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A fourth-order degenerate parabolic equation with a viscous term:
is studied with the initial-boundary conditions u_{x}=w_{x}=0
on {-1,1}×(0,T)
, u(x,0)=u_{0}(x)
in (-1,1)
. It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)
and the viscosity term. The existence of nonnegative weak solution is obtained.

In recent years, the research of nonlinear fourth-order degenerate parabolic equations has become an interesting topic. The typical examples include the Cahn-Hilliard equation and the thin film equation. The Cahn-Hilliard equation can describe the evolution of a conserved concentration field during phase separation. It (see [

The thin film equation can analyze the motion of a very thin layer of viscous incompressible fluids along an inclined plane or model the fluid flows such as draining of foams and the movement of contact lenses. The thin film equation belongs to a class of fourth order degenerate parabolic equations (see [

In this paper, we study the following initial and boundary value problems for the viscous thin film equation:

{ u t − ( m ( u ) w x ) x = 0 in Q T , w = − u x x + ν u t in Q T , u x = w x = 0 on Γ , u ( x , 0 ) = u 0 ( x ) , (1)

where T > 0 , m ( u ) = u , Ω = ( − 1 , 1 ) , Q T = Ω × ( 0 , T ) and Γ = ∂ Ω × ( 0 , T ) .

Formally, if we substitute the second equation into the first one, we can get another form for this question:

{ u t + ( m ( u ) ( u x x − ν u t ) x ) x = 0 in Q T , u x = u x x x = 0 on Γ , u ( x , 0 ) = u 0 ( x ) . (2)

Our main result is the following theorems.

Theorem 1. Let u 0 ∈ L 2 ( Ω ) and ν > 0 . Then there exists at least one pair ( u , w ) of (1) satisfying

1) u ∈ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ L 2 ( 0 , T ; H 2 ( Ω ) ) ∩ C ( [ 0 , T ] ; L 2 ( Ω ) ) , w ∈ L 2 ( 0, T ; H 1 ( Ω ) ) u t ∈ L 2 ( Q T ) ;

2) For any test function ϕ ∈ L 2 ( 0, T ; H 1 ( Ω ) ) , it has

∬ Q T u t ϕ d x d t + ∬ Q T u w x ϕ x d x d t = 0,

∬ Q T w ϕ d x d t = − ∬ Q T u x x ϕ d x d t + ν ∬ Q T u t ϕ d x d t .

3) u ( x ,0 ) = u 0 ( x ) .

Theorem 2. Let u 0 ∈ L 2 ( Ω ) and ν > 0 . Then there exists at least one pair ( u , w ) of (2) satisfying

1) u ∈ L ∞ ( 0 , T ; H 1 ( Ω ) ) ∩ L 2 ( 0 , T ; H 2 ( Ω ) ) ∩ C ( [ 0 , T ] ; L 2 ( Ω ) ) , u t ∈ L 2 ( Q T ) ;

2) For any test function ϕ ∈ L 2 ( 0, T ; H 2 ( Ω ) ) with ϕ x ( − 1 , t ) = ϕ x ( 1 , t ) = 0 , it has

∬ Q T u t ϕ d x d t + ∬ Q T u x x u x ϕ x d x d t − ν ∬ Q T u u t ϕ x x d x d t = 0.

3) u ( x , 0 ) = u 0 ( x ) .

The following lemmas are needed in the paper:

Lemma 1. (Aubin-Lions, see [

1) Let F be bounded in L p ( 0, T ; X ) where 1 ≤ p < ∞ , and ∂ F ∂ t = { ∂ f ∂ t : f ∈ F } be bounded in L 1 ( 0, T ; Y ) . Then F is relatively compact in L p ( 0, T ; B ) ;

2) Let F be bounded in L ∞ ( 0, T ; X ) , and ∂ F ∂ t = { ∂ f ∂ t : f ∈ F } be bounded in L r ( 0, T ; Y ) where r > 1 . Then F is relatively compact in C ( [ 0, T ] ; B ) .

Lemma 2. (see [

In this paper, C is denoted as a positive constant and may change from line to line. The paper is arranged as follows. The existence of solutions to the approximate problem will be proved in Section 2. In Section 3, we will take the limit for small parameters δ → 0 .

For any 0 < δ < 1 , we consider the following approximate problem. In order to apply existence theory better, we transform (1) into a system:

{ u δ t − ( m δ ( u δ ) w δ x ) x = 0 in Q T , w δ = − u δ x x + ν u δ t in Q T , u δ x = w δ x = 0 on Γ , u δ ( x , 0 ) = u δ 0 ( x ) (3)

with u δ 0 ( x ) = u 0 ( x ) + δ , m δ ( u δ ) = u δ + + δ and u δ + = max { u δ , 0 } .

Lemma 3. There exists at least one solution u δ to (3) satisfying

1) w δ ∈ L 2 ( 0, T ; H 1 ( Ω ) ) , u δ ∈ L 2 ( 0, T ; H 2 ( Ω ) ) ∩ L ∞ ( 0, T ; H 1 ( Ω ) ) ∩ C ( [ 0 , T ] ; L 2 ( Ω ) ) , u δ t ∈ L 2 ( Q T ) and u δ ( x , 0 ) = u δ 0 ;

2) For any test function ϕ ∈ L 2 ( 0, T ; H 1 ( Ω ) ) , it has

∬ Q T u δ t ϕ d x d t + ∬ Q T m δ ( u δ ) w δ x ϕ x d x d t = 0,

∬ Q T w δ ϕ d x d t = − ∬ Q T u δ x x ϕ d x d t + ν ∬ Q T u δ t ϕ d x d t .

Proof. We apply the Galerkin method to prove this Lemma and so we choose { ϕ i } i = 1 , 2 , 3 , ⋯ as the eigenfunctions of the Laplace operator with Neumann boundary value conditions such that − ϕ i x x = λ i ϕ i . Moreover, we can suppose that the eigenfunctions are orthogonal in the H^{1} and L^{2} spaces. We use ( ⋅ , ⋅ ) to denote the scalar product in L^{2} space and we can normalize ϕ i such that ( ϕ i , ϕ j ) = δ i j = { 1 , i = j , 0 , i ≠ j . Besides, we can choose λ 1 = 0 and ϕ 1 = 1 .

For any positive integer M, we define u δ M ( x , t ) = ∑ i = 1 M c i ( t ) ϕ i ( x ) , u δ M ( x , 0 ) = ∑ i = 1 M ( u 0 , ϕ i ) ϕ i , w δ M ( x , t ) = ∑ i = 1 M d i ( t ) ϕ i ( x ) . Now we consider the following ordinary differential equations system:

d d t ( u δ M , ϕ j ) = − ( m δ ( u δ M ) w δ x M , ϕ j x ) , (4)

( w δ M , ϕ j ) = − ( u δ x x M , ϕ j ) + ν d d t ( u δ M , ϕ j ) , (5)

for

with

Taking

Thus, we have

Therefore, for any

Since

By taking

By integrating over

which yields

There exists a subsequence of

where the last estimate is from Lemma 1. By (13)-(17), we can perform the limit

In the section, we will perform the limit

Moreover, the function

By applying this function, we can get the following estimates.

Lemma 4. There exist some constants C independent of

1)

2)

3)

4)

5)

6)

Proof. Taking

Thus, it yields the results 1 - 3. We can prove 4 and 5 from (8). By choosing

We have completed the proof of this lemma.

Lemma 5. There exists a pair

1)

2)

3)

4)

5)

Proof. By Lemma 4, we can get the results 1 - 2 and 4 directly. Lemma 1 yields 3. By applying the definition of

It yields

Letting

Proof of Theorem 1 and Theorem 2. Taking

which yields Theorem 1.

On the other hand, by integrating by parts, it implies

Thus, it has

It gives Theorem 2.

Through this paper, two forms of a viscous thin film equation are studied (see the Equations (1) and (2)) and we give the corresponding existence theorems of weak solutions (see Theorem 1 and Theorem 2). For any test function

Since the thin film equation is a degenerate parabolic equation, it is hard to give the existence of strong solutions. On the another hand, the viscous term affects the regularity of solutions and we have shown that

We can expect that we can show that the existence results would be true with some conditions in high-dimensional space.

The work was supported by the Education Department Science Foundation of Liaoning Province of China (No. JDL2016029) and the Natural Science Fund of Liaoning Province of China (No. 20170540136).

The authors declare that they have no competing interests.

Qiu, Y. and Liang, B. (2018) Existence of Solutions to a Viscous Thin Film Equation. Journal of Applied Mathematics and Physics, 6, 2119-2126. https://doi.org/10.4236/jamp.2018.610178