Limit process

We show that under the assumption u₀, v₀ ∈ H¹(Ω) the approximate solutions converge to a weak solution, whereas assumingu0, v0 ∈H²(Ω) yields a weak in space solution. The stepping stone in the proof of the convergence is the following energy estimate.

Proposition 4.2.1. Let h < 1 and u0, v0 ∈ H¹(Ω). Then under the assumptions of Section 4.2.1, the approximate solution satisfies

ku^h_t(t)k²L²(Ω)+k∇u¯^h(t)k²L²(Ω) ≤2(1 +h²)

kv0k²H¹(Ω)+k∇u0k²L²(Ω)

≤CE (4.2.15) for almost every t∈(0, T), where CE is a constant independent of h.

Proof. We select the test functionψn = (1−θ)un+θun−1,θ ∈(0,1). Noting thatψn ∈ K and by the minimality property we get

0 ≤ 1

θ(Jn(ψn)−Jn(un))

= 1

2h² Z

Ω

2(un−1−un)(un−2un−1+un−2) +θ(un−un−1)² dx +1

2 Z

Ω

2∇un∇(un−1−un) +θ|∇(un−1−un)|² dx.

Passing to the limit as θ→0+, 0 ≤ − 1

h² Z

Ω

(un−un−1)(un−2un−1+un−2)dx+ Z

Ω∇un∇(un−1−un)dx

≤ 1 2h²

Z

Ω

(un−1−un−2)²−(un−un−1)²dx+ Z

Ω

|∇un−1|²− |∇un|²

2 dx.

Thus, after summing up from n = 2 tok = 2,3, . . . , N, we arrive at 1

h²kuk−uk−1k²L²(Ω)+k∇ukk²L²(Ω) ≤ 1

h²ku1−u0k²L²(Ω)+k∇u1k²L²(Ω).

Recollecting (4.2.11) and u1 =u0+hv0, this is already the desired estimate. 2 Using this result, we can derive a subsequencehi, such thatu^hi converges in a certain topology to be mentioned in Lemma 4.2.1. Using the simplified notation without indeces of h (see Chapter 3), we thus have

Lemma 4.2.1. There exists a subsequence of {h → 0+} and a function u belonging to L^∞(0, T;H₀¹(Ω))∩W^1,∞(0, T;L²(Ω)) such that

u^h_t * ut weakly^∗ in L^∞(0, T;L²(Ω)),

∇u¯^h * ∇u weakly^∗ in L^∞(0, T;L²(Ω)), u^h,u¯^h → u strongly in L²(QT).

Proof. First two convergences are an immediate consequence of estimate (4.2.15). The last convergence for u^h holds by Rellich theorem because u^h − u0 ∈ H¹(QT) vanishes on ([0, T]×∂Ω)∪({0} ×Ω), and k∇u^hkL²(Ω) and ku^h_tkL²(Ω) are uniformly bounded (see (3.1.16)). Since

ku¯^h−u^hk^L2(QT) ≤hku^h_tk^L2(QT),

(see (3.1.14)), we conclude that ¯u^h and u^h converge to the same function. 2 A finer estimate for more regular initial data is stated in the following lemma.

Lemma 4.2.2. Let u0, v0 ∈ H²(Ω). Then the approximate weak solution u^h obeys the estimate

Z

Ω

u^h_t(t)−u^h_t(t−h) h

2dx+ Z

Ω|∇u^h_t(t)|²dx≤C_E⁰ for a.e. t∈(h, T), (4.2.16)

where constant C_E⁰ is independent of h.

Moreover, there is a utt ∈ L^∞(0, T;L²(Ω)) and a subsequence to the sequence from Lemma 4.2.1 such that

u^h_t(t)−u^h_t(t−h)

h * utt weakly^∗ in L^∞(0, T;L²(Ω)). (4.2.17) Proof. We recall the identityu₁ =u₀+hv₀. Let us further set u₋₁ =u₀−hv₀+h²∆u₁ ∈ L²(Ω). This function may not satisfy the volume constraint but since

u1−2u0+u−1

h² = ∆u1, we have for every ζ ∈H₀¹(Ω) the relation

Z

Ω

u1−2u0+u−1

h² ζ dx+ Z

Ω∇u₁∇ζ dx= 0 = Z

Ω

λ₁ζ dx.

The last equality follows from λ1 =

Z

Ω

u1−2u0+u−1

h² u1+|∇u1|² dx=

Z

Ω

(∆u1u1+|∇u1|²)dx = 0.

Let us use the notationd_n=u_n−u_n−1,n = 0,1, ..., N, and subtract equation (4.2.9) with n replaced by n−1 from (4.2.9) itself. This corresponds to discrete differentiation of the equation with respect to time. We get

Z

Ω

dn−2dn−1+dn−2

h² ζ dx+ Z

Ω∇dn∇ζ dx= Z

Ω

(λn−λn−1)ζ dx, n = 2,3, . . . , N.

We chooseζ =d_n−d_n−1 and since this function has zero volume, that is, Z

Ω

ζ dx= Z

Ω

(dn−dn−1)dx= Z

Ω

(un−2un−1+un−2)dx= 0, (4.2.18) we find that

Z

Ω

dn−2dn−1+dn−2

h² (dn−dn−1)dx+ Z

Ω∇dn∇(dn−dn−1)dx= 0,

n= 2,3, . . . , N. (4.2.19) Note that the property (4.2.18) is very important because it enables us to get rid of the nonlinear termsλn.

After summing up from n= 2 to k = 2,3, . . . , N, and applying (3.1.10), this yields 1

h² Z

Ω

(dk−dk−1)²dx+ Z

Ω|∇dk|²dx≤ 1 h²

Z

Ω

(d1−d0)²dx+ Z

Ω|∇d1|²dx, fork = 2, . . . , N, which is, after dividing by h², the same as

Z

Ω

u^h_t(t)−u^h_t(t−h) h

2dx+ Z

Ω|∇u^h_t(t)|²dx≤ Z

Ω|∆u1|²dx+ Z

Ω|∇v0|²dx

for a.e. t ∈ (h, T). If we extend the definition (4.2.11) of approximate functions for t ∈ (−h,0), (4.2.16) holds for a.e. t ∈(0, T). Hence, there is a function v ∈ L^∞(0, T;L²(Ω)) so that

u^h_t(t)−u^h_t(t−h)

h * v weakly^∗ in L^∞(0, T;L²(Ω)).

On the other hand, taking ϕ∈C₀^∞(QT), which is an allowed test function for the above weak^∗-convergence, we have

Z T 0

Z

Ω

u^h_t(t)−u^h_t(t−h)

h ϕ dx dt

= Z T

0

Z

Ω

u^h_t(t)

h ϕ dx dt− Z T−h

−h

Z

Ω

u^h_t(t)

h ϕ(t+h)dx dt

= Z T

0

Z

Ω

u^h_t(t)ϕ(t)−ϕ(t+h)

h dx dt−

Z 0

−h

Z

Ω

u^h_t(t)

h ϕ(t+h)dx dt +

Z T T−h

Z

Ω

u^h_t(t)

h ϕ(t+h)dx dt

= Z T

0

Z

Ω

u^h_t(t)ϕ(t)−ϕ(t+h)

h dx dt−

Z h 0

Z

Ω

(v0−h∆u1)ϕ dx dt

→ − Z T

0

Z

Ω

utϕtdx dt as h→0 +.

This shows that v =utt in the sense of distributions, and (4.2.17) follows. 2

Remark 4.2.1. The proof of Lemma 4.2.2 suggests that we can get any regularity of u with respect to t, if only the initial conditions are sufficiently regular. The regularity of initial conditions allows to define appropriate approximations u−n for negative times t∈ (−nh,−(n−1)h) satisfying the approximate equation. Then we obtain relation (4.2.19) for difference quotients d_n of arbitrary order.

By energy estimate (4.2.15) and the strong convergence of u^h we immediately get Lemma 4.2.3. The limit function ufrom Lemma 4.2.1 satisfies the homogeneous bound-ary condition (in the sense of traces) and the volume constraint (4.2.6).

Now, we prove the main result for initial functions fromH²(Ω).

Theorem 4.2.1. Letu0, v0 belong toH²(Ω). Then the approximate weak solutions defined by (4.2.11) converge to the unique weak in space solution of the original problem (4.2.1)-(4.2.4).

Proof. Let φ∈C^∞((0, T);C₀^∞(Ω)). Then Z T

h

Z

Ω∇u¯^h∇φ dx dt→ Z T

0

Z

Ω∇u∇φ dx dt as h→0 +. (4.2.20)

Moreover, we have Z T

h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h φ dx dt→

Z T 0

Z

Ω

uttφ dx dt, as h→0 +. (4.2.21) If the right-hand side of (4.2.12) converges to RT

0

R

Ωλφ dx dt, we arrive at the definition of weak in space solution to our problem. The multiplier ¯λ^h has the form

λ¯^h = 1 V

Z

Ω

u^h_t(t)−u^h_t(t−h)

h u¯^h+|∇u¯^h|²

dx. (4.2.22)

It can be extended to t ∈ (0, h), when one uses the definition of u−1 introduced in the proof of Lemma 4.2.2. By the estimates (4.2.15) and (4.2.16) we find that the approximate Lagrange multipliers are bounded inL²(0, T) independently of h:

Z T 0

(¯λ^h)²dt≤ 2 V²

Z T 0

hZ

Ω

u^h_t(t)−u^h_t(t−h) h

2

dx· Z

Ω

(¯u^h)²dxi dt + 2

V² Z T

0

Z

Ω|∇u¯^h|²dx 2

dt

≤C(T).

In fact, by the same reason, we have even the uniform estimate ¯λ^h(t) ≤ C for a.e. t ∈ (0, T). The above estimate shows that there exists a function κ ∈ L²(0, T), such that

¯λ^h * κ weakly in L²(0, T). Passing to the limit as h → 0+ in (4.2.12), we have by (4.2.20) and (4.2.21),

Z T 0

Z

Ω

(uttφ+∇u∇φ)dx dt= Z T

0

Z

Ω

κφ dx dt for φ ∈L²(0, T;H₀¹(Ω)).

For everyt ∈(0, T) we select φ(t, x) =

(u(t, x), t∈[0, t), x∈Ω, 0 t ∈[t, T], x∈Ω, to obtain

Z t 0

κ dt= 1 V

Z t 0

Z

Ω

(uttu+|∇u|²)dx dt= Z t

0

λ dt ∀t∈(0, T).

This shows thatκ=λ almost everywhere in (0, T). Hence, Z T

0

Z

Ω

(uttφ+∇u∇φ)dx dt= Z T

0

Z

Ω

λφ dx dt.

Thus, in particular, Z

Ω

(uttφ+∇u∇φ)dx= Z

Ω

λφ dx ∀φ∈H₀¹(Ω), for a.e. t ∈(0, T).

The function ubelongs toW^2,∞(0, T;L²(Ω)) and thus also to C¹([0, T];L²(Ω)), justifying the strong formulation of initial conditions. Applying Mazur’s theorem to u−u0 and u_t −v₀, we infer that the initial conditions are satisfied.

The uniqueness follows, as in Remark 4.1.1, from the fact that after subtracting the above equation corresponding to two different solutions and testing by the difference of the solutions, the multiplier term disappears. Then we can use standard technique for uniqueness of solutions to hyperbolic equations (see, e.g., [8], Section 7.2). 2 We now explain how to obtain a weak solution for initial data belonging only toH¹(Ω).

In the sequel, we shall need the following identity.

Lemma 4.2.4. Let v be any smooth function independent of t, satisfying boundary con-ditions and volume constraint (4.2.6). Let ξ belong to L²(0, T;H₀¹(Ω)). By X we denote the integral R

Ωξ dx, which is a function of time only. Then it holds Z T

h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h u¯^hX+|∇u¯^h|²X

dx dt (4.2.23)

= Z T

h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h vX+∇u¯^h∇vX

dx dt.

Proof. The equation is obtained from (4.2.12) by putting φ = (¯u^h −v)X, which is a

function from L²(0, T;H₀¹(Ω)). 2

Theorem 4.2.2. Let u0, v0 ∈ H¹(Ω). Then the approximate weak solutions defined by (4.2.11) converge to the unique weak solution u of the original problem (4.2.1)-(4.2.4).

Proof. We can pass to limit in the left-hand side of (4.2.12). Selecting an arbitrary φ ∈C₀^∞([0, T)×Ω), we find that

Z T h

Z

Ω∇u¯^h∇φ dx dt= Z T

0

Z

Ω∇u¯^h∇φ dx dt− Z h

0

Z

Ω∇u¯^h∇φ dx dt

→ Z T

0

Z

Ω∇u∇φ dx dt as h→0 +. (4.2.24) Moreover, we have (see Chapter 3 for details)

Z T h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h φ dx dt

= Z T

h

Z

Ω

u^h_t(t)

h φ dx dt− Z T−h

0

Z

Ω

u^h_t(t)

h φ(t+h)dx dt

= Z T

0

Z

Ω

u^h_t(t)φ(t)−φ(t+h)

h dx dt−

Z h 0

Z

Ω

u^h_t(t)

h φ dx dt +

Z T T−h

Z

Ω

u^h_t(t)

h φ(t+h)dx dt

→ − Z T

0

Z

Ω

utφtdx dt− Z

Ω

v0φ(0)dx as h→0 +. (4.2.25)

The convergence of ¯λ^h is not obvious and is proven below by the application of Lemma 4.2.4. Let Φ =R

Ωφ dx. Then, due to (4.2.23), we have Z T

h

λ¯^h Z

Ω

φ dx

dt = Z T

h

λ¯^hΦdt

= 1

V Z T

h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h u¯^hΦ +|∇u¯^h|²Φ

dx dt

= 1

V Z T

h

Z

Ω

u^h_t(t)−u^h_t(t−h)

h vΦ +∇u¯^h∇vΦ

dx dt.

Here we proceed similarly as in (4.2.24) and (4.2.25), writingvΦ instead of φ : Z T

h

¯λ^hΦdt→ 1 V

Z T 0

Z

Ω

(−utvΦt +∇u∇vΦ) dx dt− 1 V

Z

Ω

v0vΦ(0)dx. (4.2.26) Using (4.2.24)–(4.2.26), we can pass to limit in (4.2.12) to arrive at

Z T 0

Z

Ω

(−utφt+∇u∇φ)dx dt− Z

Ω

v0φ(0)dx

= Z T

0

Z

Ω

(−utvΦt+∇u∇vΦ)dx dt− Z

Ω

v0vΦ(0)dx.

By density argument, we see that in the above equation we can choose againφ = (u−v)Ψ withv as in Lemma 4.2.4 and Ψ = R

Ωψ dx, ψ ∈C₀^∞([0, T)×Ω) arbitrary. We get Z T

0

Z

Ω −u_t(uΨ)_t+|∇u|²Ψ

dx dt− Z

Ω

v₀u(0)Ψ(0)dx (4.2.27)

= Z T

0

Z

Ω

(−u_tvΨ_t+∇u∇vΨ) dx dt− Z

Ω

v₀vΨ(0)dx.

We have used the fact that the limit function u satisfies the volume condition. Using (4.2.26) and (4.2.27) for the test function φ ∈ C₀^∞([0, T) ×Ω) introduced in (4.2.24) instead ofψ, we obtain

Z T h

¯λ^hΦdt→ 1 V

Z T 0

Z

Ω

(−ut(uΦ)t+|∇u|²Φ)dx dt− 1 V

Z

Ω

v0u(0)Φ(0)dx.

Combining this fact with (4.2.24) and (4.2.25), together with standard technique for initial conditionu(0) =u₀ (see Chapter 3) proves our statement. The uniqueness follows by the

same reasoning as in the proof of Theorem 4.2.1. 2

In this section, a mathematical model for surface vibration preserving volume was derived strictly from the mathematical point of view by introducing a series of variational functionals and making use of the essential properties of their minimizers. Mathematical analysis of the convergence of approximate solution is also given, leading to the proof of existence of weak solutions to a hyperbolic partial differential equation and the derivation of concrete expression for Lagrange multiplier. We realized the approximate solution by numerical computation for a physical phenomenon of lifting a film, comparing the volume-preserving and non-preserving cases (see Section 7.2).

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