ= 0on(0, T)×∂Ω. Furthermore, usatisfies conservation of mass.

Theorem 1.3(Strictly positive solutions). Assume0< α <2 and n >2 +_α+1² . There exists a set P ⊂ (0, T)such that | (0, T)\P |= 0 and the solution u constructed as in Theorem 1.1 satisfiesu(t, .)∈C^0,β(Ω)for allt∈P and for allβ < min{1,^α+1₂ }andu(t, .)is strictly positive inΩ. Furthermore, u is a solution of

ut+∂xJ = 0 in D^′(Ω) where

J(t, .) =uⁿ∂xI(u)∈L¹(Ω) for all t∈P.

Organization of the paper

The paper is organized as follows: in Section 2, we define the nonlocal operatorI by using the spectral decomposition of the Laplacian and we write an integral representation for it. Then we prove two important Propositions used in the proofs. In Section 3, we study a regularized problem before proving our Theorems in Section 4.

Notation

In this work, we denote Ω = (0,1) andQ= (0, T)×Ω. The spaceH_N^s(Ω) is the functional space defined in [15, Section 3.1] by

H_N^s(Ω) = (

u=

∞

X

k=0

ckϕk;

∞

X

k=0

c²_k(1 +λ^s_k)<+∞ )

where{λk, ϕk}^k≥0 are the eigenvalues and corresponding eigenvectors of the Laplacian operator in Ω with Neumann boundary conditions on∂Ω with the norm

kuk²H_N^s(Ω)=

∞

X

k=0

c²_k(1 +λ^s_k), (13)

equivalently to

kuk²H_N^s(Ω)= Z

Ω

udx 2

+kuk²_H^.s

N(Ω) (14)

where the homogeneous norm is given by kuk²_H^.s

N(Ω)=

∞

X

k=0

c²_kλ^s_k. (15)

Note thatH_N^s(Ω) =H^s(Ω) for all 06s < ³₂ (see [1]) with equivalent norms. Indeed, kuk²H^s(Ω)=kuk²L²(Ω)+kuk²_H^.s_(Ω)

and since we are in dimension 1 we have for these values ofs kuk_H^.s

N(Ω)=kuk_H^.s_(Ω) Note also that we have

R

Ωudx6C(Ω)kuk² (H^..older inequality), kuk²26C(Ω)k(−∆)^s2uk²26ckuk²_H^.s

N(Ω)(fractional Poincar´e’s inequality).

Finally, as usuals+= max{0, s}.

2 Preliminaries

2.1 Operator I

Spectral definition. We define the operator I by I:

∞

X

k=0

ckϕk−→ −

∞

X

k=0

ckλ_k^α2ϕk which mapsH_N^α(Ω) ontoL²(Ω)

where{λk, ϕk}k≥0 are the eigenvalues and corresponding eigenvectors of the Laplacian operator in Ω with Neumann boundary conditions on∂Ω:







−∆ϕk=λkϕk in Ω,

∂νϕk = 0 on∂Ω, R

Ωϕ²_kdx= 1.

Integral representation. The operatorI can also be represented as a singular integral oper- ator. We will prove the following.

Proposition 2.1. Consider a smooth functionu: Ω→R. Then for all x∈Ω, I(u)(x) =

Z

Ω

(u(y)−u(x))K(x, y)dy whereK(x, y)is defined as follows. For allx, y ∈Ω

K(x, y) =cα

X

k∈Z

1

|x−y−2k|^1+α + 1

|x+y−2k|^1+α

wherecαis a constant depending only onα.

Proof. Let’s replace Ω by (−1,1) and uby its even extension to (−1,1). Then let’s extend u periodically toRand let ¯ube this extension. Forx∈Ω,

I(u)(x) =−(−∆)^α2u(x) =¯ cα

Z

R

(¯u(y)−u(x))¯ dy

|y−x|^1+α

=cα

X

k∈Z

Z 1+2k

−1+2k

(¯u(y)−u(x)) dy

|y−x|^1+α

=cα

Z 1

−1

(¯u(y)−u(x)) X

k∈Z

1

|y+ 2k−x|^1+α

!

dy because ¯uis 2-periodic

=cα

Z 1 0

(u(y)−u(x))X

k∈Z

1

|x−y−2k|^1+α + 1

|x+y−2k|^1+α

because ¯uis even.

Now we can easily conclude the following Corollary.

Corollary 2.1. Consider two smooth functionsu, ϕ: Ω→R. Then Z

Ω

I(u)(x)ϕ(x)dx= Z

Ω

u(x)I(ϕ)(x)dx (16)

2.2 Important identities

As [15, Section 3], the semi-normsk.k_H^.α2(Ω),k.k_H^.α

N(Ω),k.k_H^{. α}2+1

N (Ω)andk.k_H^.α+1

N (Ω)are related to the operatorI by important and very useful equalities.

Proposition 2.2. 1. For allu∈H^α2(Ω),we have −hI(u), ui=kuk^2.

H

α2(Ω).

2. For allu∈H_N^α(Ω), we have kuk²_H^.α

N(Ω)=R

ΩI(u)²dx.

3. For allu∈H_N^α2+1(Ω),we have kuk^2.

H

α 2+1

N (Ω)=−R

ΩI(u)xuxdx.

4. For allu∈H_N^α+1(Ω), we havekuk^2.

H^α+1_N (Ω)=R

ΩI(u)²_xdx.

Proof. Note that ifu∈H^α2(Ω) thenI(u)∈H^−α2(Ω) and hI(u), vi_H^−α2(Ω),H^α2(Ω)=−

∞

X

k=0

ckλ_k^α2dk

wherev=P∞

k=0dkϕk∈H^α2(Ω) andu=P∞

k=0ckϕk, so

− Z

uI(u) =

∞

X

k=0

c²_kλ_k^α2 =kuk^2.

H

α2(Ω).

The second equality is actually very easy to prove sinceI(u) =−P∞

k=0ckϕ_k^α2. Indeed, Z

Ω

I(u)²dx=

∞

X

k=0

c²_kϕ^α_k =kuk²_H^.α

N(Ω).

In order to prove the other equalities, we note that (∂xϕk)k form an orthogonal basis of L²(Ω). We write

ux=

∞

X

k=0

ck∂xϕk inL²(Ω).

and∂xI(u) =−

∞

X

k=1

ckλ_k^α2∂xϕk in L²(Ω) so

− Z

Ω

I(u)xuxdx=

∞

X

k=0

c²_kλ_k^α2 Z

Ω

∂xϕ²_kdx=

∞

X

k=0

c²_kλ_k^α2 Z

Ω

ϕk(−∂xxϕk)dx

=

∞

X

k=0

c²_kλ_k^α2 Z

Ω

λkϕ²_kdx=

∞

X

k=0

c²_kλ_k^α2+1=kuk^2.

H

α2+1 N (Ω). For the last equality,

Z

Ω

I(u)²_xdx=

∞

X

k=1

c²_kλ^α_k Z

Ω

∂xϕ²_kdx=

∞

X

k=0

c²_kλ^α+1_k =kuk^2.

H^α+1_N (Ω).

2.3 The problem − I(u) = g

We consider the following problem

(For a giveng∈L²(Ω),findu∈H_N^α(Ω)such that

−I(u) =g. (17)

Since R

ΩI(u)dx= 0 for allu∈H_N^α(Ω), we must assume that R

Ωg(x)dx = 0 otherwise (17) has no solution.

Proposition 2.3. For all g ∈ L²(Ω) such that R

Ωgdx = 0, there exists a unique function u∈H_N^α(Ω) such that

−I(u) =g in L²(Ω) and Z

Ω

udx= 0.

Furthermore if g∈H¹(Ω), then u∈H_N^α+1(Ω).

Proof. Letg∈L²(Ω). Forg=P∞

k=1dkϕk withP∞

k=1d²_k <∞, we consider u=I⁻¹(g) =

∞

X

k=1

dk

λ_k^α2ϕk∈H_N^α(Ω) and verify Z

Ω

udx= 0.

Since (ϕk)k form an orthogonal basis ofL²(Ω), the solution is the unique satisfyingR

Ωudx= 0.

It is clear that every further regularity on g will imply a further regularity on ushifted by an α.

We thus conclude the following Corollary which will be used to prove the existence of solutions for the stationary problem.

Corollary 2.2. For allg∈L²(Ω), there exists a unique functionv∈H_N^α(Ω) such that

−I(v) + Z

Ω

vdx=g. (18)

Furthermore if g∈H¹(Ω), then u∈H_N^α+1(Ω)and the map g→uis bijective.

Proof. Letm=R

Ωgdxandg^′ =g−m. Theng^′∈L²(Ω)(since Ω is bounded) andR

Ωg^′dx= 0.

From Proposition 2.3, there exists a functionu∈H_N^α(Ω) such that

−I(u) =g^′ and Z

Ω

udx= 0.

Letv=u+m. ThenR

Ωvdx=mand

−I(v) =−I(u) =g^′=g−m=g− Z

Ω

vdx.

For the uniqueness, consider two solutionsv1 andv2 then Z

Ω

v1dx= Z

Ω

v2dx= Z

Ω

gdx

andw=v1−v2satisfies−I(w) = 0. Hence,w= 0 from the uniqueness given by Proposition 2.3.

3 Regularized problem

We consider the following regularized problem







∂tu+∂x(fǫ(u)∂xI(u)) = 0 forx∈Ω, t >0,

∂xu= 0, fǫ(u)∂xI(u) = 0 forx∈∂Ω, t >0, u(x,0) =u0(x) forx∈Ω,

(19)

wherefǫ(s) =sⁿ₊+ǫ, ǫ >0 and 0< α <2.

To prove Theorem 1.1 and 1.2, we need to prove the existence of a solution for the regularized problem. Let us pass with the followingstationary problem

Forτ >0, g∈H^α2(Ω), findu∈H_N^α+1(Ω) s.t.

(u+τ ∂x(fǫ(u)∂xI(u)) =g in Ω,

∂xu= 0, ∂xI(u) = 0 on∂Ω. (20) Once we get a solution for (20), we can prove the existence of a solution for (19).

3.1 Stationary problem

Proposition 3.1. For all g ∈H^α2(Ω), there exists u∈ H_N^α+1(Ω) such that for all ϕ∈H¹(Ω) we have

Z

Ω

uϕdx−τ Z

Ω

fǫ(u)∂xI(u)∂xϕdx= Z

Ω

gϕdx. (21)

Furthermore,uverifies

Z

Ω

u(x)dx= Z

Ω

g(x)dx (22)

and

kuk^2.

H

α2(Ω)+ 2τ Z

Ω

fǫ(u)∂xI(u)²dx6kgk^2.

H

α2(Ω). (23)

If R

ΩGǫ(g)dx <∞ whereGǫ is a nonnegative function such thatG^′′_ǫ(s) = _f¹

ǫ(s), then Z

Ω

Gǫ(u)dx+τkuk^2.

H

α 2+1 N (Ω)6

Z

Ω

Gǫ(g)dx. (24)

Remark 3.1. Note that we can considerGǫ(s) =Rs 1

Rt

1G^′′_ǫ(r)drdt, soGǫis a non- negative convex function for allǫ >0 satisfyingGǫ(1) =G^′_ǫ(1) = 0.

Proof. Thanks to Corollary 2.2, we can recover all test functions fromH¹(Ω) by considering ϕ=−I(v) +

Z

Ω

vdx

for some functionv∈H_N^α+1(Ω). So equation (21) becomes

− Z

Ω

uI(v)dx + Z

Ω

udx Z

Ω

vdx

+τ Z

Ω

fǫ(u)∂xI(u)∂xI(v)dx

=− Z

Ω

gI(v)dx+ Z

Ω

gdx Z

Ω

vdx

. (25)

Now, we consider the nonlinear operator A defined by A(u)(v) =−

Z

Ω

uI(v)dx + Z

Ω

udx Z

Ω

vdx

+τ Z

Ω

fǫ(u)∂xI(u)∂xI(v)dxforu, v∈H_N^α+1(Ω).

We prove that this is a continuous, coercive and pseudo-monotone operator. Note that the functionalTg defined by

Tg(v) =− Z

Ω

gI(v)dx+ Z

Ω

gdx Z

Ω

vdx

forv∈H_N^α+1(Ω).

is a linear form onH_N^α+1(Ω). So our problem reduces to the following











LetV =H_N^α+1(Ω).

A:V →V^∗ coercive, continuous and pseudo-monotone.

Tg∈V^∗.

Findu∈H_N^α+1(Ω) such thatA(u) =Tg inV^∗.

(26)

The theory of pseudo-monotone operators [17] implies the existence of a solution for (26) so there existsu∈H_N^α+1(Ω) such that

A(u)(v) =Tg(v) for allv∈H_N^α+1(Ω).

It remains to prove thatAis a continuous, coercive and pseudo-monotone operator onH_N^α+1(Ω).

The reader can find the proof in [15, Appendix A] forV =H_N²(Ω) but this proof can be easily adapted for our caseV =H_N^α+1(Ω).

By using Corollary 2.2 we deduce thatusatisfies (21) for allϕ∈H¹(Ω).

For the properties of u, first by taking ϕ= 1 as a test function in (21) we obtain mass conser- vation (22). Secondly, takev =u−R

Ωudxin (25), by using Proposition 2.2 we have kuk^2.

H

α 2(Ω)+τ

Z

Ω

fǫ(u)∂xI(u)²=− Z

Ω

gI(u)dx 6kgk_H^{. α}2(Ω)kuk_H^{. α}2(Ω)

6 1 2kgk^2.

H

α2(Ω)+1 2kuk^2.

H

α2(Ω)

which (23)(Note that the high regularity ofgis solely used in this inequality, otherwiseg∈L²(Ω) is sufficient to prove the existence above). Finally, note that G^′_ǫ is smooth with G^′_ǫ andG^′′_ǫ are bounded, and Ω is bounded so we can takeϕ=G^′_ǫ(u)∈H¹(Ω) as a test function in (21),

Z

Ω

uG^′_ǫ(u)dx−τ Z

Ω

fǫ(u)∂xI(u)∂xuG^′′_ǫ(u)dx= Z

Ω

gG^′_ǫ(u)dx.

So by using Proposition 2.2 and the fact thatG^′′_ǫ(s) = _f¹

ǫ(s) we get τkuk^2.

H

α2+1 N (Ω)=

Z

Ω

G^′_ǫ(u)(g−u)dx6 Z

Ω

(Gǫ(g)−Gǫ(u))dx becauseGǫ is convex and we deduce (24).

3.2 Implicit Euler scheme

We construct a piecewise constant function

u^τ(t, x) =u^k(x) fort∈[kτ,(k+ 1)τ), k∈ {0, ..., N−1}

whereτ =_N^T and (u^k)k∈{0,...,N−1}is such that

u^k+1+τ ∂x(fǫ(u^k+1)∂xI(u^k+1)) =u^k.

The existence of theu^k follows from Proposition 3.1 by induction onkwithu⁰=u0. We deduce the following

Corollary 3.1. For anyN >0andu^ǫ₀∈H^α2(Ω), there exists a functionu^τ ∈L^∞(0, T;H^α2(Ω)) such that

1. t−→u^τ(t, x) is constant on[kτ,(k+ 1)τ), k∈ {0, ..., N−1} andτ= _N^T. 2. u^τ =u0 on[0, τ)×Ω.

3. For allt∈(0, T),

Z

Ω

u^τ(t, x)dx= Z

Ω

u0(x)dx. (27)

4. For allϕ∈C_c¹(0, T;H¹(Ω)), Z Z

Qτ,T

u^τ−Sτu^τ

τ ϕdxdt= Z Z

Qτ,T

fǫ(u^τ)∂xI(u^τ)∂xϕdxdt (28) whereSτu^τ(t, x) =u^τ(t−τ, x)and Qτ,T = (τ, T)×Ω.

5. For allt∈(0, T), ku^τ(t, .)k^2.

H

α2(Ω)+ 2 Z T

0

Z

Ω

fǫ(u^τ)∂xI(u^τ)²dxdt6ku0k^2.

H

α2(Ω). (29)

6. IfR

ΩGǫ(u0)dx <∞, then for all t∈(0, T) Z

Ω

Gǫ(u^τ(t, x))dx+ Z t

0 ku^τ(s, .)k^2.

H

α2+1 N (Ω)ds6

Z

Ω

Gǫ(u0)dx. (30)

3.3 Existence of solution for the regularized problem

Now we are able to prove the existence of a solution for the regularized problem.

Proposition 3.2. Let0< α <2. For all u0∈H^α2(Ω)and for allT >0, there exists a function u^ǫ such that

u^ǫ∈L^∞(0, T;H^α2(Ω))∩L²(0, T;H_N^α+1(Ω)) satisfying

Z Z

Q

u^ǫ∂tϕdxdt+ Z Z

Q

fǫ(u^ǫ)∂xI(u^ǫ)∂xϕdxdt=− Z

Ω

u0ϕ(0, .)dx (31)

for allϕ∈C¹(0, T;H¹(Ω)) with support in [0, T)×⁻Ω.

The functionu^ǫ satisfies for almost everyt∈(0, T) Z

Ω

u^ǫ(t, x)dx= Z

Ω

u0(x)dx (32)

and

ku^ǫ(t, .)k^2.

H

α2(Ω)+ 2 Z T

0

Z

Ω

fǫ(u^ǫ)∂xI(u^ǫ)²dxdt6ku0k^2.

H

α2(Ω). (33)

Finally, ifR

ΩGǫ(u0)dx <∞then for almost every t∈(0, T), Z

Ω

Gǫ(u^ǫ(t, x))dx+ Z t

0 ku^ǫ(s, .)k^2.

H

α2+1 N (Ω)ds6

Z

Ω

Gǫ(u0)dx. (34)

Proof. We consider the sequence (u^τ) constructed in Corollary 3.1 and letτ →0. Bound (29) and (27) implies that (u^τ) is bounded inL^∞(0, T;H^α2(Ω)) and (∂xI(u^τ)) is bounded inL²(Q).

Case 0< α61. Note that

u^τ−Sτu^τ

τ =∂x(fǫ(u^τ)∂xI(u^τ)).

Sincen>1, the functionfǫis Lipschitz and so (fǫ(u^τ)) is bounded inL^∞(0, T;H^α2(Ω)) thus by the Sobolev embedding theorem, we deduce that (fǫ(u^τ)) is bounded in L^∞(0, T;L^p(Ω)) for all p < _1−α² . We know that (∂xI(u^τ)) is bounded inL²(0, T;L²(Ω)) sofǫ(u^τ)∂xI(u^τ) is bounded in L²(τ, T;L^r(Ω)) where ¹_r = ¹₂+_p¹. We deduce that

∂x(fǫ(u^τ)∂xI(u^τ)) is bounded inL²(τ, T;W^−1,r(Ω)) Sinceα61, we have the following embedding

H^α2(Ω)֒→L^p(Ω)→W^−1,l(Ω)

for allp < _1−α² and for alll >2 (because Ω is bounded and we have a Sobolev space of negative regularity). Aubin’s lemma implies that (u^τ) is relatively compact in C⁰(0, T;L^p(Ω)) for all p < _1−α² . Note that (∂xI(u^τ)) is bounded in L²(Ω) and (u^τ) is bounded in L^∞(0, T;L¹(Ω)) (because 1<_1−α² ). Hence, (u^τ) is bounded inL²(0, T;H_N^α+1(Ω)). Since

H_N^α+1(Ω)֒→H_N^α2+1(Ω)→W^−1,l(Ω),

we deduce that (u^τ) is relatively compact inL²(0, T;H_N^α2+1(Ω)). So we can extract a subsequence, also denoted (u^τ), such that whenτ tends to zero we have

• u^τ →u^ǫ∈L^∞(0, T;H^α2(Ω)) almost everywhere inQ,

• u^τ →u^ǫ inL²(0, T;H_N^α2+1(Ω)) strongly,

• ∂xI(u^τ)⇀ ∂xI(u^ǫ) inL²(Q) weakly.

Now let us pass to the limit in (28). We have Z Z

Qτ,T

u^τ−Sτu^τ

τ ϕdxdt= 1 τ

Z T 0

Z

Ω

u^τ(t, x)ϕ(t, x)dtdx− Z τ

0

Z

Ω

u^τ(t, x)ϕ(t, x)dtdx

− Z T−τ

0

Z

Ω

u^τ(t, x)ϕ(t+τ, x)dtdx

= Z T

0

Z

Ω

u^τ(t, x)ϕ(t, x)−ϕ(t+τ, x)

τ dxdt

−1 τ

Z τ 0

Z

Ω

u^τ(t, x)ϕ(t, x)dtdx+1 τ

Z T T−τ

Z

Ω

u^τ(t, x)ϕ(t+τ, x)dtdx

−→τ→0− Z Z

Q

u^ǫ∂tϕdxdt− Z

Ω

u^ǫ(0, x)ϕ(0, x)dx+ 0.

For the nonlinear term, we integrate by parts Z Z

Q

fǫ(u^τ)∂xI(u^τ)∂xϕ=− Z Z

Q

fǫ(u^τ)I(u^τ)∂_xx² ϕ− Z Z

Q

n(u^τ)ⁿ⁻¹∂xu^τI(u^τ)∂xϕ. (35) We have

u^τ→u^ǫin L²(0, T;H_N^s(Ω)) for alls <1 +α.

So

I(u^τ)→I(u^ǫ) inL²(0, T;H^s′(Ω)) for alls^′ <1 and

∂xu^τ→∂xu^ǫin L²(0, T;H^s′′(Ω)) for alls^′′< α.

So we deduce the following convergences

I(u^τ)→I(u^ǫ) inL²(0, T;L^q(Ω)) for allq <∞. u^τ_x→u^ǫ_xin L²(0, T;L^p(Ω)) for allp < 2

1−α.

Furthermore, sinceu^τ →u^ǫ inC⁰(0, T;L^p(Ω)) for allp < _1−α² andfǫis lipschitz then fǫ(u^τ)→fǫ(u^ǫ) inC⁰(0, T;L^p(Ω)) for allp < 2

1−α. For the term (u^τ)ⁿ⁻¹, ifn>2 then the functions→sⁿ⁻¹is lipschitz and

(u^τ)ⁿ⁻¹→(u^ǫ)ⁿ⁻¹ inC⁰(0, T;L^p(Ω)) for allp < 2 1−α. Ifn <2 then _n−1^p >1 and

(u^τ)ⁿ⁻¹→(u^ǫ)ⁿ⁻¹in C⁰(0, T;L^n−1p (Ω)) for allp < 2 1−α.

Thus we can pass to the limit in (35) and reverse the integration by parts to obtain Z Z

Q

fǫ(u^τ)∂xI(u^τ)∂xϕ→ − Z Z

Q

fǫ(u^ǫ)I(u^ǫ)∂²_xxϕ− Z Z

Q

n(u^ǫ)ⁿ⁻¹∂xu^ǫI(u^ǫ)∂xϕ

= Z Z

Q

fǫ(u^ǫ)∂xI(u^ǫ)∂xϕ.

For the properties ofu^ǫ, first sinceu^τ →u^ǫ inL^∞(0, T;L¹(Ω)) mass conservation equation (32) follows from (27).

Secondly, we note that (u^τ) is bounded in L^∞(0, T;H^α2(Ω)) so (u^τ) weakly converges tou^ǫ inH^α2(Ω) and

ku^ǫk_H^{. α}2(Ω)≤lim inf

τ→0 ku^τk_H^{. α}2(Ω). Note that estimate (29) implies thatp

fǫ(u^τ)∂xI(u^τ) is bounded inL²(0, T;L²(Ω)) thus it weakly converges inL²(0, T;L²(Ω)) and the lower semicontinuity permits us to conclude (33).

Finally, to derive (34) we note thatGǫ(u^τ)→Gǫ(u^ǫ) almost everywhere and Fatou’s lemma implies for almost everyt∈(0, T)

Z

Ω

Gǫ(u^ǫ(t, x))dx≤lim inf

τ→0

Z

Ω

Gǫ(u^τ(t, x))dx.

Furthermore, (u^τ) is relatively compact inL²(0, T;H_N^α2+1(Ω)) thus Z t

0 ku^ǫ(s)k^2.

H

α 2+1 N

ds= lim

τ→0

Z t

0 ku^τ(s)k^2.

H

α 2+1 N

ds.

Hence (30) implies (34).

Case 1< α <2. Note that

u^τ−Sτu^τ

τ =∂x(fǫ(u^τ)∂xI(u^τ)).

We have (u^τ) is bounded inL^∞(0, T;H^α2(Ω)) so by the Sobolev embedding theorem, we deduce that (u^τ) is bounded inL^∞(0, T;C^0,α−12 (Ω)). Thus (fǫ(u^τ)) is bounded inL^∞(0, T;L^∞(Ω)). We know that (∂xI(u^τ)) is bounded inL²(0, T;L²(Ω)) so (fǫ(u^τ)∂xI(u^τ)) is bounded inL²(τ, T;L²(Ω)).

We deduce that

∂x(fǫ(u^τ)∂xI(u^τ)) is bounded inL²(τ, T;W^−1,2(Ω)).

Sinceα >1 we have the following embedding

H^α2(Ω)֒→C^0,α−12 (Ω)→W^−1,2(Ω).

Aubin’s lemma implies that the sequence (u^τ) is relatively compact inC⁰(0, T;C^0,α−12 (Ω)). Since (∂xI(u^τ)) is bounded inL²(Ω) and (u^τ) is bounded inL^∞(0, T;L¹(Ω)) then, (u^τ) is bounded in L²(0, T;H_N^α+1(Ω)). Using the following embedding

H_N^α+1(Ω)֒→H_N^α2+1(Ω)→W^−1,2(Ω),

we deduce that (u^τ) is relatively compact inL²(0, T;H_N^α2+1(Ω)). So for a subsequence we have

• u^τ →u^ǫ locally uniformly,

• ∂xI(u^τ)⇀ ∂xI(u^ǫ) inL²(Q)-weakly,

• u^τ →u^ǫ inL²(0, T;H_N^α2+1(Ω)) strongly.

Let us pass to the limit in (28). As in the first case Z Z

Qτ,T

u^τ−Sτu^τ

τ ϕdxdt−→

τ→0− Z Z

Q

u^ǫ∂tϕdxdt− Z

Ω

u^ǫ(0, x)ϕ(0, x)dx.

For the nonlinear term, since

u^τ →u^ǫ locally uniformly, Then

fǫ(u^τ)∂xϕ→fǫ(u^ǫ)∂xϕin L²(0, T;L²(Ω))−strongly.

Furthermore

∂xI(u^τ)⇀ ∂xI(u^ǫ) inL²(0, T;L²(Ω))−weakly.

Hence

Z Z

Q

fε(u^τ)∂xI(u^τ)∂xϕdxdt→ Z Z

Q

fǫ(u^ǫ)∂xI(u^ǫ)∂xϕdxdt and the proof is complete.

For the properties of u^ǫ, the proofs of estimates (32), (34) and (33) are the same as in the first case.

4 Proofs of main results

4.1 Proof of Theorem 1.1

Consider the sequence (u^ǫ) such that u^ǫ ∈ L^∞(0, T;H^α2(Ω))∩L²(0, T;H^α2+1(Ω)) solution of (19). Our goal is to pass to the limitǫ→0.

Note that (33) and (32) imply that (u^ǫ) is bounded inL^∞(0, T;H^α2(Ω)). Sincefǫis Lipschitz then fǫ(u^ǫ) is bounded in L^∞(0, T;H^α2(Ω)). So by using the Sobolev embedding theorem, we deduce thatfǫ(u^ǫ) is bounded in L^∞(0, T;L^p(Ω)) for allp < _1−α² .

Furthermore, (33) also implies that fǫ(u^ǫ)¹²∂xI(u^ǫ) is bounded in L²(0, T;L²(Ω)). Thus fǫ(u^ǫ)∂xI(u^ǫ)is bounded inL²(0, T;L^r(Ω)) where 1

r =1 2 + 1

2p. Hence

∂tu^ǫ=−∂x(fǫ(u^ǫ)∂xI(u^ǫ)) is bounded inL²(0, T;W^−1,r(Ω)).

Since

H^α2(Ω)֒→L^1−2α(Ω)→W^−1,l(Ω),

Aubin’s lemma implies that (u^ǫ) is relatively compact inC⁰(0, T;L^p(Ω)) for allp < _1−α² . So we can extract a subsequence such that

• u^ǫ→uin C⁰(0, T;L^p(Ω)) for allp < _1−α² .

• u^ǫ→u∈L^∞(0, T;H^α2(Ω)) almost everywhere in Q.

Let us pass to the limit in (31). Let ϕ∈ D([0, T)×Ω) satisfying¯ ∂xϕ= 0 on (0, T)×∂Ω.

Sinceu^ǫ→uinC⁰(0, T;L¹(Ω)), we have Z Z

Q

u^ǫ∂tϕdxdt→ Z Z

Q

u∂tϕdxdt.

Remark that (33) implies that ǫ

Z Z

(∂xI(u^ǫ))²≤c.

The Cauchy-Schwarz inequality implies ǫ

Z Z

∂xI(u^ǫ)∂xϕdxdt6c(ϕ)√ ǫ(√

ǫk∂xI(u^ǫ)k2)→0.

Estimate (33) also gives that (u^ǫ)₊ⁿ²∂xI(u^ǫ) is bounded in L²(0, T;L²(Ω)). For the term (u^ǫ)ⁿ², we consider two cases, ifn>2 then the function s→sⁿ² is Lipschitz and ((u^ǫ)ⁿ²) is bounded in L^∞(0, T;L^p(Ω)) for allp < _1−α² . We deduce that ((u^ǫ)ⁿ₊∂xI(u^ǫ)) is bounded inL²(0, T;L^m(Ω)) where _m¹ = ¹₂+¹_p. Ifn <2 then ((u^ǫ)ⁿ²) is bounded inL^∞(0, T;L^2pn(Ω)) for allp < _1−α² (in this case ^2p_n >1). We deduce that ((u^ǫ)ⁿ₊∂xI(u^ǫ)) is bounded inL²(0, T;L^m(Ω)) where _m¹ = ¹₂+_2pⁿ, hence

h^ǫ:= (u^ǫ)ⁿ₊∂xI(u^ǫ)⇀ hinL²(0, T;L^m(Ω)) weakly.

Passing to the limit we obtain Z Z

Q

u∂tϕdxdt+ Z Z

Q

h∂xϕdxdt=− Z

Ω

u0ϕ(0, x)dx.

It remains to show that

h=uⁿ₊∂xI(u) in the following sense

Z Z

Q

hϕdxdt=− Z Z

Q

nuⁿ⁻¹₊ ∂xuI(u)ϕdxdt− Z Z

Q

uⁿ₊I(u)∂xϕdxdt (36) for all test functionsϕsuch thatϕ= 0 on (0, T)×∂Ω, that is

h=∂x(uⁿ₊I(u))−nuⁿ⁻¹₊ ∂xuI(u) inD^′(Ω).

Note thatGǫ is decreasing with respect toǫ, so Z

Ω

Gǫ(u0)dx≤ Z

Ω

G(u0)dx≤c.

Thus estimate (34) implies that (u^ǫ) is bounded in L²(0, T;H_N^α2+1(Ω)). Recall that (∂tu^ǫ) is bounded inL²(0, T;W^−1,l(Ω)). Aubin’s lemma implies that

(u^ǫ) is relatively compact inL²(0, T;H_N^s(Ω)) for alls < α 2 + 1.

Hence

(∂xu^ǫ) is relatively compact in L²(0, T;H^s′(Ω)) for alls^′< α 2 and

(I(u^ǫ)) is relatively compact inL²(0, T;H_N^s′′(Ω)) for alls^′′<1−α 2. Thus we can extract a subsequence such that

u^ǫ→uin C⁰(0, T;L^p(Ω)) for allp < 2 1−α, I(u^ǫ)→I(u) in L²(0, T;L^q(Ω)) for allq <∞,

∂xu^ǫ→∂xuin L²(0, T;L^p(Ω)) for allp < 2 1−α. We write

Z Z

Q

hǫϕdxdt= Z Z

(u^ǫ)ⁿ₊∂xI(u^ǫ)ϕdxdt

=− Z Z

n(u^ǫ)ⁿ⁻¹₊ ∂xu^ǫI(u^ǫ)ϕdxdt− Z Z

(u^ǫ)ⁿ₊I(u^ǫ)∂xϕdxdt.

Using these convergences and the fact thatI(u^ǫ) converges inL²(0, T;L^q(Ω)) for all q <∞we can pass to the limit and obtain (36). Note that for the terms (u^ǫ)ⁿ and (u^ǫ)ⁿ⁻¹ we consider two cases n > 2 and n <2 and we proceed as above. In the first case the functions s → sⁿ and s →sⁿ⁻¹ are Lipschitz and then (u^ǫ)ⁿ →uⁿ and (u^ǫ)ⁿ⁻¹ → uⁿ⁻¹ in C⁰(0, T;L^p(Ω)) for all p < _1−α² . If n < 2 then ^p_n > 1 and _n−1^p > 1 and (u^ǫ)ⁿ → uⁿ in C⁰(0, T;L^np(Ω)) and (u^ǫ)ⁿ⁻¹→uⁿ⁻¹ inC⁰(0, T;L^n−1p (Ω)) for allp < _1−α² .

For the properties ofu, passing to the limit in (32) implies mass conservation equation (8).

Since (u^ǫ) is bounded in L²(0, T;H_N^α2+1(Ω)) thenu^ǫ⇀ uand kuk_L2(0,T;H

α2+1

N (Ω)) ≤lim inf

ǫ→0 ku^ǫk_L2(0,T;H

α2+1 N (Ω)). Note that

Gǫ(u^ǫ)→G(u) almost everywhere andGǫ(u0)≤G(u0).

Then by Fatou’s lemma estimate (10) follows from (34).

Remark that estimate (33) implies that gǫ = (u^ǫ)₊ⁿ²∂xI(u^ǫ) weakly converges in L²(Q) to a functiong and the lower semi-continuity of the norm implies (9). It remains to prove that

g=∂x(u+ⁿ²I(u))−n

2u+ⁿ²⁻¹∂xuI(u) inD^′(Ω). (37)